The Development of Modern Logic
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The Development of Modern Logic
Edited by
Leil...

Author:
Leila Haaparanta

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The Development of Modern Logic

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The Development of Modern Logic

Edited by

Leila Haaparanta

2009

Oxford University Press, Inc., publishes works that further Oxford University’s objective of excellence in research, scholarship, and education. Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With oﬃces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam

Copyright © 2009 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data The development of modern logic / edited by Leila Haaparanta. p. cm. Includes bibliographical references. ISBN 978-0-19-513731-6 1. Logic—History. I. Haaparanta, Leila, 1954– BC15.D48 2008 160.9—dc22 2008016767

9 8 7 6 5 4 3 2 1 Printed in the United States of America on acid-free paper

Preface

This volume is the result of a long project. My work started sometime in the 1990s, when Professor Simo Knuuttila urged me to edit, together with a few colleagues, a volume on the history of logic from ancient times to the end of the twentieth century. Even if the project was not realized in that form, I continued with the plan and started to gather together scholars for a book project titled The Development of Modern Logic, thus making a reference to the famous book by William and Martha Kneale. Unlike that work, the new volume was meant to be written by a number of scholars almost as if it had been written by one scholar only. I decided to start with thirteenth-century logic and come up with quite recent themes up to 2000, hence, to continue the history written in The Development of Logic. My intention was to ﬁnd a balance between the chronological exposition and thematic considerations. The philosophy of modern logic was also planned to be included; indeed, at the beginning the book had the subtitle “A Philosophical Perspective,” which was deleted at the end, as the volume reached far beyond that perspective. The collection of articles is directed to philosophers, even if some chapters include a number of technical details. Therefore, when it is used as a textbook in advanced courses, for which it is also planned, those details are recommended reading to students who wish to develop their skills in mathematical logic. In 1998, we had a workshop of the project with most of the contributors present. It was a ﬁne beginning, organized by the Department of Philosophy at the University of Helsinki and by the Philosophical Society of Finland. We got ﬁnancial support from the Academy of Finland and from the Finnish Cultural Foundation, which I wish to acknowledge. I moved to the University of Tampere in the fall of 1998. Unlike logic perhaps, life sometimes turns out to be chaotic. As we were a large group, it was no surprise that various personal and professional matters inﬂuenced the process of writing and editing. Still, we

vi

Preface

happily completed the volume, which became even larger than was originally intended. I wish to thank the contributors, from whom I have learned a great deal during the editorial process. It has been a pleasure to cooperate with them. Renne Pesonen and Risto Vilkko kindly assisted me with the editorial work. I am very grateful to my colleagues for useful pieces of advice. There are so many who have been helpful that it is impossible to name them all. My special thanks are due to Auli Kaipainen and Jarmo Niemelä, who prepared the camera-ready text for publication. Jarmo Niemelä also assisted me with compiling the index. I wish to thank Peter Ohlin, editor at Oxford University Press, who has been extremely helpful during the process. I have beneﬁted considerably from the help of my editors, Stephanie Attia and Molly Wagener, of Oxford University Press. The ﬁnancial support given by the Academy of Finland is gratefully acknowledged. I have done the editorial work at the University of Tampere, ﬁrst at the Department of Mathematics, Statistics and Philosophy and then at the Department of History and Philosophy. Finally, I wish to express my deep gratitude to my mother and to my husband, whose support and encouragement have been invaluable. L. H.

Contents

Contributors

ix

1. Introduction 3 Leila Haaparanta 2. Late Medieval Logic 11 Tuomo Aho and Mikko Yrjönsuuri 3. Logic and Philosophy of Logic from Humanism to Kant Mirella Capozzi and Gino Roncaglia

78

4. The Mathematical Origins of Nineteenth-Century Algebra of Logic Volker Peckhaus 5. Gottlob Frege and the Interplay between Logic and Mathematics Christian Thiel 6. The Logic Question During the First Half of the Nineteenth Century Risto Vilkko 7. The Relations between Logic and Philosophy, 1874–1931 Leila Haaparanta

159 196 203

222

8. A Century of Judgment and Inference, 1837–1936: Some Strands in the Development of Logic 263 Göran Sundholm 9. The Development of Mathematical Logic from Russell to Tarski, 1900–1935 318 Paolo Mancosu, Richard Zach, and Calixto Badesa

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Contents

10. Set Theory, Model Theory, and Computability Theory Wilfrid Hodges 11. Proof Theory of Classical and Intuitionistic Logic Jan von Plato

471

499

12. Modal Logic from Kant to Possible Worlds Semantics Tapio Korte, Ari Maunu, and Tuomo Aho

516

Appendix to Chapter 12: Conditionals and Possible Worlds: On C. S. Peirce’s Conception of Conditionals and Modalities Risto Hilpinen

551

13. Logic and Semantics in the Twentieth Century 562 Gabriel Sandu and Tuomo Aho 14. The Philosophy of Alternative Logics Andrew Aberdein and Stephen Read

613

15. Philosophy of Inductive Logic: The Bayesian Perspective Sandy Zabell 16. Logic and Linguistics in the Twentieth Century Alessandro Lenci and Gabriel Sandu 17. Logic and Artiﬁcial Intelligence Richmond H. Thomason

724

775

848

18. Indian Logic 903 J. N. Mohanty, S. R. Saha, Amita Chatterjee, Tushar Kanti Sarkar, and Sibajiban Bhattacharyya Index

963

Contributors

Andrew Aberdein, Humanities and Communication, Florida Institute of Technology, [email protected]ﬁt.edu. Tuomo Aho, Department of Philosophy, University of Helsinki, [email protected]ﬁ. Calixto Badesa, Department of Logic, History and Philosophy of Science, University of Barcelona, [email protected] Sibajiban Bhattacharyya, Department of Philosophy, University of Calcutta (died in 2007). Mirella Capozzi, Department of Philosophical and Epistemological Studies, University of Rome “La Sapienza,” [email protected] Amita Chatterjee, Department of Philosophy and Center for Cognitive Science, Jadavpur University. Leila Haaparanta, Department of History and Philosophy, University of Tampere, [email protected]ﬁ. Risto Hilpinen, Department of Philosophy, University of Miami, [email protected] Wilfrid Hodges, Department of Mathematics, Queen Mary, University of London, [email protected] Tapio Korte, Department of Philosophy, University of Turku, [email protected]ﬁ. Alessandro Lenci, Department of Computational Linguistics, University of Pisa, [email protected] ix

x

Contributors

Paolo Mancosu, Department of Philosophy, University of California, Berkeley, [email protected] Ari Maunu, Department of Philosophy, University of Turku, [email protected]ﬁ. J. N. Mohanty, Department of Philosophy, Temple University, Philadelphia. Volker Peckhaus, Department of Humanities, University of Paderborn, [email protected] Stephen Read, Department of Philosophy, University of St. Andrews, [email protected] Gino Roncaglia, Department of Humanities, University of Tuscia, Viterbo, mc3430[email protected] S. R. Saha, Department of Philosophy, Jadavpur University. Gabriel Sandu, Department of Philosophy, University of Helsinki; Department of Philosophy, Sorbonne, [email protected]ﬁ. Tushar Kanti Sarkar, Department of Philosophy, Jadavpur University. Göran Sundholm, Department of Philosophy, University of Leiden, [email protected] Christian Thiel, Department of Philosophy, University of Erlangen, [email protected] Richmond H. Thomason, Department of Philosophy, University of Michigan, [email protected] Risto Vilkko, Department of Philosophy, University of Helsinki, [email protected]ﬁ. Jan von Plato, Department of Philosophy, University of Helsinki, [email protected]ﬁ. Mikko Yrjönsuuri, Department of Social Sciences and Philosophy, University of Jyväskylä, [email protected]ﬁ. Sandy Zabell, Department of Mathematics, Northwestern University, [email protected] Richard Zach, Department of Philosophy, University of Calgary, [email protected]

The Development of Modern Logic

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1

Introduction Leila Haaparanta

1. On the Concept of Logic When we state in everyday language that a person’s logic fails, we normally mean that the rules of valid reasoning, which ought to guide our thinking, are not in action for some reason. The word “logic” of our everyday language can usually be analyzed as “the collection of rules that guide correct thinking or reasoning.” That collection is assumed to be known naturally; a rational human being follows those rules in normal circumstances, even if he or she could not formulate them, that is, express them in language. When the word “logic” (in Greek logos “word,” “reason”) refers to one subﬁeld of philosophy or of mathematics, it usually means the discipline concerning valid reasoning or the science that studies that kind of reasoning. In his logical studies, Aristotle (384–322 b.c.) considered inferences, which are called syllogisms. They consisted of two premises and a conclusion, and the validity of the argument of a syllogistic form was determined by the structure of the argument. If the premises of a syllogism were true, the conclusion was also true. According to Aristotle, the basic form of a judgment is “A is B,” where “A” is a subject and “B” is a predicate. Forms of judgments include “Every A is B,” “No A is B,” “Some A is B,” and “Some A is not B.” Unlike Aristotelian logic, modern formal logic is called symbolic or mathematical, as it studies valid reasoning in artiﬁcial languages. Until the nineteenth century logic was mainly Aristotelian. Following Aristotle, the main focus was on judgments that consisted of a subject and a predicate and that included such words as “every,” “some,” and “is” in addition to letters corresponding to the subject and the predicate. The Stoics, for their part, were 3

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interested in what is nowadays called propositional logic, in which the focus is on such words as “not,” “and,” “or,” and “if–then.” It was not until the nineteenth century that symbolic logic, which had its model in mathematics, became a serious rival of Aristotelian logic. The grammatical analysis of judgments was challenged in the late nineteenth century by logicians who took the model of analysis from mathematics. The words “function” and “argument” became part of the vocabulary of logic, and predicates that expressed relations as well as quantiﬁers were included in that vocabulary. In the new logic, which was mostly developed by Gottlob Frege (1848–1925) and Charles Peirce (1939–1914) and which was codiﬁed in Principia Mathematica (1910, 1912, 1913), written by A. N. Whitehead (1861–1947) and Bertrand Russell (1872–1970), the rules of logical inference received a new treatment, as the pioneers of modern logic tried to give an exact formulation of those rules in an artiﬁcial language. Except for the collection of the rules of valid reasoning and the discipline or the science that focuses on those rules, the word “logic” means a speciﬁc language that fulﬁlls certain requirements of preciseness. It also means a ﬁeld of research that focuses on such a language or such languages. Since the seventeenth century, it has been typical of the ﬁeld called logic to construct and study a formal language or formal languages called logic or logics. The old Aristotelian logic heavily relied on natural language. Aristotle and his followers thought that natural language reﬂects the forms of logical inference and other logical relations, even the form of reality. The pioneers of modern logic sought to construct an artiﬁcial language that would be more precise than natural languages. In the twentieth century those languages called logics have been used as models of natural languages; hence, modern logic that rejected the grammatical analysis of judgments has, among other things, served as a tool in linguistic research. It is important to note that the pioneers of modern logic, such as G. W. Leibniz (1646–1716) and Frege, did not intend to present any tools of studying natural languages; they wished to construct a symbolic language that would overcome natural language as a medium of thought in being more precise and lacking ambiguities that are typical of natural language. As the views of the tasks and the aims of logic have varied in history, we may wonder whether Aristotle and the representatives of modern logic, for example, Frege, were at all interested in the same object of research and whether it is possible to talk about the same ﬁeld of research. In spite of diﬀerences, we may name a few common interests whose existence justiﬁes the talk about research called logic and the history of that ﬁeld. In each period in the history of logic, researchers called logicians have been interested in concepts or terms that are not empirical, that is, whose meanings are not, or at least not incontestably, based on sensuous experience, and that can be called logical concepts or terms. What concepts or terms have been regarded as logical has varied in the history, but interest in them unites Aristotle, William of Ockham, Immanuel Kant, and Frege as well as logicians in the twentieth and twenty-ﬁrst centuries. Other

Introduction

5

points of interest have been the so-called laws of thought, for example, the law of noncontradiction and the law of excluded middle. A third theme that unites logicians of diﬀerent times is the question of the validity of reasoning. In several chapters of the present volume, the question concerning the nature and the scope of logic is discussed in view of the period and the logicians that are introduced to the reader.

2. What Is Modern Logic? The starting point of modern logic is presented in textbooks in various ways depending on what features are regarded as the characteristics of modernity. Some say modern logic started together with modern philosophy in the late Middle Ages, while others think that it started in the seventeenth century with Leibniz’s logic. Still others argue that the beginning of modern logic was 1879, when Frege’s Begriﬀsschrift appeared. If the beginning of modern logic is dated to the seventeenth century, its pioneers include Leibniz, Bernard Bolzano (1781–1848), Augustus De Morgan (1806–1871), George Boole (1815–1864), John Venn (1834–1923), William Stanley Jevons (1835–1882), Frege, Peirce, Ernst Schröder (1841–1902), Giuseppe Peano (1858–1932), and Whitehead and Russell. Unlike many contemporary logicians, modern logicians believed that there is one and only one true logic. Leibniz was the most important of those thinkers who argued that the terms of our natural language do not correspond to the objects of the world in a proper way and that therefore we have to construct a new language, which mirrors the world correctly. Following Leibniz, modern logicians sought to construct an artiﬁcial language that would be better than natural languages. If we think that this kind of eﬀort is an important feature of modern logic, then we may say that modern logic started with Leibniz. The idea of calculus has also been an important feature of modern logic. Logic has been considered a system which consists of logical and nonlogical vocabulary, formation rules, and transformation rules; the formation rules tell us what kind of sequences of symbols are well formed, and the transformation rules are the basis on which logical reasoning is performed like calculating. Many early pioneers of modern logic relied on the grammatical subjectpredicate analysis in analyzing sentences that was also part of traditional logic, as mentioned above. It was not until Frege’s logic that this division was rejected. The division between arguments and functions thus became central in logic. Frege also stressed that it was the distinction between individuals and concepts that he wants to respect. If we stress that feature, we may say that the philosophical ideas of modern logic can be found in medieval nominalists, but that they did not become codiﬁed in formal languages until the latter half of the nineteenth century in Frege’s and Peirce’s discoveries. Those two logicians also made quantiﬁers into the basic elements of logic. As modern thinkers, many late medieval philosophers were interested in individuals, but

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The Development of Modern Logic

the distinction between an individual and a concept was not taken into account in logic until Frege’s and Peirce’s discoveries. Frege regarded his logic as an axiomatic theory. That feature can also be considered a typical feature of modern logic. As was said before, it is often thought that Frege’s Begriﬀsschrift gave birth to modern logic. In that book there were many logical discoveries, such as the theory of quantiﬁcation and the argument-function analysis. Frege’s book was both philosophical and mathematical. Later, in the ﬁrst volume of his Grundgesetze der Arithmetik (1893), Frege states that he is likely to have few readers; all those mathematicians stop reading who, when seeing the words “concept,” “relation,” and “judgment” think: “It is metaphysics, we do not read it,” and those philosophers stop reading, who, when seeing a formula, shout: “It is mathematics, we do not read it” (p. xii). Charles Peirce discovered the logic of relatives in the 1870s. That logic was inspired by Boole’s algebra of logic and De Morgan’s theory of relations. Peirce’s articles “The Logic of Relatives” (1883) and “On the Algebra of Logic: A Contribution to the Philosophy of Notation” (1885) contain the ﬁrst formulation of his theory of quantiﬁcation that he calls his general algebra of logic. Peirce’s algebra diﬀered from that of Boole’s especially in that Peirce introduced signs that refer to individuals in addition to signs that signify relations. Second, he introduced the quantiﬁers “all” and “some.” Frege only used the sign for generality and deﬁned existence by means of generality and negation. Both the logicians rejected Boole’s idea that judgments are formed by combining subjects and predicates. Frege and Peirce, who made their important discoveries independently of each other, Peirce maybe with his group of students and Frege alone, had common features. They were both philosophers and mathematicians and could combine philosophical ideas with technical novelties in their logical thought. Frege and Peirce both invented a notation for quantiﬁers and quantiﬁcation theory almost simultaneously, independently of each other. Therefore they can be regarded as the principal founders of modern logic. However, as many scholars have emphasized, most notably Jean van Heijenoort in his paper “Logic as Calculus and Logic as Language” (1967), Jaakko Hintikka in his papers “Frege’s Hidden Semantics” (1979) and “Semantics: A Revolt Against Frege” (1981), and Warren Goldfarb in his article “Logic in the Twenties: The Nature of the Quantiﬁer” (1979), the two logicians seem to be far apart philosophically. The division between the two traditions to which the logicians belong has also been emphasized by a number of authors of the present volume. The distinction between the two conceptions of logic, namely, seeing logic as language versus seeing it as calculus, has been suggested from the perspective of twentieth-century developments, but the origin of the division has been located in nineteenth-century logic. Diﬀerent interpretations of the history of logic follow depending on how the distinction is understood. According to van Heijenoort, Hintikka, and Goldfarb, those who stressed the idea of logic as language thought that logic speaks about one single world. It is certain

Introduction

7

that Frege held that position. He thought that there is one single domain of discourse for all quantiﬁers, as he assumed that any object can be the value of an individual variable and any function must be deﬁned for all objects. On the other hand, those who supported the view that logic is a calculus gave various interpretations or models for their formal systems. That was Boole’s and his followers’ standpoint. Several other features of the two traditions are mentioned in the chapters of the present volume. The volume titled Studies in the Logic of Charles Sanders Peirce (1997) introduces another pair of traditions, which are mathematical logic and algebraic logic and which are also touched upon in the present collection of articles. Ivor Grattan-Guinness states in his contribution to the volume on Peirce that the phrase “mathematical logic” was introduced by De Morgan in 1858 but that it served to distinguish logic using mathematics from “philosophical logic,” which was also a term used by De Morgan. However, in Grattan-Guinness’s terminology, De Morgan’s logic was part of the algebraic tradition; using algebraic methods in logic would be typical of what he calls algebraic logic. The most common phrase used in the nineteenth century was “the algebra of logic” or sometimes “logical algebra.” In the ﬁgure which Grattan-Guinness presents to us, Boole, De Morgan, Peirce, and Schröder belong to the tradition of algebraic logic, while Peano and Russell belong to the tradition of mathematical logic. It seems that many of those who belong to the tradition of logic as calculus belong to the tradition of algebraic logic in Grattan-Guinness’s division, and that many of those who think that logic is a language belong to what Grattan-Guinness calls the tradition of mathematical logic. Grattan-Guinness gives us a few typical features of the two traditions that he discusses. In algebraic logic, laws were stressed, while in mathematical logic axioms were emphasized. Moreover, he states that in mathematical logic, especially in the logicist version represented by Russell, logic was held to contain all mathematics, while in algebraic logic it was maintained that logic had some relationship with mathematics. In Grattan-Guinness’s view, algebraic logic used part-whole theory and relied on a basically extensionalist conception of a collection, while in mathematical logic the theory of collections was based on Cantor’s Mengenlehre. In addition, there was, in his view, an important diﬀerence between the traditions concerning quantiﬁcation; the interpretation of the universal and existential cases as inﬁnite conjunctions and disjunctions with the algebraic analogies of inﬁnite products and sums was typical of the algebraic tradition. Grattan-Guinness also notes that the questions addressed in mathematical logic were more speciﬁc than those addressed in algebraic logic. Frege’s and Peirce’s logical views are discussed in several chapters of the present volume. Many contributors also touch on the more general question concerning the borderline between traditional and modern logic, the divisions between the traditions of modern logic, and the shift from the modern logic of the late nineteenth century and the early twentieth century till twentiethcentury logic. The periods of Western logic that are studied in the present

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collection of articles extend from the thirteenth century to the end of the twentieth century. Unlike the rest of the contributions, the chapter on Indian logic covers several schools whose history reaches far back in the history but which are also living traditions in contemporary Indian logic.

3. Logic and the Philosophy of Logic Besides the term “logic,” the terms “philosophical logic” and “philosophy of logic” have various uses. Philosophy of logic can be understood as a subﬁeld of philosophy that studies the philosophical problems raised by logic, including the problem concerning the nature and the scope of logic. Those problems also include metaphysical, or ontological, and epistemological questions of logic, problems related to the speciﬁc features of logical formal systems (e.g., related to the basic vocabulary of logic) and logical validity, questions concerning the nature of propositions, judgments, and sentences, as well as theories of truth and truth-functions, and the questions concerning modal concepts and the alternatives of classical logic, which some call by the name “deviant logics.” The term “philosophical logic” is often used as a synonym of “philosophy of logic”; occasionally it means the same as “intensional logic,” or it is used as an opposite to “mathematical logic.” By metalogic, one normally means the study of the formal properties of logical systems, such as consistency and completeness, and thus distinguishes it from the philosophy of logic, which studies their philosophical aspects. The present volume deals with the history of modern logic and pays attention both to the core area of logic and to the philosophy of logic. Such terms as “classical logic,” “modal logic,” “alternative logics,” and “inductive logic” are also used and explained in the chapters of the volume. The variety of logics raises the problem of demarcation that is essential to the philosophy of logic: which formal systems belong to the objects of logical research, and which ought one to exclude from the ﬁeld of logic? For example, the program of logicism, which was supported by Frege, among others, was a position taken in the discussion concerning the demarcation of logic. Logic and philosophy have complicated relations. Nowadays logical tools are often used as the methods of philosophy. Logical discoveries have also been motivated by philosophical views, and philosophers have changed their opinions because of logical discoveries. Logic can be said to have a philosophical basis, and likewise there are philosophical doctrines that rely on developments of logic. The present collection of articles studies some of those relations. To some extent, it also pays attention to the relations between logic and mathematics and logic and linguistics. Logic and rationality are often tied together, but the concept of rationality has many uses in everyday language and in philosophical discussion. We talk about logical or argumentative rationality and refer to one’s ability to reason or to give arguments, and we also think that one who is rational is able to

Introduction

9

evaluate various views critically and independently of authorities; in this latter meaning, logic is considered to play a signiﬁcant role. Moreover, rationality is both theoretical and practical, the latter form of rationality being related to a person’s actions, and philosophers also tend to regard one’s ability to control one’s volitional and emotional impulses as a sign of rationality. There is no one concept or “essence” of reason that can be detected in philosophical or in everyday discussion. However, what we can ﬁnd in most uses of the concept is the general idea of control (control of thought, actions, passions, etc.), which is also central in logical rationality. Even if rationality as control or as rule-following seems to be crucially important, rationality as a faculty of judgment is also in everyday use in the practice of logicians as in all science. In the tradition of logic, it has been important both to be able to follow rules or repeat patterns and to be able to evaluate the commands and prohibitions. It is important both to be able to think inside a given system and to be able to evaluate the very system from the outside. The history of modern logic is a history of these two huge projects. Philosophers and logicians have used the volume titled The Development of Logic by William and Martha Kneale (1962) for decades. The ambitious idea behind the present work was to write a book on the development of modern logic that would bring the history of modern logic till the end of the twentieth century and would also pay attention to the philosophy of logic and philosophical logic in modern times. The idea was not to bring about a handbook but a volume that would be as close as possible to a one-author volume, that is, a balanced whole without serious gaps or overlaps. It was taken for granted in the very beginning that that goal cannot be reached in all respects. Each author has chosen his or her style, some wish to give detailed references, others are happier with drawing the main lines of development with fewer details; some express their ideas in many words, while others prefer a concise manner of writing. However, what has been reached is a story that covers a number of themes in the development of modern logic. The history begins with late medieval logic and continues with logic and philosophy of logic from humanism to Kant, that is, with two chapters whose scope is chronologically determined. Chapters 4–7 cover the nineteenth century and early twentieth century in certain respects, namely, they focus on the emergence of symbolic logic in two ways, ﬁrst, by paying attention to the relations between logic and mathematics, second, by emphasizing the connections between logic and philosophy. That discussion is completed by a chapter that focuses on the themes of judgment and inference from 1837 to 1936. The volume contains an extensive chapter of the development of mathematical logic 1900–1935, which is continued by a discussion on main trends in mathematical logic after the 1930s. The subﬁelds of logic that are called modal logic and philosophical logic are discussed in two separate chapters, one dealing with the history of modal logic from Kant until the late twentieth century and the other discussing logic and semantics in the twentieth century. Separate chapters are reserved for the philosophy of alternative logics, for the

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The Development of Modern Logic

philosophical aspects of inductive logic, for the relations between logic and linguistics in the twentieth century, and for the relations between logic and artiﬁcial intelligence. Eastern logic is not covered, but the main schools of Indian logic are presented in the last chapter of the volume. While the former part of the volume is chronologically divided, the chapters of the latter part follow a thematic division.

Note I have used extracts from my article “Peirce and the Logic of Logical Discovery,” originally published in Edward C. Moore (ed.), Charles Peirce and the Philosophy of Science (University of Alabama Press, Tuscaloosa, 1993), 105–118, with the kind permission of University of Alabama Press. The chapter also contains passages from my review article “Perspectives on Peirce’s Logic,” published in Semiotica 133 (2001), 157–167, which appear here with the kind permission of Mouton de Gruyter.

References Frege, Gottlob. [1879] 1964. Begriﬀsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. In Frege (1964), 1–88. Frege, Gottlob. 1964. Begriﬀsschrift und andere Aufsätze, ed. Ignacio Angelelli. Hildesheim: Georg Olms. Frege, Gottlob. 1893. Grundgesetze der Arithmetik, begriﬀsschriftlich abgeleitet, I. Band. Jena: Verlag von H. Pohle. Goldfarb, Warren D. 1979. Logic in the Twenties: the Nature of the Quantiﬁer. Journal of Symbolic Logic 44: 351–368. Grattan-Guinness, Ivor. 1997. Peirce between Logic and Mathematics. In Studies in the Logic of Charles Sanders Peirce, ed. Nathan Houser, Don D. Roberts, and James Van Evra, 23–42. Bloomington: Indiana University Press. Hintikka, Jaakko. 1979. Frege’s Hidden Semantics. Revue Internationale de Philosophie 33: 716–722. Hintikka, Jaakko. 1981. Semantics: A Revolt against Frege. In Contemporary Philosophy, vol. 1, ed. Guttorm Fløistad, 57–82. The Hague: Martinus Nijhoﬀ. Kneale, William and Martha Kneale. [1962] 1984. The Development of Logic. Oxford: Clarendon Press. Peirce, Charles Sanders. 1931–1935. Collected Papers of Charles Sanders Peirce, vols. 1–6, ed. Charles Hartshorne and Paul Weiss. Cambridge, Mass.: Harvard University Press. van Heijenoort, Jean. 1967. Logic as Calculus and Logic as Language. Synthese 17: 324–330. Whitehead, Alfred North, and Bertrand Russell. [1910, 1912, 1913] 1925–1927. Principia Mathematica I–III. London: Cambridge University Press.

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Late Medieval Logic Tuomo Aho and Mikko Yrjönsuuri

1. The Intellectual Role and Context of Logic Our aim is to deal with medieval logic from the time when it ﬁrst had full resources for systematic creative contributions onward. Even before that stage there had been logical research and important logicians. The most original of them, Abelard, achieved highly signiﬁcant results despite having only a very fragmentary knowledge of ancient logic. However, we shall concentrate on the era when the ancient heritage was available and medieval logic was able to add something substantial to it, even to surpass it in some respects. A characterization such as this cannot be adequately expressed with years or by conventional period denominations; we hope though that the grounds for drawing boundaries will become clearer during the course of our story.

1.1. Studies It was characteristic of later medieval logic that it was pursued as an academic discipline, as a major component in an organized whole of studies. Indeed, after the Middle Ages, logic has never been allotted so large a share in the activities of the universities. Moreover, logic was connected to certain classical texts that were seen as natural foundations of this science. Thus, it is reasonable ﬁrst to say something about the system of studies in general and about the nature of these works in particular. Ever since Rome, school teaching had always centered on the trivium of grammar, rhetoric, and dialectic. When schools developed and the most prominent clusters of schools began to turn into universities, these disciplines 11

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found their place in the faculty of arts (artes). Dialectic, the art of arguing and reasoning, was largely concerned with logical issues, and was often taken to be the most important art of the trivium. Thus, the outcome was that every student had to take extensive courses in logic. Perhaps the dialectical background can throw some light upon the linguistic and semantic tone of medieval logical thought. The faculty of arts was always much bigger than the higher faculties (theology, law, medicine). If there was a theological faculty in a university, it was associated with advanced studies and required a preliminary education in arts. But philosophical and logical research was pursued by theologians even after proceeding to the higher faculty; in fact, the most competent scholars often preferred the privileged higher faculty. Thus the history of logic must take into account the production of both faculties. Many commentaries on Peter Lombard’s theological Sentences contain important passages on logic, and topics related to logic are often dealt with in the so-called quodlibetal disputations, to mention just two examples. We cannot pay much attention to the history of universities, though we can say that the process of university education started in Italy in the twelfth century, Bologna being the oldest university. Paris, however, was undoubtedly the most important university for philosophy, and it received its oﬃcial statutes in 1215. Paris was a permanent international center for current philosophical and theological discussion. Another place where logical research was often especially popular was Oxford. These were the two capitals of medieval logic, although the center of gravity shifted to Italy in the less innovative period toward the end of the fourteenth century. During the fourteenth century, universities spread to the east and to the north. There were 15 universities in 1300, 30 in 1400, and about 60 in 1500, naturally of very diﬀerent size and quality, though one component of studies was standard everywhere, and that was logic.

1.2. The Growth of Logic Medieval philosophers normally made use of an array of authoritative classical texts, which were taken to be trustworthy, though not infallible. The curriculum was organized around these texts, and very often the problems discussed were put forward as questions of interpretation and explication of the texts. Hence the general breakthrough of Aristotelianism in the thirteenth century represented a great change, establishing Aristotle as the main source of academic studies. But in logic Aristotle had even before that been regarded as the greatest of authors, and anti-Aristotelian reactions did not seriously extend to logic. Rather than being rejected in the Middle Ages, Aristotle’s own work in logic was built upon and developed ever further toward the end of the period. The famous standard translation for most of Aristotle’s texts was that by William Moerbeke. With the logical works the case was diﬀerent: Though

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Moerbeke translated some of them in the 1260s, the authority of the old Roman translation by Boethius (c. 480–524) remained unquestioned. The Organon that late medieval logicians used was the Latin text of Boethius. (For Posterior Analytics, no translation by Boethius is known; its standard rendering was made by James of Venice before 1150.) These translations are actually quite accurate, although written in a very formal and literal idiom. Three concise basic works belonged to the kernel of logic throughout the Middle Ages. These were Aristotle’s own short Categories and De interpretatione, and Porphyry’s introduction, Isagoge. In addition to these, the so-called old logic (logica vetus) used Boethius’s logical works and a few minor ancient texts (by, e.g., Apuleius and Augustine). The shape of logic changed considerably when Aristotle’s complete works of logic became known in the middle of the twelfth century. That opened the way for the new logic, logica nova, and in a relatively short time the corpus of logica vetus was practically replaced by new works. Even Boethius’s treatises on syllogisms fell into disuse. Except for Aristotle and Porphyry, the only work that retained its place was Liber sex principiorum, a treatise explaining the categories that Aristotle himself does not dwell upon. The period of logica nova used as its authoritative corpus all the six works in Aristotle’s Organon: Categories, De interpretatione, Prior Analytics, Posterior Analytics, Topics, and Sophistici elenchi. At ﬁrst, dialecticians were especially fascinated by fallacies and sophisms (Soph. el.), but gradually the investigation turned more toward the formal theory of syllogism (Pr. Anal.). During the thirteenth century, they encountered problems that could not be answered by straightforward Aristotelian principles, and were thus drawn to new ﬁelds of logic. After the introduction of such new subjects, logic came to be called logica moderna, in contrast to logica nova, now called logica antiqua. This way of speaking, however, did not imply any break with the earlier Aristotelian tradition, only an expansion of investigation. The ﬁrst complete handbooks of logica moderna date from the second quarter of the thirteenth century. The earliest known overview is Introductiones in logicam by William of Sherwood from the 1230s, but the greatest success of all was the Tractatus, also called Summulae logicales, by Peter of Spain (probably from the 1240s). This comprehensive work maintained its status as a famous standard textbook throughout the later Middle Ages and the Renaissance, even in the time of printed books. It also served as the source for numerous shorter courses. Similar ambitious textbooks were written by Roger Bacon (Summulae dialectices, 1250s) and Lambert of Auxerre (Logica, 1250s). In a way, these works can be seen as a synthesis of the founding period of logica moderna: On the one hand, they were the ﬁrst systematic presentations of whole logic, on the other hand, they completed the new so-called terminist logic. Simultaneously a more profound philosophical discussion was started by the inﬂuential Robert Kilwardby, who wrote one of the ﬁrst commentaries on Prior Analytics (1240s).

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As to the logical content in the overall presentations of logic, signiﬁcant advance comes only much later, in the generation of Walter Burley (c. 1275– 1344) and William Ockham (c. 1285–1347). Ockham’s Summa logicae is from the early 1320s, Burley’s De puritate artis logicae from the late 1320s. These works manifest a turn in logical literature toward new problems and to a more theoretical way of thinking. The greatest representative of the next period is John Buridan (c. 1300–1361?); a comprehensive picture of his teaching in Paris is given in his Summulae de Dialectica. In the latter half of the fourteenth century, logic was already highly technical. In particular, a series of Englishmen distinguished themselves, among them William Heytesbury (d. 1372?), Ralph Strode (d. 1387?), and Richard Lavenham (d. 1399). A kind of summary of this stage is the enormous Logica magna (c. 1400) by the Italian Paul of Venice (c. 1369–1429).

1.3. Non-Latin Traditions Our account will be only about the Latin West. The signiﬁcance of Arabic philosophy must be emphasized, and yet we shall not discuss the Arabic logic per se, as it had its creative phase long before the time of Western late medieval philosophy. Aristotle’s Organon was translated into Arabic in the ninth century in Baghdad, and a commentary tradition started soon after that. Logic was honored as a kind of grammar of reasoning, and for example, al-Farabi (c. 870–950) underscored its importance as the “forecourt of all philosophy.” Avicenna (Ibn S¯ın¯a, 980–1037), on the other hand, was already a brilliant, independent exception: During his time, logical research was already declining, and commentaries were replaced by handbooks. His work had a profound inﬂuence on Western theories of meaning. In the twelfth century, the Spanish Arabic school revived commentaries, and the last commentator, Averroes (Ibn Rushd, 1126–1198), was also the greatest. The works of Averroes, “the Commentator,” were soon translated and became highly appreciated in Europe. In logic he was not as dominant as in metaphysics or in natural philosophy, but undoubtedly his works belong to the background that was always present. Averroes’s thought survived mainly in the West. In the Islamic world, logic was integrated into studies of theology and law, and even handbooks were gradually replaced by more or less elementary textbooks. During the period we describe, from the thirteenth century on, Arabic logic no longer produced anything but new versions and editions of established textbooks. On the other hand, a rich tradition of Jewish philosophy was alive in Europe through the late Middle Ages. Logic was not its favorite ﬁeld, but some Jewish authors paid considerable attention to logical questions. However, these studies had little interaction with Latin logic, and thus had to rely solely on Aristotle as commented by Averroes and al-Farabi. Still, there were some innovations, the most interesting ﬁgure being Gersonides (1288–1344). Writing in a rigorous manner, he made a number of criticisms of traditional doctrines; among other

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things, he rejected the old Averroistic construction of modal syllogistics in Prior Analytics (thus paralleling contemporaneous Latin developments). Even today, very little is known about Byzantine logic. Apparently an uninterrupted interest in logic, “the instrument of philosophy,” existed among Byzantine scholars. It produced mainly Aristotelian commentaries, often in the neo-Platonist spirit. Its independent progress was severely hindered by a conservative, philological approach to Greek sources, and occasionally also by religious scruples against the pagan heritage.

1.4. Texts Aristotle was the essential basis of later medieval logicians, but other classical ideas also played their part. First, Greek Aristotelian commentators had discussed various problems in Aristotle’s logic and its correct systematization, and their work became partly known (either directly or through Arabic sources). Second, the Stoics had argued that Aristotelian predicate logic was insuﬃcient and required some background from the propositional logic that they studied as the real logic. No complete Stoic works were preserved, but these Stoic themes were transmitted, for example, by Augustine’s Dialectica and by Boethius. We shall meet similar problems in the medieval theory of demonstration in topics and consequences. Because of the Stoic inﬂuence, medieval logicians were always in a diﬀerent position from the ancient Peripateticians in that they were aware of the necessity of essentially nonsyllogistic inference. Furthermore, logic was obviously inﬂuenced by classical grammar, which provided it with categories like nouns, verbs, and other parts of speech, as well as central syntactical notions, the main authority being Priscian’s grammar of Latin. Finally, some logical material had found its way into the work of famous ancient authors, among them Cicero, and the Christian fathers. From the middle of the thirteenth century, there was a rapid increase both in logical studies and in Aristotelian studies in general. Soon the obligatory logical curriculum included the whole of the Organon. Aristotle’s text is so concise and diﬃcult that it was always accompanied by commentaries and explanatory texts. It was required that students mastered this material thoroughly, and practical logical exercises became very popular as a supplement to lectures. A major and growing part of studies was dedicated to logic. If we understand logic in the widest sense, it appears that more than half of the program of an arts faculty could be about logic. At least we may note that logic had an undisputed place in medieval learning, and that it was not a specialist subject since almost all leading philosophers wrote about logic. Most logical works were closely connected to university teaching. The usual teaching method in medieval universities was that a text was lectured on and explained in detail. The intention was to build a consistent interpretation of the text, to eliminate ambiguities and to resolve the problems and conﬂicts the text gave rise to, and a typical medieval method of study included disputations where some theses were argued for and against. The character of university

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teaching goes some way to explaining the literary types that became widespread. In addition to simple lecture drafts, there were all kinds of commentaries, ranging from elementary glosses to large systematic books. There were treatises (tractatus), that is, manuals or more advanced surveys of some ﬁeld, which gradually became more independent of the underlying texts. The liking for argumentation and disputation produced quaestiones, analytic works where some speciﬁc question is resolved or a thesis defended. (Later, systematic studies were organized in the form of a series of questions even though they were often referred to as commentaries.) We must also remember that logical subjects are often encountered as digressions in other works, for example, in the extensive sentence commentaries of theologians. There has been quite a decisive improvement in the accessibility of medieval logic over the last three decades, when numerous texts have been published. However, a large amount of material still remains unpublished and even completely unstudied. In fact, it is quite possible that our whole view of the outlines of medieval logic will undergo a change; indeed, such changes have occurred before, and systematic historical research of this logic is still a very young enterprise.

1.5. Interdisciplinary Relations Obviously, logic had a well-established place in the system of disciplines in the Middle Ages. But what kind of interaction did logic have with the other sciences? Unfortunately, it is not easy to say anything deﬁnite about this. First of all, formal philosophy of science was studied by logicians in connection with the Posterior Analytics, which discusses the correct form and nature of deductive theories. In this way, the methodology and philosophy of science were a part of medieval logic. Also, the occasional attempts to create calculative scientiﬁc speculations used heavy logic, but in general there was little concrete connection of logic to particular natural sciences, which took care of their own subjects. On the other hand, metaphysics—universally considered a real science—was always relevant for logic. Thus, semantic theory, so prominent in medieval logic, is immediately bound to metaphysical questions. Just as early supposition theory employs a metaphysical basis, so in the latemedieval nominalist trend it is impossible to separate logical from ontological thought. The role of theological matters is less transparent. Obviously theology needed systematic thought and conceptual analysis, and was hence favorable to logic. The conceptual examples and diﬃculties that logicians examined were very often drawn from theology. Generally, the signiﬁcance of theology for logic must have been positive. Their union was made problematic, however, when many philosophers began to think that some mysteries of faith, such as the Trinity, were not only inscrutable but literally beyond logic—even if their exact formulation could be a task for logic. Thus Ockham and Buridan thought that certain theological notions had to be explicitly declared unsuitable for use as

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substitution instances in ordinary logical principles, and a few authors were even more radical on this point. On the other hand, there was—of course—some religious hostility which regarded logical reasoning as an unhealthy method in theological matters.

2. Language as the Subject Matter of Logic Thinkers in the Middle Ages were anxious to discuss the correct system and classiﬁcation of sciences. Since their philosophy of science was realist, they believed that the classiﬁcation should be based on the order of nature. Logic, however, clearly has special features that make its place in this scheme problematic. Is it a science that has as its subject matter some part or aspect of reality? Or is it merely the art of using linguistic idioms? Or is its function something else altogether?

2.1. A Science “of Words” or “of Reason” The ﬁrst known medieval textbook of logic, Garlandus Compotista’s Dialectica from the late eleventh century, already sets the discussion of this topic on a track that was to have crucial inﬂuence on the kinds of innovations that were to be achieved in medieval logic. Throughout the Middle Ages, logical theories had a very intimate relation to actual language use. According to Garlandus, logic is concerned with actual utterances (voces). After Garlandus, Abelard, for example, continues on the same track, but reﬁnes the position: As he sees it, statements are not built from mere spoken sounds but from words that have a signiﬁcation (sermones). Thus, they also constitute the subject matter of logic. Logic is “a science of words” (scientia sermonicalis). It seems that well into the thirteenth century the idea that logic studies actual language use remained basically unchallenged. Teachers and students of logic considered that their studies helped in the acquisition of argumentative skills for actual scientiﬁc disputations. Given the status of Latin as the language of all medieval learning, it was natural to make the appropriate logical distinctions from the viewpoint of spoken Latin. This gave an important status to essentially linguistic structures even in the later developments of medieval logic. In approaching many particular features of medieval logic, it is crucial to remember this pragmatic way of looking at the subject matter of logic. In the Middle Ages, the art of logic was not taken to be concerned with abstract structures in the way modern logic and modern mathematics are, but with actual linguistic practices of reasoning. It was generally accepted that logic is, at least in some sense, a practical science giving advice on how to understand and make assertive statements and how to argue and reason in an inferential manner—though opinions varied whether this practical characterization of logic was accurate in any deeper sense.

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In the Arab world logic was thought of in a diﬀerent manner, and thus toward the thirteenth century under Arabic inﬂuences the Latin world became aware of a diﬀerent way of looking at the character of logic as a ﬁeld of research. According to al-Farabi, the logos (in Arabic, al-nutq) discussed in logic occurs on two levels, one inscribed in the mind, and the other existing externally in spoken sounds. Thus, we may even separate diﬀerent senses of the Greek word logos in accordance with the level of discourse at issue. Avicenna was also inﬂuenced by al-Farabi’s discussion, and gave even further impetus to the idea that logic is concerned with intellectual structures rather than with what we do in spoken discourse. Thus, logic should be called “a science of reason” (scientia rationis), as the Latin world translated the idea. In the thirteenth-century Latin tradition, both the idea of logic as “a science of words” and as “a science of reason” had a foothold. In his major classiﬁcation of all the university disciplines, De ortu scientiarum, Robert Kilwardby (c. 1215–1279) gave a deﬁnition of the nature of logic that combined the two views. It is worth taking a closer look at his deﬁnition, because it also clariﬁes the medieval way of locating branches of logic in terms of Aristotle’s logical works in the Organon. According to Kilwardby, logic is “a science of words” (scientia sermonicalis) in the sense that “it includes grammar, rhetoric and logic properly so-called.” But as Kilwardly immediately points out, “in the other sense, it is a science of reason,” and in this sense it is “distinguished from grammar and rhetoric.” It may seem that here Kilwardby would be demarcating two diﬀerent disciplines both ambiguously called “logic.” But this is not really his intention, as he hastens to explain: Logic properly so-called must in his opinion be listed as one of the “sciences of words”; it is the science of words that attends to their rational content. As he sees it, logic does not study arguments as mere words nor as mere rational structures, but as rational structures presented in linguistic discourse. The grammatical and rhetorical features of these arguments, for example, do not pertain to the art of logic. Logic studies the rational structures expressed and understood in linguistic discourse—neither rational structures as such, nor linguistic structures as such. The core of logic can in Kilwardby’s view be found in Aristotle’s Prior Analytics. This is because at its core, logic is concerned with reasoning, and this is the main topic of Prior Analytics and its system of syllogistic reasoning. It is of some interest to note that Kilwardby is very Aristotelian in claiming that all forms of valid reasoning can be reduced to the categorical syllogism discussed in Prior Analytics. This was not the received view at the time, and Kilwardby’s position did not win unconditional approval. Abelard had discussed the theory of conditional inference and clearly would not have accepted such a principle. Indeed, conditional inferences were throughout the Middle Ages a standard part of logical curriculum. Soon after Kilwardby, toward the end of the thirteenth century, the theory of consequences (consequentiae) grew into a self-conscious general theory of inference that had no speciﬁc reference to the syllogistic system; syllogism was increasingly presented as a special case of inference.

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Kilwardby pushes aside one aspect of Aristotle’s discussion in the Prior Analytics. According to Kilwardby, only dialectical and demonstrative syllogisms are relevant to logic, while the rhetorical syllogisms discussed by Aristotle fall out of the scope of logic because they take “the form that is suited to the singular, sensible things considered by the orators.” Logic as a science is concerned with universal rational structures as captured in discursive reasoning. In Kilwardby’s presentation of the structure of logic, the system developed in the Prior Analytics is put to further use in the Posterior Analytics and the Topics. As Kilwardby sees it, the division into diﬀerent works is based on the matter to which the syllogistic structures are applied. The Posterior Analytics discusses the way in which the syllogistic form is applied to “speciﬁc matter” and yields scientiﬁc demonstrations. For its part, Topics is concerned with “common matter” and shows how we can construct good inferences relying on generic considerations. Aristotle’s Sophistici Elenchi, for its part, plays in Kilwardby’s view the role of considering what can go wrong in constructing an inference. As Kilwardby shows, the role of De interpretatione and Categoriae can also be considered in terms of the syllogism. De interpretatione considers the propositional structures that are essential for constructing syllogisms. A syllogism must be construed so that it has a middle term, and for this purpose it is necessary to see how assertive statements usable as premises can be built to consist of two terms conjoined aﬃrmatively or disjoined negatively. Categoriae goes into the structure even more deeply, considering the terms and their signiﬁcation in reality. From the mid-thirteenth century onward, Avicenna’s conception of logic as a science of reason gained increasing currency in philosophical discussions on the subject matter of logic. To some extent this happened at the expense of the earlier view of logic as a science of words. As we have already seen, Kilwardby restricts the meaning in which logic is a science of words so that it no longer carries much weight. Albert the Great’s general position concerning the nature of logic is similar, but in the beginning of his commentary on Aristotle’s Categories he takes the explicit position that logic is strictly speaking not a “science of words” at all. Rather, logic is concerned with argumentation, and argumentation should be referred to reason rather than to words. Albert’s student Thomas Aquinas (1224?–1274) followed him in this matter. In his more elaborate system, the subject matter of logic consists of three conceptual operations of the mind, namely, formation of concepts, of judgments, and of inferences. This systematization can be traced back to Plotinus and the neo-Platonic commentators of Aristotle’s logic in a more explicit way than Kilwardby’s system. The ﬁrst two operations are discussed, respectively, by Aristotle’s Categories and De interpretatione, and the third by the other four works included in the Organon. As Aquinas saw it, making a judgment—and, in fact, anything that logic is concerned with—requires an intellectual act of understanding. Thus, making a judgment is not primarily to be understood as a speech act but as a mental act. According to Aquinas, externally

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expressed linguistic structures should be seen as results and representations of intellectual acts, and only in this intermediate way does logic come to be concerned with linguistic structures. In the ﬁrst place, logic is concerned with the intellectual operations by which the universal features of material reality are understood. The detailed structure of Aquinas’s presentation of the subject matter of logic has the crucial feature that it relies heavily on the Aristotelian idea that all inferences can be presented as syllogisms. As Aquinas saw it, all of logic can be understood in terms of syllogistic structures. Since he thought that logic deals with the three basic operations of the intellect, any inference would have to be based on them. However, there are understandably quite stringent limitations on the extent to which logic can be derived from these basic operations. For example, with the claim that all assertions are made by the composition of a predicate with a subject, Aquinas was almost forced to reject conditionals as assertions. However, hypothetical propositions had a long tradition deriving from Stoic logic and had been dealt with already by Boethius, and thus Aquinas was compelled to comment on them. As he put it in the ﬁrst section of his commentary on De interpretatione, hypothetical propositions “do not contain absolute truth, the understanding of which is needed in demonstration . . . but they signify something to be true on condition.” According to Aquinas’s logic, conditionals could not be used as premises in scientiﬁc demonstrations. Neither Albert nor Aquinas worked much with the actual details of logical systems, and their discussion has more of the character of the philosophy of logic. However, the distinctive ﬂavor of medieval logic showed itself in its close connections to actual language use, and it incorporated analysis of a much wider variety of linguistic structures than the simple predications included in the syllogistic presented in the Prior Analytics. Moreover, Abelard’s work had already made medieval logicians acutely aware of a concept of inferential validity that was essentially unconnected to the syllogistic structure. While Kilwardby, Albert the Great, and Aquinas defended the strong Aristotelian program of reducing all inferential validity and thereby all logic to an analysis of the syllogistic system, actual work in logic was taking another course. In the subsequent development, Aquinas’s three operations of the mind were often referred to, but usually understood in a loose and suggestive manner. It became standard to treat logic with the organizational principle that Categories studies concepts and De interpretatione propositions, while Prior Analytics and the three subsequent Aristotelian works concentrate on inferences. It was, furthermore, commonly accepted that there are many traditional logical genres inherited from the twelfth century that do not ﬁt into this basic scheme. For example, there was an abundance of literature on the so-called syncategorematic terms, analyzing the logical properties of words such as “except” (praeter), “begins” (incipit), “whole” (totum) and “twice” (bis). Such problems had little connection to the development of syllogistic systems. Furthermore, it remained

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a problem to explain how such logical genres could ﬁt into the description of logic as a science of reason, because many of them were quite clearly motivated by the analysis of linguistic structures.

2.2. Mental Language An interesting alternative way of characterizing logic as concerned with mental concepts rather than Latin words was being developed at the time Aquinas was working, and it gained momentum among logicians in the latter half of the thirteenth century. It was based on quite a diﬀerent understanding of the workings of the human mind from that of Aquinas’s Aristotelian outlook. Roger Bacon (c. 1214–c. 1293) rejected the idea that the human understanding works only with real universals existing intentionally in the mind. Rather, the mind should be understood in terms of a discourse consisting of singular acts of intellection whereby diﬀerent singular things are understood in diﬀerent ways. According to Bacon, logic is not concerned with an external discourse but with the internal discourse of the mind, with “mental expressions and terms” (dicciones et termini mentales). In other words, Bacon posits a mental language to serve the role of the subject matter of logic. As we shall see, this approach was to play a major role in later developments. First, however, we must take a closer look at the content of Bacon’s suggestion. One of the central classical texts that Bacon refers to was Boethius’s distinction between three levels of discourse (oratio): intellectual, spoken, and written. In making the distinction, Boethius was commenting on Aristotle’s De interpretatione 16a10, and Boethius’s way of reading the passage was well known in the late Middle Ages, but it remained a debated issue how one should understand the intellectual level of discourse and how one should relate logic as a discipline to these levels. It seems clear, though, that Bacon understood the intellectual discourse in a way that can with good reason be called linguistic. He even takes pains to show how word order functions in this discourse. Without going into details, it is suﬃcient here to point out that he looked at the structure of mental sentences in terms of Aristotelian predication: The subject comes ﬁrst, then the predicate, both with their “essential determinations.” They are then followed with the various “accidental parts” of the composition. Especially his way of dealing with these “accidental parts” shows how looking at thought as a linguistic phenomenon gives Bacon a clear advantage in comparison to Aquinas from the logician’s point of view. Through his theory of mental language, Bacon is able to attribute considerably more logically relevant linguistic structure to the intellectual level. One of the aims of this enterprise was—as is evident to any logician—to show how to solve ambiguities of scope arising in Latin through the relatively loose rules concerning word order. In this way, Bacon worked toward a theory of an ideal language to serve logical functions as early as the 1240s, if the current scholarly opinion of the date of his Summa de sophismatibus et distinctionibus is correct.

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The mental discourse that Bacon was after is abstracted from spoken languages like Latin and overcomes their arbitrariness. However, there is also the other side of the coin: He speciﬁcally wanted to ﬁnd the basis for certain logically relevant Latin structures from mental discourse. Although he thought that grammatical gender has no correlate in mental discourse, the subject-predicate structure and many syncategorematic expressions have. Indeed, Bacon seems to ﬁnd from the mental discourse even more than a logician would need. In many issues, it becomes apparent that he was working more as a linguist than as a logician. His aim was a universal grammar rather than a universal language suitable for logic. Commentators have, accordingly, connected Bacon to the movement of speculative grammar emerging in the latter half of the thirteenth century. The approach to linguistic analysis employed by this school is often called “modist.” The label reﬂects the speciﬁc use of a threefold series of concepts: “Modes of being” (modi essendi) in reality were paralleled in language by “modes of signiﬁcation” (modi signiﬁcandi) and in the mind by “modes of understanding” (modi intelligendi). The movement was more closely connected to language theory than logical theory, and accordingly we will only discuss it brieﬂy here. The main idea of modist theory was to approach Latin expressions as generated from a universal grammatical structure accurately reﬂecting the structure of reality. That is, they thought that grammar is (in the words of Bacon) “substantially one and the same in all languages, although varied in its accidents.” Other central ﬁgures of this movement include Boethius of Dacia, Martin of Dacia, and Radulphus Brito. At the beginning of the fourteenth century, the program lost ground, although much of the terminological innovations, including the term “mode of signifying,” survived until the Renaissance in the standard vocabulary of logicians. According to the modists, all words have two levels of meaning. Words have in addition to their own speciﬁc meanings certain more general meanings, or so-called modes of signifying. To be more exact, a phonological construction gains a special meaning when it is connected to a referent that it “is imposed” (imponitur) to mean (in the so-called ﬁrst imposition). Furthermore, the word is also “imposed” (in the second imposition) to mean its referent in a certain grammatical category with certain modes of signifying. For example, pain can be referred to by a variety of Latin words in diﬀerent grammatical categories: dolor refers to it as a noun, doleo as a verb, dolens as a participle, dolenter as an adverb, and heu as an interjection. In all these words the special signiﬁcation is the same, but the modes of signifying are diﬀerent. The modists found no theoretical use for the most central logical term of the terminists, “supposition” (suppositio; it will be described with more detail in the next section). In their view, the varieties of ways in which words are used in sentential contexts are based on modes of signifying contained in the words, and thus they were not willing to admit that the sentential context as such would have an eﬀect on how the term functions—which is one of the leading principles

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of the supposition theorists. Rather, their approach was generative in the sense that the sentences were to be generated from words that have their signiﬁcation independently. This approach made it unnatural to distinguish the sentential function of a term from its signiﬁcation. It may, however, be noted that the term “consigniﬁcation,” meaning the function of syncategorematic terms in the terminist approach, was used by modists to express the way in which phonological elements of actually used words mean modes of signiﬁcation: For example, the Latin ending -us “consigniﬁes” nominative case, singular number, and masculine gender. The thirteenth-century grammarians recognized the congeniality of syncategorematic terms and modes of signifying: Both are understood as the elements of discourse that show how the things talked about are talked about and what in fact is said about them. From the viewpoint of the history of logic, it is important to recognize that from the twentieth-century viewpoint, the modist conception of grammar can be characterized as making the subject a “formal science.” The criteria of congruence were taken to depend solely on the grammatical structure, or the consigniﬁcations of the elements of the sentence, regardless of the special signiﬁcations of the terms used in the sentence. Modists thought of the generation of language as putting semantically signiﬁcant elements into grammatical structures. It seems that at least the Parisian master Boethius of Dacia wanted to develop also logic into this direction and wanted to make a distinction somewhat like the twentieth-century distinction between logical form and semantic content. Nevertheless, it was only some decades later at the time of John Buridan that the substance of logic was thoroughly reconsidered from this viewpoint.

2.3. The Universality of Logic From the viewpoint of practicing logicians, the debate concerning the subject matter of logic at the end of the thirteenth century probably seemed like a search for a credible account of the universal basis of the invariable features of argumentation found in the logical analysis of actual use of language. That is, what is the universal basis on which the validity of an inference formulated in a particular language is grounded? It was accepted as relatively clear that logic is about actually or potentially formulated tokens of terms, propositions, and arguments that are linguistic in some sense of the word. It was clear that such discursive arguments existed in such external media as spoken or written Latin expressions. However, logic aimed at, and appeared to have found, some kind of universality, and such universality apparently could not be achieved if logic was tied to a particular spoken language. Instead, thirteenth-century discussions converged in ﬁnding the universality of logic in intellectual operations. But what are these intellectual operations? Can we speak of a mental language serving as the domain of logic? In particular, is a mental proposition linguistic in any relevant sense? And because it was assumed that an aﬃrmative predication is

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based on or performs a composition, one had to ask what exactly does this composition put together. At the turn of the fourteenth century, we ﬁnd diﬀerent logicians giving diﬀerent answers to these speciﬁc questions. At that time, the most common answer was the one inspired by Bacon. It was based on looking at the mental discourse from the viewpoint of “imagined spoken words,” and accepting it as the privileged medium of logical arguments. This kind of explanation is straightforward and relatively acceptable from the metaphysical viewpoint, but is, of course, less satisfactory in explaining the kind of universality achieved in logic. If mental language is nothing but imagined Latin words, there seems to be little reason for assuming it to have any more universal status than Latin has. Yet that appears to be what Bacon wished to propose. The realist Walter Burley seems to have approached the problem from the viewpoint of the universality of logic. Given his realist metaphysics, it is understandable that he contributed the concept of “real proposition” (propositio in re). He aimed at explaining mental propositions as consisting of real external things, which are conceived and propositionally combined in the mind. This model of the metaphysical basis of mental language of course works only if conceptual essences are understood in a realist way without separating them from the things themselves. Also, such “real propositions” are not very language-like. The nominalist William Ockham formulated the most innovative and by far the most inﬂuential theory of mental language. He ridiculed the position of Burley by asking how it could be that the subject of a proposition formulated in Oxford is in Paris while the predicate is in Rome. A suitable example of such a proposition would be “Paris is not Rome.” Ockham seems to have gone back to Bacon’s theory, but with the awareness of some of its shortcomings. With his nominalist metaphysical outlook, he strongly held the view that all the metaphysically real things involved in mental propositions are particular mental acts or states. But the substantial logical strength of his theory of mental language was really that it was formulated in a way that was suﬃciently neutral from the metaphysical point of view. Indeed, Ockham himself started with the idea that mental language consists of ﬁcta (that is, of intellectually imagined objects of thought that do not have any kind of existence outside the mind but are simply “made up” by the mind) but ended with the view that mental language is better understood as consisting of intellectual acts intentionally directed at real or possible things. At one stage of his career he was working on the theory of mental language without being able to make up his mind which of these two rather diﬀerent views would provide the appropriate metaphysical foundations. In the ﬁrst chapters of Summa Logicae, William Ockham addresses the Boethian idea of three levels of language. In opposition to Aquinas’s treatment of the same topic, Ockham claims that written language is subordinated to spoken language rather than signiﬁes it. Similarly, spoken language is subordinated to mental language rather than signiﬁes it. That is, according

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to Ockham, all three languages similarly signify things in the external world. They are, furthermore, all equally languages. Written language is inscribed on external material things, and spoken language exists as a continuum of sounds. Similarly, mental language consists of real qualities of the thinking mind. Furthermore, in Aquinas’s picture intellectual acts were the signiﬁcations of linguistic expressions and by their nature could not serve as a medium of communication. For Ockham, mental language could by its nature serve equally as a medium of communication if only there were beings who could perceive its expressions apart from the “speaker” him- or herself. In fact, Ockham thought that we have every reason to suppose that the angels described in the Christian doctrine communicate in the same language in which we think. The main diﬀerence between mental language and the two other kinds of language is the naturalness of mental language. Unlike spoken ordinary languages, which we nowadays call natural, Ockham’s mental language is natural in the sense of not being conventional. The expressions of mental language have their signiﬁcations naturally, without explicit or implicit consent or any other kind of conventionality involved. A mental word is capable of signifying only the things it really signiﬁes, and it signiﬁes exactly those things to all competent users of the word. (It may be noted that Ockham admits that in angelic communication some mental expression may be unfamiliar to the perceiver and thus unintelligible to him.) In principle, there are no ambiguous terms in the mental language. This is one of the central features that make Ockham’s mental language an ideal language, which is then suitable for the purposes of a discipline like logic. There are also two other senses in which Ockham aims at description of an ideal universal language. On the one hand, he tries to describe in general terms what must be required of any language that is used for thinking, and assumes that mental language has only such necessary features without any accidental “ornaments of speech.” Since these features are necessary requirements of thought, all thought must comply with them. Thus, Ockham constructs a theory of a language that is universal in the sense of being used by all intellects that think discursively. On the other hand, according to Ockham, mental language is directly related to the constitution of the world. It reﬂects accurately mind-independent similarities between real things. Thus, a fully developed mental language would be universal in its expressive power: There cannot be any feature of the world that could be conceived by an intellectual being but not expressed in mental language. Everything that can be thought can also be cast in terms of mental language. From this principle it also follows that all linguistic diﬀerences between expressions of spoken languages that result in diﬀerent truth values (which are not “ornaments of speech”) have their correspondents on the level of mental language. From the logical point of view, perhaps the most interesting ideal feature of Ockham’s mental language is its compositionality, which makes it a recursive system. Complex expressions get their meaning from their constituent parts

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in a systematic way. In this respect, mental language shows similarities to twentieth-century formal calculi, although it is much more complex. According to Ockham, the expressions of mental language consist of categorematic and syncategorematic parts with speciﬁc linguistic roles (we will return to this distinction more fully in the next section). A categorematic term (e.g., “animal”) signiﬁes real individuals and refers to them as the other elements of the propositional context determine. A syncategorematic term (e.g., “every”) does not signify any external things but rather, as Ockham puts it, “performs a function with regard to the relevant categorematic term.” Typically, syncategorematic terms aﬀect the way in which the signiﬁcations of the categorematic terms result in reference (or suppositio) in the sentential context. We may say that the categorematic terms of a sentence determine which things are talked about, whereas the syncategorematic terms determine how they are talked about and what is actually said about them. The number of basic categorematic terms of the ideal mental language accords to the variety of things that could exist in the world; they express the natural kinds of possible things. Ockham’s view of the number and selection of syncategorematic terms is more diﬃcult to determine. On the one hand, it is clear that he is thinking of a much wider variety of such logical constants than twentieth-century logic used. On the other hand, it is equally clear that most of his logical rules concern the eﬀects of syncategorematic terms on logical relations between sentences. Because the compositional characteristics of mental language depend on the distinction between categorematic and syncategorematic terms, Ockham’s mental language seems to conform to the twentieth-century ideal principle of logical formalism, namely, the idea that all sentences directly reveal their logical form. This seems to be one of the features of the mental language that Ockham is most interested in, and much of his logic is devoted to systems elaborating on the functions of syncategorematic expressions. However, Ockham’s theory has interesting details that reﬂect a conscious decision not to accept logical form (as we nowadays understand it) as the guiding universal principle in determining the logical validity of an inference. The theory of mental language was also discussed and developed after Ockham, but without major revisions. The most important innovator was John Buridan, who altered much of the terminology used in deﬁning language and gave a rather diﬀerent account of how the simple terms of language are learned, but these revisions resulted in few changes that would be relevant to our purposes here. After Buridan, some minor topics like the role of proper names and individual terms, and the nature of word order as explanatory of issues of scope were discussed. These can hardly be called revolutionary with regard to the nature and purposes of logic. At the peak of its success, medieval logic had thus found a deﬁnition of its subject matter that provided a relatively reasonable explanation both of its universality and of its dependency on discursive linguistic structures. For the logicians of the second quarter of the fourteenth century, logic was the art

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of constructing and using mental propositions. It studied the basic syntactic features of mental language, the ways and forms of assertions that can be produced in it, and the ways these assertions can be organized in inferential relations. Because mental language was understood as capable of expressing all possible universal structures of discursive thought, logic studied the universal art of reasoning.

3. Terms 3.1. The Notion of a Term Textbooks of “traditional logic” used to divide their material into three sections: the doctrines of terms, propositions, and inferences. This practice is based on ancient grounds, of course, but Aristotle nowhere says that all logic should be so divided, and medieval logic did not at ﬁrst do so. In thirteenth-century logic books, often the chapters are still relatively independent, or at least not organized according to such a general plan. But then, at the turn of the century, this idea soon became dominant. We ﬁnd it, for example, in both Burley and Ockham, in spite of their sharp disagreement. We are going to follow this familiar order, starting with terms. Everybody agreed that terms were the ultimate units of discourse. In a way this is obvious, but the emphasis on this fact in logical contexts also has a nontrivial sense which shows the Aristotelian character of medieval logic. For the logic that Aristotle had developed was term logic, unlike that of the Stoics. But Aristotle gave two diﬀerent explanations of terms. In Categories he speaks about noncomposite expressions (“such as ‘man’, ‘ox’, ‘runs’, or ‘wins’ ”). In Prior Analytics he says (24b16–18): “I call that a term into which a proposition is resolved, i.e., the predicate or that of which it is predicated, when it is asserted or denied that something is or is not the case.” These explanations lead to very diﬀerent uses of the word “term.” In the ﬁrst sense, a term is simply any word. Many medieval logicians mentioned even meaningless words, like “ba,” “bu,” but only to concentrate on ordinary words. In this sense, which is that of grammarians, it is only required that a term is a noncomposite signiﬁcant element of the language. Or it can be a composite expression signifying one thing. In the second sense, which is more exciting for the logicians, a term is something that can stand as a subject or a predicate of a proposition. This excludes wide classes of words from the status of terms. According to the strictest deﬁnition, a term is only that type of nominal expression that can ﬁgure as S or P in a categorical proposition “S is P.” This leads to a question concerning the structure of terms because S and P can be complex expressions. Buridan, for instance, took a strictly propositional view and argued that a simple proposition has exactly two terms. In this usage a term is identiﬁed with an extreme (extremum) of a proposition. But

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since the various words occurring within such terms can be extremes in other propositions, authors kept on saying that propositions can have complex terms that are composed of simple terms. The theory of terms is obviously connected to grammar, and Priscian’s classiﬁcations had a strong inﬂuence on earlier writers. But logically it was important to eliminate Latin contingencies and consider as general cases as possible. However, that is a problematic requirement regarding terms: What could be those language-independent terms? Diﬀerent ways to tackle this question systematically were oﬀered ﬁrst by so-called speculative grammar, and then by the mentalistic interpretation of language, which was ﬁnally victorious, but both approaches emphasized the universality of language. For late medieval logicians, the terms were in the ﬁrst place mental terms that occurred in mental propositions.

3.2. Categorematicity A distinction that is especially important for logic was made between categorematic and syncategorematic terms. This distinction was well known to all logicians, and they usually introduced it immediately after the deﬁnitions of terms. The source of these notions was in grammar, but logicians gave them a new function, following a hint from Boethius. Priscian had written about “syncategorematic, i.e., consigniﬁcant, parts of speech”: Most words are grammatically categorematic since they can occur as subjects or predicates, but for instance, conjunctions, prepositions, adverbs, and auxiliary verbs cannot. They are syncategorematic and signify only together with other words. Logicians proceeded from this picture to distinguish two ways of meaning and to describe the logical behavior of philosophically interesting syncategorematic words. Syncategorematic words were ﬁrst studied in special treatises. This genre of Syncategoremata was popular from the last quarter of the twelfth century to near the end of the thirteenth century. Well-known treatises of this kind were written by Peter of Spain, William of Sherwood, Nicholas of Paris, and even the famous metaphysician Henry of Ghent. Later, the subject was incorporated into general textbooks of logic. The distinction itself had its systematic place at the outset of the exposition of the theory of terms, since it was utilized in many questions; particular syncategoremata were then discussed in their due places. Even in the fourteenth century most authors apparently based their deﬁnitions of syncategoremata on diﬀerent ways of signifying. According to Ockham, “categorematic terms have a deﬁnite and determinate signiﬁcation. . . . Examples of syncategorematic terms are ‘every’, ‘no’, ‘some’, ‘all’, ‘except’, ‘so much’, and ‘insofar as’. None of these expressions has a deﬁnite and determinate signiﬁcation.” Buridan states: “Syncategorematic terms are not signiﬁcative per se, as it were, but only signiﬁcative with another.” Paul of Venice still defended this view against “a common deﬁnition” that a syncategorematic term cannot be the subject or the predicate or a part of either. Such a purely syntactical criterion had been supported by Albert of Saxony (1316–1390).

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Syncategorematic expressions were usually counted as terms. A theoretical reason for this was a slogan that was in use at least from Peter of Ailly onward: A term is a sign that in a proposition represents something or somehow. Syncategoremata, indeed, signify “somehow” (aliqualiter), for thirteenth-century treatises had already pointed out that syncategoremata serve to show how the categorematic terms ought to be understood. It is thus essential for their signifying that they are joined with other terms to elaborate their meanings. Present-day readers will easily associate syncategorematic terms with logical constants. This is partly correct but must not be taken too literally. For one thing, the class of syncategoremata of language is much wider than the small sets of logical constants nowadays. However, the medievals ignored most syncategoremata and studied only those which seemed to be philosophically interesting. These were just words with special logical peculiarities, and hence, for these terms, the comparison with logical constants may be justiﬁed. The lists of diﬀerent logicians varied greatly, but several dozens of words were thus discussed. Among them belonged sentential connectives; words like “only” and “except”; quantiﬁers; modal operators; words like “whole” and “inﬁnite”; some verbs like incipit and desinit (“begins” and “ends”); and the copula est, that is, the copulative use of the verb “to be,” esse. General textbooks listed them but did not usually go into details of particular syncategoremata. In the fourteenth century, such closer study often took place by means of sophismata: In this literature it was typical to analyze sentences that were problematic or ambiguous because of syncategorematic words (see section 6). Buridan expressly said that the matter of a proposition consists of purely categorematic terms while syncategoremata belong to its form. From this point of view, it is interesting to notice that the notion of syncategorematicity proved diﬃcult because it did not determine a precise class. Thus Buridan had trouble with attitude operators: Verbs like “to know” and “to promise” clearly have a formal function and yet they are independently meaningful. The two criteria, the semantical and the grammatical, did not always coincide, and Peter of Ailly suggested that they should be wholly separated. A term could therefore be syncategorematic either “by signiﬁcation,” or “by function,” or in both ways.

3.3. Predicables In a proposition something is said of something, as Aristotle taught. It is therefore logically important to have some idea of the various types of things that can be thus predicated, the predicables (praedicabilia). Medieval logicians based their classiﬁcation here on Porphyry and Boethius. Obviously, a predicable is something that can be said (predicated) of something else, but in a stricter sense, it is only a universal term that can be predicated of many things. This distinction was made already in thirteenth-century textbooks, and it is easy to see that predicables have a close connection to the most famous medieval metaphysical problem, the problem of universals.

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Explaining Aristotle’s Categories, Porphyry mentioned ﬁve types of universal terms: species, genus, diﬀerentia, proprium, and accidens. These were the “ﬁve universals” (quinque voces) that recur in medieval discourses. They reveal various relations of the predicate to the subject: What kind of information does the predicate give us about the subject? When it is said that S is P, the predicate P may express a species to which S belongs, or a genus to which every S belongs, or a characteristic essential feature of them (diﬀerentia), or a nonessential but necessary property of every S and only them (proprium), or their accidental feature (accidens). (The P of species is a somewhat obscure case here because it can be predicated of individuals, too, unlike the others.) Added to a genus, a speciﬁc diﬀerence (diﬀerentia speciﬁca) deﬁnes a species, which in turn can be a genus for lower subspecies. In this way, the famous “Porphyrian tree” is generated, ranging from uppermost genera down to individuals. The doctrine of predicables was a standard part in medieval logic texts, and it was a relatively unproblematic part: The diﬃculty, of course, is metaphysical and concerns the essential, necessary, and accidental qualities. Logicians, however, used the ﬁve universals as metalinguistic tools to classify predicates. A more ontological question is that of categories, or praedicamenta, as logicians preferred to call them. The ﬁrst category is substance; the other categories are ways in which something can belong to a substance. Aristotle studied quality, quantity, and relation in his Categories, and more brieﬂy he discussed even place, time, position, habit (having), passion, and action. With some variants, medieval praedicamenta treatises give the same list of 10 members. As Buridan says, “this treatise is found in many summulae, but in many it is not.” Indeed, it is not obvious why logicians need to discuss a question that seems purely metaphysical. But there was a motive for those who included this treatise in their summulae—an assumption of the parallelism between predication and being. Except substance, all categories both “are said of things” and “are in things.” Thus, a classiﬁcation of ways of being in a substance also produces a classiﬁcation of questions and answers that can be made concerning an entity, and this is a logically relevant achievement. Later, nominalists give up the assumed parallelism and analyze categories simply metalinguistically, as classes of terms. Ockham, for instance, has a long discussion in which he wishes to show how terms of other categories are secondary to substance and quality. We may note in passing that predicables and categories have a very diﬀerent role among the speculative grammarians of the late thirteenth and early fourteenth centuries. For them, terms are intelligible because they manifest the same characters and structures as the entities of the world; the “modes of signifying” belonging to grammatical features of lexical meaning and inﬂection are functions that reﬂect categorial features of objects. Such an approach leads to a special view of metalinguistic issues. Hence it is also natural that these authors, the modists, concentrated on rather abstract lexical contents and were not very interested in the semantic properties of concrete occurrences of terms in particular sentences.

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3.4. Signiﬁcance The main body of the theory of terms consisted of proprietates terminorum. The tireless analysis of these “properties of terms” displays the intense interest in philosophical semantics that was characteristic of later medieval philosophy. This is a ﬁeld that seems to be a medieval invention. In Aristotle and other ancient sources, there were only scattered remarks on semantic questions, and it can hardly be said that they attempted to establish any self-conscious theory of semantics. On the other hand, after scholastic philosophy these problems were often considered futile, and explicit philosophical semantics was largely rejected. But the medieval theory has had a striking revival in the latter half of the twentieth century, when philosophical semantics has again grown into a complex discipline, often struggling with questions that bear an obvious resemblance to medieval themes. Undoubtedly the two most important properties of terms are signiﬁcation and supposition. They have often been compared to present-day “meaning” and “reference,” but this comparison must not be taken literally. For one thing, the emphasis was on the words and signs: Unlike many accounts of meaning and reference, the medieval doctrine viewed signiﬁcation and supposition mainly as something that the words do or as something that is done by means of words. Let us start with signiﬁcation. Logicians were aware of the ambiguity of this word. Usually, instead of interpreting signiﬁcation as a signiﬁed entity of some sort, they started from “acts of signifying” and assumed that terms had a property of being signiﬁcant. (In this respect, terms diﬀered from other words which had no signiﬁcation by themselves.) A word signiﬁes, or has signiﬁcation, because of its “institution,” or according to another common account, because of its “use” in language. In short, signiﬁcation is the role of the term in language. The same idea acquires a new slant with the introduction of mental terms. It then becomes standard to claim that spoken words have their signiﬁcations because of linguistic conventions, whereas the mental terms are natural signs that have their signiﬁcations necessarily, without any stipulation. Signiﬁcation is generally connected to mental acts of understanding: A linguistic term signiﬁes that of which it makes a person (a speaker or a hearer) think, a mental term is itself an act of thinking of something, a representation. (To quote John Aurifaber: “signifying is an accident of the intellect, but a word is the thing by means of which the intellect signiﬁes.”) The thing thus signiﬁed has “intentional being.” Even before the mentalistic turn, it was usual to ﬁnd the essence of words in signiﬁcation. Thus Thomas Aquinas said that “signiﬁcation is like the form of a word”—the matter was the phonological shape, the form was its signifying capacity. Later, it was said that mental concepts have their signiﬁcations “formally,” and spoken and written words essentially function as instruments of this signiﬁcation. Signiﬁcation is the deﬁning property of all terms; thus it is natural that it can be deﬁned no further. Late medieval philosophers seem to agree that

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signiﬁcation is a basic notion that can only be explained by illustration. For them, it was obvious that a term signiﬁes something, but there was a great, partly metaphysical controversy about what this something is. Boethius had already said that words signify concepts, that is, corresponding mental entities. This gave an impulse for the view that words signify concepts immediately and objects indirectly. (Such a “semiotic triangle” had been discussed earlier by Greek Aristotelian commentators.) This opinion became prevalent among the Thomists. Aquinas himself had pointed out that a term signiﬁes a general nature that is abstracted from individual entities. The later Thomists emphasized that the concepts were signs, too: Thus the words do signify objects “principally” (most important), although they do it only “indirectly” (through the concepts). A contrary position was championed by Bacon, and it won general support at the end of the thirteenth century. It started from the obvious fact that terms are used in propositions, and the propositions are about objects and not concepts. Thus all terms must signify objects. However, nonexistent objects cause problems which compel logicians to make reservations concerning that general principle. What is signiﬁed is, for instance, the object “regardless of its being or not being” (Kilwardby), or the object “secundum quod the intellect perceives it by itself” (Duns Scotus). Moreover, spoken words and mental terms signify the same objects. According to Ockham, it is a basic fact that words are “subordinated” to the corresponding mental terms in such a way that they signify the same things. He apparently did not think that this use of language could be further explained. Buridan was not satisﬁed with this kind of answer and again interpreted the subordination as a type of signiﬁcation: Words also signify concepts, in some sense. Later discussion became rather complex when diﬀerent positions were combined and reﬁned. Admitting then that terms signify something extramental, it is still not clear what this signiﬁcatum is for general terms. The question is inevitably connected to the theory of universals. The realist answer is that the term signiﬁes something general; “man” signiﬁes a universal, a species, a property, or a common nature of “man in general.” The nominalist answer is that the term signiﬁes all relevant individuals; “man” signiﬁes each man. Both answers cause trouble, which shows the uneasy union of signiﬁcation and denotation. For it was assumed, after all, that a term signiﬁes what it is true of, and this characterization would better suit denotation. Syncategorematic words have no signiﬁcation in the strict sense. However, most logicians were not as rigorous as Ockham, who said that they do not signify at all. Even Buridan was willing to admit that they did not signify things but ways of thinking. And both realists and nominalists agreed that syncategorematic words could “consignify,” that is, participate in forming signiﬁcant wholes. There is even another sense of the word “consigniﬁcation.” In addition to its basic signiﬁcation, a word can have some consigniﬁcation that further determines its content. Especially thirteenth-century authors often use this

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approach to explain the role of features like case endings and—the most discussed example—tenses. The idea is that the actual occurrences of words get richer contents than bare lexical words.

3.5. Supposition Denotation was ﬁrst discussed by means of appellation, a notion borrowed from the “appellative nouns” of grammar. Appellation is the relation between a general term and the things actually belonging under it at the moment of utterance. Often this notion was applied only to the predicates of propositions, but at least from William of Sherwood onward it had unrestricted use. The “property of terms” that caused the most extensive study was supposition. The word derives from grammatical contexts. According to Priscian, a word has a supposition when it is placed as the subject of a proposition. This meaning was usual in the twelfth century. On the other hand, grammar had also formed the idea that a word supposits because it refers to an individual. Gradually this became the central aspect, and the supposition of terms was their way to denote individuals. As the supposition theory expanded, logicians had to seek for suppositions even for other terms than the subjects of propositions—for predicates and parts of complex terms. The question of supposition began to concern the denotation of terms quite generally, and at the same time appellation lost much of its importance, turning into a special case of supposition. Supposition theory was a challenging subject especially because the supposition of terms depends on their position in a proposition. Each word that is not equivocal always has the same signiﬁcation, but its supposition varies in diﬀerent propositions. As Ockham said, “supposition is a property of a term, but only when it is in a proposition.” This compelled the logicians to develop classiﬁcations for the several kinds of supposition. As many scholars have pointed out, precisely this propositional approach was characteristic of the theory of supposition. It must, however, be noted that Peter of Spain admitted even a “natural supposition” (suppositio naturalis) that belongs to a term immediately because of its own signiﬁcation, and this idea was preserved by many Parisian logicians. Thirteenth-century terminist textbooks already include a detailed and clearly developed doctrine of supposition. In Paris during the second half of the century, this tradition had to give way to the modistic inﬂuence, but it survived largely undisputed in Oxford. Subsequently, the ideology of mental terms made it again generally accepted in the beginning of the fourteenth century. After this it became part of the permanent apparatus of late medieval logic. “To supposit” is obviously a technical term; it means something like “to stand for,” and this indeed was an alternative expression. Early terminists like Sherwood thought that supposition belongs only to substantives that are posited as subjects (i.e., subposited under predicates), whereas the denotative

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function of verbs and adjectives is copulation. Soon, however, it became the rule to merge these cases and connect supposition to every categorematic term. The deﬁnition of this general supposition is not evident. Perhaps it is clearest simply to quote concise deﬁnitions from two authors: “When a term stands for something in a proposition in such a way that we use the term for the thing and the term (or its nominative case, if it is in an oblique case) is truly predicated of the thing (or a pronoun referring to the thing), the term supposits for that thing” (Ockham). “All and only those terms supposit which, when something is pointed out by the pronoun ‘this’ or several things by the pronoun ‘these’, can truly be aﬃrmed of that pronoun” (Buridan). We shall try to sketch an overview of the divisions of supposition. First of all, in some cases the supposition is “improper” because the word is used in a nonliteral or metaphoric way; let us concentrate on “proper supposition” only. The deﬁnition of its various types displays both semantic and syntactic factors. It seems that the suppositum of a word can be of three fundamentally diﬀerent semantic kinds, and the supposition is accordingly called either material, simple, or personal.

supposition

material

simple

personal

A term has material supposition (suppositio materialis) when it stands for itself. It must be kept in mind that people in the Middle Ages did not use quotation marks, and material supposition is an alternative way to cope with some problems of use and mention. Sherwood notes that material supposition can be of two types: The word supposits itself either as a sheer utterance or as something signiﬁcant. His examples are “man” in “Man is monosyllabic” and “Man is a noun.” The supposition is simple (simplex) when a word stands for a concept. The classical elementary example is “Man is a species.” To realists, the suppositum then should be equated with some extramental conceptual signiﬁcatum. “If ‘man is a species’ is true, the term ‘man’ supposits its signiﬁcatum. . . . The word ‘man’ does not primarily signify anything singular; thus it signiﬁes primarily something general, and this is a species” (Walter Burley). According to nominalists, the simple suppositum is a mental entity, such as an intention. In the most common case, the word supposits some things that it signiﬁes. For historical reasons, this was called by the surprising name of personal supposition (suppositio personalis). Because both simple and personal supposition are related to the meaning, unlike material supposition, they were often together called formal supposition. On the other hand, nominalists liked to reduce concepts to mental words, so in a sense Buridan and Peter of Ailly are

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more straightforward than Ockham when they do not admit simple supposition as an independent class, counting it as material. The main task is the classiﬁcation of personal supposition, and here syntactic matters interfere. Let us start by providing the next diagram, representing the early state of the classiﬁcation, and then proceed to explanations of its titles.

personal supposition

idscrete

common

edterminate

merelyconfused

confused

confusedandidstrib utiv e

This scheme was in fact given by William of Sherwood, except that he makes the diﬀerence between common and discrete supposition in another context. Discrete supposition (suppositio discreta) belongs to discrete terms: that is, to proper names and demonstrative expressions, like “this man” or “this.” Then the suppositum is the unique object that is signiﬁed. All other terms have suppositio personalis communis. This common supposition is further divided into determinate and confused kinds. The supposition is determinate (determinata) when it allows instantiation, as we might say. But medieval logicians had no such notion. Early authors thought that a determinately suppositing term stands for one determinate object. Ockham improved on this, saying instead that determinate supposition supports descent to singulars, that is, to sentences that are got by substituting singular terms in place of a general term. Thus, in “A man is running” the term “man” supposits determinately because we can legitimately infer that “This man is running or that man is running or. . . ,” and each member of this disjunction in its turn allows ascent back to the original sentence. The supposition is merely confused (confusa tantum) if the proposition does not allow instantiation but is instead implied by its particular instances. As Ockham puts it, “in the proposition ‘Every man is an animal’, the word ‘animal’ has merely confused supposition; for one cannot descend to the particulars under ‘animal’ by way of a disjunctive proposition. The following is not a good inference: every man is an animal, therefore, every man is this animal or every man is that animal or every man is. . .” But it is also worth noticing that “Every man is this animal or that animal or that. . .” does indeed follow. Finally, the word has confused and distributive supposition (confusa distributiva) if it allows descent to all singulars but does not support any ascent. Here the reference concerns “distributively” each and every one of the individuals. For example, in “Every man is an animal” the term “man” has confused

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distributive supposition. It is correct to infer “This man is an animal and this man is an animal and. . .”; or as we write nowadays, “The man a is an animal and the man b is an animal and. . .” On the other hand, none of these singular propositions implies the original sentence. For confused supposition—and especially for the descent to singulars—it is important to decide what the adequate class of individuals ought to be. There was some debate on this point about the correct formulation until it was agreed that the terms had to be duly ampliated, in other words, extended from the basic case of all individuals presently belonging under the concept, such as all actual men, to include all past and future men as well, and in later logic even all possible instances (all possible men). So terms could be examined either with their actual supposition or with an extended supposition. It is not obvious what the motive behind supposition theory really was. Early authors possibly just wanted to capture various kinds of referring. But when Ockham and his followers started to build a more complex theory, with rules of descent and ascent, they probably did pursue something else. Thus, the supposition theory has been compared to the modern framework of quantiﬁcation theory, and clearly it has something to do with the problems of multiple quantiﬁcation and scope—problems that had no explicit place in Aristotle’s logic. Also, it can be seen as an attempt to warrant certain inference types, like those of descent and ascent. The interpretation here is still a matter of controversy.

4. Proposition The core of the medieval theory of judgment centers around the standard deﬁnition of proposition (propositio), deriving from the late ancient period through Boethius. The deﬁnition runs as follows: A proposition is an expression that signiﬁes something true or false. Propositio est oratio verum falsumve signiﬁcans. This deﬁnition accords with the classical theory of deﬁnition. It consists of the generic part (expression) and the distinguishing characteristic (signifying something true or false). For our purposes, however, it seems more useful to divide it into three parts and look at the concepts of truth and falsity separately from the problem of what it is that the proposition exactly signiﬁes.

4.1. Propositions Are Expressions As we have already seen, medieval authors understood logic as a discipline whose subject matter is linguistic discourse. It is well in line with this general approach that they also thought of the propositions studied in logic as sentences actually uttered in some language, typically either spoken or written. As we saw in section 2, a central issue in the determination of the subject matter

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of logic was whether (and in what sense) we could distinguish a special class of mental propositions. Thus, thoughts can be propositional only in so far as they have linguistic structure. A proposition, as the medievals thought of it, is something that is put forward as a sentence, and thus it has actual existence in time and typically also in space. As we will soon see in more detail, it was not the case that medieval logicians would have failed to make a distinction between the actual utterance and that which it expresses. Rather, they simply thought of propositional truth as an issue that comes up in connection with claims actually put forward, not as a property of abstract entities. From the viewpoint of twentieth-century logicians, this feature of medieval conceptual practice has some implications which are worth pointing out, although they are ultimately superﬁcial. According to medieval parlance, a proposition has to exist (i.e., has to be actually put forward in some language) to have a truth value, and it has its truth value in respect to some speciﬁc instant and context. Thus, a proposition like “there are no negative propositions” cannot be true, since it falsiﬁes itself, though it is clear that the case could be as it claims. Also, the same proposition can have diﬀerent truth values in diﬀerent situations. The truth value of “Socrates is seated” varies when Socrates either stands up or sits down. Furthermore, the truth of “this is a donkey” varies depending on what the demonstrative pronoun refers to. Indeed, all logical properties that a proposition has presuppose that it exists; thus medieval logicians often pointed out that their study applies to propositions, not eternally, but on all occasions in which they are put forward.

4.2. Propositions Carry Truth Values Not all signiﬁcative expressions are propositions. Boethius’s textbook distinguishes between “perfect” and “imperfect” expressions with the idea that an imperfect expression does not make complete sense but the hearer expects something more. More important, Boethius continues by listing questions, imperatives, requests, and addresses in addition to indicative sentences that make an assertion and count as propositions. This listing of the kinds of expressions is based on grammatical categories, and similar strategies were also followed in subsequent discussions. It may be of some interest to note that Buridan, for example, takes it to be worth an argument to reject Peter of Spain’s claim that sentences in the subjunctive mood (like “if you were to come to me, I would give you a horse”) do not count as propositions. It seems that medieval logicians disagreed on whether a proposition that is just mentioned without being asserted carries a truth value. The distinction between apprehensive and judicative uses of a propositional complex was rather standard. Ockham, for example, argues that it concerns propositions so that even an apprehended proposition has a truth value, although no stance is taken to it in the apprehension. Judgments, as he sees it, take stances on truth values, but propositions have them by themselves. Buridan, for his part, seems to rely on similar considerations to show that the sentential complex at issue

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is not a proposition. He seems to have thought that sentences are able to carry truth values through being asserted. We will return to this issue in connection with molecular propositions. For the most part, medieval logicians accepted the laws of noncontradiction and the excluded middle. Thus, every proposition has one and only one truth value. But neither of the two principles remained unchallenged. Aristotle’s famous sea battle in De interpretatione chapter 9 was widely discussed and within that debate it was also suggested that contingent propositions about the future perhaps do not yet have a truth value. This did not become the standard view. Similarly, in the widespread discussions concerning limitdecision problems and particularly the instant of change, it was suggested that perhaps contradictories are both true at the instant of change. Instead of accepting this, the standard line was to provide elaborate analyses of limit decision relying on mathematical considerations concerning inﬁnitesimal magnitudes. As is well known, in the more philosophical discussions concerning the nature of truth medieval logicians often put forward the principle of correspondence: Truth is adaequatio rei et intellectus. This deﬁnition was not, however, much used in the speciﬁc context of logic. There the term “truth” was mostly used with the more limited meaning of propositional truth, and it proved diﬃcult to exemplify from the real world anything that corresponded to a propositional complex. Thus, truth could hardly be explained as a relation between a real thing and a proposition. In the Aristotelian approach, things are referred to by using simple terms, and no simple expression—a mere term—can have a truth value. Truth rather arises from “composition” or “division” of terms in a predication, and depends on how this composition or division accords with how things really are. In his Syncategoreumata, Peter of Spain gives an elaborate suggestion that there is some kind of real composition, typically explicable with reference to the way in which everything in the world is composed of matter and form. According to Peter’s suggestion, the truth of a sentence depends on whether this “real composition” is expressed adequately. The standard Aristotelian dictum, “it is because the actual thing exists or does not that the statement is called true or false” (Cat. 12; 14b21–22), was not always understood in this manner. A typical way of explicating the claim that a proposition is true was to say that it “signiﬁes as it is” (signiﬁcat sicut est) or something to the same eﬀect. By such formulas logicians tried to avoid committing themselves to positing any real entity with which the true proposition would have direct correspondence. Instead, the expression often worked in a way analogous to what has lately been called “disquotational”: allowing transformation of the claim “p is true” into the simple claim “p.” Ockham’s Summa logicae (I, 43) contains an interesting discussion of in what sense truth is predicated of a proposition. In his opinion, it is not a real quality of the proposition. This can be proved by the fact that a proposition may change from truth to falsity by fully external change. For example, when something ceases to move, the truth value of the proposition “this thing

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is moving” changes without the proposition itself changing. According to Ockham’s explanation, “true” is a connotative term signifying that things are as the proposition signiﬁes. However, this remark leaves open the issue of what it is for things to be as the proposition signiﬁes.

4.3. Are There Any Propositional Signiﬁcates? Stoic logic, and in its wake important early medieval authors like Boethius and Peter Abelard, made a distinction between a declarative sentence and its dictum, or that which “is said.” Thus, the dictum expresses, or it simply is, the content of the proposition without being itself a proposition. For example, the proposition “Socrates is seated” (Socrates sedet) says or puts forward the dictum “that Socrates is seated,” which in Latin is typically expressed as an accusative plus inﬁnitive construction (Socratem sedere). Over the centuries, many logicians discussed the status of the dictum. Also, the related distinction between a proposition (as an expression) and its total signiﬁcate (in distinction from the separate signiﬁcates of its constituents) became a topic of an interesting dispute toward the second quarter of the fourteenth century. In his early work, Commentary on the Sentences, Ockham puts forward a theory according to which belief always concerns a proposition formulated in mental language. That is, when a person assents to something, he has to formulate a mental proposition expressing that which he assents to. He then reﬂexively apprehends the proposition as a whole and assents to it. It seems that Ockham’s motivation for this theory was the view that there is no way to grasp propositional content apart from formulating a proposition in mental language. Thus, if objects of beliefs are true or false, they must be formulated in mental language. Several contemporaries of Ockham did not straightforwardly accept the idea that the object of belief must always be an actually formulated proposition. Even Ockham himself shows some hesitation toward this theory in his Quodlibetal questions, which he composed later. It seemed to many authors that when one believes, for example, that God exists or that a man is running, the object of belief is somehow out in the world and not merely a proposition in the mind. The idea is that people do not always believe in sentences, but at least sometimes it should rather be said that they believe things to exist in a certain way. This consideration made medieval logicians search for something like propositional content outside the mind and a number of diﬀerent theories of how it could be found emerged. In Walter Chatton’s theory, the object of the assent has to be some extramental thing. If you believe that a man is running, the object of your belief is the man at issue. Thus, the signiﬁcate of the proposition “a man is running” is the man. Chatton recognized that his theory has the problematic consequence that the simple term “a man” and the propositional complex “a man is running” signify the same thing. As Chatton saw it, the diﬀerence in these two expressions is not in what they signify but in how they signify it.

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The terminology he used in this connection refers back to modist grammatical theories. It seems that both Ockham and a younger contemporary, Adam Wodeham, reacted against Chatton’s theory. In his Quodlibetal disputations, Ockham makes a further distinction concerning propositional assent, in eﬀect allowing it to be the case that you give assent without reﬂexively considering a mental proposition. In such a case, you simply form the proposition and give your assent in an unreﬂective way as connected to rather than directed at the mental proposition. As Ockham curiously points out, this kind of assent is not at issue in scientiﬁc knowledge, only in beliefs of ordinary life. According to Ockham’s obscure remarks, nothing is the object of this kind of assent. Wodeham seems to have continued from this basis in his theorizing. He wanted, though, to allow that even the nonreﬂective kind of assent is about something, and the signiﬁcate of the proposition appeared to be a suitable candidate for an object. However, its metaphysical status seemed quite unclear to the medieval mind. According to Wodeham, the signiﬁcates of propositions need to be categorically diﬀerent from the signiﬁcates of the terms. As he put it, propositions do, of course, signify the things signiﬁed by their terms, but no thing or combination of things is the adequate total signiﬁcate of the propositional complex. The adequate signiﬁcates of propositions are such that they can only be signiﬁed by propositions; even further, they do not belong to any of the Aristotelian categories nor can they be called things. Wodeham’s theory became known as a theory endorsing “complex signiﬁables” (complexa signiﬁcabilia). Such entities were rejected by most subsequent logicians, including major ﬁgures like John Buridan, but accepted by some, most famously by Gregory of Rimini—in subsequent discussion, the theory became known as his theory. In the third quarter of the fourteenth century, discussion of what propositions signify was abundant. Is it something like a mode of being? Or just a mental act of composition? Do propositions in fact signify anything more than just the things denoted by the terms, or perhaps even just the thing denoted by the subject? The fourteenth-century discussion concerning complex signiﬁables seems to have made it clear to late medieval logicians that their logic was based on a metaphysical view of the world as consisting of things and not of states of aﬀairs. The constituents of the world could be referred to by terms, but to make claims about the world, a diﬀerent kind of mental act was needed. Paradigmatically, one had to construct a complex expression asserting a composition of multiple entities.

4.4. Predication In Aristotelian logic, the ground for all judgments is laid by the predicative structure, where two terms are either joined or disjoined as the subject and the predicate. After Boethius, it remained customary in the Middle Ages to treat aﬃrmative predication and negative predication as two diﬀerent kinds

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of statement, and also to take negation simply as “destroying the force of the aﬃrmation.” Thus, it is not necessary here to treat negative predication distinctly from the basic aﬃrmative case. The aﬃrmative predication consists, as already Boethius recognized, not only of the two terms but also of the copula. Thus, when Aristotle remarks that a predication can be constructed either with a verb (e.g., “a man runs,” homo currit) or with a participle (e.g., “a man is running,” homo currens est), this was normally interpreted as meaning that the latter form is to be taken as primary. In the latter, the copula “is” was said to be added as a third part (tertium adiacens). In Latin, the copula was of course the standard verb “to be” (esse), which was also used in the simple existential claim “a man exists” (homo est). This use of est as secundum adiacens had to be explained since it appeared to lack either the copula or the predicate. As Boethius saw it, the verb serves here a double role. This solution was accepted in the Middle Ages, and thus there was no need to see it as an altogether diﬀerent kind of statement. Buridan even argued against ordinary linguistic practice that logically one should prefer the formulation “a man is a being” (homo est ens). Given that the copula joins the two terms into a predicative proposition and gives the sentence its assertive character, it still remains unclear exactly how it joins the terms together. It seems that this was one of the most fundamental points of disagreement among medieval logicians. For modern scholars it has proved rather diﬃcult to ﬁnd a satisfactory description of how the simple predication was understood in the Middle Ages. One crucial nontrivial issue seemed clear, though. Throughout the Middle Ages, it was commonly assumed that in the absence of speciﬁc contrary reasons, the verb “to be” even as the copula retains its signiﬁcation of being. Thus, all aﬃrmative predications carry some kind of existential force, while negative predications do not. In an aﬃrmation, something is aﬃrmed to exist; a negation contains no such aﬃrmation of existence. But beyond this simple issue, interpretations of the nature of predication seem to diverge widely. Most of the twentieth century discussions of the exact content of the diﬀerent medieval theories of predication have been based on the Fregean distinction between the diﬀerent senses of “to be.” Scholars have distinguished between inherence theories and identity theories of predication, despite the evident threat of anachronism in such a strategy. For want of a better strategy, we also have to rely on that distinction here. But instead of trying to classify authors into these two classes, let us simply look at the motivations behind these two rather diﬀerent ways of accounting for what happens in a predication. The idea of the inherence theory is that the subject and the predicate have crucially diﬀerent functions in the predication. While the function of the subject is to signify or pick out that which is spoken of, the function of the predicate is to express what is being claimed of that thing. The idea is, then, that the Aristotelian form signiﬁed by the predicate inheres in the thing signiﬁed by the subject. Peter of Spain seems to defend this kind of theory

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of predication when he tries to show that the copula signiﬁes that relation of inherence obtaining between matter and form, or between a subject and its accident. Aquinas seems to follow this account. Scholars have disagreed about Abelard’s theory, and it indeed seems that his rich discussion of the topic provided grounds for several kinds of diﬀerent subsequent theories. On the one hand, he seems to lay the basis for the inherence theory. On the other, he defends the idea that to look at the exact truth conditions of a predication like “a man is white” (homo albus est), it should be analyzed into a fuller form “that which is a man is that which is white” (idem quod est homo est id quod album est; Logica ingredientibus 60.13). With such a formulation he seems to have in mind the idea that for the aﬃrmative predication to be true, the subject and the predicate must refer to the same things. This is commonly called the identity theory of predication. Abelard’s “that which is” (quod est) formulation remained part of the actual practice of logical writing for several centuries. It can be found commonly from logical texts throughout the Middle Ages, although it was not always oﬀered as an explanation of the truth conditions of predication in general. The formulation has the special feature that it appears to give the subject and the predicate of a predication a similar reading. Both are to be understood as referring to some thing, and then the assertion put forward in the proposition would be the identity of these two things. This seems to amount to the identity theory of predication in Fregean terms. In the fourteenth century, both Ockham and Buridan seem to have quite straightforwardly defended the idea that the Aristotelian syllogistic is based on identity predications. As they put it, the simple predication “A is B” is true if and only if A and B supposit for the same thing. For the most part, truth conditions of diﬀerent kinds of propositions can be derived from this principle. Somewhat interestingly, Ockham nevertheless recognizes the need of basic propositions expressing relations of inherence. For Ockham, the predicate “white” is a so-called connotative term, and therefore a somewhat special case. According to his analysis, the predication “Socrates is white” (Sortes est albus) should be analyzed into “Socrates exists and whiteness is in Socrates” (Sortes est et Sorti inest albedo). In his metaphysical picture, Ockham allows both substances and qualities to be real things, and if one is allowed to use only so-called absolute terms that supposit in a sentence only those things which they signify, the relation of inherence (inesse) is not expressible with an identity predication. Qualities inhere in substances, but they are not identical with substances. The whiteness at issue in the claim “Socrates is white” is not Socrates, it is a quality inhering in Socrates. Socrates is not whiteness even if he is white. In his Summa logicae, Ockham has some special chapters on propositions involving terms in oblique cases (in cases other than the nominative). The just-mentioned proposition “whiteness is in Socrates” is a paradigm case of

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such a proposition (in Latin, the subject has to be in the dative case Sorti; in English, the eﬀect of the case is represented with the preposition “in”). Furthermore, all propositions involving the terms that Ockham calls “connotative” require in their logical analysis that oblique cases are used. The main claim of the short chapters of Summa logicae addressing propositions containing such terms is that their truth conditions cannot be given by the simple rule of thumb that the subject and the predicate must supposit for the same thing in an aﬃrmative sentence. Consequently, the rules for syllogisms formulated with such propositions are also abnormal. In eﬀect, Ockham excludes propositions with oblique terms from the ordinary syllogistic system, thus leaving a surprising gap in his logical system. In his logic, Buridan proceeds diﬀerently. For the purposes of the syllogistic system, he requires that all propositions should be analyzed into a form where truth conditions can be given through variations of the rule that in aﬃrmative sentences the subject and the predicate supposit for the same thing. This allows him to apply the standard syllogistic system to all propositions. The solution is at the price of greater semantic complexity. Buridan has to allow so-called connotative terms (including, e.g., many quality terms like “white”) as logically simple terms despite their semantic complexity. Both Ockham and Buridan apparently thought that identity predication is the logically privileged kind of predication. Nevertheless, they also both accepted the Aristotelian substance-accident ontology to such an extent that they had to ﬁnd ways of expressing the special relation of inherence. While Ockham allowed exceptions to the syllogistic through irreducible propositions expressing inherence, Buridan opted for a syllogistic system with obviously complex terms expressing inherential structures.

4.5. Negations As the medieval logicians saw it, the simple predication “A is B” contains altogether four diﬀerent places where a negation can be posited: 1. It is not the case that A is B. 2. A is not B. 3. Not-A is B. 4. A is not-B. It is of course clear that 1 is closest to the negation used in twentieth-century logic. In it, the negation is taken to deny the whole proposition. According to Boethius’s commonly accepted formulation, the force (vis) of the predication is in the copula, and hence denying the copula denies the whole proposition. Thus, the negation in 2 has the same eﬀect as in 1. (As Buridan notes, for quantiﬁers and other modiﬁers, the location of the negation may still make a diﬀerence.)

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2 is the standard negation of medieval logic. It is the direct contradictory of the corresponding aﬃrmative predication. In particular, it is noteworthy that this negation does not carry any existential presuppositions. Thus, “a chimera is not an animal” is true simply because no chimeras exist. 3 and 4 are aﬃrmative statements containing an inﬁnite term, as terms of the type “not-A” were called. In these cases, the negation is connected directly to a term and not to a proposition. An inﬁnite term was taken to refer to those things to which the term itself does not refer. Thus, not-man refers to anything that is not a man. Because these negations do not make the proposition negative, 3 and 4 carry existential content: Some B must exist for 3 to be true, and some A for 4 to be true. Although the syntactic idea of attaching a negation to a term was universally accepted in the Middle Ages, logicians seem to have disagreed about whether the term-negation should be taken to be essentially the same negation as the propositional one but in a diﬀerent use. In his Syncategoreumata, Peter of Spain seems to reject this idea. He presents the two negations as genuinely diﬀerent in themselves. His discussion is connected to a theory where even simple names and verbs signify in a composite sense. Thus, the idea is that “man,” for example, means a composition of a substance with a quality, a substance having the quality of being human. Thus, the inﬁnite term “not-man” signiﬁes a substance that has not entered into a composition with the quality of being human. Ockham, for his part, preferred to reduce negating a term to ordinary propositional negation, claiming that the meaning of “not-man” can be explained as “something which is not a man.” Buridan allows inﬁnite terms a signiﬁcant role in his syllogistic system, and thus seems to go back to thinking that the negation involved in them is fundamentally distinct from that which he calls “negating negation”—that is, the propositional negation that has power over the copula. Given that negations can be put in many places even in a simple predication, medieval logicians gained skill in handling combinations of diﬀerent negations. The idea that two negations cancel each other (provided that they are of the same type and scope) was also well known.

4.6. Quantiﬁers Aristotelian predications typically have so-called quantity. Medieval logic commonly distinguished between universal (“every A is B”), particular (“some A is B”) and indeﬁnite propositions (“A is B”). As Boethius already pointed out, the indeﬁnite predication that lacks any quantiﬁer is equivalent to the particular one. Some logicians did specify certain uses that violate this rule of thumb, but such exceptions are rare. In addition to quantiﬁed and indeﬁnite predications, singular predications were also discussed (e.g., “Socrates is running”). They had a subject term that was a proper name or some suitable demonstrative pronoun.

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Basic quantiﬁed predications were given vowel symbols as mnemonic labels from the ﬁrst two vowels of the Latin verbs “aﬃrm” (aﬃrmo) and “deny” (nego). Thus, the universal aﬃrmative was shortened as AaB, where A is the subject and B the predicate. Similarly, the particular aﬃrmative was AiB, the universal negative AeB and the particular negative AoB. These four predications were further organized into the so-called square of opposition to show their interrelations.

AaB

AeB

AiB

AoB

The upper two, the universal aﬃrmative and the universal negative, were called contraries; they cannot be true simultaneously, but could both turn out to be false. Similarly, the particular aﬃrmative and particular negative were subcontraries; they cannot be false simultaneously, but both could turn out true. The relation between the universal and the particular was called subalternation on both sides; the particular follows from the universal but not vice versa. The propositions in the opposite corners were called contradictories, since one of them had to be true and the other false. In the Middle Ages, a substantial amount of ink was used discussing whether a universal aﬃrmative could be true when only one thing of the relevant kind exists. The paradigm example was “every phoenix exists,” and many logicians rejected it with the requirement that there must be at least three individuals to justify the use of “every.” Toward the end of the thirteenth century this discussion seems to disappear, apparently in favor of the view that one referent is enough; the existential presupposition was never dropped, however. Another issue of detail that was also widely discussed later was the case of universal predications of natural sciences, which capture some invariable that does not appear to be dependent on the actual existence of the individuals at issue. A suitable example is “every eclipse of the moon is caused by the shadow of the Earth.” According to a strict interpretation of the existential presupposition, such predications prove false most of the time—which seems somewhat inconvenient. Two fundamentally diﬀerent suggestions for a better reading of the predication were put forward. Ockham seems to favor the idea that what really is at issue here is the conditional proposition “if the moon is eclipsed, the eclipse is caused by the shadow of the Earth.” This solution draws on the traditionally recognized idea that the conditional is implied by the universal aﬃrmation. However, Buridan opted for another solution. As he reads the universal aﬃrmation at issue, its verb should be read in a nontemporal sense. In such a reading, past and future eclipses also provide instances satisfying the diluted existential presupposition.

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4.7. Complex Terms Medieval logicians allowed that not all terms of a predication are simple. A predication can, of course, always be divided into the subject and the predicate together with the copula and the appropriate quantitative, qualitative, and modal modiﬁers. But the two terms may possibly be further analyzable (e.g., “a just man is talking,” where the subject “just man” consists of two parts), and indeed this was a topic that attracted much attention during the Middle Ages. Toward the middle of the fourteenth century, discussion on this topic resulted in a detailed theory on the interaction between diﬀerent kinds of combinations of categorematic and syncategorematic elements that can be found in a predication. To tackle with issues of scope an elaborate system of word order rules was introduced for the technical Latin used by logicians. It seems that thirteenth-century logicians did not take it to be a serious problem that complex predications do not behave in ways that suit the needs of syllogistics. Following Aristotle’s remark (Analytica priora I, 36; 48b41–49a5), syllogisms with oblique terms in the various cases were usually discussed separately, and thus it seems that the thirteenth-century logicians probably thought that more complicated predications do not necessarily ﬁt into the ordinary syllogism. As we already noted, Ockham makes this slight inconvenience clear in his Summa logicae. It seems that Ockham fully understood that the traditional syllogistic logic does not always work if actually used linguistic structures are given full logical analyses. Also, he explicitly allows that there is no general way of giving the truth conditions of the various kinds of complex predications; in particular, he points out that even as simple a construction as the genitive case makes the standard truth conditions of identity predications inapplicable. “The donkey is Socrates’s” is an aﬃrmative predication. However, its truth requires, but it is not suﬃcient for it, that the subject and predicate supposit for diﬀerent things (“donkey” for a domestic animal owned by a person, and “Socrates” for the owner of the animal). More generally, Ockham thought that mere identity predications are not suﬃcient to explain the expressive power of the actually used language. A richer variety of propositions had to be accounted for, but in fact they found no place in syllogistic logic. Thus, syllogistic logic was not a complete system covering all valid inferences. After Ockham, Buridan took another approach. As he saw it, all categorical propositions can be reformulated as straightforward Aristotelian predications ﬁtting the needs of the ordinary syllogism and having the rule of identity or nonidentity of supposition as the criterion of truth. For this purpose, he had to modify the traditional systems of combining diﬀerent categorematic and syncategorematic elements so that they appear as geared toward building up terms whose suppositions can be decided. Perhaps most important, he saw that he could not assume that standard Aristotelian predications would be found as the end results of logicolinguistic analysis. Rather, he understood the building blocks of the syllogistic system—identity predications—to be more

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or less artiﬁcial constructions built from complex terms. For example, for the purposes of syllogistic logic the sentence “the donkey is Socrates’s” must be read as “the donkey is Socrates’s thing,” although the predicate “Socrates’s thing” clearly is not a simple term of the ideal mental language. Buridan did not assume that all mental or spoken propositions would be identity predications. Rather, he assumed that for the purposes of syllogistic logic, any proposition could be transformed into an equivalent identity predication. By such means, syllogistic logic could serve as a complete system containing all inferences. Buridan’s strategy involved, therefore, a massive expansion of the syllogistic system toward incorporating increasingly complex terms. Whereas logicians up to Ockham had accepted that a wide variety of propositions are nonstandard from the viewpoint of syllogistic logic, Buridan builds rules on how the content of these nonstandard propositions can be expressed by standard structures involving very complex terms. Buridan provides elaborate rules concerning complex terms. The idea is to show how nouns and verbs interact with diﬀerent syncategorematic expressions and produce terms that ﬁt into standard Aristotelian predications. In Buridan’s view, all propositions can be transformed so that the truth conditions can be expressed through the criteria of an identity predication. In aﬃrmative sentences, the terms must supposit for the same thing, while in negative sentences, they must not supposit for the same thing. To see the full strength of Buridan’s new system, let us consider a somewhat more complicated example. Buridan analyzes “Each man’s donkey runs” (cuiuslibet hominis asinus currit) in a new way. Traditionally, this Aristotelian sentence was understood as a universal aﬃrmation consisting of the subject “man” in the genitive case, and a complex predicate. This analysis makes the subject supposit for men, and the predicate for running donkeys so that the assertion cannot be read as an identity predication. Thus, standard syllogistics are not applicable to a proposition like this. Most logicians up to and including Ockham seem to have been satisﬁed with the implied limitations of the syllogistic system. Buridan, however, analyzes the proposition as an indeﬁnite aﬃrmation. It has a complex subject “each man’s donkey,” which includes two simple categorematic terms (“man,” “donkey”), a marker for the genitive case (the genitive ending “’s”), and a quantiﬁer (the universal sign “each”). The quantiﬁer does not make the proposition universal, because it has only a part of the subject in its scope and must therefore be understood as internal to the subject term. As a whole, the subject supposits for sets of donkeys such that each man owns at least one of the donkeys in the set. The predicate of the proposition is a simple term, “running.” It supposits for sets of running things. Construed in this way, the predication can be evaluated with the standard criteria of truth, and standard syllogistics can be applied to it. It seems clear that Buridan took seriously the programmatic idea that the Aristotelian syllogistic system should provide a universal logical tool which did not allow major exceptions to behave in nonstandard ways. But instead of analyzing complex propositions into combinations of predications with simple

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terms, Buridan provides elaborate rules concerning the ways in which complex terms are built.

4.8. Hypothetical Propositions In the Middle Ages, not only conditionals but also conjunctions and disjunctions were called hypothetical (hypothetica) propositions. Otherwise the treatment of conditionals and disjunctions causes no surprises to a modern reader familiar with basic propositional logic. Walter Burley, for example, gives the following account. Conjunctions are propositions consisting of two further propositions that are joined with the conjunction “and” or something equivalent. Their truth conditions require that the propositions thus joined are true. Negating a conjunction makes reference to another type of hypothetical, namely disjunction, because denial of a conjunction requires only that one or the other of the conjuncts is denied. Disjunction, for its part, is deﬁned in the inclusive manner: Its truth conditions require that one of the parts is or both of them are true. Denial of a disjunction produces a conjunction, and as Burley notes, denial of a disjunction of contradictories (e.g., “Socrates runs or Socrates does not run”) produces a conjunction which includes contradictories. Certain interesting issues are raised in more detailed discussions of conjunctive and disjunctive propositions. One such is the nature and exact content of conjunctive and disjunctive terms used in propositions like “every man runs or walks.” Are they reducible to conjunctive and disjunctive propositions and why not exactly? How ought they be accounted for in inferential connections? Another, more philosophical issue was the question of whether the parts of conjunctions and disjunctions are strictly speaking propositions. As Buridan notes, the “force of the proposition” (vis propositionis) in a disjunction is in the connective, and thus not in either of the disjuncts. Hence, it is only the whole and not the parts that carry truth value in the composition. When someone utters a disjunctive proposition consisting of contradictories, he does not, according to Buridan, say anything false, although one of the parts would be false if uttered as a proposition. Thus, hypothetical propositions do not, strictly speaking, consist of categorical propositions but of linguistic structures exactly like categorical propositions. It seems that medieval logicians treated conjunctions and disjunctions in a straightforwardly truth-functional manner. It seems equally clear that their treatment of conditionals diﬀers from the twentieth-century theory of material implication. Indeed, in the Middle Ages theory of conditionals was mainly developed in connection with a general theory of inference, under the label “consequences.” Conditionals were taken to express claims concerning relationships of inferential type. Medieval logicians also distinguished further types of hypothetical propositions. In Buridan’s discussion (1.7), altogether six kinds of hypothetical propositions are accounted for, including conditional, conjunctive, disjunctive,

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causal (“because the sun shines above the Earth, it is daytime”), temporal (“Socrates runs when Plato disputes”), and local ones (“Socrates runs to where Plato disputes”). Buridan even vaguely suggests that perhaps other Aristotelian categories may also give rise to hypothetical propositions in a way similar to temporal and local hypotheticals. It is clear that this approach to hypothetical propositions relies on other ways of combining propositions than just the truth-functional ones. The connective may express something more than just a truth function.

4.9. Modal Operators Logical issues connected to possibility and necessity, which in the twentieth century have been studied as alethic modal logic, were a central research topic in late medieval logic. These modal terms were usually discussed together with other modiﬁers operating in similar syntactic roles. For example, the twentiethcentury ﬁelds of study known as deontic logic (dealing with permissibility and obligation) and epistemic logic (dealing with concepts of knowledge and belief) have their counterparts in the Middle Ages, where these issues were discussed together with possibility and necessity. Most medieval logicians discussed altogether four modal operators crucial to modern alethic modal logic: possibility, impossibility, contingency, and necessity. These were deﬁned in relation to each other so that the necessary was usually taken to be possible but not contingent, whereas the impossible was taken to be neither possible nor contingent. Like the square of opposition of simple predications, modal predications were often organized into a square of modal opposition following Aristotle’s presentation in De interpretatione (ch. 13). It is particularly noteworthy that following Aristotle’s model, the square of modal opposition typically contained just the modal operators, not complete sentences. In a somewhat schematized way, the basic square can be illustrated as follows:

N ¬M¬

N¬ ¬M

¬N¬ M

¬N M¬

In this square, the relations of contrariety, subcontrariety, subalternation, and contradiction were said to behave as they would in the basic square of simple predications. Following Aristotle, medieval logicians made a distinction between two ways of understanding a modal predication to make sense of examples like the

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possibility that someone sitting walks. Understood de dicto, there is no such possibility. The sentence “someone sitting walks” is impossible. But understood de re, there is such a possibility, since the person who is sitting may be able to walk. Thus, if the modal predicate “can walk” is understood de re, or as concerning the person who is actually sitting now, the sentence might be true. All subsequent major logicians discussed this distinction in some form or other. In the sections concerning modal propositions in Ockham’s Summa logicae, it is clear that the de dicto reading is given logical priority. Using another traditional terminology, Ockham prefers to call it the composite sense (sensu composito) and does not oppose it to a de re reading but to the roughly similar divided sense (sensu diviso). Ockham apparently thinks that modality is a property of propositions rather than terms, and aims at reducing readings sensu diviso to sensu composito through analyzing modal propositions in sensu diviso into propositions sensu composito. There are three main models by which modern scholars have been able to account for the way in which medieval logicians understood what it means to say that something is possible: the statistical model, the potency model, and the consistency model. During the medieval period, the modal concepts used by particular logicians typically ought not to be explained through reference to a single model. Rather, these three diﬀerent strands of thought have inﬂuenced to varying degrees the modal thinking of diﬀerent medieval logicians. The basic intuition explained by the statistical model is that all and only those things seem to be possible which sometimes occur. If something never happens, it means that it can’t happen. The potency model, for its part, explains the intuition that whether something is possible depends on whether it can be done. For something to be possible it is required that some agent has the potency to realize it, though it is not required that the thing is actually realized. However, because normally there are no generic potencies that remain eternally unrealized (why should we say that humans can laugh if no one ever did?), this model becomes clearly distinct from the statistical model only when God’s omnipotence is understood to reach wider than just actual reality. God could have created things or even kinds of things which he never did, and these things remain therefore eternally unrealized possibilities. It seems that throughout the Middle Ages, God’s omnipotence was thought to be limited only by the law of noncontradiction. Contradictions are not real things, and therefore God’s power is not limited, although we can say that he cannot realize contradictions. This consideration seems to have been one of the reasons why logicians in the thirteenth century increasingly used the criterion of consistency to judge claims about possibility. But it seems that the development of syntactic logical techniques also made it natural to demarcate a class of propositions that are impossible in the traditional sense but nevertheless seem to involve no contradiction (e.g., that man is irrational, or that man is not an animal). Especially the traditional technique of laying down a false or even impossible thesis for an obligational disputation (see the

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following) seems to have encouraged consideration of consistent propositions or sets of propositions that are in some sense impossible. Some authors, like Boethius of Dacia, even use the special expression compossibilitas to refer to this kind of concept of consistency as distinct from possibility. As is well known, from Duns Scotus onward, several logicians made this kind of concept of consistency crucial for possibility in general. The medieval discussion can be characterized as aiming at ﬁnding a way to account for these rather distinct intuitions of what it means to say that something is possible. A certain shift in emphasis is visible. Whereas earlier authors pay more attention to the statistical idea at the expense of consistency, later authors tend to neglect or argue against intuitions captured in the statistical model while emphasizing consistency as the criterion of possibility.

5. Classical Forms of Inference 5.1. Syllogisms We must next turn to the “theory of inference.” Ignoring probable inferences for now, we can say that this part of logic tries to describe how some propositions necessarily follow from others, from their premises. The propositions of a certain sequence have such properties that the last one must follow necessarily from its predecessors. An important type of inference is the syllogism—the inference on which Aristotle concentrated in his Prior Analytics. The syllogism was the best-known and paradigmatic type of inference throughout the Middle Ages. In the thirteenth century, when logicians studied demonstrative inference, they were almost exclusively concerned with syllogistics; but afterward, when a more general inference theory developed, the policies of various authors diﬀered widely. Thus Ockham still devotes the main part of his inference theory to a detailed analysis of syllogistics, and so does Buridan, whereas Burley regards it as a well-known special case of the more interesting subject of inference in general. The syllogism is probably the most famous item of “traditional” logic, but actually it has a not very dominant place in the works of medieval logicians. (For instance, in the Logica magna of Paul of Venice it is the subject of only one of 38 treatises.) However, it is systematically and historically so important that we must discuss it in relatively more detail. All authors start by presenting or elaborating the highly condensed deﬁnitions in the beginning of the Prior Analytics. The often-quoted characterization in An. Pr. 24b19–20 says: “A syllogism is a discourse (oratio) in which, certain things having been supposed, something diﬀerent results of necessity because these things are so.” In a broad sense, any formally valid inference could be called a syllogism. But in the stricter sense, a syllogism has precisely two “things supposed,” two premises. There has been quite a lot of discussion on

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whether Aristotelian syllogisms are better understood as conditionals (“if p and q then r”) or as deductive inferences (“p, q; therefore r”). The latter interpretation is perhaps more popular nowadays, and apparently it is also the correct way to see medieval syllogistics, at least in its classical stage. (Obviously the two things have a systematic correspondence, the relation that we nowadays call the deduction theorem, and many medieval authors were fully aware of it.) This means that syllogisms are like natural deductions of present-day logic. Though medieval syllogistics followed Aristotle closely, there were some formal diﬀerences. Thus Aristotle—for special reasons—had formulated his syllogistic propositions as “Y belongs to X,” “mortal belongs to man.” This manner was never adopted in Latin; the medieval logicians wrote just “X is Y ,” “man is mortal.” Aristotle himself had brought the theory of nonmodal syllogistics to such perfection that there was little room left for initiative or disagreement. However, medieval texts produced a more systematic form for the theory, obviously aiming at didactic clarity. A syllogism consists of two premises and one conclusion; the ﬁrst premise is called major, and the second premise minor. Each proposition has two terms, a subject and a predicate, connected by a copula. But the two premises have a term in common, the so-called medium, and the terms of the conclusion are identical with the two other terms of the premises. Syllogisms can then be classiﬁed according to their conﬁguration Subj–Pred into four diﬀerent ﬁgures as follows:

major minor

I

II

III

IV

M –B A–M

B–M A–M

M –B M –A

B–M M –A

Further, each syllogistic proposition belongs to its type a, e, i, or o because of its quality and quantity: They are aﬃrmative or negative, universal or particular. If we proceed by deﬁning that the conclusion must always have the structure A–B, then it is obvious that each ﬁgure includes 43 = 64 alternative combinations, and the total number is 256. But this is not exactly the classical method, so let us have an overview of the syllogism as it was usually presented. Medieval logicians have a full and standard apparatus for syllogistics as early as the ﬁrst terminist phase. They list the same valid syllogisms, usually in the same order, and also call them by the same names. The textbooks of William of Sherwood and Peter of Spain supply these names, which stem from some unknown earlier source and even the famous mnemonic verse composed on them. The names have three syllables, one for each sentence, containing the logical vowels a, e, i, and o. (In the following list of syllogisms, we mention these names that have recurred in all later logic.) The valid syllogisms were known as moods.

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The ﬁrst ﬁgure includes the four famous syllogisms from which Aristotle starts: every M is B, every A is M , therefore every A is B

(Barbara)

no M is B, every A is M , therefore no A is B

(Celarent)

every M is B, some A is M , therefore some A is B

(Darii)

no M is B, some A is M , therefore some A is not B

(Ferio)

The second ﬁgure has four moods: no B is M , every A is M , therefore no A is B

(Cesare)

every B is M , no A is M , therefore no A is B

(Camestres)

no B is M , some A is M , therefore some A is not B

(Festino)

every B is M , some A is not M , therefore some A is not B

(Baroco)

Furthermore, the third ﬁgure contains six moods: every M is B, every M is A, therefore some A is B

(Darapti)

no M is B, every M is A, therefore some A is not B

(Felapton)

some M is B, every M is A, therefore some A is B

(Disamis)

every M is B, some M is A, therefore some A is B

(Datisi)

some M is not B, every M is A, therefore some A is not B

(Bocardo)

no M is B, some M is A, therefore some A is not B

(Ferison)

After the Renaissance, logicians continue by giving the ﬁve moods of the fourth ﬁgure: Bramantip, Camenes, Dimaris, Fesapo, and Fresison. That, however, is not the orthodox Aristotelian way. Aristotle knew inferences like these but did not include a fourth ﬁgure in his theory. Instead, he wanted to place these syllogisms into the ﬁrst ﬁgure. Following his remarks, Theophrastus developed a clear account of the matter, and it was well known in the Middle Ages through Boethius. In Theophrastus’s account, the major term need not be the predicate in the conclusion, which can also have the inverted order B–A. This gives us the ﬁve so-called indirect moods of the ﬁrst ﬁgure: every M is B, every A is M , therefore some B is A no M is B, every A is M , therefore no B is A every M is B, some A is M , therefore some B is A

(Baralipton) (Celantes) (Dabitis)

every M is B, no A is M , therefore some B is not A

(Fapesmo)

some M is B, no A is M , therefore some B is not A

(Frisesomorum)

This method can replace the fourth ﬁgure, though it does introduce a certain unsatisfactory asymmetry.

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The problem of the missing ﬁgure has caused much scholarly debate that we cannot enter into here. Medieval logicians were quite aware of the problem since they had seen at least Averroes’s comments on the fourth ﬁgure. Arguments were often given to refute “objections” questioning the suﬃciency of three ﬁgures. Apparently medieval authors were unanimous in thinking that the fourth ﬁgure could be eliminated with the indirect moods of the ﬁrst ﬁgure. They either said that there were only three ﬁgures, or more precisely, like Albert of Saxony, that the fourth is superﬂuous. It is noteworthy that they did not regard the order of premises as essential. Thus there are 19 valid syllogistic moods. A small addition was obtained by allowing the ﬁve “subaltern” moods, which yield a particular conclusion though a universal one would be valid too. For example, Barbari instead of Barbara leads to “some A is B.” This step would not be accepted in modern logic where universal implications have no existential import, and it indicates clearly that medieval syllogistics assumed that every term really had existential reference. Aristotle had only implicit allusions to singular propositions in syllogisms, and it was a good achievement that medieval logicians constructed a full and systematic theory of singular syllogisms. Ockham was the most active worker here. He emphasizes that the singularity of terms makes no diﬀerence for the validity of inference. This amounts to a considerable reinterpretation of the whole notion of a propositional term. Moreover, he gives explicit cases of singular syllogisms in each ﬁgure, for example, the third ﬁgure “expository syllogisms” like “x is B, x is A, therefore some A is B.” (For nominalists like him, the question had special epistemological relevance because of the basic status of truths about individuals.) Some later Ockhamists even drew a dichotomy across the whole syllogistics between expository syllogisms and those with general mediums.

5.2. Theory of Syllogistics Syllogistics, undoubtedly, is just a small portion of logical inferences, but systematically it is extremely important. The unique thing in classical syllogistics is that it was a formal theory. Its results are not separate truths achieved by trial and error; instead, they are derived in a deductive manner. This had largely been achieved already in the Prior Analytics and continued by ancient commentators. Medieval logicians were very interested in this project. The most important tool here is conversion. It is a completely general method that pertains to all propositions of the S–P form, but it ﬁnds good use in syllogistic theory. Brieﬂy, in a conversion the subject and the predicate change places, and conversion rules tell when such a transposition is legitimate. The following set of (nonmodal) conversion rules was universally accepted. First, in simple conversion AeB converts with BeA, and AiB converts with BiA. In other words,

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some A is B if and only if some B is A, and no A is B if and only if no B is A. Second, in conversion per accidens AaB implies BiA, and AeB implies BoA (this negative one is the only rule that was not in Aristotle): if every A is B then some B is A, and if no A is B, then some B is not A. These are only per accidens, because they change the quantity and do not hold in the opposite direction. Third, ever since Boethius even contraposition was taken as a type of conversion. It preserves the quality and quantity but “changes the ﬁnite terms into inﬁnite ones.” For example, “if every A is B then every nonB is non-A.” Fourteenth-century logicians noticed that contraposition need not be valid when any of the terms is empty—an existential assumption is required. With conversions, some syllogistic moods can be derived from others. The idea is that if certain syllogisms are selected as basic, others can be derived from them by a clever use of ﬁxed methods. Aristotelians called this process “reduction,” present-day logicians would call it proof. Conversion was the most important method of reduction. The other method was reductio ad impossibile: A mood is valid because the negation of the conclusion leads to the negation of a premise. With these methods, all syllogistic moods could be reduced to the direct moods of the ﬁrst ﬁgure—in fact even further, to Barbara and Celarent. This was basic stuﬀ in all textbooks, and the consonants in the names of moods refer to the methods of reduction. (S: convert simply; P: convert per accidens; M: transpose the premises; C: reduce ad impossibile.) These privileged syllogisms are cases of dici de omni et nullo, in which the conclusions can be seen as immediate corollaries of simple aﬃrmation or negation. As Buridan explains, “dici de omni applies when nothing is taken under the subject of which the predicate is not predicated, as in ‘Every man runs’. Dici de nullo applies when nothing is taken under the subject of which the predicate is not denied.” So direct ﬁrst ﬁgure syllogisms are immediately self-evident, and medieval logicians, like Aristotle, called them “perfect.” Others are imperfect in the sense that their validity needs to be shown. The growth of syllogistic theory naturally leads to the philosophical question of its foundations. Such a problem can arise from two perspectives: One may wonder about the status of syllogistics in the totality of logic, or one may ask if particular syllogistic inferences depend on some other principles. a. The question about the general status of syllogistics became current when the theory of consequence developed in the beginning of the fourteenth century (see section 6). Aristotle had started from syllogisms and proceeded to a brief discussion of other inferences; now logicians took the opposite direction. In the thirteenth century, some logicians’ attitude seems to be that all strict demonstrative logic is syllogistical, but the more people were concerned with logical research, the clearer it became that other inferences are valid, too; and

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this was then explicated by means of the concept of consequence. However, the relation between syllogistics and consequences is not very clear. Syllogistics is a part of consequence theory, in the sense that one particular type of consequences are “syllogistic consequences.” (This is especially clearly said by Buridan, whereas Ockham prefers to keep the titles unconnected.) And syllogisms hold because they are good or solid consequences, in our words, logically valid ones. But does syllogistics depend somehow on other parts of the theory? It seems that medieval logicians did not think so. They were aware of the importance of propositional logic—after all, the Stoic heritage had survived—but they did not work in the present-day fashion and build ﬁrst a propositional calculus, then a predicate logic on it. Burley is an interesting case here: He really starts from the simple consequences of propositional logic. But he had no followers in this respect, and contrary to what has been suggested, even he does not apparently aim at any stratiﬁcation of logics here. b. More concretely, one might ask if the validity of a particular syllogistic mood is based on some principles, or if a syllogism involves the use of other logical laws. This problem does not appear in terminist manuals, but it is discussed in the 1240s by Robert Kilwardby. He insists that the necessity of dici de omni et nullo is of such a self-evident nature that it cannot be regarded as a genuine inference step. Many logicians agreed with him. Kilwardby even asks if syllogisms presuppose separate inferences of conversion, and argues that it is not so. Suppose that no B is A; just add “every A is A” as the second premise, and you get the converted sentence, “no A is B,” by Cesare. Similarly in other cases, we see that conversion reduces to syllogism. This idea was not generally accepted, but conversion was occasionally considered so immediate a transformation that it could not be called an inference at all. Soon, however, an alternative view was articulated. About 1270, Peter of Auvergne refers to loci, the governed steps of argumentation theory, and says that “every syllogism holds because of a locus from a more extensive whole to its part.” Simon of Faversham, Radulphus Brito, and others then developed this thought that a syllogism must involve a “principle of consequence.” The conclusion is somehow included in the premises. But the remarks are brief and obscure. In any case, they anticipate the fourteenth-century view of logically necessary consequence relation that is not peculiar to syllogisms.

5.3. Modal Syllogisms Aristotle devotes a large part of his Prior Analytics to modal syllogisms. But unlike nonmodal syllogistics, this area remains in a very unsatisfactory state. The modalities he there studies are necessity, impossibility, and contingency. He wishes to produce a complete set of syllogisms in which some propositions have such modalities; further, he tries to systematize these syllogisms like the nonmodal ones. Here he needs conversion, reductio, and a third method, ekthesis, based on deﬁning new predicates. Medieval logicians replaced ekthesis with a more elegant method of expository syllogisms.

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The main problem is that Aristotle’s theory looks incoherent. His set of accepted syllogisms might be the outcome if all modal propositions were read de re only, as concerning the modal properties of individuals. But then the conversion rules do not hold: Obviously “every A is something necessarily B” does not convert to “some B is something necessarily A.” Moreover, his choice of valid syllogisms contains some oddities. Ancient commentators struggled with these puzzles, and medieval Aristotelians could not avoid them. Peter of Spain’s Summulae does not really discuss modal logic, but Kilwardby, Lambert, and Albert the Great try to save Aristotle’s doctrine. They resort to a very strong interpretation of necessity, proposed by Averroes, which concerns only necessities which hold per se because of essences. Even this technique demands some arbitrary decisions, and in any case it amounts to a severe restriction of modal syllogistics. A similar approach seems to have continued through the thirteenth century. The ﬁrst known work that introduces new methods is the commentary by Richard of Campsall, written about 1308. Campsall’s own theory is conservative, since he wants to maintain the Aristotelian syllogisms and conversions by means of a strict and somewhat confused de re reading. But the novelty is that he makes a systematic distinction between divided and composite readings. It is connected to the idea, initiated by Duns Scotus, of simultaneous alternative states of aﬀairs. This new semantics of modal notions made possible a new and diﬀerent approach to modal logic. From this point of view, modal logic was seen to be much wider than the part that Aristotle had developed, and the relations of modes could be systematized in a new way. The basic notions were now necessity and possibility, which could be understood as realization in all and some alternatives respectively. The ﬁrst exact presentation of the resulting syllogistics was the very thorough account in Ockham’s Summa logicae. In Paris, the orthodox Aristotelian model survived much longer, but Buridan’s Tractatus de consequentiis (1335) provides a modern theory, which is almost as full as Ockham’s. A third and more concise classical text is in Pseudo-Scotus’s commentary on Prior Analytics (c. 1340). The new modal logic gave plenty of room for the notion of contingency, and it caused some disagreement, but for simplicity we bypass this and concentrate on the syllogistics of possibility and necessity. The composite and divided readings of them were strictly distinguished. The composite readings are easier, and accordingly they were less discussed. They were indeed de dicto in the sense that strictly speaking they only make a singular nonmodal claim about a dictum; for example, “it is necessary that some A is B” is interpreted as “the dictum ‘some A is B’ is necessary.” The syllogistic for such propositions follows from the general consequence theory. Ockham and Buridan agree that in every mood, if both premises are preﬁxed with necessity N, the conclusion is necessary too. On the other hand, a syllogism MMM, with all the three propositions modalized as possible, does not hold because the premises need not be compatible. Ockham also remarks on NMM and MNM.

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Much more problematic were divided premises, that is, propositions with genuinely modalized copulas. The main device for dealing with them was ampliation (see section 3.5), which extends the subject term to refer to supposita that occur in alternative nonactual states of aﬀairs; thus “every A is possibly B” will be read “everything which is or possibly is A is possibly B.” But ampliation may be blocked by adding quod est A, “what (actually) is” A. Now it is striking that ampliation was understood in two diﬀerent ways. Ockham assumed that ampliation is good for possibilities (and contingencies)—but he did not accept it for necessities. In other words, only actual things could be said to have necessary properties. The reasons for this are not clear; perhaps he thought that necessities always involve some existence postulate. Buridan, in his turn, said clearly that all modalities amplify the subject in the same way, and this became the common view, that is, if the subject of a modality is not explicitly restricted to what is, it is freely ampliﬁed. (We must therefore be cautious if we wish to use present-day possible world apparatus here.) Buridan drew an octagonal diagram of the propositions “Every/Some A is necessarily/possibly B/not B” and analyzed all the 56 logical relations between them. This made the map of modalities much clearer. Combinations of syllogistic moods, modalities, and restrictions produce a huge number of cases, and logicians could not mention every case explicitly, although they did pursue a full theory of them. They also comment on cases where some propositions are nonmodal. We can only sketch some outlines now. In the direct ﬁrst ﬁgure, everybody accepted MMM syllogisms as valid. Buridan and Pseudo-Scotus accept NNN, MNM, and NMN. The seemingly surprising NMN here shows the eﬀect of ampliation. (For instance, every M is necessarily B, some A is possibly M , therefore some A is necessarily B.) Ockham accepts NNN only when restricted to actuals; for Buridan’s school this is another valid syllogism, like several other moods resulting from a restriction of subjects of N or M. Buridan also accepts, for example, _NM with an assertoric major. In the second ﬁgure, Buridan mentions NNN, NMN, and MNM (and Pseudo-Scotus mistakenly adds MMM). These again have restricted versions (in the style of: if every actual B is necessarily M and every actual A is possibly not M , then no A is B). But Ockham allows no valid syllogisms here. In the third ﬁgure, all accept MMM. Buridan and Pseudo-Scotus accept NNN, NMN, and MNM, while Ockham accepts only restricted versions of these. Some of them, not precisely the same ones, are in Buridan. Ockham’s theory looks somewhat unﬁnished: His view of ampliation causes trouble, and he derives a great number of results by discussing individual examples one by one. Buridan, on the other hand, uses a very elegant deductive method with, for example, cleverly formulated conversion rules. His theory is the summit of medieval modal logic. His pupils Albert of Saxony and Marsilius of Inghen continued to give comprehensive accounts of modal syllogistics, with some usually unsuccessful innovations, but after them modal syllogisms seem to have fallen out of fashion.

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5.4. Topics and Methodology An important part of medieval logic was topics. The dialectics of the old trivium mostly belonged to it. The ultimate source was Aristotle’s Topics, but a second and simpler authority that replaced it for a long time was Boethius’s De diﬀerentiis topicis. The main subject in this inquiry was loci, locus being Latin for Aristotle’s topos (literally “place,” here something like “consideration”). Aristotle does not deﬁne his topos, whereas Boethius gives two meanings for locus. It can be a “maxim,” a self-evident sentence that needs no further proof, but it can also be a logically relevant feature that distinguishes two sides. Confusingly, the distinction can be between sentences, like aﬃrmative and negative, antecedent and consequent, or between concepts, like genus and species, part and whole. For example, the distinction between genus and species supports the maxim: What belongs to the genus belongs to the species. Boethius’s double notion of loci long guided medieval topics. On the other hand, Aristotle emphasized an aspect which was not so prominent in Boethius: Topics concerns dialectical argumentation, the ﬁnding, testing, and examining of plausible theses. Hence it is not restricted to methods of demonstrative scientiﬁc proof of necessary results. Treatises as early as the eleventh century discuss topics, and this interest culminates with the Aristotelian revival of the thirteenth century. Thus, Peter of Spain gives a detailed list of various loci which follows Boethius closely. An important idea in such lists is that loci are supposed to guarantee the validity of an inference or argument that was not immediately valid because of its form. For instance, Peter’s inference “The housebuilder is good, therefore the house is good” is surely not formally valid—and not even quantiﬁed—but it is “conﬁrmed” by the locus of cause and eﬀect: “That whose eﬃcient cause is good, is itself also good.” We see that the result is still not conclusively proved, but the addition connects the argument to syllogistics. This need of support is characteristic of “enthymematic” arguments, demonstrative or not. Nowadays we are accustomed to think that they are valid because of some suppressed deductive premises, but medieval authors did not always see the matter so. Often they thought that the support came from a rule and not from an implicit premise. The terminists were inclined to think that all valid arguments are reducible to syllogisms; topics gives metalogical directions for ﬁnding suitable middle terms for the reduction. After the early terminists, topics was still constantly discussed. After all, the Topics was a big book in the Organon and belonged to the obligatory courses, at least in part. But the heyday of topics was over when logica moderna was developed. It was no longer a really inspiring ﬁeld, although it undoubtedly had some importance: Topics apparently inﬂuenced the growth of consequence theory (see section 6.1), and the doctrine of loci was also relevant in discussions concerning the foundation of syllogisms.

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When the consequentia theory developed, both syllogisms and nonsyllogistic inferences could be seen as cases of the same general patterns. As a result, topics lost an important function. The arguments that were formerly studied in topics were, in the fourteenth century, normally included in consequences. Also, it is signiﬁcant that topics was no longer connected to enthymemes but to dialectical arguments, that is, its special character was seen as epistemic. Usually, the leading logicians no longer treated topics as a separate subject at all—Ockham, for instance, studied topical arguments only as a relatively uninteresting special case. On the other hand, Buridan still painstakingly devoted a whole treatise to topics. Later the interest in topics diminished even more; Paul of Venice did not speak of it. However, commentaries on Aristotle’s Topics were written throughout the fourteenth and ﬁfteenth centuries, but no new ideas were presented. The Aristotelian theory of science was highly abstract; while it had little contact to concrete problems, it did have a close connection to logic. The basic source for medieval discussion was Posterior Analytics, though direct commentaries on this diﬃcult work were not very common. In the Aristotelian picture, developed for example by Aquinas, an ideal science consists of a system of demonstrative syllogisms. Their premises must be true, necessary, and certain. Premises can be derived by other syllogisms, but ultimately they rest on evident necessities. As Kilwardby says, “the demonstrator considers his middle term as necessary and essential, and as not possibly otherwise than it is; and so he acquires knowledge, which is certain cognition that cannot change.” Science is thus a system of syllogisms about causes and essences; it can use logical principles, but logic itself is obviously not a science. Much of this grandiose view later had to be given up, when ﬁrst Scotus problematized the notion of necessity and then Ockham problematized the notion of evidence.

6. New Approaches to Inferences During the thirteenth century, four new domains of logical research broadly falling into the scope of propositional logic emerged: consequences, obligations, insolubles, and sophisms. In overall treatments of logic like Ockham’s Summa logicae and Buridan’s Summulae dialectica, these new branches of logic were discussed in the place traditionally occupied by treatments of dialectical topics in the sense in which they referred to what Aristotle discusses in his Topics. This is not to say that the traditional theory of dialectical topics, for which Cicero and Boethius had provided the classical texts, had disappeared altogether. Nor can we say that these new areas of logic had replaced the tradition of dialectical topics. Rather, the purposes aimed at by research in these new areas were seen to be approximately similar to those traditionally aimed at by the theory of dialectical topics, and consequently the new ﬁelds were taken to complement traditional discussions. In modern terms, we can say that the point of gravity was moving from the theory of argumentation toward formal logic.

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Let us start with consequences, considering four diﬀerent issues pertaining to this crucial area of logic. Late medieval discussion of consequences aimed at giving clear and speciﬁc determinations of (1) what is a consequence, (2) the deﬁnition of the validity of a consequence, (3) how they should be classiﬁed, and (4) rules concerning valid consequences.

6.1. What Is a Consequence? In general, late medieval treatments of consequences understood them as inferences. That is, they were not called “true” (vera) or “false” (falsa), but rather were said to be “good” (bona), or simply “to be valid” (valeo) or “to hold” (teneo), or in the opposite case “to fail” (fallo). Despite an acknowledged close connection to conditional propositions, consequences were usually discussed separately as belonging to a diﬀerent place in the overall structure of logic. Ockham, for example, discusses conditionals within his theory of propositions, and turns to consequences as a theory of nonsyllogistic inference in the beginning of III, 3 in Summa logicae: “After treating syllogism in general and demonstrative syllogism, we now have to turn to the arguments and consequences that do not apply the syllogistic form.” The genre of logical writings on theory of consequences seems to have arisen in the thirteenth century from recognition of the fact that a general theory of inferential validity can be formulated in addition to, or as an extension of, the traditional syllogistic system. As such, medieval logicians had been aware of the idea at least since Abelard’s work, and Boethius had already composed a special treatise on what he called “Hypothetical syllogisms,” that is, on propositional logic. Nevertheless, Walter Burley’s De puritate logicae seems to have been the ﬁrst overall presentation of logic to discuss the theory of inference systematically starting from general issues of consequences and moving toward more particular issues after that, allowing syllogistic only the minor position of a special case. That most medieval logicians saw consequences as inferences and not as propositions is reﬂected in the fact that they aimed at formulating general rules (regulae) of valid inferences; traditional dialectical topics were also seen to belong to this set in addition to a number of more formal ones. The outstanding exception in this picture is John Buridan and his Tractatus de consequentiis. He explicitly deﬁnes consequences as hypothetical propositions consisting of two parts, the antecedent and the consequent, joined by a connective like “therefore” (ergo). Thus, Buridan’s consequences amount to conditional propositions with speciﬁc content. He treats consequences as pieces of discourse that assert the validity of an inference from the antecedent to the consequent: “One follows from the other” (una sequatur ad aliam). Accordingly, Buridan does not discuss or lay down metalinguistic rules (regulae) of consequences in this treatise, but instead asserts “conclusions” (conclusiones) concerning what can be truly said about the kinds of sentences following from each other.

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It seems clear that all prominent medieval logicians saw the distinction between the acceptability of performing an inferential step and the assertion that a valid inferential relation obtains. Whereas most logicians thought that consequences should be understood as inferences, Buridan made the opposite decision. For him, a consequence was a proposition, a conditional claim concerning an inferential relation between the antecedent and the consequent. He seemed to have had no followers in this opinion, but because of his prominent position in late medieval logic, his surprising stand has caused a number of misunderstandings concerning the issue both for medieval authors and for modern commentators.

6.2. Criteria of Validity The simplest way to formulate the deﬁnition of inferential validity was to ground it on the idea that it is impossible for the antecedent to be true and the consequent false. Indeed, it seems that all late medieval deﬁnitions of validity can be seen as variously qualiﬁed or modiﬁed versions of this principle. In the ﬁrst known treatise directly dedicated to consequences, Burley’s De consequentiis, we ﬁnd the deﬁnition that a consequence is valid if “the opposite of the consequent is repugnant to the antecedent.” The problem with this deﬁnition is that it seems unclear in which sense we are to take the word “repugnant,” since it is often used in a way that already contains reference to inferential connections. Indeed, Burley elsewhere opts for alternative deﬁnitions closer to the modal criterion. In Buridan, we ﬁnd the following list of three alternative descriptions concerning when some proposition “is an antecedent to another” or, in other words, a consequence is valid: (a) “that is antecedent to something else which cannot be true while the other is not true” “illa alia non existente vera”; (b) “that proposition is antecedent to another proposition which cannot be true while the other is not true when they are formed simultaneously” “illa alia non existente vera simul formatis”; (c) “that proposition is antecedent to another which relates to the other so that it is impossible that howsoever it signiﬁes, so is the case, unless howsoever the other signiﬁes, so is the case, when they are formed simultaneously” “sic habet ad illam quod impossibile est qualiterqumque ipsa signiﬁcat sic esse quin qualiterqumque illa alia signiﬁcat sic sit ipsis simul propositis.” Buridan ﬁnds each of these three descriptions problematic, but accepts the last, if it is understood in a suitably loose manner. The problem with the ﬁrst deﬁnition is related to the standard medieval requirement that a proposition must be actually formulated to be true. This makes it clear that almost no consequence would be valid according to the ﬁrst criterion, since the consequent need not be

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formulated when the antecedent is. The second aims at correcting this problem through the simple addition “when they are formed simultaneously,” but falls prey to it as well. A consequence like “no proposition is negative, therefore no donkey runs” should be invalid, but turns out to be valid on criterion (b) as well as on (a), because the antecedent is never true when it is actually put forward. Thus, it cannot be true without the consequent being true even if they were simultaneously formed. With criterion (c), Buridan takes another strategy. He recognizes that the consequential relation should not be seen to obtain with the sentences themselves, not even between potentially formulated ones, but rather between their contents. However, Buridan did not believe that such sentential contents would exist (see the section about propositional signiﬁcates, complexa signiﬁcabilia), and therefore the formulation of the criterion (c) makes problematic ontological commitments. Apparently he could not ﬁnd a formulation that would avoid them, and thus we are left without a satisfactory description of inferential validity. It seems, nevertheless, that Buridan’s strategy of transporting criteria of validity from the actual sentences to their signiﬁcations or contents became a generally accepted one. In some interesting sense, which still puzzles modern scholars, Buridan’s further discussions on the topic take a “mentalistic” turn in the conception of logical validity. He considered that logical validity depended on the mind in a more crucial sense than many of his predecessors. Some formulations by his followers made this mentalistic turn even more obvious in ways that we shall see in the next section.

6.3. Classiﬁcations of Valid Consequences The most traditional medieval distinction among kinds of valid consequences was the distinction between those valid “as of now” (ut nunc) and those valid “simply” (simpliciter). Validity ut nunc was taken to mean something like validity given the way things now are: From “every animal is running,” it follows ut nunc that Socrates is running, at the time in which Socrates exists as an animal. After his death, the consequence ceases to be valid. Simple validity, on the other hand, meant validity in all circumstances. In this sense, from “every animal is running,” it follows that “every human is running.” It is noteworthy that validity ut nunc also contains some kind of necessity, and thus it cannot be compared to twentieth-century material implication. Late medieval logicians put their main interest in two other, philosophically more interesting distinctions. Somewhat confusingly the concepts “form” and “matter” were used in both distinctions, so that when we come to Paul of Venice, a consequence may be, for example, “formally formal” or “materially formal,” since he combines the two distinctions into one systematic presentation. In both distinctions the issue was to separate a class of consequences that were valid in a privileged manner: not only valid, but “formally valid.” In one sense, formal validity meant a substitutional kind of validity, where a consequence is formally valid, if it “is valid for all terms” (tenet in omnibus

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terminis), and only materially valid if its validity is based on the special content of some of the terms used in the inference. In this sense, the paradigm examples of formally valid inferences were syllogisms in the Aristotelian ﬁgures, but also examples like modus ponens could be put forward. In the other sense, an inference was called formally valid only if the consequent was “formally included” (includit formaliter) in the antecedent or in the “understanding” (intellectus) of the antecedent; this kind of formal validity was often called “natural” or “essential” validity. It seems that the roots of both distinctions can be traced back to the early Middle Ages. At least Kilwardby gives ground for both distinctions. Nevertheless, the two distinctions seem to have had a somewhat diﬀerent history. Furthermore, the concept of material validity remained in most treatments rather obscure. It seems, however, that especially as related to the latter deﬁnition of formal validity based on inclusion, material validity was often understood as having to do with certain properties of the propositions used. The paradigm cases of materially valid inferences followed the rules “from the impossible anything follows” and “the necessary follows from anything.” Let us ﬁrst look at the latter kind of formal validity, the one based on the idea that the antecedent must “formally include” the consequent. The concept of “formal inclusion” seems to have been developed by late thirteenthcentury theologians, such as Henry of Ghent, Godfrey of Fontaines, and Duns Scotus. In many texts the topic comes up in a discussion concerning the role of the third person in the divine Trinity, employing the special technique of obligations (see following). These discussions resulted in elaborated theories of what it means to say that a concept is included in another concept, or that an assertion conceptually includes and thus entails another claim. The primary examples studied by medieval logicians included inferences like “a human exists, therefore, an animal exists,” and the explanation of their “formal” validity was based on the necessary conceptual or essential relation between the species “human” and the genus “animal.” The concept “human” was said to “formally include” the concept “animal,” and thus the inferences based on this relation were said to be “formally valid.” In twentieth-century terms, we would rather describe them as analytically valid inferences. William Ockham was aware of this discussion and aimed at bringing the results into the systematic context of logical theory. In the classiﬁcation of Summa logicae, a consequence is formally valid if it is valid by general rules of a speciﬁc kind. They must concern the syntactic features of the propositions (forma propositionis) involved in the consequence. Also, the rules must be self-evident (per se nota). This part of the deﬁnition is in eﬀect identical with, or at least comes very close to, the substitutional type of deﬁnition of formal validity. But on the same page Ockham also admits as formally valid consequences that are valid by something he calls an “intrinsic middle.” His example is “Socrates does not run, therefore a man does not run,” which is valid by the “intrinsic middle,” “Socrates is a man.” It seems that Ockham wanted to present this type of formal validity to allow also inferences based

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on something like conceptual inclusion within this group, the inclusion being expressible as an intrinsic middle. Some 10 years later, Ockham’s student Adam Wodeham explicitly distinguished between two diﬀerent ways of understanding the concept of formal validity. One of them uses only the substitutional criterion, while the other accepts as formally valid all consequences based on truths known in themselves (per se nota). Insofar as Wodeham’s per se nota refers to all analytic truths and not only conceptual inclusion, the deﬁnition is wider than that derived from the traditional slogan “formally includes,” but it is clearly on the same track. Material validity is deﬁned by Ockham with reference to something he calls “general conditions of the propositions,” and he gives the ex impossibile quodlibet rule as an example. In Ockham’s case this is strange, since he clearly knew that from a contradiction it is possible to derive anything with rules which he allows to be formal. Do we, thus, have inferences that are both material and formal? In his deﬁnition of formal validity, Buridan presents only the substitutional principle, without mentioning the idea of conceptual inclusion. His examples of inferences which are valid but not formally so, however, show that he was aware of the criterion but did not want to use it. He straightforwardly claims that those inferences, which are valid so that all substitutions of the categorematic terms with other terms are also valid, are formally valid. Among formally valid inferences, Buridan explicitly counts inferences from contradictions, though of course not from weaker impossibilities like from “a man is not an animal.” These he classiﬁes as material. It seems that in the latter half of the fourteenth century, Buridan had few followers in his classiﬁcation principles. Only Albert of Saxony seems to have accepted the substitution principle as the sole criterion of formal validity. The majority of logicians seem to have wanted to develop an idea which is closer to what was later in the twentieth century called analytic validity. The criterion of formal validity was, therefore, formal or conceptual inclusion of the conclusion in the premises. As an interesting special case, Paul of Venice presents a system that uses both concepts of formal validity, thus producing a very elaborate system.

6.4. Rules of Consequences Usually medieval discussions of the theory of consequences also included a selection of rules warranting valid inferences. Instead of anything close to a complete listing of such rules, we must here satisfy ourselves with a look at the types of such rules presented in the medieval discussions. We have already encountered two such rules: “from the impossible anything follows” and “the necessary follows from anything.” These rules were practically never completely rejected in the later Middle Ages. However, their applicability in speciﬁc contexts was often limited, and as they were typically classiﬁed as

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materially valid, they were understood as belonging to a somehow inferior kind. Medieval authors knew the main rules of so-called classical propositional logic. For example, detachment is put by Burley as concisely as possible with a term variable: “If A is, B is; but A is; therefore B is.” Transitivity rule is presented by Burley as the consequence “from start to ﬁnish” (a primo ad ultimum). He also discusses other examples of basic propositional logic of the kind, but when we turn to the later fourteenth century, the selection of rules of this type leaves nothing to be hoped for. One type of rules of consequences that seems to have interested medieval authors quite widely is based on epistemic operators. These were often discussed by direct comparison with modal rules; if something is necessary, it is the case, and if something is known, it is the case. More interesting (and more disputable) examples of relevant inference schemes are more complicated. The rule “if the antecedent is known, the consequent is also known” was often held to be valid only on the further condition that the consequence itself is known. The ﬁrst rule of consequences in Ockham’s Summa logicae is that “there is a legitimate consequence from the superior distributed term to the inferior distributed term. For example, ‘Every animal is running; hence every man is running’.” All medieval logicians accepted this example as valid, though they often formulated the rule diﬀerently, and the explanation of the kind of validity varied. Buridan, who relied on the substitution principle, thought that the consequence is valid in a standard syllogistic mood with the help of a suppressed premise. But almost all other logicians thought that something like the rule given by Ockham suﬃces for showing the validity. The reference to the relation between a superior and an inferior term given in the rule was understood in terms of the criterion on conceptual containment. It seems that in twentieth-century terms, the rules of this type could be characterized as regulating analytic validity. These types of rules already bring us close to Aristotle’s program in the dialectical topics presented in the Topics. This work was indeed much used in compilations of the listing of the rules for consequences. Also, in many works the lists contain rules that have more of the character of the theory of argumentation than of formal logic. The rules for consequences are indeed one of the places where the diﬀerences between modern and medieval conceptions of logic are most clearly visible.

6.5. Obligations The genre of late medieval logical literature that has perhaps been the most surprising for modern commentators carries the title obligations (obligationes). The duties or obligations at issue in the treatises carrying this title were of a rather special kind. The basic idea was based on the Socratic question/answer game as described and regulated by Aristotle in the Topics. In the speciﬁc medieval variant of the game the opponent put forward propositions that had

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to be granted, denied, doubted, or distinguished by the respondent. In giving his answers, the respondent was expected to pay recognition to the truth, but especially to some special obligation given to him in the beginning of the exchange by the opponent. This duty was understood to override the general duty of following the truth, but not the general logical duty of respecting arguments and avoiding contradictions. Here we cannot go into details concerning the diﬀerent variants of the system, although a number of interesting logical issues arose through the study of the particular kinds of possible duties. The main type of an obligational disputation, as medieval authors knew it, was based on a positum, a sentence put forward by the opponent in the beginning as something that the respondent has to grant. This sentence was typically false, and often even impossible in some way not directly implying a contradiction (conceptually impossible, naturally impossible, etc.). Then the opponent put forward further propositions, and in answering them the respondent had to pay attention to inferential relations between the positum and these later proposita. Altogether four main alternative sets of exact rules of how the inferential connections ought to be recognized were developed in the Middle Ages. According to one late thirteenth-century system, described by the Parisian logician Boethius of Dacia in his commentary on Aristotle’s Topics, the respondent must grant everything that the opponent puts forward after the positum, with the sole exception of propositions that are inconsistent (incompossibile) with the positum or the set of posita, if there are several. Boethius divides propositions into “relevant” and “irrelevant” ones with the criterion of an inferential connection to the positum. Those inconsistent with the positum are called repugnant (repugnans), and those following from it are called sequent (sequens). The repugnant ones must be denied and the sequent ones must be granted. Others are irrelevant, and Boethius claims that the respondent must grant them, since this implies nothing for the positum. In his discussion, Boethius relied on an already traditional terminology, but not all of the earlier authors would have agreed with his rules. The early fourteenth-century discussion took place mainly in England, and there a diﬀerent set of rules came to be accepted as the traditional system. According to these rules, the respondent should of course grant the positum and anything following from it. Similarly, he should deny repugnant propositions. But he should grant true irrelevant propositions and deny false ones. After having granted or denied such propositions, he should take them into account in the reasoning. He should grant anything that follows from the positum together with propositions that have been granted earlier or whose negations have been denied. Thus, the respondent must keep the whole set of his answers consistent, but otherwise follow the truth. Duns Scotus claimed that in an obligational disputation based on a false positum one need not deny the present instant, but one can understand the counterfactual possibility at issue in respect to the present instant. (Unlike many of his predecessors, Scotus denied the principle “what is, is necessary,

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when it is.”) After Scotus, it became customary to think of the set of answers after the disputation as a description of some consistently describable situation. This brought obligational disputations close to counterfactual reasoning and thought experiments. Richard Kilvington suggested in his sophismata an interesting revision of the rules apparently based on the idea that the disputation ought to describe the situation that would obtain if the false positum were true. He claimed that this principle ought to be taken as the rule guiding answers, giving the respondent a duty to grant what would be true and deny what would be false if the positum was true. Kilvington’s suggestion did not gain many followers. Most authors kept to the traditional rules, probably because Kilvington’s rules seemed too vague. Formally valid inferential connections were taken to provide a better foundation for obligational disputations. But another revision was also suggested, and for some time it gained more followers. Roger Swineshed suggested that all answers ought to be decidable solely on the basis of the positum without recognition of any subsequent exchange. Swineshed’s suggestion was that the respondent ought to grant the positum and anything following from it, and deny anything repugnant with it. Other propositions were to be taken as irrelevant, and they were not to be respected in the reasoning. This had the implication that irrelevant propositions would have to be kept separate from the mainline of the disputation, as a kind of second column in the bookkeeping. As Swineshed explicitly recognized, contradictions between the two columns could arise so that, for example, a conjunction may be denied when one of its conjuncts is granted as the positum and the other is granted as true and irrelevant. The main logical topic studied in obligational disputations was logical coherence. The disputations were in essence structures allowing propositions to be collected together into a set, with evaluation of the coherence of the set as the crucial issue at each step. The diﬀerent rules formulated the alternative exact structures for such a procedure.

6.6. Insolubles Early treatises on obligations are often connected with treatises carrying the title “insolubles” (insolubilia). In these treatises something is laid down in a way similar to how the obligational positum is laid down, but the crux of the discussion is that the given propositions appear to describe a possible situation and yet they entail a contradiction. The case is thus paradoxical. As a common example from the obligational treatises themselves, we may mention the rule that the respondent ought not accept “the positum is false” as his positum. The case is, of course, closely analogous to what is nowadays known as the Liar Paradox. It is not clear that obligational disputations were the original context of the genre of logic that came to be called insolubilia, since the ﬁrst treatments of such paradoxes in their own right seem to be equally early and have other

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sources, too. But the way medieval logicians formulated their versions of the Liar Paradox comes with an obligational terminology and context. If we turn to the mature treatises of the early fourteenth century, the paradigmatic insoluble is the proposition “Socrates is saying what is false,” and the assumed situation is that Socrates utters this and only this proposition. Then it is shown that if the sentence is true it is false (because if it is true, what it signiﬁes is the case), and if it is false it is true (because it signiﬁes that it is false, and that was assumed to be the case). Because these results cannot stand together—every proposition is true or false but not both—a contradiction seems to follow from what is clearly possible, for the only assumption seems to be that Socrates makes a simple understandable claim. Medieval logicians discussed a wide variety of carefully formulated analogous paradoxes, and it seems that some of them were formed to counter speciﬁc purported answers to the paradox. For example, if the paradox is claimed to result from direct self-reference, we may be asked to consider other examples. For example, medieval logicians considered cases where two or more people make assertions about the truth or falsity of each other’s claims and thus produce a paradoxical circle. A paradox reminiscent of the Liar Paradox can be produced without any proposition referring to itself—the paradox is not dependent on direct self-reference. It is also interesting to note that some practical analogs of the paradox were considered. Assume, for instance, like Buridan, that Plato is guarding a bridge when Socrates wants to cross it. Then Plato says, “If you utter something false I will throw you into the river, and if you utter something true I will let you go.” Socrates replies, “You will throw me into the river.” Now, what should Plato do? Cervantes makes Sancho Panza face a similar problem when he is the fake governor of an island, and indeed, Cervantes probably got the paradox from some medieval treatment of logic. The variety and the history of the diﬀerent solutions of the insolubles is too wide and complicated to be even summarized here. Some main alternative solutions presented in the medieval discussion must suﬃce for now. In the early discussions, the so-called nulliﬁers (cassantes) claimed that the one who utters a paradoxical sentence “says nothing.” If Socrates says only the sentence “Socrates says what is false,” he has not really uttered a proposition at all, and thus no truth value is needed. The problem, of course, is to explain precisely why the utterance fails to be a proposition. Some authors gave the reason that a part, like “false,” cannot refer to its whole; but this thesis is too generalized. In his Sophistical Refutations, Aristotle mentions the case where somebody says something that is simultaneously both true and false. This remark occurs in connection to the fallacy of confusing truth in a certain respect and absolute truth (secundum quid and simpliciter). Thus applications of this fallacy were often tried in solving insolubles, but understandably the results were not very convincing. One related suggestion was that insolubles were to be treated not as cases of genuine self-reference but instead as cases where a certain shift of reference (transcasus) takes place. When Socrates says that he is lying, he

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simply cannot mean that very utterance itself, and therefore we must look for some other utterance in the immediate vicinity. In the assumed case, this approach makes the insoluble false simply because there is nothing else that Socrates says. Fourteenth-century logicians found all these suggestions too simple-minded. In the early 1320s, Thomas Bradwardine used symbolic letters for propositions and assumed that every proposition a signiﬁes, in addition to its ordinary signiﬁcation, even “a is true.” (Strictly speaking, this was formulated as a general doctrine only later.) Substituting a = “a is false,” we get “a is false and a is true,” a contradiction that shows that a is false. A similar strategy is further reﬁned by William Heytesbury (1335). He puts the issue within the framework of obligation theory, discussing cases where insolubles are pressed on the respondent. All insolubles turn out to be false, but he admits that there is no general solution; what is needed is a careful study of what exactly is extraordinary in the signiﬁcation of each relevant sentence. Some authors, like Swineshed round 1330, argued that an insoluble proposition “falsiﬁes itself.” This requires a new opinion about truth: For the truth of a proposition, it does not suﬃce that it signiﬁes what is the case, but it also must not falsify itself. This fundamental novelty may have been one reason why the theory was not generally accepted—and, moreover, its applications soon lead to obscurities. Later, Gregory of Rimini and Peter of Ailly tried to utilize the doctrine of mental language in this context. The complex theory that Peter developed (in the 1370s) argues that spoken insolubles correspond to two conﬂicting mental propositions, whereas a mental proposition cannot ever be insoluble. This idea became well known but did not gain general acceptance. To sum up, we may say that the common view was that certain propositions were called insoluble not because of logical puzzles that could not be solved but because providing a solution “is diﬃcult,” as many authors remark. It was generally agreed that insolubles were false. Only a few authors took seriously the possibility that the paradox might be a genuine one, one that did not allow any satisfactory solution. But even they did not think of insolubles as a threat to the system of logic as a whole. Insolubles were not considered to undermine the foundations of logic but simply to be one interesting branch of logical studies. One might surmise that this can derive from the idea of looking at logic as an art dealing with the rational structures embedded in the mental basis of ordinary language, rather than as a calculating system based on special foundations.

6.7. Sophismata Buridan’s Summulae de Dialectica concludes with an almost 200-page section containing sophisms (sophismata), which are examples construed in a rather distinct way so that they make the need of logical distinction clearly visible. Buridan’s work is no exception; diﬀerent kinds of collections of such sophisms

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are commonly found in medieval logic manuscripts. It seems that they were used in medieval logic teaching as exercises to show how general logical systems could be applied in practical contexts. But often they also contain interesting material that is not discussed in systematic treatises. Separate collections of sophisms circulated throughout the Middle Ages. Perhaps the most famous of the early examples of such aids of teaching was known as the magister abstractionum. Little is known of the person, and he may not have been a single person. It is possible that we simply have a collection of examples which circulated among teachers of logic, who would each add their own examples and drop out others. Later, many authors of logical textbooks compiled their own collections of sophisms. This is what we ﬁnd for example in the case of Buridan. In early fourteenth-century Oxford, such a textual genre gained new signiﬁcance by assuming a relatively speciﬁc independent role not only in the university curriculum (where undergraduate students in their ﬁrst years of university were called “sophists” [sophistae]) but also in logical study. The collections of sophisms composed by Heytesbury, Kilvington and some other members of the so-called group of Oxford calculators were an important locus of logicolinguistic and mathematical study providing important results that were later used by pioneers of early modern science. A sophism in this sense of the word consists of (1) the sophisma sentence; (2) a casus, or a description of an assumed situation against which the sophisma sentence is evaluated; (3) a proof and a disproof of the sophisma sentence based on the casus; and (4) a resolution of the sophism telling how the sophisma sentence ought to be evaluated and how the arguments to the contrary should be countered. In the discussion of sophisma 47 in his collection, Kilvington assumes that the procedure in solving a sophisma must abide with the rules of obligational disputations. That is, the casus is to be understood as having been posited in the obligational sense, and thus anything following it would have to be granted and anything repugnant to it would have to be denied. From this viewpoint, the proof and disproof can be articulated as obligational disputations. Although an explicit commitment to using obligational rules such as Kilvington’s is rare in the collections of sophisms in general, obligational terminology is omnipresent. In many sophisms, the problematic issue was to show how the sophisma sentence was to be exactly understood. For this reason, sophismata became an especially suitable place for determining exact rules of scope and the interpretation of words serving important logical roles. Indeed, this is the context where late medieval logicians developed the exactitude in regulating logical Latin that was ridiculed by such Renaissance humanists as Juan Luis Vives. Heytesbury’s Rules for Solving Sophisms (1335) is a good representative of the genre and can thus be used as an example here. It consists of six chapters. The ﬁrst is on a topic we have already mentioned, so-called insolubles. The second discusses problems of epistemic logic with sophisms based on the words

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“to know” and “to doubt.” The third tackles problems connected to the use of pronouns and their reference. In the remaining three parts, Heytesbury turns to problems that may be better characterized as natural philosophy rather than logical analysis of language. The fourth part considers a traditional topic, the verbs “to begin” and “to cease,” and thereby issues connected to limit decision problems and temporal instants. The ﬁfth part, on maxima and minima, continues on the same tract from a diﬀerent viewpoint. The sixth and ﬁnal part is dedicated to “three categories,” referring to the Aristotelian categories of place, quantity, and quality. Especially this last part and its discussions of speed and acceleration proved very fruitful in the early development of modern science despite the fact that all the cases studied in it are purely imagined and lack any sense of experiment. For example, instead of real bodies in motion, medieval logicians considered imagined bodies in motion. In fact, this chapter and others of its kind show how the medieval secundum imaginationem method, relying only on logicolinguistic analysis, was able to provide results that have often been misguidedly attributed to experimental scientists working centuries later. One of the speciﬁc techniques used in solving sophisms deserves treatment of its own in a history of logic. In early thirteenth-century texts, a sentence like “Socrates begins to be pale” was analyzed as something like “Socrates was not pale and Socrates will be pale.” The analysis was accompanied with a discussion on which of the two conjuncts in the particular kind of change at issue should be given in the present tense, and how one should formulate the continuity requirement that Socrates, say, will be pale immediately after the present instant, even before any given determinate future instant. Such an analysis became a standard technique used in a large variety of cases and was called “exposition” (expositio). Without going into the particulars of the speciﬁc verb “to begin,” it is worth pointing out here that the idea in such an analysis is to break down the sentence containing the problematic “exponible term” into a conjunction or a disjunction that is equivalent in its truth conditions. For fourteenth-century logicians, it was a commonly accepted doctrine that there is a large number of terms that admit, or in the contexts of a sophism, demand such an analysis. Furthermore, this kind of analysis was taken to be necessary for practically all philosophically central terms if there was a need to treat them in a logically exact manner.

7. The End of the Middle Ages 7.1. Later University Logic Undoubtedly, the main plot of medieval Aristotelian logic lies in the development that began from the early terminists and led to the stage of Burley and Ockham, and then had its academic culmination in the systematic work of Buridan. But the discipline of logic survived after that, and some new special features appeared and new developments took place in the late Middle Ages.

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It is probably true to say that logicians were no longer very original during this time. But here it is necessary to emphasize that logic was a widespread and multiform discipline; the volume of material is very great, and much of it is still unexamined. A gradual change happened in philosophy in general during the fourteenth century, a change whose background is hard to explain. It has been pointed out that the whole cultural climate was no longer the same: The fourteenth century included great political upheavals; the Church had diﬃculties that led to the great schism; various protest movements appeared, and so on. All this contributed to the loss of the previous unity. It is customary to start the “autumn of the Middle Ages” from 1350, but this demarcation is largely symbolic; the only concrete thing that can support it is the Black Death, which killed many philosophers in 1349. After 1350, philosophy was still practiced in the old style, and logic has hardly ever been as prominent a part in philosophy as in the latter half of the fourteenth century. However, the overall authority of philosophy and logic started to diminish. Let us try to sketch an overview of the historical development. Ockham, a political dissident, had never made an uncontested breakthrough—in fact, he was considered an extremist even among nominalists. In logic, however, his thought had a wide inﬂuence. Buridan, then, had more indisputable prestige, and as regards logic, his inﬂuence became dominant in Parisian philosophy during the 1340s. In this ﬁeld he had two extremely competent pupils, Albert of Saxony (d. 1390) and Marsilius of Inghen (d. 1396). After the generation of Buridan’s students, the position of Paris weakened, although it was still the most famous university. England underwent a quite distinctive process. In the beginning of the fourteenth century the best logicians were English, and even after them there were original ﬁgures in Oxford, like Bradwardine, Heytesbury, and Billingham. Then, after 1350, logic turned to great technical sophistication but little essentially original appeared in the works of logicians such as Hopton, Lavenham, Strode, Feribrigge, and Huntman. Soon after 1400, a complete collapse took place in England, and only some elementary texts were produced during the ﬁfteenth century. But English logic was, however, very inﬂuential in the late Middle Ages on the Continent. English works of the fourteenth century were studied and commented on in Italy. Particularly Ralph Strode’s logic achieved great fame. Paul of Venice had studied at Oxford, and he transmitted the comprehensive English tradition to the Italian logicians of the ﬁfteenth century: Paul of Pergula, Gaetano of Thiene, and others. Moreover, the ﬁfteenth century is the era of the triumph of the university, which also involved a geographical expansion of philosophical studies. Hence we meet a number of new active centers of logic emerging in Central Europe, in universities like Prague, Cracow, and Erfurt. A typical feature in ﬁfteenth-century philosophy is a conscious turn toward old masters. Thus, philosophy formed into competing schools with their own

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clear-cut doctrines; this process was promoted by the commitment of religious orders to their oﬃcial authorities and by the allotment of chairs in philosophy. These Thomist, neo-Albertist, Scotist, and nominalist currents were not very innovative in logic, though some of their leaders were ﬁrst-rate logicians (like the Scotist Tartaretus). The form of logical works changed gradually. Instead of voluminous commentaries, two other types of work became popular: shorter discussions of individual subjects, and more general summulae expositions. A far-reaching step was the innovation of printing, which led to the promotion of textbooks in particular. (The ﬁrst printed logical book was the Logica parva by Paul of Venice, in 1472.) On the whole, we can say that logic was no longer very creative; there were few original results, and perhaps they were not even actively pursued. We can feel some signs of the later sentiment that the science of logic had already been completed. In statements like this, we must remember, though, that there has been particularly little historical research on ﬁfteenth-century logic. Attention was often concentrated on earlier results; thus there was much interest in all kinds of special cases and counterexamples, which we cannot discuss here. Generalizing crudely, we might say that the exponibilia, the sophismata, and the insolubilia became especially popular themes, whereas the fundamental questions of terms, propositions, and inferences were less debated. Modal logic seems to disappear, though it has a surprising revival at the end of the ﬁfteenth century (with Erfurtians like Trutvetter). At the same time there is also a revival of philosophical logic in Paris (e.g., Scotsmen around John Mair). The strictly formal part of older logic, such as syllogistics, was still taught everywhere, and occasionally even cultivated in so far as there was an opportunity to develop it. A famous example is the innovation of the so-called pons asinorum. Fifteenth-century authors formulated clearly this virtually mechanical method for ﬁnding a suitable minor premise by means of which a given conclusion can be syllogistically inferred from a given major. In less formal matters, we encounter an interesting line by examining the widely read Speculum puerorum (1350s) by Richard Billingham. He discusses the probatio, literally “proof” but also meaning “trial,” of propositions and concludes that it is only possible by a further probatio of its terms. “Immediate” terms are simple, but others can be submitted to some of the three forms of such a treatment. First, “exponible” terms can be replaced by several occurrences of simpler terms in a conjunction of simpler propositions which is equivalent to the original one. Thus “only a man” is exponible in the proposition “only a man runs,” and its exposition leads to the equivalent “a man runs and nothing but a man runs.” “Resoluble” terms are replaceable by simple terms, leading (not to equivalents but) to truth grounds; thus “a man” is resoluble to “this” since “a man runs” has a truth ground “this runs and this is a man.” And for “oﬃcial” terms, it can be shown that they hold an oﬃce together with dicta, like the modal and attitudinal operators do. By means of the probatio of terms,

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the proposition ought to acquire a logically elementary form, and problems arising from diﬃcult constructions can then be handled. Finally Billingham gives some grammatical rules for advancing without error in the probatio. Similar ideas can be found in a number of later English and Italian authors who discuss such basically non-Aristotelian themes. The attention turned to logical grammar. Logic courses often started from logica vetus, continued with material from the Aristotelian Prior and Posterior Analytics, and then concentrated on the new themes. This feature can be seen as a mark of a shift from logic in the strict sense toward conceptual analysis of logically diﬃcult items: problematic concepts, ambiguous linguistic constructions, and so on. Accordingly, much attention was awarded to questions of grammatical deep structure and its accurate expression by means of variants in lexical forms and word order. The serious nature of these problems can now be appreciated again, in the light of present-day grammatical theory, but it is of course true that ﬁfteenth-century authors did not have a suﬃcient technical apparatus for mastering their Latin sentences. It is also easy to understand that these undertakings seemed useless and annoying to many critics.

7.2. Reactions It is common to speak about “medieval logic,” and one easily thinks of it as a monolithic totality. Perhaps we have managed to say that the truth is much more complex. But all the authors we have discussed so far had a solid Aristotelian background. There were, however, even other tendencies, which started to grow during the ﬁfteenth century. We might prepare the way for the novelties by mentioning earlier dissidents. The Aristotelian methodology in science was rather restrictive, and for a long time repeated attempts had been made to ﬁnd a place for something more innovative. Bacon is perhaps the most famous among these authors: He showed great curiosity in matters of empirical science and made initiatives in the philosophy of science. But his logic seems to follow well-known Aristotelian lines. A much more perplexing case is Raimundus Lullus (Ramón Llull, c. 1235–1316). Having no academic training, he did not care about logica moderna; instead, he sought to create an original way of argumentation that would undeniably prove Christian dogmas to inﬁdels. This so-called Ars magna, to which he gave several formulations, uses various basically neo-Platonic sources. As basic concepts, he chooses some central divine attributes and cross-tabulates them with certain logicometaphysical aspects. This ought to produce, in the way of multiplication tables, a scheme of interesting manifestations. Lullus also suggested that concepts should be written on concentric circles and arguments performed by rotation of the circles. In fact, Lullus never achieved any logical results, and his program rests heavily on theological premises. But he introduced the idea of purely combinatorial procedure (with symbolic letters), and this was something that fascinated many later authors. “Lullists” reappeared during the ﬁfteenth century, and even Leibniz was interested in Lullus.

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Late medieval university logic acquired a respected opponent when Italian humanists began to propagate their new ideals. In the middle of the fourteenth century, Petrarch had violently attacked scholasticism and particularly logic, making it clear that professional logic was a corrupt and useless discipline that could not beneﬁt a literary civilization. His leading followers, such as Bruni and Bracciolini, were more detailed in their criticisms. According to them, what is sensible in logic is delivered through the studies of language and dialectics, whereas university logic is mostly incomprehensible sophistry. They also pointed out, correctly, that medieval logic consisted of additions made by barbarians to the classical heritage. The early humanists mainly expressed nothing but their discontent, but a more substantial alternative logic was developed by the famous philologist Lorenzo Valla (1407–1457) in his Dialecticae disputationes. He argued that a lot of the scholastic problems were actually illusory and resulted from obscure and abstract misinterpretation of questions that were essentially linguistic. Valla admitted that a small kernel of elementary logic was needed, as ancient Romans had already admitted, but for him formal validity was not as interesting as the informal convincing power of arguments. Thus he focused on the dialectical theory of reasoning and discussion, emphasizing matters of grammar and style. His work anticipates the revival of topics in a new form. A similar nonscholastic development was continued by many other authors. Gradually the humanist inﬂuence extended outside Italy to the whole of Europe, and there grew a conscious eﬀort to form a simple logic free of tradition. In this process, the new logic also found a place in the academic environment and much logical literature turned to dialectical issues, new ancient sources became known, and logic deﬁnitely entered the era of printed books. All this amounts to a basic transformation, and the next part can well start with it.

Selected Further Readings Ashworth, E. J. 1974. Language and Logic in the Post-Medieval Period (Synthese Historical Library 13). Dordrecht: Reidel. Ashworth, E. J. 1978. The Tradition of Medieval Logic and Speculative Grammar from Anselm to the End of the Seventeenth Century: A Bibliography from 1836 Onwards (Subsidia Mediaevalia 9). Toronto: Pontiﬁcal Institute of Mediaeval Studies. Bäck, Allan. 1996. On Reduplication: Logical Theories of Qualiﬁcation (Studien und Texte zur Geistesgeschichte des Mittelalters 49). Leiden: Brill. Biard, Joël. 1989. Logique et théorie du signe au XIV siècle. Paris: Vrin. Boh, Ivan. 1993. Epistemic Logic in the Later Middle Ages. London: Routledge. Braakhuis, Henk A. G., C. H. Kneepkens, and L. M. de Rijk, eds. 1981. English Logic and Semantics from the End of the Twelfth Century to the Time of Ockham and Burleigh (Acts of the Fourth European Symposium of Medieval Logic and Semantics). Leiden: Ingenium Publishers. Ebbesen, Steen, ed. 1995. Sprachtheorien in Spätantike und Mittelalter (Geschichte der Sprachtheorie, Teil 3). Tübingen: Narr.

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Green-Pedersen, Niels J. 1984. The Tradition of the Topics in the Middle Ages. München: Philosophia Verlag. Jacobi, Klaus, ed. 1993. Argumentationstheorie: Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns (Studien und Texte zur Geistesgeschichte des Mittelalters 38). Leiden: Brill. Knuuttila, Simo. 1993. Modalities in Medieval Philosophy. London: Routledge. Kretzmann, Norman, Anthony Kenny, and Jan Pinborg, eds. 1982. Cambridge History of Later Medieval Philosophy. Cambridge: Cambridge University Press. Kretzmann, Norman, and Eleonore Stump, eds. 1988. Cambridge Translations of Medieval Philosophical Texts, vol. 1: Logic and the Philosophy of Language. Cambridge: Cambridge University Press. Kretzmann, Norman, ed. 1988. Meaning and Inference in Medieval Philosophy. Studies in Memory of Jan Pinborg. Dordrecht: Kluwer. Lagerlund, Henrik. 2000. Modal Syllogistics in the Middle Ages (Studien und Texte zur Geistesgeschichte des Mittelalters 70). Leiden: Brill. Nuchelmans, Gabriel. 1973. Theories of the Proposition. Amsterdam: North Holland. Nuchelmans, Gabriel. 1980. Late Scholastic and Humanist Theories of the Proposition. Amsterdam: North Holland. Panaccio, Claude. 1999. Le discours intérieur de Platon à Guillaume d’Ockham. Paris: Seuil. Perler, Dominik. 1992. Der propositionale Wahrheitsbegriﬀ im 14. Jahrhundert (Quellen und Studien zur Philosophie 33). Berlin: Walter de Gruyter. Pinborg, Jan. 1972. Logik und Semantik im Mittelalter. Stuttgart: FrommannHolzboog. Pironet, Fabienne. 1997. The Tradition of Medieval Logic and Speculative Grammar: A Bibliography (1977–1994). Turnhout: Brepols. Read, Stephen, ed. 1993. Sophisms in Medieval Logic and Grammar (Nijhoﬀ International Philosophy Series 48). Dordrecht: Kluwer. de Rijk, Lambertus Marie. 1982. Through Language to Reality. Studies in Medieval Semantics and Metaphysics. Northampton: Variorum Reprints. Rosier-Catach, Irène. 1983. La grammaire spéculative des modistes. Lille. Spade, Paul Vincent. 1988. Lies, Language and Logic in the Late Middle Ages. London: Variorum Reprints. Spade, Paul Vincent. 1996. Thoughts, Words and Things: An Introduction to Late Mediaeval Logic and Semantic Theory, available at http://pvspade.com/Logic/ noframes/index.shtml. Stump, Eleonore. 1989. Dialectic and its Place in the Development of Medieval Logic. Ithaca N.Y.: Cornell University Press. Yrjönsuuri, Mikko, ed. 2001. Medieval Formal Logic: Obligations, Insolubles and Consequences (New Synthese Historical Library 49). Dordrecht: Kluwer.

3

Logic and Philosophy of Logic from Humanism to Kant Mirella Capozzi and Gino Roncaglia

1. Humanist Criticisms of Scholastic Logic The ﬁrst impression of a reader who “crosses the border” between medieval and Renaissance logic may be that of leaving an explored and organized ﬁeld for a relatively unexplored and much less ordered one. This impression is emphasized by the fact that while in the medieval period we can assume, despite relevant theoretical diﬀerences, some consensus about the nature and purpose of logic, such an assumption cannot be made with reference to the postmedieval and Renaissance period: The many “logics” coexisting and challenging each other were often characterized by deeply divergent assumptions, articulations, and purposes. As far as logic is concerned, we could almost be tempted to use this “explosion of entropy” as the very marker of the shift between the medieval and the Renaissance period. The development of humanism, with its criticism of the late medieval logical tradition, is not the only factor contributing to this situation, but surely is a relevant one. Excessive and artiﬁcial subtlety, lack of practical utility, barbarous use of Latin: These are the main charges that humanist dialecticians made against scholastic logic. Such charges do not simply point out formal deﬁciencies that could be eliminated within a common logical framework, but call for a change of the logical paradigm itself. The eﬀort to address such charges had a deep inﬂuence on the evolution of logic and resulted in a variety of solutions, many of which were based on contaminations between selected but traditional logical theories, on the one hand, and mainly rhetorical or Though we decided on the general structure of this chapter together, sections 1–4 and 8 are by Gino Roncaglia, while sections 5–7 and 9–11 are by Mirella Capozzi.

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pedagogical doctrines on the other. But the charges themselves were initially made outside the ﬁeld of logic: One of the very ﬁrst invectives against scholastic logic came from Francesco Petrarca (1304–1374), hardly to be considered a logician (Petrarca 1933–42, I, 7). The central point at issue is the role of language. The late medieval scholastic tradition used language as a logical tool for argumentation, and favored the development of what J. Murdoch (1974) aptly called “analytical languages”: highly specialized collections of terms and rules which—once applied to speciﬁc and deﬁnite sets of problems—should help guarantee the formal precision of reasoning. In this tradition, the use of a simpliﬁed and partly artiﬁcial Latin could help the construction of sophisticated formal arguments. The humanists, on the contrary, privileged the mastery of classical Latin. For them, language—together with a few simple and “natural” arguments taken from ancient rhetoric—was a tool for an eﬀective and well-organized social and pedagogical communication. Besides the diﬀerent theoretical standpoints, there is a social and cultural gap between two diﬀerent intellectual ﬁgures. Scholastic-oriented teachers are usually university professors who tend to consider logic, philosophy, and theology as specialized ﬁelds. For them, knowledge is reached through a self-absorbing (and largely self-suﬃcient) intellectual activity, whose formal correctness is regulated by logic. Many humanist dialecticians, on the contrary, do not belong to and do not address themselves to the academic world: They consider logic a tool to be used whenever language is used with rhetorical or practical purposes, and regard a broad “classical” culture more important than a specialized and abstract one (see Jardine 1982, 1988). One should be careful, however, in assessing the reasons for the privilege humanists accorded to rhetoric. For the humanists, logic—or rather dialectic, to use the term that, already present in the Ciceronian tradition and in the Middle Ages (see Maierù 1993), was preferred by most humanist and Renaissance authors—has to do with the use of arguments. But to be practically eﬀective, such arguments have to be natural, aptly chosen, easily stated and grasped, expressed in good, classical Latin. And they don’t need to be demonstrative arguments: Probable arguments are also included within the scope of dialectic. One should also be careful in considering humanism as a monolithic movement aimed at banishing all reminiscence of medieval logic. Humanism is not chronologically subsequent to scholasticism, and many humanists knew late scholastic logical texts fairly well, such as those by Paul of Venice. Some even praised them (Vasoli 1968, 20–23; Perreiah 1982, 3–22; Mack 1993, 14–15). Nevertheless, formally correct and truth-preserving arguments were considered as only some of the tools available to a good dialectician. The latter’s aim is to master the art of using language (ars bene disserendi), the Ciceronian disserendi diligens ratio, and this requires not only demonstrative skills but also the ability to persuade, to construct probable arguments, to obtain consensus. The deﬁnition of dialectic provided by Rudolph Agricola (1444–1485)—one of the many Renaissance variations on Cicero’s own—is representative of this

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point of view. According to it, dialectic is the “ars probabiliter de qualibet re proposita disserendi” (art of speaking in a probable way about any proposed subject). The explication of “probabiliter” clariﬁes the broad scope of the term (see Mack 1993, 169–173, where “probabiliter” is translated as “convincingly”): “probable (probabile) in speaking is not only what is actually probable, that is, as Aristotle states, what is accepted by all, or by the most part, or by the learned. For us, probable is what can be said about the proposed subject in an apt and adequate way” (Agricola 1967, 192). This meaning of the concept is broad enough to include good old-fashioned demonstrative arguments in the ﬁeld of dialectic (Risse 1964–70, I, 17–18), but they are no longer the only kind of arguments a dialectician should take into account. A ﬁrst introduction to sources, principles, and precepts of humanist-oriented logic is provided by the works of the prominent humanist dialectician Lorenzo Valla (1407–1457), who, signiﬁcantly, received his cultural training mostly within the humanist circles of the papal curia. While some of the earlier humanists were content with a dismissal of scholastic logic—Petrarca’s and Bruni’s invectives against the barbari britanni being the most often quoted testimony of this attitude (Garin 1960, 181–195; Vasoli 1974, 142–154)—in his Repastinatio dialecticae et philosophiae (Valla 1982), Valla added to heavy criticism of traditional logical doctrines a complete and systematic reassessment of the nature and purpose of dialectic from a humanistic point of view. According to Valla, dialectic deals with demonstrative arguments, while rhetoric deals with every kind of argument—demonstrative as well as plausible ones. Therefore dialectic is to be considered as a part of rhetoric, and rhetoric has to provide the widest spectrum of argumentative tools to all branches of learning. Moreover, dialectic should be simple and disregard all the questions that, though discussed by logicians with technical logical tools, actually pertain to Latin grammar. During the Middle Ages the relation between logic and grammar had been closely investigated by the so-called modist logicians. They worked at a sophisticated speculative grammar, based on an ontologically grounded correspondence between ways of being, ways of thinking, and ways of signifying. Valla’s grammar, on the contrary, is based on the Latin of classical authors, and therefore on a historically determined consuetudo in the use of language. Valla thus carries out what has been described as a “deontologization” of language (Camporeale 1986; Waswo 1999). Valla devotes the ﬁrst of the three books of his Repastinatio to the foundations of dialectic and to a discussion of the Aristotelian doctrine of the categories. Here, too, Valla applies his general rule: simpliﬁcation through reference to concrete uses of Latin, rather than to an abstract metaphysical system. The 10 Aristotelian categories are thus reduced to 3—substance, quality, and action—and examples are given to show how the remaining categories can be reduced to quality and action. Similarly, the transcendental terms, which according to the medieval tradition “transcend” the division among the 10 categories and are reciprocally convertible, are reduced to the only term “res.” The reason why Valla prefers the term “res” to the traditional

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“ens” is that “ens” in classical Latin is not a noun but a participle that can be exposed as “that thing (res) which is.” Therefore the term “res” is the true fundamental one. This example shows how Valla explains problematic terms or sentences by oﬀering a reformulation considered more precise and easier to analyze. The practice of explanation through reformulation was familiar to medieval logicians under the name of expositio, but Valla uses expositio to reach linguistic, rather than logical clariﬁcation. Valla’s second book is devoted to proposition and addresses the question whether all propositions should be reduced to the basic tripartite form: subject– copula–predicate (“A est B”). This question was the object of a long debate, continued during the whole period we are dealing with (Roncaglia 1996), and had usually been investigated under the assumption that it was the logical structure of the proposition at issue. Valla, on the contrary, perceives the problem as related to the grammatical structure of the proposition, and accordingly oﬀers a negative answer, since in the use of Latin the construction “est + participle” (Plato est legens) is not equivalent to the use of an indicative form of the verb (Plato legit). The Spanish humanist Juan Luis Vives (1492– 1540) will share the same attitude (see Ashworth 1982, 70). To support his contention, Valla considers propositions like Luna illuminatur, which—in Latin—can be transformed into a tripartite form only through a shift in meaning. A further argument is drawn from the idea that the participle form of the verb may be seen as somehow derivative with respect to the indicative form. Therefore—if something is to be reduced at all—it should be a participle like legens, to be reduced to qui legit (Valla 1982, 180). Logicians should not superimpose their logical analysis to the “good” use of language, but should rather learn from it. Language should be studied, described, and taught, rather than “corrected” from an external point of view. Valla did not consider the study of modal propositions as pertaining to logic (hence his complete refusal of modal syllogistic). This refusal—common to most humanistic-inﬂuenced Renaissance philosophy—is once again defended on linguistic rather than purely logical grounds. Why should we attribute to terms like “possible” and “necessary” a diﬀerent status from that of grammatically similar terms like “easy,” “certain,” “usual,” “useful,” and so on? (Valla 1982, 238; see Mack 1993, 90; Roncaglia 1996, 191–192.) Valla’s third book, devoted to argumentation, preserves the basic features of Aristotelian syllogistic, but dismisses the third ﬁgure and, as already noted, modal syllogisms. Owing to his desire to acknowledge not only demonstrative but also persuasive arguments, Valla pays great attention to hypothetical and imperfect syllogisms and to such nonsyllogistic forms of argument as exemplum and enthymemes. The ﬁnal section of Valla’s work is devoted to sophistic argumentations. Medieval discussions of sophisms allowed logicians to construct interesting, complex, and borderline situations to test the applicability and the eﬀectiveness of their logical and conceptual tools. Valla is fascinated by the persuasive and literary strength of “classical” problematic arguments, such as the sorites (a

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speech proceeding through small and apparently unavoidable steps from what seems an obvious truth to a problematic conclusion) or the dilemma, in which all the alternatives in a given situation are considered, only to show that each of them is problematic. Valla does distinguish “good” and “bad” uses of these kinds of “arguments,” but his criterion is basically that of practical usefulness in persuasive rhetoric. Valla’s Repastinatio is also a typical example of the importance humanists assigned to the “invention” (inventio) of arguments, connected with topics. Renaissance dialecticians considered Aristotle’s Topica as a systematic treatment of practical reasoning, and complemented it with Cicero’s Topica and with the treatment of topics included in Quintilian’s Institutio oratoria, which— rediscovered in 1416—had become one of the most popular textbooks on rhetoric by the end of the century, while Boethius’s De diﬀerentiis topicis, widely used in the Middle Ages (Green-Pedersen 1984), had only few Renaissance editions (Mack 1993, 135). Both Cicero’s and Quintilian’s treatment of topics helped shift the focus from “formal” disputations to rhetorical and persuasive ones. The most complete and inﬂuential Renaissance study of topics is contained in Agricola’s De inventione dialectica (Agricola 1967, 1992). Agricola grounds his conception of topics on his realist conception of universals (Braakhuis 1988). In his opinion, things are connected by relations of agreement and disagreement, and topics are orderly collections of common marks, which help us organize and label relations, and ﬁnd out what can or cannot be said about a given thing in an appropriate way. While being systematically arranged, topics, according to Agricola, are not a closed system: The very possibility of viewing things from diﬀerent angles and perspectives, of relating them in new ways, not only enables us to draw or invent arguments but also allows us to ﬁnd new common marks. We have already considered Agricola’s deﬁnition of dialectic. In his opinion, topics are the method of dialectical invention, while the discourse (oratio) is its context. There are, however, two diﬀerent kinds of dialectical discourse: exposition (expositio) and argumentation (argumentatio). The former explains and clariﬁes, and is used when the audience doesn’t need to be convinced, but only enabled to understand what it is said. The latter aims at “winning” assent, that is, at persuading. Although argumentation is connected with disputation, necessary arguments are not the only way to win a disputation: Plausible and even emotionally moving arguments should be considered as well. Agricola’s concept of argumentation is thus connected with rhetoric, a connection strengthened by the fact that both use natural language. This explains why Agricola has no use for the kind of highly formalized, analytical language used by medieval and late medieval logic. However negative Valla’s and Agricola’s attitude toward the logical tradition, it was never as negative as that of Petrus Ramus (Pierre de La Ramée, 1515–1572). According to his biographer Freigius, Ramus’s doctoral dissertation (1536) defended the thesis: “everything that Aristotle said is misleading

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(commentitium).” This does not imply—as many assumed—that Ramus considers all Aristotelian theories to be false: In his opinion, Aristotle is guilty of having artiﬁcially complicated and corrupted the simple and “natural” logic which Aristotle’s predecessors—notably Plato—had devised before him (Risse 1964–70, I, 123–124). Scholastic logic is obviously seen by Ramus as a further step in the wrong direction. Various versions of Ramus’s logic (including the 1555 Dialectique, in French: Ramus 1996; for a survey of the diﬀerent editions of his works and of the stages marking the complex development of Ramus’s dialectic, see Bruyère 1984) were published between 1543 and 1573. After his conversion to Protestantism in 1561, his library was burned, and he had to ﬂee from Paris. Ramus died on August 26, 1572, killed on the third day of the St. Bartholomew’s massacre. His being one of the Huguenot martyrs undoubtedly boosted the fortune of his already popular works in Calvinist circles. Ramus’s concept of dialectic is based on three main principles: Dialectic should be natural (its foundations being the “eternal characters” which constitute, by God’s decree, the very essence of our reasoning), it should be simple (it deals with the correct way of reasoning, but disregards metaphysical, semantic, and grammatical problems as well as unnecessary subtleties), and it should be systematically organized, mainly by means of dichotomic divisions. Therefore, Ramus’s books extensively used diagrams, usually in the form of binary trees: A feature that may be connected—as argued by Ong (1958)—with the new graphical possibilities oﬀered by printed books, and that will inﬂuence a huge number of sixteenth- and seventeenth-century logic textbooks, not only within the strict Ramist tradition. The ﬁrst and foremost division adopted by Ramus is Cicero’s division between invention (inventio) and judgment (iudicium or dispositio). They are the ﬁrst two sections of logic. A third section, devoted to the practical and pedagogical exercise of dialectic (exercitatio), is present in the ﬁrst editions of Ramus’s logical works but disappears after 1555. The inventio deals with the ways arguments are to be found. Because arguments are to be found and classiﬁed by means of topics, according to Ramus, the treatment of topics should precede, rather than follow (like in Aristotle), that of judgment. Ramus’s table of topics, organized by means of subsequent dichotomic divisions, is strongly inﬂuenced by Agricola and by Johannes Sturm (1507–1589), who taught dialectic and rhetoric in Paris between 1529 and 1537 and greatly contributed to the popularity of Agricola in France. Ramus’s treatment of judgment is also unconventional. While in traditional logic this section presupposes an extensive treatment of proposition, Ramus deals with this subject in a sketchy way and adds an independent (albeit short) section on the nature and structure of proposition only in the 1555 and successive editions of his work. In the last edition Ramus follows Cicero in using the term axioma to refer to a categorical proposition (having used earlier the term enuntiatio or enuntiatum), while he always gives the more speciﬁc meaning of major premise of a syllogism to the term propositio.

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Syllogism and its various forms (including induction, example, and enthymeme) constitute the core of the “ﬁrst judgment”: the ﬁrst of the three sections in which Ramus divides his treatment of judgment in the earlier editions of his dialectic. Ramus’s explicit eﬀort is that of simplifying Aristotelian syllogistic, but during the years between the 1543 edition of the Dialecticae institutiones and his death, his syllogistic underwent so many changes that it is impossible to give a faithful account of it in a few pages. Typical of Ramus’s syllogistic is his use of the terms propositio, assumptio, and complexio to refer to the major premise, minor premise, and conclusion of a syllogism, and his tendency to favor a classiﬁcation of syllogisms according to the quantity of the premises, considering as primary moods those with two universal premises. In the earlier editions of his dialectic, Ramus held that all moods with particular premises should be reduced to universal moods. He admitted some of them later on, but banned the reductio ad impossibile used to reduce second and third ﬁgure moods to the ﬁrst ﬁgure. But Ramus’s better known innovation in the ﬁeld of syllogistic is the so-called Ramist moods: syllogisms in which both premises are singular, accepted on the ground that individuals could be seen as (lowest) species. The discussion about Ramist moods will keep logicians busy for most of the subsequent century. The second section of Ramus’s treatment of judgment (called “second judgment” in the earlier editions of his work) deals with the ways to connect and order arguments by means of general principles. Ramus attributes great importance to this “theory of method,” which he further develops in the later editions of his logical works, and which in his opinion shapes the whole system of science (also oﬀering the conceptual foundation for an extensive use of dichotomies). According to Ramus, the dialectical method (methodus doctrinae) goes from what is most general to what is most particular. This is done by means of divisions that, in turn, are drawn on the base of deﬁnitions expressing the essence of the concepts involved. Division and deﬁnition are thus the two main tools of method. The opposite route, going from particular instances to more general concepts (methodus prudentiae), might be used when either the lack of a more general conceptual framework or reasons of practical convenience force us to dwell on single or partial pieces of information. However, it cannot guarantee certainty; and is therefore mainly used in rhetorical discourse aiming at persuasion, rather than in demonstrative reasoning. In Ramus’s opinion, however, the distinction between methodus doctrinae and methodus prudentiae does not imply that we have two methods: We have only one method—based on an ideal “knowledge space” organized by means of deﬁnitions and divisions— that, in given and concrete situations, also allows for tentative and partial bottom-up routes. Thus conceived, the dialectical method is governed by three laws, which constitute the Ramist counterpart of the Aristotelian-Scholastic de omni, per se and universaliter primum principles. Ramus calls them the laws of truth, justice and wisdom: in the ﬁeld of science every statement (i) should be valid in all its instances; (ii) should express a necessary (essential) connection of

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the concepts involved; (iii) should be based on subject and predicate that are proper and proportionate (allowing for simple conversion). Ramus’s logic was very inﬂuential in the second half of the sixteenth and in the ﬁrst half of the seventeenth century (Feingold, Freedman, and Rother 2001). However, “pure” Ramist scholars—mostly active in the Calvinist areas of Germany, in Switzerland, in Holland, and in England—were to face an almost immediate opposition not only in Catholic but also in Lutheran universities, and saw their inﬂuence decrease after the beginning of the seventeenth century. Much more inﬂuential (and more interesting) were the many “eclectic” logicians who either tried to reconcile Ramus’s and Melanchthon’s logical views (PhilippoRamists) or introduced some Ramist themes within more traditional (and even Aristotelian) contexts.

2. The Evolution of the Scholastic Tradition and the Inﬂuence of Renaissance Aristotelianism Despite humanist criticisms, the tradition of scholastic logic not only survived during the sixteenth and seventeenth centuries but evolved in ways that are much more interesting and articulated than most modern scholars suspected until a few decades ago. Our knowledge of this evolution is still somehow fragmentary, but the scholarly work completed in recent years allows some deﬁnite conclusions. We can now say that in this evolution of the late scholastic logical tradition, six factors were particularly relevant: (i) the work of a group of Spaniards who studied in Paris at the end of the ﬁfteenth and at the beginning of the sixteenth century and later taught in Spanish universities, inﬂuencing the development of logic in the Iberian peninsula; (ii) a renewed attention toward metaphysics, present in the Iberian second scholasticism and most notably in the works of Francisco Suárez (1548–1617), whose Disputationes Metaphysicae (Suárez 1965) inﬂuenced many authors all across Europe; (iii) the crucial role of the newly formed (1540) Society of Jesus, whose curriculum of studies (Ratio Studiorum) was to shape institutional teaching in all of Catholic Europe; (iv) the complex relations with humanism, and the inﬂuence of logicians like Agricola, whose doctrines, while taking as their starting point a humanist conception of logic, were nevertheless susceptible of somehow being absorbed or integrated within a more traditional framework; (v) the “new Aristotelianism” of authors like Jacopo Zabarella (1533–1589) and Bartholomaeus Keckermann (1572?–1609); and (vi) the renewed interest in scholastic logic, discernible in reformed Europe (and most notably in Germany) as a consequence of the doctrinal and theological conﬂicts with the catholic ﬁeld and within the reformed ﬁeld itself. In the following pages, we provide some details on this complex development. At the end of the ﬁfteenth century and in the ﬁrst decades of the sixteenth, the Paris college of Montaigu became a center of logical research in which the late medieval logical (especially nominalist) tradition survived and to

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some extent ﬂourished. A group of Spanish and Scottish logicians, lead by the Spaniard Jeronimo Pardo (d. 1505) and by the Scottish John Mair (1467/9– 1550), debated themes such as the nature of supposition and signiﬁcation, the distinction between categorematic and syncategorematic terms, the role of beings of reason (entia rationis), the nature of proposition (further developing the late medieval discussions on mental propositions), modality, and the theory of consequences. Somehow connected to this Paris group, or active there at the beginning of the sixteenth century, were the Spaniards Antonio Núñez Coronel (d. 1521), Fernando de Encinas (d. 1523), Luis Núñez Coronel (d. 1531), Juan de Celaya (1490–1558), Gaspar Lax (1487–1560), Juan Dolz (ﬂ. 1510), the Frenchman Thomas Bricot (d. 1516), the Belgian Pierre Crockaert (Pierre of Brussels, d. 1514), and the Scot George Lokert (d. 1547). Particularly interesting is their discussion about the nature of complexe signiﬁcabile (propositional complex), a subject already debated by medieval logicians. The medieval defenders of this theory, associated with the name of Gregory of Rimini (c. 1300–1358), held that the object of science is not the proposition itself but what is signiﬁed by it (and determines its truth or falsity); such total and adequate meaning of the proposition is neither a physical nor a purely mental being and is not reducible to the meaning of its parts. It is rather similar to a state of aﬀairs, which can be signiﬁed only by means of a complex (the proposition) and is therefore called complexe signiﬁcabile. The discussion on the nature (and usefulness) of the complexe signiﬁcabile was connected to the discussion on the role of the copula, since the copula was usually considered as the “formal” component of the proposition, “keeping together” subject and predicate. The copula was thus considered as a syncategorematic term: a term that does not possess an autonomous meaning but helps determine the meaning of the proposition as a whole. The defenders of a “strict” complexe signiﬁcabile theory did not need a separate discussion of the mental copula, because in their opinion the complexe signiﬁcabile is a unity and cannot be analyzed in terms of its parts. But many authors—among them John Buridan (c. 1295–1356)—assigned a much more relevant role to the copula, seen as the (syncategorematic) mental act that, in connecting subject and predicate, establishes the proposition. It is this very theory that was discussed by many of the above-mentioned late ﬁfteenth- and early sixteenth-century Paris-based logicians (see Ashworth 1978, 1982; Muñoz Delgado 1970; Nuchelmans 1980; Pérez-Ilzarbe 1999). Pardo’s position in this discussion was the most original. In his opinion, the copula is not purely syncategorematic: It is subordinate to a conceptual schema that represents something (i.e., the subject) as related in a certain way to something else (i.e., the predicate) or to itself (Nuchelmans 1980, 49). In this way the copula, while retaining its formal function, also signiﬁes something (aliquid), that is, the subject, as considered in a given way (aliqualiter), namely as modiﬁed by the relation with its predicate. The idea of the copula signifying aliquid aliqualiter, and not simply aliqualiter, and the special relevance attributed to the subject in determining the meaning of

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the copula and of the proposition as a whole, were discussed, and generally criticized, by Pardo’s successors. They especially investigated the role of impossible propositions, as well as propositions with a negative, privative, or impossible subject, and the problem of whether a quasi-syncategorematic nature could be attributed to the proposition as a whole. The Iberian Peninsula was one of the strongholds of Catholicism. Moreover, as we have seen, it inherited many features (as well as textbooks and Paristrained professors) from Parisian late scholasticism. This made the inﬂuence of the humanist movement—albeit discernible—less radical than elsewhere. Therefore, the Iberian Peninsula was the ideal context in which Catholic logicians—dwelling on the scholastic (chieﬂy Thomist) logical and philosophical tradition—could pursue the work of doctrinal and pedagogical systematization that was required by the struggle against the reformed ﬁeld. The Carmelite universities of Salamanca (Salamanticenses) and Alcalà (Complutenses) and the Jesuit university of Coimbra (Conimbricenses) each produced a complete philosophical course, including speciﬁc volumes devoted to logic. Of these the most inﬂuential was probably the Coimbra Logic, compiled by Sebastian Couto (1567–1639) but partially dependent on Pedro da Fonseca (1528–1599), who had been teacher at that university. Fonseca, the “Portuguese Aristotle,” published the Institutionum Dialecticarum Libri VIII (Fonseca 1964) in 1564, a logical treatise built on the model of Peter of Spain and widely read throughout Europe. Fonseca’s logic interprets the traditional emphasis on terms by giving a theoretical priority to the conceptual moment over the judicative one (truth and falsity are in concepts rather than in judgment) and among concepts, to singulars over abstracts and universals. To reconcile God’s foreknowledge and human free will, and to handle the problem of future contingents—a theme of special interest for all Iberian philosophers—Fonseca developed, independently from Luis de Molina (1535–1600), a theory of the scientia media, or, as he says, of “conditioned futures,” by which God foreknows all the consequences of any possible free decision. Placing Fonseca’s theories within a wider and more systematic treatment, the Coimbra logic oﬀers a translation and a detailed commentary of Aristotle’s Organon, which, in the form of questions, includes a discussion of most of the topics debated by sixteenth- and seventeenth-century logicians. The Conimbricenses reject the idea that beings of reason are the object of logic (in the scholastic tradition logical concepts such as “genus” and “species” were considered to be entia rationis, and the Thomist tradition considered them as the formal object of logic): Dwelling on the idea of logic as ars disserendi, they prefer to characterize it as a “practical science” dealing with the construction of correct arguments. Argumentation is, therefore, the ﬁrst and main object of logical enquiry. Particularly interesting is the long section devoted to the nature of signs at the opening of the commentary on Aristotle’s De Interpretatione (see Doyle 2001). The concept of sign is here taken in a broad meaning, as to include not only spoken, written, and mental “words,” but also iconic

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languages and arithmetical signs. It is to be remarked that the inﬂuence of Coimbra logic was not limited to Europe: Jesuit missionaries used it in Latin America and even in China. If the teaching of logic in Coimbra is connected to Fonseca, another important ﬁgure of Iberian logic and philosophy, Domingo de Soto (1494/5–1560), is connected to Alcalà and Salamanca, where he taught. Soto made important contributions to a plurality of ﬁelds, so much so that it was said qui scit Sotum, scit totum (who knows Soto, knows everything). Despite his endorsement of Thomism—testiﬁed by his defense of the theory that the object of logic are beings of reason—Soto was open to Scotist, nominalist, and even humanist inﬂuences, and his commentary on Aristotle’s logic (Soto 1543) criticizes the “abstract sophistries” of the late scholastic logical tradition. This, however, did not prevent him from discussing and adopting many late scholastic logical theories, including large sections of medieval theories of terms. His Summulae (Soto 1980) are a commentary on one of the key works of medieval logic, Peter of Spain’s Tractatus (best known as Summulae Logicales; see previous chapter), and include an ample discussion of signiﬁcation, supposition, and consequences (see d’Ors 1981; Ashworth 1990; Di Liso 2000). Soto adopts an apparently Ciceronian deﬁnition of dialectic, considered as the art of discussing probabiliter. As remarked by Risse (1964–70, I, 330), however, this should not be considered a rhetorical attempt to establish apparent plausibility, but rather as an attempt to establish rational assertibility. Among the interesting points of the Summulae are the treatment of induction in terms of ascensus (the passage from a conjunction of singular propositions—or from a proposition with a copulative term as subject or predicate—to a universal proposition, or to a proposition with a general term as subject or predicate) and a complex square of modalities, which takes into account the quantity of the subject. Soto’s discussion of second intentions oﬀers what has been interpreted as a sophisticated theory of higher-level predicates (Hickman 1980). One of Soto’s students in Salamanca was Franciscus Toletus (1533–1596), who later taught both in Zaragoza and Rome, in the Jesuit Collegium Romanum, and was the ﬁrst Jesuit to be appointed cardinal. Toletus wrote both an Introduction and a Commentary on Aristotle’s logic (Toletus 1985). Like Soto, Toletus adopts some humanist theories—he takes the deﬁnition of logic as ratio disserendi from Boethius and divides it into invention and judgment— but his logic is actually a synthesis of Aristotelianism and Thomism, deeply inﬂuenced by the late medieval logical tradition. He considers beings of reason as formal objects of logic—thus partly endorsing the Thomist position—but maintains that logic’s material object is constituted by our concepts of things and, ultimately, by things themselves, for logical beings of reason are only second intentions, based on ﬁrst-order concepts—thus partly endorsing the position of Arab commentators of Aristotle (Ashworth 1985b, xli). Of special interest is his extensive use of physical and geometrical examples within the discussion of categories, and his long discussion of contingent futures within the commentary on De Interpretatione.

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The most important Jesuit philosopher working in Spain at the end of the sixteenth century was Francisco Suárez (1548–1617). His Disputationes Metaphysicae, ﬁrst published in 1597 (Suárez 1965), constituted a reference text and a model for further works both in the Catholic and in the Reformed ﬁelds. According to Suárez, metaphysics oﬀers a general and uniﬁed theory of real being (ens reale) and of its divisions, whereas logic deals with the way of knowing and explaining such divisions. Though the Disputationes Metaphysicae is not a logic textbook, it discusses many issues relevant to the philosophy of logic. Suárez pays great attention to relations, subdivided into real relations (only conceptually and not really distinct from the things on which they are grounded, but nevertheless to be considered as a category of beings) and conceptual relations, which are only a product of the mind and as such do not have any ontological status. Suárez’s detailed discussion of both kinds of relations helps to explain the special interest that many scholastic-oriented logicians devoted to this topic in the seventeenth century. The last of the Disputationes—disputation LIV—is devoted to a subtle discussion about beings of reason (entia rationis) and relations of reason. According to Suárez, beings of reason are not “real” (actual or possible) beings and do not share a common concept with real beings; their only reality is that of being object of the understanding (they only have objective existence in the intellect). Therefore, they are not to be included within the proper and direct object of metaphysics. They can nevertheless be dealt with within the context of metaphysical research, given their nature of “shadows of being” (Suárez 1996, 57) and given their usefulness in many disciplines, especially logic and natural philosophy. Suarez’s opinion on entia rationis is thus diﬀerent both from that of those—like the Scotist Francis of Mayronnes (1280?–1327?)—who simply denied their existence, and from that of those—like many Thomists, including Cardinal Cajetanus (Tommaso de Vio, 1469–1534)—who thought that there is a concept common to them and to real beings. Suárez included impossible objects in the range of entia rationis: His discussion is thus especially relevant to the history of the logical and ontological status of impossible entities (Doyle 1987–88, 1995). The discussion on the nature of entia rationis was a lively one in sixteenth-century Spain and was bound to continue in Catholic Europe during most of the seventeenth century. An interesting example is that of the Polish Jesuit Martinus Smiglecius (1564–1618). In his opinion, the opposition between ens reale and ens rationis is not grounded on the fact that the ens rationis is not a form of being, but on the fact that it is by deﬁnition a being which is not, and cannot possibly be, an ens reale. A being of reason is thus, according to Smiglecius, one whose essence implies the impossibility of its real existence. The fact that entia rationis cannot have real existence is, according to Smiglecius, a logical and not just a physical impossibility. They, however, can have conceptual (and hence intentional) existence. In the later Middle Ages, English logicians had been famous for their subtleties: The logical, physical, and epistemic sophisms discussed by the socalled calculatores, working at Merton College in Oxford, deeply inﬂuenced late

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fourteenth- and early ﬁfteenth-century logic both in Paris and in Italy, and were exactly the kind of logical subtleties rejected by humanist logicians. During the ﬁfteenth and in the ﬁrst decades of the sixteenth century, however, the English logical tradition declined (see Giard 1985). This did not prevent a slow penetration of humanist ideas, testiﬁed by the 1535 statutes or the university of Cambridge, recommending the reading of Agricola and Melanchthon as substitutes for late medieval scholastic texts, and by the Dialectica published in 1545 by the Catholic John Seton (c. 1498–1567). The latter oﬀers a drastically simpliﬁed treatment of traditional topics such as signiﬁcation, supposition, categories, syllogism, but liberally uses nonformal arguments and literary examples, divides dialectic into invention and judgment, adopts Agricola’s deﬁnition of dialectic as well as his classiﬁcation of topics, and quotes, beside Cicero and Quintilian, modern humanists like Erasmus and Vives. In the last decades of the sixteenth century, the debate on Ramism was to shake both English and continental universities. In England, Ramus found in William Temple (1555–1627) a learned defender and commentator, who, despite the strong opposition of his fellow Cambridge teacher and former master Everard Digby (1550–1592), managed to make of Cambridge, albeit for a short time, a stronghold of Ramism. The penetration of Ramism in Oxford was less substantial, and by the beginning of the new century the anti-Ramist positions were predominant in both universities. The defeat of Ramism was accompanied by the propagation of Aristotelianism—tempered by humanist-oriented attention toward classical literary examples rather than purely logical ones and toward rhetorical practices such as the declamatio— and by the circulation of the leading logic books published in the continent (among them Zabarella and Keckermann). The Logicae Artis Compendium by the Oxford professor Robert Sanderson (1587–1663; Sanderson 1985) is a good example of this new situation. Sanderson abandons the division of logic into invention and judgment, favoring a threefold division according to the three acts of the mind: The ﬁrst, dealing with simple concepts, is associated with the treatment of simple terms; the second, dealing with composition and division, is associated with propositions; and the third, dealing with discourse, is associated with argumentation and method. Though this threefold division is present in the medieval and late medieval tradition and is discussed by the Conimbricenses, Zabarella, and Keckermann, Sanderson and other Oxford logicians seem to have been among the ﬁrst to use it as the main division for logic textbooks (Ashworth 1985b, xli). In his logic, Sanderson includes medieval topics such as the theory of supposition and consequences, but their presentation is straightforward and not very elaborated. His discussion of method is more articulate and gives a foremost role to pedagogical concerns. We have already mentioned the Padua professor Jacopo Zabarella, who, advocating a renewed, “pure,” and philologically accurate Aristotelianism, absorbs both some humanist instances—visible in the pedagogical organization of his works and in the inclusion of Aristotle’s Rhetoric and Poetic within a broad treatment of logic, on the ground of their dealing with probable

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arguments, such as rhetorical syllogisms and examples—and some features of the so-called Paduan Aristotelianism: a distinctive attention to the Arab interpretations of Aristotle (notably Averroes) and to Galen’s concept of science. Zabarella wrote commentaries on Aristotle’s logical works as well as autonomous logic tracts: Among the latter are the De Natura Logicae (in Zabarella 1966), the De Methodis, and the De Regressu (both in Zabarella 1985). According to Zabarella, logic deals with second intentions, that is, with the (meta)concepts produced by our intellect in reﬂecting on the ﬁrst notions, those derived from and referring to real things. Because second intentions are products (and ﬁgments) of our intellect, logic is not a science but an instrument, or, to be more precise, an instrumental intellectual discipline, aimed at devising conceptual tools for correct reasoning and for discriminating truth and falsity. Because of its instrumental nature, logic is somehow similar to grammar: Just as grammar provides the tools needed to write and speak in an appropriate way, logic provides the tools needed to reason in an appropriate way. According to Zabarella, order (ordo) and method (methodus) are among the main tools oﬀered by logic: The ﬁrst organizes the subject matter of a discipline and the knowledge we have acquired; the second gives the rules and procedures to be followed to acquire new knowledge, going from what we know to what we do not know (on the Renaissance concept of method, see Ong 1958; Gilbert 1960). In dealing with contemplative sciences, the ordo goes from the universal to the particular and to the singular (“compositive order”), while in dealing with practical and productive arts it goes from the desired eﬀects to the principles that produce them (“resolutive order”). Although order concerns a discipline as a whole, method always has to do with the handling of speciﬁc problems, of speciﬁc “paths” going from what is known to what is unknown. Those paths are basically syllogistic demonstrations: The method is thus somehow a special case of syllogism. And since a syllogism can only go from cause to eﬀect (compositive method or demonstratio propter quid) or from eﬀect to cause (resolutive method or demonstratio quia), the same will hold for method. The resolutive method is used in the “hunt” for deﬁnitions, and is most needed in natural sciences; the compositive method is used in mathematics, where we start from already known, general principles and try to demonstrate all their consequences. Both methods presuppose necessary connections and are therefore only valid within contemplative sciences: Practical and productive arts, dealing with contingent truths, will have to content themselves with rhetorical and dialectical arguments, which, being only probable, are not subject to a rigorous application of method. However, even in contemplative sciences (especially in natural sciences), our knowledge of eﬀects and of their causes is often far from clear, and we need a process of reﬁnement, which Zabarella calls regressus and which involves the use of both compositive and resolutive methods: (1) We ﬁrst use the resolutive method to go from a confused knowledge of the eﬀect to a confused knowledge of its cause; (2) we then examine and clarify the knowledge of the cause (examen)—an activity that Zabarella connects to a speciﬁc ability of the human mind, interpreted by

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some modern scholars in terms of the construction of a model; and (3) we ﬁnally use the compositive method to go from a clear knowledge of the cause to a clear knowledge of the eﬀect. This last stage is the highest sort of demonstration (demonstratio potissima), a notion already present in the Thomist tradition.

3. Logic in Reformed Europe: From Humanism to “Protestant Scholasticism” It is unfortunate that most historical accounts of logic devote relatively little attention to Philipp Melanchthon (1497–1560), “Germany’s teacher” (praeceptor Germaniae), prominent reformer and close collaborator of Luther. Actually, in the overall context of European logic in the mid-sixteenth century, the role played by Melanchthon is one of the highest signiﬁcance. From 1520, when at the age of 23 he published his Compendiaria dialectices ratio (Melanchthon 1520), to 1560—the year of his death—there are records of more than 60 different editions of his logic works. The last version of Melanchthon’s dialectics, the Erotemata dialectics (Melanchthon 1846), was to be the standard reference for protestant logic until the beginning of the seventeenth century. What makes Melanchthon’s logic interesting and explains its inﬂuence is above all the very evolution of his works. In the Compendiaria Dialectices Ratio, a young, strongly antischolastic Melanchthon oﬀers a simpliﬁed and rhetorically oriented treatment of dialectic, purged of many “superﬂuous” scholastic subtleties. Like many humanist dialecticians, here Melanchthon rejects the third syllogistic ﬁgure (which he considers “remote from common sense”) and the treatment of modality (the scholastic theories on modality are considered “tricky rather than true”). A few years later, however, Melanchthon’s opinions on both matters (as well as on many others; see Roncaglia 1998) radically changed. In the De Dialectica libri IV (Melanchthon 1528) the third ﬁgure is accepted and discussed at length, and Melanchthon bitterly criticizes Valla for rejecting it, while in the Erotemata (Melanchthon 1846) the discussion of modal propositions is considered to be “true and perspicuous, useful in the judgment of many diﬃcult questions.” The evolution of Melanchthon’s logic is thus marked by a progressive rejection of humanistic-rhetorical models and by a return to the Aristotelian and scholastic tradition. Two further aspects of the evolution of Melanchthon’s dialectic deserve attention: the gradual shift from a bipartite toward a tripartite conception of the structure of the proposition, and the growing interest in fallacies. In 1520, Melanchthon endorses the theory that every proposition has two main components: subject and predicate. In 1528, the question is seen from a grammatical perspective, and noun and verb are considered as being the two main components of the proposition. The verb, however, is further subdivided: It may be a proper verb or a construct made up of the substantive verb (the copula “est”) and a noun. The copula acquires a fully autonomous role in the Erotemata, where every proposition is seen as having not two but three main

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parts: the subject, the predicate, and an (explicit or implicit) copula, seen as the formal sign of the connection between subject and predicate. Such a theory will play an important role in subsequent logic, because the copula will be considered to be not only the logical “glue” of the proposition but the actual bearer of its modal and quality modiﬁcations (see Nuchelmans 1980, 1983; Roncaglia 1996, 2003). The discussion of fallacies also testiﬁes Melanchthon’s increasing use of scholastic doctrines. Absent in 1520, a short section on fallacies appears in 1528, accompanied, however, by the observation that anyone who has fully understood the precepts supplied for the construction of valid arguments doesn’t need special rules to avoid paralogisms. But in the following editions of his dialectics, Melanchthon systematically adds new divisions and new examples. He distinguishes between fault of matter and fault of consequence, corresponding to the traditional division of fallacies in dictione and extra dictionem, and presents the principal fallacies of both sorts. In the Erotemata, fallacies undergo a still closer scrutiny within a systematic framework clearly derived from scholasticism. Many elements indicate that there was one main and primary reason for this return to scholasticism: the perception that the Reformed ﬁeld—engaged in the sharp debate with Catholic theologians, and in the equally sharp debate among diﬀerent Reformed confessions—desperately needed eﬀective logical tools. Rhetoric could be useful in winning popular support, but was much less eﬀective in winning subtle theological debates. In the complex theological and political struggle that was under way in Europe, universities were to become crucially relevant players. Logic was to become a weapon in the theological struggle, and Melanchthon was probably the ﬁrst to perceive that clearly. Melanchton’s works on dialectic, together with the Dialectica by Johannes Caesarius (1460–1550), another interesting and inﬂuential mixture of humanist and Aristotelian elements, had thus the ultimate eﬀect of paving the way that was to be followed by Protestant logicians: endorsement of some humanist doctrine (ﬁrst and foremost the pivotal role of topics and inventio), and great attention to the pedagogical organization of their work, but within a context that retained many tracts of traditional logic; and that—given the relevance of logic for the theological debate—was to devote a renewed attention even to some of the once deprecated scholastic subtleties. It is therefore hardly surprising that the attempt, made by the so-called Philippo-Ramist logicians, to conjugate Ramus’s drive for simplicity and for systematic, method-oriented classiﬁcation, with Melanchthon’s humanistinﬂuenced but somehow more conservative treatment of logic, was not destined to have a long success. Given the renewed role of logic in the interconfessional theological debate, the two paths were bound to diverge and the Ramist component was to succumb: At the end of the sixteenth century, the antiRamist pamphlet was to become a well-established literary genus in the logic production of Protestant Germany. A ﬁerce battle against Ramism was led by Cornelius Martini (1568–1621), among the founders of the so-called Protestant

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Scholastic. Martini endorses Zabarella’s deﬁnition of logic as mental habitus dealing with “second notions,” that is, concepts used to represent and classify, rather than immediately derived from perception. Martini divides logic into formal and material, with formal logic seen as dealing with the pure form of (syllogistic) consequences. Zabarella’s inﬂuence is also apparent in the work of Bartholomaeus Keckermann (ca. 1571–1608). In his Systema logicae (1600, in Keckermann 1614), the eﬀort to organize logic as a discipline (largely on the basis of topics) is clear from the very deﬁnition of logic, which can be considered as a human ability—and is then to be regarded as a mental habitus—but can also be considered as the corpus of doctrines resulting from the use of this ability (ars externa): that is, as a system. In this perspective, knowledge of the historical constitution of this doctrinal corpus becomes important: Therefore, it is not by chance that the short section on the history of logic, present in many sixteenthand seventeenth-century treatises, acquires in Keckermann status, accuracy, and completeness. Keckermann’s interest in the history of logic is also connected with the eclectic tendency of many early seventeenth-century logicians: Given that in a Zabarella-oriented perspective logic is a human activity (and is also the systematically arranged, historical product of this activity), it is natural to try to collect the “logical tools” developed by diﬀerent logicians in diﬀerent times and contexts. This eclectic tendency is usually implicit, and is not necessarily connected with the endorsement of Zabarella’s positions (some aspects of which were actually criticized by many systematic-oriented or eclectic logicians), but it is clearly present in the encyclopedism of authors such as Johann Heinrich Alsted (1588–1638) or Franco Burgersdijk (1590-1636), whose Institutionum logicarum libri duo (1626) was the standard logic handbook in the Netherlands, and, like Keckermann, included a large section on the history of logic (see Bos and Krop 1993). A remarkable feature of this eclecticism is the tendency to reabsorb, within a context usually marked by Renaissance Aristotelianism, even some of Ramus’s doctrines, notably the emphasis on the practical utility of logic and on the need of a well-arranged, easily graspable, and pedagogically oriented method. In the ﬁrst half of the seventeenth century, in reformed Europe, despite the terrible destruction of the Thirty Years War, the university system was expanding, and acquiring a political relevance that was bound to transform any doctrinal diﬀerence in the occasion of sharp conﬂicts (see Wollgast 1988b). This complex situation enhanced logical research and produced some new and interesting theories. In discussing the structure of the proposition, the Berlin-based Johannes Raue (1610–1679) proposed a new theory of the nature and role of the copula. In his opinion, the standard proposition of the form “S is P” should be analyzed as “that what is S is that what is P” (id quod est S est id quod est P), that is, as having three copulas. The role of the main copula (the middle one, which Raue calls “real copula”) is then diﬀerentiated from that of the auxiliary ones: It can be used only in the present tense, while time and modal modiﬁcations are seen as operating on the auxiliary copulas.

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The pronoun “id” stands for the “third common entity” (tertium commune) in which subject and predicate are joined, and it has been observed that “Raue delights a Fregean reader when he emphasizes that ‘S’, the subject, . . . in his analysis is predicated of the tertium commune just as the predicate ‘P’ ” (Angelelli 1990, 188). This “newest theory,” of which Raue is very proud, was criticized by Johannes Scharf (1595–1660): a polemical exchange that was well known to Leibniz. Leibniz had the highest opinion of another famous logician of the time, Joachim Jungius (1587–1657). Jungius’s Logica Hamburgensis, one of the most clear and complete logical works of the seventeenth century, deals at length with such relevant and “advanced” topics as the theory of relations and the use of nonsyllogistic consequences (Jungius 1957, 1977). Jungius is not the only one to deal with such theories, which were considered useful in theological disputations, but his treatment of them is always clear and insightful. This is especially true of his investigation of the inversio relationis (from “David is the father of Solomon” to “Solomon is the son of David”) and of the consequence a rectis ad obliqua (from “the circle is a ﬁgure” to “he who draws a circle draws a ﬁgure”). Jungius’s discussion of the latter—which he considers a simple consequence (the consequent is inferred from the antecedent without the need of a middle term)—was the subject of a detailed analysis in the correspondence between Leibniz and Jungius’s editor Johannes Vagetius (1633–1691), who tried to oﬀer a formal representation of its structure (see Mugnai 1992, 58–62 and 152–153).

4. Descartes and His Inﬂuence When I was younger I had studied, among the parts of philosophy, a little logic, and, among those of mathematics, a little geometrical analysis and algebra. . . . But, in examining them, I took note that, as for logic, its syllogisms and the greater part of its other teachings serve rather to explain to others the things that one knows, or even, like the art of Lull, to speak without judgment about those of which one is ignorant, than to learn them. . . . This was the reason why I thought that it was necessary to seek some other method, which, comprising the advantages of these three, were free from their defects. (Descartes 1994, 33–35) This passage, from René Descartes (1596–1650) Discours de la méthode (1637), oﬀers a good synthesis of Descartes’s attitude toward traditional logic. Descartes’s criticism of syllogism does not concern its validity but its power as a tool for scientiﬁc research, and is clearly expressed in his Regulae ad directionem ingenii: “dialecticians are unable to devise by their rules any syllogism which has a true conclusion, unless they already have the whole syllogism, i.e. unless they have already ascertained in advance the very truth which is deduced in that syllogism” (Descartes 1964–1976, X, 406). The core of the argument is a classic one, advanced in diﬀerent forms at least since

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Sextus Empiricus (see Gaukroger 1989, 6–25): Syllogism is a circular form of reasoning, since it only holds if both its premises are already known to be true, but if both premises are already known to be true, the conclusion is already known to be true, too. Therefore, to discover something new, we cannot depend on syllogism. Pierre Gassendi (1592–1655) advanced a similar criticism. He observed that the evidence needed to accept one of the premises of a syllogism is provided or presupposed by its conclusion. Thus, in the Barbara syllogism “All m are p, all s are m, therefore all s are p,” the truth of “all m are p” can only be established by generalization of the fact that all instances of m—including s—are p: The truth of the conclusion is presupposed by, rather than inferred from, the truth of the premises. Descartes discusses a further argument against syllogism: The validity of a syllogism does not guarantee the truth of its conclusion, which depends on the truth of the premises. The syllogism alone—while giving us the false impression of dominating the concepts we are dealing with—cannot establish it. This argument too is fairly traditional; in the period we are dealing with, we ﬁnd a similar one in Francis Bacon (1561–1626): We reject proofs by syllogism, because it operates in confusion and lets nature slip out of our hands. For although no one could doubt that things which agree in a middle term, agree also with each other (which has a kind of mathematical certainty), nevertheless there is a kind of underlying fraud here, in that a syllogism consists of propositions, and propositions consist of words, and words are counters and signs of notions. And therefore if the very notions of the mind (which are like the soul of words, and the basis of every such structure and fabric) are badly or carelessly abstracted from things, and are vague and not deﬁned with suﬃciently clear outlines, and thus deﬁcient in many ways, everything falls to pieces. (Bacon 2000, 16) While in Bacon this argument is used to advocate the need of “true induction” (progressive generalization accompanied by the use of his “tables of comparative instances”), in Descartes it is used to advocate the role of intuition. According to Descartes, the process of knowledge acquisition depends on (1) intellectual intuition, that is, the intellectual faculty that allows a clear, distinct, immediate, and indubitable grasp of simple truths; and (2) deduction, that is, the grasp of a connection or relation between a series of truths. According to Descartes, deduction is therefore not to be seen as an inferential process governed by logical rules, but rather as the exercise of an intellectual faculty that is ultimately based on intuition. The process of mastering a long or complex deduction is a sort of intellectual exercise, consisting in the recursive application of intuition over each of its steps. The aim of this process is certainty: The idea of degrees of certitude or probability is totally alien to Descartes’s intuitionbased conception. And since for Descartes intellectual intuition is a natural

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faculty, there are no abstract rules or inference patterns governing intuition or deduction: We can only give precepts—like the well-known four regulae given in the Discours—helping us in the better use of this faculty. One aspect of Descartes’s concept should be stressed: The combined use of intuition and deduction allows us to attain knowledge, but does not suﬃce by itself to guarantee that the knowledge we attain is true. If a proposition p is intuitively clear and evident for us, we are entitled to claim that it is true. But while this claim is justiﬁed, its correctness is not grounded on the fact that p is perceived by us as clear and evident, because a deceptive God could give us a clear and distinct intuition of something that is not true (i.e., something that does not correspond to reality; from this point of view, Descartes is now generally considered as holding a correspondence theory of truth; see Gaukroger 1989, 66). Therefore, the well-known cogito argument is needed to ensure God’s external guarantee of our knowledge. According to Descartes, we only need (and we can only achieve) this external guarantee: God’s knowledge is not a model for our knowledge, and there is no set of eternal truths binding God’s knowledge and ours in the same way, since eternal truths themselves result from the joint action (or rather from the uniﬁed action) of God’s will and understanding. Descartes’s resort to intellectual intuition as ultimate foundation of certainty is somehow at odds with his work in the ﬁeld of algebra and geometry and with his discussion on the relevance of analysis. In Descartes’s opinion, analysis is associated with the discovery of new truths (while synthesis has to do with presenting them in such a way as to compel assent), and its function is apparent in mathematics and in analytical geometry, when we use variables (general magnitudes) instead of particular values. Descartes, however, doesn’t seem to perceive the possible connection of this method with deductive reasoning: on the contrary, he seems to associate deduction with the less imaginative, painstaking word of synthetically computing individual magnitudes. The inﬂuence of scholasticism on Descartes’s philosophy is greater than one might suspect at ﬁrst sight (see already Gilson 1913). For instance, hints at a “facultative” concept of logic (see section 6) were present in authors (among them Toletus, Fonseca, and the Conimbricenses) he knew. But Descartes’s concept of deduction diﬀered very much from traditional logic. This did not prevent some Cartesian-Scholastic logicians to reconcile them. Johann Clauberg (1622–1665), in his aptly named Logica vetus et nova (Clauberg 1658), defended Descartes’s methodical rules against the charge of being too general or useless, attributing them the same kind of rigor and strength of Aristotle’s logical rules. Johann Christoph Sturm (1635–1703) made a similar attempt. More articulate was the position of Arnold Geulincx (1624–1669), who published a Logica Fundamentis suis restituta (1662), and a logic more geometrico demonstrata, the Methodus inveniendi argumenta (1663; both in Geulincx 1891–1893). Like most Dutch logicians, Geulincx was deeply inﬂuenced by the eclectic Aristotelianism of Burgersdijk (see section 3). He thus merged late scholastic, Aristotelian, and Cartesian themes in a logic that, with some hyperbole, he labelled “geometric.” Its treatment includes the so-called De Morgan

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rules (well known in medieval scholastic logic, but less frequently dealt with by Renaissance logicians). He also devised a “logical cube” whose faces represented all the axioms and argument forms of his logical system. Descartes’s inﬂuence is also evident in the Port Royal Logic, which we will discuss in the next section. Before dealing with it, however, there are two authors—somehow diﬃcult to classify by means of traditional historiographic labels—which are worth mentioning: the French Jesuit Honoré Fabri (1607– 1688) and the Italian Jesuit Gerolamo Saccheri (1667–1733). Neither of them was an “academic” logician, and they both had wide-ranging interests. Fabri corresponded with most of the major philosophers and scientists of the time (including Descartes, Gassendi, and Leibniz); was interested in philosophy, mathematics, astronomy (he discovered the Andromeda nebula), physics, and biology; and wrote on calculus and probabilism (his book on this subject was condemned by the Church). His Philosophia (1646) is inﬂuenced by Descartes, but the section on logic is pretty original: He developed a combinatorial calculus which allowed him to classify 576 syllogistic moods in all the four ﬁgures; he also used a three-valued logic (based on truth, falsity, and partial falsity) which he applied to the premises and conclusions of syllogisms, and used disjunctions to express hypothetical judgments. Wide-ranging were also the interests of Saccheri, who, besides working on logic, also wrote on mathematics and geometry. In trying to prove the parallel lines postulate, in the Euclides ab Omni Naevo Vindicatus (1733) he hints—against his will—to non-Euclidean geometries. Both in logic and in geometry he makes use of the consequentia mirabilis (well known to the mathematicians of the time): If p can be deduced from non-p, then p is true. In his Logica demonstrativa (published anonymously in 1697)—a treatise on logic organized “more geometrico”—he applies the consequentia mirabilis to syllogistic. One of his proofs refers to the rejection of AEE syllogism in the ﬁrst ﬁgure. Saccheri shows that this very rejection (stated in E-form: “no AEE syllogism in ﬁrst ﬁgure is valid”) can be the conclusion of a ﬁrst ﬁgure AEE syllogism with true premises. If such a syllogism is not valid, then it constitutes a counterexample to the universal validity of AEE syllogisms (which are thus to be rejected). If it is valid, the truth of its premises implies the truth of its conclusion. Such an elegant demonstration has been correctly seen as the mark of an argumentation strategy based on the skillful use of confutations and dilemmas (see Nuchelmans 1991, 133–137).

5. The Port-Royal Logic A mixture of ancient and new doctrines characterizes the Logique ou l’Art de penser published anonymously in 1662 but written by Antoine Arnauld (1612– 1694) and Pierre Nicole (1625–1695). The authors belonged to the Jansenist movement of Port-Royal, hence the current denomination of their work as the Port-Royal Logic (Arnauld and Nicole [1683]). According to widespread

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opinion, the authors endorse Descartes’s philosophy. This is in many respects true, especially as regards the origin of ideas and the account of the scientiﬁc method. Indeed, in the 1664 edition of the book, the authors declare that the section on the analytic and synthetic method is based on the manuscript of Descartes’s Regulae ad directionem ingenii. However, the Port-Royal Logic is not a straightforward Cartesian logic because it relies on many sources. Apart from the inﬂuence of Augustine and Pascal (1623–1662), the authors, though condemning scholastic subtleties, acknowledge the utility of some scholastic precepts and are not always adverse to Aristotle. True, they reject the Aristotelian categories and topics, but describe these doctrines and make them known to their readers. The Port-Royal Logic is also diﬀerent from a humanistic ars disserendi, and even more from an ars bene disserendi, as Ramus would have it, for it is intended to be an ars cogitandi, an art for thinking. The authors maintain that, since “common sense is not so common a quality as people think” (First Discourse 17, trans. 6), people ought to educate themselves to be just, fair, and judicious in their speech and practical conduct. Such an education should be oﬀered by logic, but traditional logic pays too much attention to inference, whereas it should concentrate on judgment because it is in judging that we are liable to make errors compromising our rational and practical conduct. So, because judgment is a comparison of ideas, a detailed study of ideas must precede it. In the Port-Royal Logic, “idea” is an undeﬁned term: “The word “idea” is one of those that are so clear that they cannot be explained by others, because none is more clear and simple” (I, i, 40, trans. 25). “Idea,” therefore, is a primitive term that can only be described negatively. Accordingly, the authors maintain that ideas are neither visual images nor mere names, and are not derived from the senses, because, although the senses may give occasion to forming ideas, it is only our spirit that produces them. Once ideas are produced, logic investigates their possible relations and the operations one can perform on them. Such relations and operations are founded on a basic property of universal or common ideas (as diﬀerent from singular ideas): the property to have a comprehension and an extension. The comprehension of an idea consists of “the attributes that it contains in itself, and that cannot be removed without destroying the idea. For example, the comprehension of the idea of a triangle contains extension, shape, three lines, three angles, and the equality of these three angles to two right angles, etc.” (I, vi, 59, trans. 39). The fact that the comprehension of “triangle” contains not only three lines but also the property proved by the theorem that the sum of its angles is equal to two right angles, gives way to speculations as to what extent humans dominate the comprehensions of their own ideas (Pariente 1985, 248ﬀ). The extension of an idea “are the subjects to which the idea applies. These are also called the inferiors of a general term, which is superior with respect to them” (I, vi, 59, trans. 40). The notion of extension is ambiguous. The subjects to which an idea applies can be intended either as the class of individuals of whom that idea can be predicated, or as the ideas in whose

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comprehension that idea is contained. In the latter sense, extension is clearly deﬁned in terms of comprehension; therefore, comprehension is considered a primitive notion in comparison with extension. The ambiguity of the notion of extension, already noticed by some interpreters (Kneale and Kneale 1962, 318–319), is somehow intended, for it serves, as we will see, to deﬁne diﬀerent properties of the operations that can be performed on ideas, as well as to solve classical problems of quantiﬁcation in the doctrine of judgment and reasoning. If we subtract an attribute from the comprehension of an idea, by deﬁnition we obtain a diﬀerent idea, in particular a more abstract one: Given the idea “man,” whose comprehension certainly contains “animal, rational,” by subtracting “rational” we destroy the idea “man” and obtain a diﬀerent and more abstract idea, which is the residual idea with respect to the original comprehension of “man.” The operation just described is abstraction that, if reiterated, produces an ascending hierarchy of increasingly abstract ideas. The operation of abstraction is the means by which the Port-Royal Logic introduces an inverse relation between comprehension and extension, often called the “Port-Royal Law” in subsequent literature. For the authors maintain that in abstractions “it is clear that the lower degree includes the higher degree along with some particular determination, just as the I includes that which thinks, the equilateral triangle includes the triangle, and the triangle the straight-lined ﬁgure. But since the higher degree is less determinate, it can represent more things” (I, v, 57, trans. 38). This means that the smaller the comprehension, the larger the extension, and vice versa. To move from a higher to a lower idea in the hierarchy we have to restrict the higher one. Restriction can be of two kinds. The ﬁrst kind of restriction is obtained by adding a diﬀerent and determined idea to a given one: If to the idea A (animal) we add the idea C (rational), so as to have a new idea composed by the joint comprehensions of A and C, and if we call B (man) the idea thus composed, then B is a restriction of A and C and is subordinated to them in the hierarchy of ideas. This restriction cannot be obtained by adding to an idea some idea it already contains in itself, for the alleged restriction would be a mere explication of the given idea: If B contains A, then by adding A to B we obtain B, that is BA = B. The second kind of restriction consists in adding to a given idea “an indistinct and indeterminate idea of part, as when I say ‘some triangle.’ In that case, the common term is said to become particular because it now extends only to a part of the subjects to which it formerly extended, without, however, the part to which it is narrowed being determined” (I, vi, 59, trans. 40). The possibility of two kinds of restriction shows that comprehension and extension do not enjoy the same properties. While the ﬁrst restriction modiﬁes the comprehension of the restricted idea so that we get a diﬀerent idea having a richer comprehension and a smaller extension, the second restriction concerns only the extension of the restricted idea with no modiﬁcation of its comprehension: It remains the same idea. But this depends on the fact that the notion of extension of the Port-Royal Logic is,

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as already noticed, ambiguous. In this second case, the extension is obviously constituted by the individuals to whom the idea applies. The Port-Royal Logic, also due to Augustine’s inﬂuence, is very attentive to the linguistic expression of ideas. The authors maintain that if reﬂections on our thought never concerned anyone but ourselves, it would be enough to examine them in themselves, unclothed in words or other signs. But we can make our thoughts known to others only by associating them to external signs, and since this habit is so strong that even in solitary thought things are presented to the mind by means of the words we use in speaking to others, logic must examine how ideas are joined to words and words to ideas. (untitled preface 38, trans. 23–24) This means that thought is prior to language and that a single thought can underlie diﬀerent linguistic forms. This view of the relation between thought and language is one of the guidelines behind the project of a universal grammar contained in the Grammaire générale et raisonnée, published in 1660 by Arnauld in collaboration with Claude Lancelot (1616–1695) (Arnauld and Lancelot [1676]). This view, which has received great attention since Noam Chomsky’s (1966) much-debated claim that it preﬁgures transformational generative grammar, is also relevant to logic (see Dominicy 1984). Given that logic studies the properties of ideas, their mutual relations, and the operations that can be performed on them, and given that ideas are designated by words which can be equivocal, the authors establish the convention that, at least in logic, they will treat only general or universal ideas (as diﬀerent from singular ideas) and univocal words (I, vi, 58, trans. 39). Such are the words associated to ideas by way of a nominal deﬁnition, meant as the imposition of a name to an idea by way of a free, public, and binding baptismal ceremony, of the kind used in mathematics and whose model is found in Pascal ([1658 or 1659], 242ﬀ). Nominal deﬁnitions make it possible to use words (particularly substantives) of ordinary language as if they were the signs of a formal language in which everything is explicit: The best way to avoid the confusion in words encountered in ordinary language is to create a new language and new words that are connected only to the ideas we want them to represent. But in order to do that it is not necessary to create new sounds, because we can avail ourselves of those already in use, viewing them as if they had no meaning. Then we can give them the meaning we want them to have, designating the idea we want them to express by other simple words that are not at all equivocal. (I, xii, 86, trans. 60) We can not only make abstractions and compositions of ideas but also compare them and, “ﬁnding that some belong together and others do not, we

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unite or separate them. This is called aﬃrming or denying, and in general judging” (II, iii, 113, trans. 82). Since judging is a comparison of ideas, and since the activity of making syllogistic inferences can be considered as a comparison of two ideas through a third one, syllogistic inferences lose much of their importance, while judging is established as the most important of our logical activities (Nuchelmans 1983, 70–87). This does not mean that the Port-Royal Logic neglects syllogisms. On the contrary, it contains an articulate doctrine of syllogism based on the fundamental principle that, given two propositions as premises, “one of the two propositions must contain the conclusion, and the other must show that it contains it” (III, xi, 214, trans. 165). It also contains a nontrivial treatment of syllogistic moods that was to be implemented by the young Leibniz in his De Arte Combinatoria (see section 8). Such a treatment is based on “the law of combinations,” applied to “four terms (such as A.E.I.O.)” giving 64 possible moods, and on a set of rules that make it possible to select the well-formed ones, so that, given rules for the valid moods in each ﬁgure, one can dispense with the doctrine of the reduction of other ﬁgures to the ﬁrst: Each mood of any ﬁgure is proved valid by itself (III, iv, 88–89, trans. 143–144). Indeed, though the ﬁrst edition of the book contained a chapter on the reduction to the ﬁrst ﬁgure, that chapter is left out of all subsequent editions. Nevertheless, though the authors seem competent in pointing out a frequent confusion between the fourth ﬁgure and the ﬁrst ﬁgure with transposed premises (III, viii, 202, trans. 155), they are very traditional in other respects. For instance, they reduce the consequentiae asyllogisticae, called complex and composed syllogisms, to the classical categorical moods, thus provoking a reproach from Vagetius in the preface to the second edition of Jungius’s Logica Hamburgensis (see section 3). The Port-Royal Logic attributes great importance to method, corresponding to the operation of the spirit called “ordering.” Method is divided into two major sections: The ﬁrst, devoted to demonstration and science, follows Descartes’s methodical rules, thus giving an outline of the methods of analysis and synthesis; the second, devoted to opinion and belief, contains interesting observations about epistemic modalities and probability, as well as the outlines of Pascal’s wager on the existence of God. By introducing the question of probability, the Port-Royal Logic breaks away from one of the major tenets of Descartes’s philosophy, and opens new perspectives for a probability not limited to games of chance but extended also to events valuable on the basis of frequencies (Hacking 1975b). The Port-Royal Logic was highly successful, as can be gathered from its numerous editions (Auroux 1993, 87). Its inﬂuence, also thanks to Latin, English, and Spanish translations (Risse 1964–70, II 79), is apparent in most of the subsequent European logical literature. Obviously some scholars still preferred the Aristotelian model. For instance, John Wallis (1616–1703), one of the best mathematicians of the time, though the Port-Royal Logic agreed with the argument he had already produced in 1638 that in syllogisms singular propositions must be considered as universal, rejects the Ramist syllogistic

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moods (Wallis 1687) on Aristotle’s authority alone. A direct reaction to the Port-Royal Logic is presented by Henry Aldrich (1648–1710), whose Artis Logicae Compendium (Aldrich [1691]), published anonymously, reprinted many times and still widely used in the Victorian era, preserves scholastic doctrines and an account of the syllogism which “is the best available” (Ashworth 1974, 237). In some concluding remarks, Aldrich criticizes the fundamental principle of syllogism of the Port-Royal Logic, which he considers as a disguised version of the dictum de omni et nullo. Yet the very existence of such a criticism implicitly proves the fame the Port-Royal Logic had achieved (Howell 1971, 54–56). As for Port-Royal’s seminal theory of probability, the young Leibniz already acknowledged its merits in 1667 (Leibniz 1923–, VI, i, 281n), while Jakob Bernoulli wrote his Ars conjectandi, published posthumously (Bernoulli [1713]), as a development of that theory and as a complement to the Ars cogitandi, the Latin version of the Art de penser (Hacking 1975b, 78; Daston 1988, 49).

6. The Emergence of a Logic of Cognitive Faculties Toward the end of the seventeenth century, many logicians developed an interest in the analysis of cognitive faculties. Descartes moved in that direction when he focused his attention on the operations of intuition and deduction, but also the Port-Royal Logic considered the reﬂection on the nature of the mind as the means for a better use of reason and for avoiding errors. The study of cognitive faculties was not simply meant to provide an expositive framework for logical doctrines. As a matter of fact, dealing with the nature and object of logic and with the justiﬁcation of the traditional partitions of logical treaties through a reference to mental operations had been a well-established practice since Aristotle’s Organon: The operations of simple apprehension, judgment, and reasoning had been mentioned as mental counterparts to the logical doctrines of concepts, judgments, and inferences (for a similar approach see Sanderson, section 2). The novelty of what has been called the “facultative logic” of the late seventeenth and eighteenth centuries (Buickerood 1985) is that the cognitive operations involved in the formation and use of ideas become a central concern of logicians. The most important author working on a logic of cognitive faculties is John Locke (1632–1704). The Essay concerning Human Understanding (Locke 1690) is often quoted as a primary example of indiﬀerence, if not contempt, for logic. This is not true if it is intended to describe Locke’s attitude to logic in general, rather than his attitude toward the doctrine of syllogism. Locke, who had been provided at Oxford with a sound scholastic logical education (Ashworth 1980), asks logicians to give up their claim that they teach humans, who are born rational, how to reason by means of syllogisms (Essay, IV, xvii, 4). Logical research should rather investigate the way we form, designate, combine and, in general, use ideas. This concept of logic attributes fundamental importance to language. If logic is the study of the faculties that produce and work with ideas, then it

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becomes impossible to ignore that we can work with ideas only in so much as we connect them to linguistic signs. This very connection of ideas to linguistic signs, although very useful, is a main source of our errors; therefore logic must provide a detailed treatment of the use and of the abuse of words. Indeed Locke compares logic to semiotic: “σημειωτικ , or the Doctrine of signs, the most usual whereof being Words, . . . is aptly enough termed also λογικ , Logick” (Essay, IV, xxii, 4). Locke does not intend to privilege the study of words over that of ideas, for he considers both ideas and words as signs, with the diﬀerence that ideas are signs of things and words are signs of ideas (see Ashworth 1984). Nevertheless it cannot be doubted that the success of Locke’s philosophy contributed to the extraordinary importance language was to have for a large section of later philosophy and logic (see Hacking 1975a). One could say that a logic centered on the study of faculties that produce ideas includes a good deal of epistemological, psychological, and linguistic research (see Hatﬁeld 1997). This is true, but one should consider that this is a consequence of a double eﬀort. On the one hand, Locke wanted to continue the old battle against the ontological basis of the Aristotelian logical tradition by eliminating all talk about natural essences. On the other hand, he wanted to win the new battle against Cartesian innatism (accepted by the Port-Royal Logic) by investigating the empirical origin of our thought. A study of such questions and of the correct use of our ideas and their signs could provide a guide to man’s intellectual conduct in the exercise of judgment, especially in an age characterized by a strong skeptical movement. But providing men with a guide in making judgments was seen as the purpose of logic: In this respect, Locke had many predecessors, notably the authors of the Port-Royal Logic. It must also be considered that it was still left to logic, once the observation of cognitive operations was completed and a careful reﬂection on them was performed, to establish norms for the correct use of those very operations. Defenders of the Aristotelian tradition, such as John Sergeant (1622–1707), criticized Locke’s approach to logic (see Howell 1971, 61–71). But many more were Locke’s admirers, and the impact of his views was impressive. Particularly successful was his refusal of innate ideas (Essay, I, ii), and his emphasis on probable knowledge, though he was far from considering probability from a quantitative point of view (see Hacking 1975b, 86–87). What interests us is that from a very early stage, Locke’s doctrines were included in logic textbooks, frequently in association with direct or indirect references to the Port-Royal Logic. This seems strange if one considers that Locke and the Port-Royalists were so far apart on the question of the origin of ideas. But this important diﬀerence was overlooked by taking into account what Locke and the PortRoyal Logic had in common: the purpose of logic as the guide for correct judgment, the idea that logicians should reﬂect on human understanding, the importance of the linguistic expression of our thoughts. Moreover, the Port-Royal Logic, whose intended readers included the students of the petites écoles of Port-Royal, who had to be prepared for the curricula of European

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universities, was a good source of information about traditional logical topics (such as syllogisms, fallacies, method) that the Essay lacked and that the Port-Royalists had treated without deference toward Aristotle. Perhaps the ﬁrst of such logic handbooks was the Logica sive ars ratiocinandi (Le Clerc 1692) written by the Swiss Jean Le Clerc (1657–1736), who adopted Locke’s doctrine of judgment, his classiﬁcation of ideas, and his philosophy of language, and proposed a mixture of Locke’s and Port-Royal Logic’s theses as concerns the question of probability. A similar approach to logic is to be found in another Swiss scholar, Jean-Pierre Crousaz (1663–1750), who published a series of books on logic, the latest of which was Logicae systema (Crousaz 1724). As it can be expected, a number of British authors published Locke-oriented logic handbooks, most of which had several editions in Britain and elsewhere. Isaac Watts (1674–1748), in his Logick (Watts 1725) and in a popular supplement to it (Watts 1741), followed Crousaz and Le Clerc in oﬀering a Lockean analysis of human nature (especially perception) with a preference for judgment and proposition, rather than syllogism. William Duncan (1717–1760), in his Elements of Logick (Duncan 1748), stated that the object of logic is the study of the faculties of the human understanding and of cognitive procedures, and divided logic into four parts according to the model of the Port-Royal Logic (see Yolton 1986). Francis Hutcheson (1694–1746) too, in his posthumous Logicae compendium (Hutcheson 1756), considered logic as the study of cognitive faculties, but also oﬀered a kind of axiomatic presentation of syllogistic. The Essay was promptly translated into Latin and French. By 1770, professors of Prussian universities were oﬃcially asked to follow Locke in their lectures on metaphysics (von Harnack [1900], I, i, 373). But much before that date Georg Friedrich Meier (1718–1777), the author of the text adopted by Kant for his logic courses (see section 11), already used the Essay in his lectures. The reception of the Essay as a book of logical content was made easier by the inclusion of Locke’s doctrines in logic handbooks such as those we have mentioned. But it was left to Locke’s posthumous Of the Conduct of the Understanding (Locke [1706] 1993), originally intended as an additional chapter to the fourth edition of the Essay, to enter directly into the ﬁeld of logical literature. For in this small book, Locke explicitly declares his views to be an improvement over the standard logic of his time (Buickerood 1985, 183). Of the Conduct of the Understanding was widely read not only in Locke’s own country (Howell 1971, 275ﬀ.) but also in Germany. In the second decade of the eighteenth century, the Thomasian philosopher Johann Jakob Syrbius (1674–1738) used it as a guide for his lectures, and Wolﬀ referred to it in his German Logic (Wolﬀ [1713], Preface). Later on Georg David Kypke translated it into German (Locke [1755]), but at the time of this translation Locke had already been favorably discussed in the incipient German literature on the history of logic (Budde 1731). In France, Locke’s views were well received by authors who also supported Port-Royal’s doctrines. This is the case of Jean Baptiste d’Argens (1704–1771)

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who, in the section devoted to logic of a philosophical treatise addressed to ladies, repeated the Lockean argument that all ideas originate from the senses, at the same time referring to the Port-Royal Logic (d’Argens 1737, Log. §§3, 1) in very positive terms. Among French scholars, Étienne Bonnot de Condillac (1714–1780), the inﬂuential promoter of a radically empiricist philosophy usually referred to as “sensationism,” deserves a special mention. In works of logical content written in the later part of his life—La Logique (Condillac [1780]) and the posthumous La langue des calculs (Condillac [1798])—he developed a concept of logic that owes much to Locke, although he proudly maintains that it is similar to no one else’s. Condillac describes logic as consisting of an analysis of experience by which we study both the origin of ideas and the origin of our own faculties (Condillac [1780], Preface). For, diﬀerently from Locke, who admitted sensation and reﬂection as sources of our ideas, Condillac admits sensation only and maintains that not only ideas, but also all our faculties originate from the use of the senses. This is possible because our senses are complemented by our fundamental linguistic capacity: We would not have complex ideas nor would we be capable of operating with them if we had no language. Consequently, Condillac maintains that any science is a well-made language and such is also the art of reasoning, a conviction that made him reverse the priority order of grammar and logic (Auroux 1993, 93). He adds that, to build a well-made language, we need a method because language, despite its role in the acquisition of our cognitive faculties, has also been used to produce a jargon for false philosophies. The method Condillac recommends for the construction of a well-made language in any science is analysis, in particular analysis as it is used in mathematics, that he considers the paradigm of a well-made science whose language is algebra.

7. Logic and Mathematics in the Late Seventeenth Century At the end of the seventeenth century, another image of the discipline began to emerge. It was borne out of a comparison of logic with mathematics, a comparison intended to prove the superiority of mathematics over logic. Some authors ascribed the superiority of mathematics to its axiomaticdeductive method. This conviction had enjoyed considerable success (see De Angelis 1964), and was enhanced by an interpretation of Descartes’s rules of method as recommendations to begin with a few simple and already known notions (axioms) and then proceed to unknown notions (theorems). Those who endorsed this interpretation seemed to draw the conclusion that the old logic centered on syllogism should be replaced by a new logic intended as a method to ﬁnd and order truths according to the model of the mos geometricus. Some authors attributed the superiority of mathematics to the problemsolving and inventive techniques of algebra. In this perspective, the search for equations relating unknown to given elements, exempliﬁed in Descartes’s

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Géométrie (1637), was interpreted as the true Cartesian logic and was absorbed into the tradition that viewed mathematics as a universal science of invention. In the seventeenth century, algebra was still a new technique, independent of logic, that many considered a rediscovery, but also an improvement of ancient mathematical doctrines. A reference to Descartes was somehow inevitable in this ﬁeld, too, since Descartes appreciated algebra as an intermediate step toward his more abstract mathesis universalis. (On the origins of algebraic thought in the seventeenth century, see Mahoney 1980.) Naturally enough, some authors suggested that algebra could be a useful model for logic. This is the case of Ehrenfried Walter von Tschirnhaus (1651– 1708). He left syllogisms and other traditional parts of logical treatises out of his logic, but, as the title of his major work declares, he held that logic must provide a Medicina mentis, a remedy against the illness represented by our errors, and an aid for the healthy art of invention (Tschirnhaus [1695]). In particular, he claimed that his method of invention would have, in all ﬁelds of knowledge, the same function of algebra in the mathematical sciences. What he actually did, however, was to give an exposition of the methods of analysis and synthesis and a comment on Descartes’s rules of method, albeit with a new attention to empirical sciences for which he envisaged a mixed method of a priori and a posteriori elements (Wollgast 1988a). The assessment of the positive role that algebra could have for logic outlived the idea that logic should imitate, or even be substituted by, the axiomaticdeductive method. The latter was either reduced to a mere synthetic (top-down) order of exposition, or was declared inadmissible outside mathematics, either because of intrinsic diﬀerences between mathematics and other sciences (and philosophy), or because it was held responsible for the degeneration of Cartesianism into Spinozism. But also the algebraic model underwent profound changes. For the algebraic model, followed by many logicians from the late seventeenth century up to almost the end of the eighteenth century, is not the same as the algebraic model used in problem solving. Algebra is no longer considered as a methodical paradigm to be followed analogically by logic to restore the latter’s function as an intellectual guide but as a tool for logic. Many logicians now try to apply algebraic techniques directly to logical objects, that is, to ideas and propositions. In other words, they try to build a logical calculus based on a symbolic representation of logical objects and on rules for manipulating signs, on the assumption that an adequate symbolism has been used. From this point of view, doctrines of ideas such as those of the Port-Royal Logic and of the emergent logic of cognitive faculties, usually considered extraneous to the development of mathematically oriented logic, instead acted as stimuli and provided a ﬁeld of application for the ﬁrst tentative logical calculi. On the one hand, as it has been pointed out (Auroux 1993, 94), a calculus of ideas needs a theory of ideas. On the other hand, scholars who had welcomed a logic of cognitive faculties professed the highest esteem for algebra. We have mentioned Condillac’s positive reference to algebra as the language of mathematics, but decades earlier Nicolas Malebranche (1638–1715)

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had claimed that “algebra is the true logic” (Malebranche [1674] 1962, VI, i, v). Also Locke, who belittled syllogistic and the axiomatic method, did not hide his admiration for algebra (Essay, IV, xii, 15). Locke, however, did not even think of applying the powerful tool of algebra to ideas. Nor did Thomas Hobbes (1588–1679), although in his De corpore, published in 1655, whose ﬁrst part is signiﬁcantly entitled Computatio sive Logica, he maintained that reasoning is computation, where computing means adding several things or subtracting one thing from another in order to know the rest (Hobbes [1839] 1961, I, i, §2). But other scholars were ready to attempt the actual construction of logical calculi. We examine some of such attempts, but ﬁrst consider a declared failure to establish a parallelism between logical and algebraic reasoning, that is, the Parallelismus ratiocinii logici et algebraici (Bernoulli [1685] 1969) of the above-mentioned Jakob Bernoulli (1655–1705). This is an academic dissertation in which Bernoulli was the Praeses (and therefore the real author) and his younger brother Johann (1667–1748) was the Respondens, a circumstance that has often brought to the attribution of the work to the “Bernoulli brothers.” The parallelism concerns the objects, the signs, and the operations of both logic and mathematics. The objects of logic are ideas of things, while the objects of mathematics are ideas of quantity. Likewise, the signs of ideas of things are words (“man,” “horse”), while the signs of quantity are letters of the alphabet: a, b, c for known quantities, and x, y, z for unknown quantities. Bernoulli does not use literal symbols for ideas of things because, on a par with the Port-Royal Logic, he assumes that every idea of thing is (at least in logic) univocally designated by a word, so that every idea of thing has its own sign. Bernoulli then introduces the operations we perform on both kinds of ideas: (1) to put together, (2) to take away, (3) to compare. 1. Ideas of quantity are put together by the sign “+”, as in “a + b.” Ideas of things are put together by the connective “and”, as in “virtue and erudition.” 2. From an idea of quantity one can take away a smaller quantity, thus obtaining the diﬀerence: Given a and b, where a is greater than b, the taking away of b from a is denoted by “a − b.” Similarly, from a complex idea of thing, one can take away one of the less complex ideas it contains, thus obtaining the diﬀerence: From the complex idea “man” one can take away “animal” and the diﬀerence is “rational.” 3. Given two ideas of quantity, if the mind perceives an equality between them, it unites them by the sign “=”, as in “a = b.” If the mind perceives an inequality between them, it uses the signs “>” and “ b,” “a < b.” Given two ideas of things, the mind can ﬁnd (1) agreement or identity between them, and in this case it will aﬃrm one of the other; (2) disagreement or diversity between them, and in this case it will deny one of the other. Aﬃrmation and negation take place thanks to an enunciation (enunciatio), and are expressed by “it is” (est) and “it is not” (non est), as in “man is animal,” “man is not brute.”

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While there is a parallelism between algebra and logic with respect to the operations of putting together and taking away, the parallelism breaks down in the case of comparison. Bernoulli ﬁrst considers the case of agreement. Two ideas of quantity agree when a common measure, applied to them the same number of times, exhausts both, that is, when they are equal. Two ideas of things agree, so that it is possible to aﬃrm a predicate of a subject, provided that a third idea, common to both, exhausts at least the predicate: It is possible to aﬃrm “theft is sin,” provided that some common idea is found in “theft” (no matter if “theft” contains some other ideas besides) and exhausts “sin.” It is clear that Bernoulli intends the comparison of ideas of things as the comparison of their comprehensions. This is conﬁrmed by the fact that “theft is sin” is an indeﬁnite proposition, that is, a proposition whose subject is not quantiﬁed. Now, Bernoulli explains, in an indeﬁnite proposition the predicate is found in the nature of the subject, which means that it cannot be taken away from the idea of the subject in which it is contained without destroying it, according to the Port-Royal Logic deﬁnition of comprehension. We have seen that for Bernoulli the agreement or identity of subject and predicate subsists even if it is incomplete, that is, if the third idea common to both exhausts the predicate without exhausting the subject. Here a problem arises: While the equality of quantities is mutual (if a = b, then b = a), an aﬃrmative indeﬁnite proposition expressing an incomplete agreement is not convertible: “Man is rational” is true, but “Rational is man” is false. Moreover, what happens if the predicate is not found in the nature of the subject but is constituted by some accidental attribute? The answer is that it would be impossible to establish even a partial agreement and, strictly speaking, it would be impossible to aﬃrm that predicate of the subject. Bernoulli decides to overcome these problems by quantifying over the subjects, that is, by taking extensions into account. In this way it becomes possible to form true aﬃrmative propositions such as “All men are sinners” and “Some men are learned,” that is, propositions that in the indeﬁnite form (“Man is sinner” and “Man is learned”) are false. The proposition “All men are sinners” is particularly interesting because it is a true universal proposition although “sinner” is not found in the nature of “man” (Jesus is [also] man, but is not sinner). Therefore, Bernoulli states that the subject of universal and particular propositions are “the species or the individuals that are contained under that [subject]” (Bernoulli [1685] 1969, §11, trans. 176). Is it practical to consider which are the essential attributes of the subject and which are the accidental ones and, in the ﬁrst case, be allowed to make indeﬁnite judgments while, in the second case, resort to quantiﬁed ones? And how to overcome the problem of the impossibility of converting true indeﬁnite propositions, which are exactly those that most resemble algebraic equations? Bernoulli suggests that one should always quantify all aﬃrmative propositions, including true indeﬁnite ones. Consequently, one will be allowed to say “Some men are learned,” which can be converted simpliciter into “Some learned beings are men,” as well as “All men are sinners” and “All men are rational,”

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which can be both converted per accidens into “Some sinners are men” and “Some rational beings are men.” Diﬀerences between algebra and logic also appear when a comparison of ideas shows their disagreement. In algebra, the disagreement of two ideas of quantity means that between them there is an inequality, a relation designated by the sign “ a. In logic, the disagreement of two ideas of things is expressed by a negative indeﬁnite proposition. But the subject and the predicate of negative indeﬁnite propositions can be converted only if the disagreement depends on the opposition of the ideas considered, as in the case of “man is not beast.” This means, as Bernoulli explained in later essays (see Capozzi 1994), that by converting “Animal is not a man” one obtains “Man is not animal,” which is false, because man and animal are not opposite ideas. Also in this case, logic has to resort to the quantiﬁed propositions of old syllogistic, but this means that there are no real logical equations between the ideas themselves. Not so in algebra, as it can be proved by the fact that every inequality is perfectly convertible. The conclusion is that no complete parallelism exists between algebra and logic. As a substitute, Bernoulli recommends the direct use of algebra in science by arguing that in science everything can be quantiﬁed and all that can be quantiﬁed can undergo algebraic treatment. His pioneering mathematical treatment of probability goes in that direction. Bernoulli’s case is instructive. It shows that this is not a lethargic period of logic, as some historians have maintained (Blanché 1996, 169–178), but it also makes one wonder what made Bernoulli fail where other logicians—at the same time or a few decades later—made progresses. In our opinion, the main reason for Bernoulli’s failure was the doctrine of ideas he choose. We have already pointed out that Bernoulli depends on Port-Royal’s view that every idea of thing can be univocally designated by a word (at least in logic), and that every idea of thing is endowed with an indestructible comprehension, conveyed by the word. In the case of aﬃrmation, this makes Bernoulli consider only the relation of containment of the predicate in the subject as basic. Consequently, he is unable to deal with possible predicates that do not disagree with the content of the subject but are not contained in its comprehension. To build a calculus it is not enough to have a rudimentary algebra and a doctrine of ideas. One has to choose a suitable doctrine of ideas.

8. Leibniz The German logician and philosopher Gottfried Wilhelm Leibniz (1646–1716) has a foremost role in the history of formal logic. However, it is almost impossible (and probably misleading) to represent his contributions to logic as a single and coherent set of theories nicely inserted within a linear path of development. There are at least three reasons that rule out such a reassuring view. In the ﬁrst place, Leibniz contributed ideas—often through scattered and

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incomplete fragments rather than through structured and polished writings—to a plurality of logically relevant subjects: from the arithmetization of syllogistic to the theory of relations, from modal logic (and semantic of modal logic) to logical grammar—and the list could be easily extended. Moreover, relevant contributions are often found in works, fragments, or letters not explicitly devoted to the ﬁeld of logic. In the second place, most of Leibniz’s writings testify to a work in progress in the deepest meaning of the expression: Diﬀerent and sometimes incompatible strategies are explored in fragments dating back to the same years or even to the same months, corrections and additions may substantially modify the import of a passage, promising and detailed analysis remain uncompleted or are mingled with sketchy hints. Nevertheless, there is a method in Leibniz’s passionate and uninterrupted research: a deeper unity that is given by a set of recurring problems and by the wider theoretical framework in which they are dealt with. Last but not least, it should always be kept in mind that Leibniz’s works known by his contemporaries and immediate successors constitute a very limited subset of his actual production. It is only during the twentieth century that Leibniz’s role in the history of logic came to be fully appreciated, and this appreciation is connected to at least two diﬀerent moments: the publication by Louis Couturat, in 1903, of the Opuscules et fragments inédits de Leibniz (Leibniz 1966; see also Couturat 1901), many of which were devoted to logic, and the progress made during the second half of the century in the publication of the complete and critical edition of Leibniz’s texts (Leibniz 1923–). Almost three centuries after his death, this edition (the so-called Akademie-Ausgabe) is, however, still to be completed. Leibniz’s interest in logic, and the amplitude of his logical background, is already evident in his youthful Dissertatio de Arte Combinatoria (Leibniz 1923–, VI, i, 163–230). This work is subdivided into 12 problemata (problems), mainly devoted to the theory of combinations and permutations, accompanied by a discussion of some of their usus (applications). Of greatest logical relevance are the combinatorial approach to syllogistic and the discussion of a symbolic language (characteristica) based on a numerical representation of concepts. In dealing with syllogistic, Leibniz takes the work done by Hospinianus (Johannes Wirth, 1515–1575) as his starting point. Like him, Leibniz considers four diﬀerent quantities—singular (S) and indeﬁnite (I) propositions are added to universal (U) and particular (P) ones—and the two traditional qualities given by aﬃrmative (A) and negative (N) propositions. Given that a syllogism consists of three propositions (the two premises and the conclusion), we have 43 possible combinations of the four diﬀerent quantities and 23 possible combinations of the two diﬀerent qualities. The number of possible diﬀerent simple moods of the syllogism, valid and invalid, is therefore according to Leibniz (43 × 23 ) = 512: the same result obtained by Hospinianus. If we take into account the four diﬀerent syllogistic ﬁgures (Leibniz includes and explicitly defends the fourth ﬁgure, which was rejected by Hospinianus), we get a total of (512 × 4) = 2048 “moods in ﬁgure.” It is still through a combinatorial

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method, based on the exclusion of the syllogisms conﬂicting with four classic rules (nothing follows from pure particulars, no conclusion can be of stronger quantity than the weaker premise, nothing follows from pure negatives, and the conclusion follows the quality of the weaker premise) and of eight further moods conﬂicting with the rules given for the four syllogistic ﬁgures, that Leibniz gets the number of 88 valid syllogistic moods. Whereas Hospinianus assimilated singular propositions to particular ones, Leibniz considers them similar to universal propositions, while indeﬁnite propositions are connected to particular ones. In this way the number of valid moods in ﬁgure can be reduced to 24 (6 in each of the four ﬁgures): the 19 “classical” ones, plus 5 new ones that are actually the result of applying subalternation to the conclusions of the 5 “classic” moods with a universal conclusion. The interest of Leibniz’s treatment of syllogistic in the De Arte Combinatoria is not to be found in radical innovations concerning the number of moods of valid syllogisms, but rather in the fact that they are obtained through the systematic use of a combinatorial calculus, used as a sort of deductive device. The syllogistic “deduction” of the rules of conversion is also part of this attempt, based—as in Ramus and in a number of sixteenth- and seventeenthcentury German logicians, including Leibniz’s former teacher Jakob Thomasius (1622–1684)—on the use of identical propositions. In a later fragment, the De formis syllogismorum mathematice deﬁniendis (Leibniz 1966, 410–416), identical propositions are used to obtain a syllogistic demonstration not only of conversion but also of subalternation. Since in this demonstration a ﬁrst ﬁgure syllogism is used, Leibniz can “deduce” all valid moods of the second and third ﬁgure using only the ﬁrst four moods of the ﬁrst ﬁgure, together with subalternation and the rule according to which if the conclusion of a valid syllogism is false and one of its premises is true, the second premise should be false, and its contradictory proposition should therefore be true (methodus regressus). The valid moods of the fourth ﬁgure can be deduced using conversion (the syllogistic proof of which only required moods taken from the ﬁrst three ﬁgures). In this way, Leibniz will complete his construction of syllogistic as a sort of “self-suﬃcient” deductive system. The second result of the De Arte Combinatoria worth mentioning is the construction of a symbolic language in which numbers are used to represent simple or primitive concepts, and their combinations (subdivided in classes according to the number of primitive concepts involved) are used to represent complex or derivate concepts. Fractions are used to simplify the representation of complex concepts, with the numerator indicating the position of the corresponding term within its class and the denominator indicating the number of the class, that is, the number of primitive concepts involved. In Leibniz’s opinion, such a language would oﬀer a solution to the main problems of the logica inventiva (logic of invention): ﬁnding all the possible predicates for a given subject, all the possible subjects for a given predicate, and all the possible middle terms existing between a given subject and a given predicate. This would also allow a mathematical veriﬁcation of the truth of propositions and of the correctness of

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syllogistic reasoning. As we shall see, this project was to ﬁnd a more developed and logically satisfactory form a few years later in 1679. The idea of constructing a symbolic language in which numbers represent concepts is not new. Such an idea was already present in a number of attempts to construct a “universal language,” attempts that were often inﬂuenced by the combinatorial works of Ramon Llull. Leibniz himself makes reference to the works by Johann Joachim Becher (1635–1682?; Character pro notitia linguarum universali, 1661) and Athanasius Kircher (1602–1680; Polygraphia nova et universalis ex combinatoria arte detecta, 1663); similar attempts were made by Cave Beck (1623–1706?; The Universal Character, 1657) and others, and are described by Kaspar Schott (1608–1666; Technica Curiosa VII—Mirabilia graphica, sive nova aut rariora scribendi artiﬁcial, 1664). In the same period, the Spanish Jesuit Sebastian Izquierdo (1601–1681; Pharus Scientiarum, 1659), aiming at a sort of “mathematization” of the ars lulliana, substituted numerical combinations for the alphabetical ones used by Llull, and something similar to a “numerical alphabet”—highly praised by Leibniz—was developed by George Dalgarno (c. 1626–1687) in his Ars Signorum (1961). In Leibniz, however, the construction of a numerical characteristica is not only a handy representational device; it is strictly connected with the idea of the inherence of the predicate in the subject in every true aﬃrmative proposition (predicate-in-subject or predicate-in-notion principle). This idea was already present in the scholastic tradition: In his Commentary on Peter of Spain’s Tractatus, Simon of Faversham (c. 1240–1306) writes that propositions “are called complex because they are founded on the inherence of the predicate in the subject, or else because they are caused by a second operation of the intellect, namely the composition and division of simples” (Simon of Faversham 1969). During the Middle Ages, however, the inherence theory of the proposition was confronted with the idea according to which “in every true aﬃrmative proposition the predicate and the subject signify in some way the same thing in reality, and diﬀerent things in the idea” (Thomas Aquinas 1888–1889, I, xiii, 12). The predicate-in-notion principle was to become a cornerstone of Leibniz’s logical work, and Leibniz was to apply it not only to analytical but also to contingent propositions: “always, in every true aﬃrmative proposition, necessary or contingent, universal or singular, the concept of the predicate is included in some way or other in that of the subject” (Leibniz 1973, 63). In the De Arte Combinatoria, however, Leibniz only deals with propositions made of general terms, and—as it has been already mentioned—the principle is mainly used for the discovery of subjects and predicates within the context of the logica inventiva. Its explicit use as a method for checking the truth of a proposition given its subject and its predicate is to be found only in the 1679 essays (see Roncaglia 1988), where Leibniz chooses to represent simple or primitive terms by means of prime numbers. The advantages of this notation were already stressed in a fragment, dated February 1678, known as Lingua generalis: “The best way to simplify the notation is to represent things using multiplied numbers, in such a way that

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the constituting parts of a character are all its possible divisors. . . . Simple elements may be prime or indivisible numbers” (Leibniz 1923–, VI, iv, 66). Just as compound (reducible) terms can be traced back—by means of deﬁnitions—to the simple, irreducible terms constituting them, the “characteristic numbers” of compound terms will be obtainable from the multiplication of the characteristic numbers of the simple terms constituting them, so that the characteristic number of a compound term can always be univocally broken down into those of the simple (relative) terms composing it. Even at the time of the De Arte Combinatoria, Leibniz was conscious of the diﬃculty in ﬁnding terms that are really simple, and he had to be satisﬁed with simple terms of a relative and provisional nature. This diﬃculty was gradually to become, for Leibniz, an actual theoretical impossibility dependent on the limits of human understanding. While we can cope with abstract systems that are the result of human stipulation (and in which simple terms are established by us), only God can handle the much more complex calculus representing the inﬁnite complexity of the actual world (and, as we shall see, of the inﬁnite number of possible worlds among which the actual one has been chosen). To give an example of his notation, Leibniz uses the deﬁnition of “homo” as “animal rationale,” to which the following characteristic numbers are assigned: animal = 2 rationale = 3 homo = (animal rationale = 2 × 3) = 6 To verify the truth of a proposition, one has just to check whether the prime factors of the characteristic number of the predicate are or not all included among those of the characteristic number of the subject. The proposition “Homo est animal” is thus true, since the characteristic number of “animal” (i.e., 2) is a prime factor of the characteristic number of “homo” (i.e., 6). A network of relations is thus established between the ﬁeld of logic and its numerical “model,” allowing an actual logical “interpretation” to be assigned to the numbers and arithmetical operations employed: Number

Term

Prime number

Simple term

Prime factorization of number

Analysis of term

Number expressed in factorial notation

“Real” deﬁnition of term by means of its component simple terms

Multiplication (calculation of the least common multiple)

Conceptual composition

Exact divisibility of a by b (where a and b are the characteristic numbers of the terms A and B)

Veriﬁcation of the truth of the proposition “A is B”

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This calculus, however, presents some diﬃculties if it is used as a device by which to verify all forms of syllogistic reasoning, as Leibniz intended to do. While it permits adequate representation of UA and PN propositions, it is clearly unsuitable—despite Leibniz’s repeated attempts (including the use of fractions and square roots) to get around the problem—to represent UN and PA propositions. The diﬃculties Leibniz encounters here are connected to the representation of the incompatibility between terms: Though it is always possible to ﬁnd the least common multiple of the characteristic numbers of two terms, it should not always be possible to construct a term that includes any two given predicates. Some predicates are simply incompatible. It is then hardly surprising that without a way to aptly “restrict” the combinations of the terms’s characteristic numbers (i.e., to restrict conceptual composition), Leibniz ﬁnds it “too easy” to verify PA propositions and “too diﬃcult” to verify UN propositions. To solve this problem, Leibniz modiﬁes his notation, making it more complex but much more powerful. Instead of using only one characteristic number for each term, he uses a pair of numbers—one positive and one negative—with no common prime factors. The use of “compound” (i.e., with a positive and a negative component) characteristic numbers allows for the following correspondences: “Compound” characteristic number

Term

Positive component of characteristic number

“Aﬃrmative” component of term

Negative component of characteristic number

“Negative” component of term

Prime factorization of “compound” characteristic number

Analysis of term

“Compound” characteristic number expressed in factorial notation, where no common prime factor is present in its positive and negative components

“Real” deﬁnition of term, demonstrating its possibility by the absence of contradictions within its deﬁnition

Presence of a common prime factor in the positive and negative components of a characteristic number

Logical impossibility of the corresponding term

Now, assuming that a(+) and a(−) represent the positive and negative components of the compound characteristic number assigned to the term A, that b(+) and b(−) have the same function with respect to the term B, and that A and B are possible terms (i.e., that no same primitive factor is present in either a(+) and a(−) or in b(+) and b(−)), the following rules for the veriﬁcation of propositions can be stated:

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Presence in a(+) and in b(−) or in a(−) and in b(+) of at least one common prime factor

Incompatibility between A and B: veriﬁcation of the proposition “no A is B” (UN)

Absence in a(+) and in b(−) or in a(−) and in b(+) of common prime factors

Compatibility of A and B: veriﬁcation of the proposition “some A is B” (PA)

Exact divisibility of a(+) by b(+) and of a(−) by b(−)

Veriﬁcation of the proposition “every A is B” (UA)

Non exact divisibility of a(+) by b(+) and of a(−) by b(−)

Veriﬁcation of the proposition “some A is not B” (PN)

This more elaborate attempt is not without ﬂaws, the most relevant being the problem of representing conceptual negation. Leibniz’s proposal to obtain the characteristic number of a negative term like “non-A” by simply changing the sign of the two components of the characteristic number of the corresponding positive term “A” is ill founded and leads to inconsistencies in the calculus. The question raised here—the nature of conceptual negation—has always raised problems for Leibniz (see Lenzen 1986), and is also connected to the more “philosophical” problem of establishing the nature of incompossibility, a problem clearly stated in a famous passage of the fragment known as De veritatibus primis: “This however is still unknown to men: from where incompossibility originates, or what can make diﬀerent essences conﬂict with each other, given the fact that all the purely positive terms seem to be compatible the one with the other” (Leibniz 1875–1890, VII, 195). Nevertheless, Leibniz’s logical essays of April 1679 represent one of the most interesting and complete attempts of arithmetization of the syllogistic, and oﬀers a well-developed—albeit not fully satisfactory—account of traditional logic by means of an intensional calculus. In a sense, they also represent a turning point in Leibniz’s logical works. The unsolved diﬃculties in ﬁnding a numerical model for his still mostly combinatorial calculus, and the problems associated with the representation of negation, probably led him to a twofold shift in his strategies. On the one hand, despite the interest that notational systems will have for him during all his life, Leibniz became increasingly aware that the research of an apt notation should be accompanied by a closer investigation of the logical laws and principles that should constitute the structure of the calculus. On the other hand, semantic acquires a deeper role: Leibniz perceives that the rules governing conceptual composition cannot be reduced to a sort of “arithmetic of concepts,” and are much more complex. Negation, modality (with special emphasis on compossibility and incompossibility among concepts), relations (with special emphasis on identity), complete concepts of individual substances—much of Leibniz’s logical and philosophical work in the following years will deal with these subjects.

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The ﬁrst tendency—a closer investigation of the logical laws and principles to be used in the calculus—is already clear in the Specimen calculi universalis and in its Addenda (both probably dating around 1680). Here, Leibniz abandons the exposition “by examples,” favoring the much more powerful algebraic notation, which uses letters to represent concepts. The perspective is still intensional, and the inclusion of the predicate in the subject remains the cornerstone of the system. According to the Specimen, the general form of a proposition is “a is b,” and a per se true proposition is of one the three following forms: “ab is a,” “ab is b,” or “a is a.” A per se valid conclusion is of the form “if a is b and b is c, then a is c” (principle of syllogism), and according to Leibniz all true propositions can be derived from (or rewritten as) per se true propositions. The Specimen also contains the ﬁrst clear formulation of one of Leibniz’s key principles, that of substitution salva veritate: “Those are ‘the same’ if one can be substituted for the other without loss of truth” (Leibniz 1973, 34; for a discussion of this principle, see Ishiguro 1991, 17–43). Among the principles used are those according to which in conceptual composition the order of terms and the repetition of a term are irrelevant (“ab = ba,” and “aa = a”). The Addenda oﬀer a short discussion of negation, which was not considered in the Specimen itself, and add some further proofs, including the theorem “if a is b, and d is c, then ad will be bc.” Leibniz calls it “praeclarum theorema” and proves it in this way: “a is b, therefore ad is bd (by what precedes); d is c, therefore bd is bc (again by what precedes), ad is bd, and bd is bc, therefore ad is bc” (Leibniz 1973, 41). In the following years Leibniz will often use the signs “+” (or “⊕”) and “−” to indicate logical composition and logical subtraction, stressing that the rules governing operations with concepts are diﬀerent from those of arithmetical addition and subtraction: Whereas in arithmetic “a + a = 2a,” in the case of conceptual composition “a+a = a.” Moreover, Leibniz will carefully distinguish between conceptual subtraction and logical negation: While in an abstract conceptual calculus it is always possible to “subtract” from the concept of man that of rationality, seen as one of its intensional components, the result of denying it (“men non-rational”) is a simple impossibility, given the fact that rationality is an essential part of the concept of man (Non inelegans specimen demonstrandi in abstractis, Leibniz 1923–, VI, iv, n. 178). These principles are among the ones governing the so-called plus-minus calculus that Leibniz developed in a number of fragments dating around 1687. The basic assumption of the plus-minus calculus is that “A + B = L” is to be interpreted as “A (or B) is included in L,” where the relation of inclusion is—as usual—the intensional inclusion between concepts. “L − A = N ” is to be interpreted as conceptual subtraction: N is the intensional content of the concept L that is not included in the concept A. If “A + B = L,” the terms A (or B) and L are said to be subalterns; two terms, neither of which is included in the other, are said to be disparate, and two terms that have a common component are said to be communicating. It should be stressed that, here as elsewhere, Leibniz uses “term” to designate not a linguistic

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entity but a concept: His calculus is thus directly an “algebra of concepts.” The calculus uses “nihil” to represent a term with empty intensional content, and the rules “A + nihil = A” and “A − A = nihil” are introduced. Leibniz also uses “nihil” as a way to obtain “privative” concepts: if “E = L − M ,” “L = nihil,” and M is a nonempty concept, E will be a privative concept. This assumption has been criticized due to the fact that it introduces inconsistencies in the calculus (Lenzen 2004), but can be seen as a further indication of the relevance that Leibniz attributed to the representation of negative or privative concepts within his logic, and of the diﬃculties connected with the diﬀerence between conceptual subtraction, arithmetical subtraction, and logical negation. The plus-minus calculus has recently been the subject of much interpretive work, also due to the publication of the long-awaited vol. VI, iv of the Akademie Ausgabe, which oﬀers the ﬁrst critical and complete edition of the relevant texts (Leibniz 1923–, VI, iv, vols. 1–3). Among the problems debated (see Lenzen 2000, 2003, 2004; Schupp 2000) are the possibility of a set-theoretical representation of the calculus, and the relation between its intensional and extensional interpretations. As we have seen, Leibniz’s approach is—in most of his logical writings—clearly intensional. However, Leibniz himself was well aware of the diﬀerence between intensional and extensional approaches, and considers the one as the reversal of the other: the method based on concepts is the contrary of that based on individuals. So, if all men are part of all animals, or if all men are included in all animals, it is true that the notion of animal is included in the notion of man. And if there are animals that are not men, we need to add something to the idea of animal to get the idea of man. Since when the number of conditions grows, the number of individuals decreases. (Leibniz 1966, 235) This thesis may be (and has been) criticized, since the number of actually existing individuals falling under a given concept is usually contingent: From a Quinean point of view, “it might just happen that all cyclists are mathematicians, so that the extension of the concept being a cyclist is a subset of the extension of the concept being a mathematician. But few philosophers would conclude that the concept being a mathematician is in any sense included in the concept being a cyclist” (Swoyer 1995, 103). Nevertheless, as Lenzen (2003) correctly observes, this criticism cannot be applied (or at least not in such a naive form) to Leibniz’s logic: The (extensional) domain of Leibniz’s logic is consistently characterized by Leibniz himself as one of possible rather than of actual individuals. The (possible) contingent coincidence of the sets of actually existing cyclists and mathematicians would by no means imply, from a Leibnizian point of view, that the two concepts have the same extension and therefore should have the same intensional content. In a later fragment, known as Diﬃcultates quaedam logicae (Leibniz 1875–1890, VII, 211–217), Leibniz will even use the idea that the domain of his logic is one of possible rather than of actual individuals to justify the conversion per accidens of UA propositions

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(from “All A are B” to “Some B is A”), thus avoiding the problem of the existential import of PA propositions. Despite the fact that they probably precede most of the texts on the plusminus calculus, the 1686 Generales Inquisitiones (Leibniz 1982) are generally considered Leibniz’s most developed and satisfactory attempt of logical calculus; Leibniz himself considered them a “remarkable progress” over his earlier works. The main feature of the Generales Inquisitiones is the attempt to oﬀer a uniﬁed framework for a calculus of terms and a calculus of propositions. As far as terms (or concepts) are concerned, Leibniz distinguishes between integral terms (terms that can be the subject or the predicate of a proposition: the categorematic terms of scholasticism) and partial terms (terms like “same” or “similar,” which are to be used only in conjunction with one or more integral terms, and specify or modify an integral term or a relation among integral terms: the syncategorematic terms of scholasticism). The introduction of partial terms and the discussion of oblique cases clearly testify to the new interest Leibniz devoted to relations. Being discussed at the term level (and therefore at the level of concepts), relations and oblique cases are clearly not considered by Leibniz as mere linguistic accidents. The problem of the possible “reduction” of partial or relational terms and of relational propositions to nonrelational ones is therefore not one of simple “surface-structure” reformulation of a spoken or written sentence, but rather one of logical analysis of the proposition and of its constituent terms. Leibniz was to devote much eﬀort and a large number of texts and fragments to this analysis, clearly inﬂuenced by the late scholastic discussion on relations and on the connection between the relation itself and its fundamenta, that is, the concepts or the things among which the relation is established. Starting with Russell (1900), who attributed to Leibniz a straightforward and uniform reductionistic approach with respect to relations, criticizing it, Leibniz’s treatment of relations has been the subject of much interpretive work. It is now clear that Leibniz oﬀered diﬀerent accounts for diﬀerent kinds of relations and that, while he consistently denied relations an extramental reality independent from that of the related concepts, he thought that at least some relations (among those involving diﬀerent individuals) are not reducible in a straightforward way to simple and nonrelational monadic predicates. However, this does not imply, according to Leibniz, the need of propositions which are not in subject-predicate form, but rather the need (1) to consider within the properties pertaining to a given subject also those expressing relational accidents, and (2) to recognize the logical role of reduplicative terms, which can be used in connecting propositions referring to the diﬀerent fundamenta of a same relation. Thus, the proposition “Paris loves Helen” can be reduced, according to Leibniz, to “Paris is a lover, and eo ipso (for this very reason) Helen is a loved one,” rather than to the simple and independent propositions “Paris is a lover” and “Helen is a loved one” (Mugnai 1992). Reduplicative terms like quatenus, eo ipso, and so on, which were already studied by scholastic and late-scholastic logicians, in this way acquire a special role within Leibniz’s logic.

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In the Generales Inquisitiones, integral terms are further subdivided into simple, complex, and derivative. The discussion of simple terms shows a clear shift when compared to the earlier combinatorial attempts: While general abstract terms like “ens,” derived from the scholastic tradition and from the discussion on transcendental terms, are still present, Leibniz adds to the class of simple terms also terms connected to individuals and perceptions, like “Ego” (“I”) or the names of colors; a passage that somehow anticipates the discussion about simple and innate ideas that will be at the core of the much later Nouveaux Essais (Leibniz 1923–, VI, vi). However, as in the earlier attempts, the choice of simple terms remains provisional and is strongly inﬂuenced by the limits—both necessary and contingent—of our knowledge. A special case is that of the privative term non-ens: Like nihil in the plus-minus calculus, non-ens corresponds here to a term with empty intensional content, and plays an important role in the axiomatization of the calculus. Complex terms are obtained by composition from simple terms, while derivative terms are obtained from partial terms “completed” by integral terms, that is, from integral terms modiﬁed or connected by means of syncategorematic and relational terms or by the use of oblique cases. In the Generales Inquisitiones, for the ﬁrst time, Leibniz includes a discussion of complex terms referring to individuals (Leibniz 1982, 58–62) in his logical calculus. According to Leibniz, they are based on complete concepts, that is, concepts that include all which can be said of that individual. Their complexity, however, is such that only God can carry out their complete analysis: Men can only rely on experience to assert the possibility of a given complete concept (i.e., the absence of contradictions within its intension) and the inclusion of a given contingent predicate within a given complete concept. Complete concepts (corresponding to individual substances) are another theoretical cornerstone of Leibniz’s philosophy, and it is no coincidence that in the very same year in which he was working at the Generales Inquisitiones Leibniz also wrote the Discours de métaphysique (Leibniz 1923–, VI, iv B, 1529–1588), the text that oﬀers for the ﬁrst time and in a structured way the philosophical framework in which the theory of complete concepts is to be placed. Much of the interpretive work done on Leibniz’s philosophy and philosophy of logic in the last three decades deals in one way or another with the discussion of complete concepts1 : from the possibility of distinguishing within them a “core set” of essential properties, which could also allow for transworld identiﬁcation of individuals across possible worlds (each complete concept, if considered in its integrity, is bound to a given possible world, and possible worlds themselves are seen by many interpreters as maximal sets of mutually compossible complete concepts), to the presence within complete concepts of relational predicates; from the discussion of Leibniz’s conception of contingency and of individual freedom to that of preestablished harmony. Shifting from terms to propositions, Leibniz states in the Generales Inquisitiones that “ ‘A is B’ is the same as ‘A is coincident with some B’ or A = BY ” (Leibniz 1973, 56), where B is part of the intensional content of A:

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a formulation close to the ones we have already discussed, which, however, can be of interest if we consider the role attributed here to “Y ,” seen as a sort of existential quantiﬁer applied to the predicate. Leibniz oﬀers a wider discussion of predicate quantiﬁcation in an undated fragment known as Mathesis Rationis (Leibniz 1966, 193–206; see Lenzen 1990), and in some of his (many) attempts of graphical representation of the basic notion of conceptual inclusion and of the four forms of categorical propositions, mainly based on the use of lines or circles (the fragment known as De formae logicae comprobatione per linearum ductus probably being the most notable among them; Leibniz 1966, 292–321). In the Generales Inquisitiones, the connection between the treatment of propositions and that of terms is, if possible, even stronger than in the preceding essays, since Leibniz observes that the four traditional forms of categorical propositions can be rewritten in the following way (Leibniz 1982, 112; his thesis is clearly indebted to the late-scholastic treatment of the passage from proposition “tertii adjecti” to propositions “secundi adjecti”): (PA)

Some A are B

AB est res

(PN) (UA) (UN)

Some A are not B All A are B No A is B

A(non-B) est res A(non-B) non est res AB non est res

Here—if the proposition is not one about contingent existence—“est res” is to be interpreted as “is possible” (Leibniz 1982, 110), and possibility is in turn to be interpreted as absence of contradiction within the intension of the composed term. A similar conception is to be found in the Primaria Calculi Logici Fundamenta, dating to August 1690 (Leibniz 1903, 232–273). This treatment of propositions leads to a term-oriented treatment both of syllogistic inferences and of hypothetical propositions. According to Leibniz, just like in a categorical proposition the subject includes the predicate, in a hypothetical proposition the antecedent includes the consequent. Therefore, an implication of the form “If p, then q” is to be interpreted as “If (A is B) then (C is D),” which in turn can be rewritten as “(A is B) is (C is D),” or “(A includes B) includes (C includes D).” This idea, already present in a fragment known as Notationes Generales probably written between 1683 and 1685 (Leibniz 1923–, VI, iv, 550–557), will return in many later texts and leads Leibniz to hold that the forms and modes of hypothetical syllogisms are the same as those of categorical syllogisms. Leibniz never devoted a detailed analysis to the logic of propositions, but in many of his works and fragments refers to propositional rules derived from the medieval tradition of consequences and from the late-scholastic discussion on topical rules; clearly, in his opinion, an “algebra of propositions” can only be grounded on the algebra of concepts. As already noted, most of the logical texts and many of the most remarkable achievements of Leibniz’s logic were not known to his contemporaries and

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his immediate successors. Nevertheless, Leibniz’s logic cannot be considered simply an isolated product of a genial mind: He had a deep knowledge of the late-scholastic logical tradition, from which he derives not only many topics he deals with but often also the approaches adopted in dealing with them. He also had a wide net of relationships—both through letters and by personal acquaintance—with many of the most prominent ﬁgures of the European learned world (among them Arnauld, Tschirnhaus, Jakob Bernoulli, and, as we will see, Wolﬀ, to name just some scholars mentioned in this chapter). His logical and philosophical theses are also the result of those interactions, and probably some of them circulated even without the support of publication. Despite the great interest of Leibniz’s logic from a contemporary point of view, Leibniz was a seventeenth-century logician, not a twentieth-century logician in disguise.

9. Logic in Germany in the First Half of the Eighteenth Century In the period we are considering, German logic deserves special attention. Since logic was a subject included in most academic curricula, it became a privileged ﬁeld of study and a great number of logical texts were published (see Risse 1965). Many German logicians enter the debate on Cartesianism, are fully aware of Bacon’s exhortation to work at a logic of empirical sciences, pay attention to the notion of probability, examine the relationship between logic and mathematics, and seem open to the suggestions of facultative logic. If one had to name a single author who takes a stand on all these questions, one should mention Leibniz. But, as already said, in this period Leibniz’s logic enters marginally in the oﬃcial picture of German logic, not only because most of his strictly logical production was unknown at that time but also because he did not belong to the academic world. What was known of Leibniz’s philosophy and logic inﬂuenced a number of German logicians of that time, but the logical scene of the ﬁrst two generations of the German Enlightenment was dominated by Christian Thomasius and Christian Wolﬀ. Christian Thomasius (1655–1728) was the son of Jakob Thomasius, Leibniz’s teacher. However, he does not share Leibniz’s view of logic, in as much as he agrees with the humanists in criticizing schoolmen for having instructed generations of students in the making of useless subtleties. At the same time he advocates the study of logic. This is less paradoxical than it sounds. While the Port-Royal Logic recommended a logical instruction because common sense is not so common as people believe, Thomasius oﬀers a moral and religious justiﬁcation. He maintains that because of the original sin, mankind has darkened its natural light (lumen naturale) and has to achieve a healthy reason through a process of puriﬁcation, so as to avoid the errors it usually makes. This cathartic process is entrusted by Thomasius to logic because logic can teach how to counteract errors in judgment and their main source, namely,

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prejudice (in particular, the prejudice of authority and the prejudice of self love). In this way Thomasius agrees with Tschirnhaus and the Port-Royal Logic that logic is a medicine and its primary aim is the correctness of judgments (Thomasius [1691b], dedication). From the Port-Royal Logic—translated into Latin (Arnauld and Nicole 1704) by one of his followers—Thomasius also borrows arguments for rejecting Aristotelian categories (Thomasius 1702, VII §25). Such a concept of logic requires that technicalities should be abolished: Only an “easy” logic can dispel prejudices and teach how to proﬁt from the few precepts needed for the rational conduct of common human beings—and not only learned scholars—in the search of truth and in the practical exercise of prudence. Thomasius’s precepts consist of two basic rules of method: (1) to proceed from what is easy and known to what is more diﬃcult and unknown, (2) to connect remote conclusions to principles only through near (propinquae) conclusions. In these two rules one can ﬁnd an echo of Descartes. But Thomasius is not a Cartesian because he opposes the doctrine of innate ideas, convinced that in the intellect there is nothing that has not been in the senses (a thesis he argues for independently of Locke). He also rejects the tradition of the mos geometricus because he holds it responsible for the degenerate Spinozistic version of Cartesianism so that, from this point of view, he diﬀers also from Tschirnhaus. Thomasius is rather an eclectic (Beck [1969], 247–256) who encourages the study of other philosophers’s ideas, thus promoting studies in the history of philosophy and also in the history of logic: From 1697 to the ﬁrst decades of the eighteenth century it is possible to register a number of essays on the latter subject (Risse 1964–70 II, 507). Thomasius’s eclecticism can be easily appreciated if one considers that, on the one hand, he adds precepts derived from the tradition of humanistic dialectic to his apparently Cartesian rules of method and, on the other hand, he insists that logic should concentrate on the problem of certainty in empirical knowledge. Like many others in this period, Thomasius believes that, although in empirical matters complete certainty is not attainable, it is still possible to avoid skepticism by working on the notion of probability. However, Thomasius’s interest for probability is not to be overestimated, since he inclines to a notion of probability still strongly connected with Aristotelian dialectic and with the doctrine of topical syllogism. Thomasius’s ideas were well received by the incipient age of the Enlightenment that looked favorably to a logic meant for ordinary people (and this favored the proliferation of textbooks) and, from a more theoretical point of view, approved of his antiskeptic battle regarding empirical knowledge. Nevertheless, Thomasius not only advocated a rigid separation between mathematics and philosophy, but also opposed any formalism in logic and deplored the enormous growth of syllogistic, convinced that the ﬁrst ﬁgure is suﬃcient, though incapable of guaranteeing the truth of the conclusions (Thomasius 1702, IX, §12; [1691a], XII, §§19–21). Many of his followers agreed on these matters, with a notable exception. In Halle, where a Thomasian circle was

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ﬂourishing, Andreas Rüdiger tried to reconcile Thomasius’s views on logic and philosophy with his own research on the nature and scope of logic. Rüdiger (1673–1731) agrees with Thomasius that, due to the original sin, mankind no longer participates in God’s archetypal logic and is prey of prejudices, thus making it necessary to conquer a recta ratio through the study of logic (Rüdiger 1722, I, i, 1). Rüdiger also agrees with Thomasius that logic should deal with probability as a response to skepticism, and to this eﬀect he produces an articulated doctrine of probability that obtained remarkable diﬀusion through the ﬁrst edition of the Philosophisches Lexicon (Walch [1726]) published by Johann Georg Walch (1693–1775). But what seems to interest Rüdiger most is that logic should be recognized as a legitimate means for ﬁnding truths and should be proved capable of attaining this purpose with a procedure as diﬀerent as possible from the procedure of mathematics. Rüdiger believes that Spinozism rests on two pillars: the doctrine of innate ideas and the illegitimate application of the mathematical method outside mathematics. Therefore, he rejects both. He is perfectly aware that mathematics is inventive, but he makes this depend on the fact that mathematical proofs can resort to sensibility. Rüdiger does not ascribe the sensibility of mathematical proofs to the use of “visual” aids, such as geometric ﬁgures, but to the demonstrative procedures based on numeration. In his opinion: (1) mathematics is the science of quantity, (2) all quantities are measurable, (3) we can measure only in so far as we can numerate, (4) “all numeration is of individuals, in so much as their terms are perceived by the senses.” His conclusion is: “Therefore all numeration is sensible: but the entire way of mathematical reasoning is numeration, then this entire way [of reasoning] is sensible Q.E.D.” (Rüdiger 1722, II, iv, 1a). The possibility to avail themselves of this kind of sensible reasoning (ratiocinatio sensualis) enables mathematicians to refer to sensible data that would escape their attention if they could rely on intellect alone, whereas sensible data oﬀer them the basis for the discovery of unknown (mathematical) truths (Rüdiger 1722, II, iv, 3c; see Cassirer 1922, 525–527). The heuristic capacity of ratiocinatio sensualis rests on Rüdiger’s theory of truth. According to Rüdiger, the truth of our judgments (which he calls logical truth) consists in the agreement of our knowledge with our sensation (“convenientia cognitionis nostrae cum sensione”; Rüdiger 1722, I, i, 12). But we can trust the agreement of our knowledge with sensation only because there is a metaphysical truth that consists in the agreement of sensation with its objects (“convenientia ipsius sensionis cum illo accidente, quod sentitur”). This means that the metaphysical truth, which makes us trust our logical truth, presupposes that our senses are not fallible (Rüdiger 1722, I, i, 11). Because our senses are not fallible (under God’s guarantee), the certainty and inventive power of mathematics can rest on sensible reasoning. Outside mathematics, however, and in particular in the ﬁeld of philosophy, we work only with ideas and cannot resort to the ratiocinatio sensualis. It is nevertheless possible to use a logical way of reasoning meant for ideas

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(ratiocinatio idealis) and as inventive as the ratiocinatio sensualis: the syllogism. By claiming that syllogism is inventive, Rüdiger openly challenges a long series of scholars, including Thomasius, who had criticized syllogism for being sterile. He argues that the inventive function of syllogism has not been appreciated because syllogism has been used for ﬁnding the premises of a given conclusion. This means that syllogism has been used analytically, whereas syllogism can be inventive if it is used synthetically, as a means for searching an unknown conclusion beginning from a given premise (Rüdiger 1722, II, vi, 1). To show how a synthetic syllogistic is possible, Rüdiger assumes that every proposition (of the four kinds that can enter into syllogisms, A, E, I, O) expresses a precise relation between the subject and the predicate, a relation belonging to a set Rüdiger carefully classiﬁes: subordination, opposition, partial diversity (he considers identical ideas as the same idea; see Rüdiger 1722, I, xii, 2, 3). On this assumption he maintains that we make a synthetic syllogism beginning with a premise whose two terms stand in one of the admissible idea-relations. We then obtain a conclusion by connecting one of the terms of the premise with any unknown idea that stands in a deﬁnite relation (included in the set of classiﬁed idea-relations) with the other term. For instance, given a universal aﬃrmative proposition “All A are B” (where A is subordinated to B) as premise, we can connect the predicate B with any idea C of which we only know the relation it entertains with the subject A. Let such a relation be that of subordination: We can validly conclude that the unknown idea C, being subordinated to A is also subordinated to B, so that, since the relation of subordination can be expressed by a universal aﬃrmative proposition, we obtain the “new” conclusion “All C are B” (Rüdiger 1722, II, vi, 55 1– 4). This single example makes it clear that Rüdiger’s synthetic syllogistic is founded on the old technique of the pons asinorum (Thom 1981, 72–75), traditionally used for ﬁnding premises, a technique that Rüdiger could ﬁnd in works of the Peripatetic tradition he knew well, pace Thomasius who had ridiculed it (Thomasius [1691a], XII, §11). Rüdiger simply reverses the pons asinorum in the search for a conclusion, as it can be appreciated from the graphical representations he gives of his synthetic syllogisms (reunited in a single representation by Schepers 1959, 99). Rüdiger expressed his views on ratiocinatio idealis in a logical environment in which they must have been unpopular. It is not surprising, therefore, that his direct followers, who approved of his separation of mathematics from logic, were no longer interested in his reason for separating them, that is, his passionate defense of the inventive capacity of syllogism. Adolph Friedrich Hoﬀmann (1703–1741) and Christian August Crusius (1715–1775) choose to refer to Rüdiger as the philosopher who did not simply denounce, like Thomasius, the ill eﬀects of the application of the mos geometricus to philosophy, but had argued that mathematics and philosophy should never be mixed because they have diﬀerent objects and diﬀerent methods of reasoning (Tonelli 1959). It may seem paradoxical, therefore, that Rüdiger’s classiﬁcation of idea-relations, supporting his doctrine of syllogism (as well as his doctrine of conversion and

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other nonsyllogistic inferences) and connected to his thesis of the independence of logical from mathematical reasoning, was to stimulate later logicians to once again take up projects of an algebraic calculus of ideas. But such projects were not resumed until the Thomasian school, including its peculiar Rüdigerian variant, was no longer dominant, due to the emergence of Christian Wolﬀ and his school. Wolﬀ (1679–1754) has much in common with the ﬁrst generation of the German Enlightenment. Like the Thomasians, he believes that philosophy should improve human life and give due importance to empirical knowledge. He also shares some of their religious motivations and their appreciation for Locke (see section 6). What divides Wolﬀ from Thomasius on logical matters is, ﬁrst, a very diﬀerent evaluation of mathematics and of the tradition of the mos geometricus: Wolﬀ had received a mathematical education and was professor of mathematics in Halle. Second, Wolﬀ acknowledges the intellectual inﬂuence exerted on him by Leibniz, with whom he exchanged a correspondence that ended only with Leibniz’s death. In one of his letters Leibniz had recommended to Wolﬀ to pay due attention to syllogism: “I absolutely never dared to say that the syllogism is not a means for ﬁnding truths” (Leibniz [1860], 18). Consequently Wolﬀ, far from contrasting syllogistic and the mathematical method, makes a double revolution with respect to the Thomasian school. He (a) assumes mathematical reasoning as an example to be followed in any research ﬁeld, (b) claims that the allegedly empty and useless syllogism is the inner fabric of any reasoning, including the exemplary mathematical one. In diﬀerent places—but especially in his logical works, the so-called German Logic of 1713 and the so-called Latin Logic ﬁrst published in 1728—Wolﬀ considers geometrical demonstrations as chains of common syllogisms in the ﬁrst ﬁgure (Wolﬀ [1713], IV, §§20–25, [1740] §551), and concludes that, where understanding and reason are concerned, there is a single rational procedure, a single method, a single logic valid for mathematics and philosophy alike: “both philosophy and mathematics derive their method from logic” (Wolﬀ [1740], Preliminary Discourse §139 note). The importance of syllogism is justiﬁed by Wolﬀ by the argument that syllogism mirrors the natural way of reasoning. But he does not ground logic on empirical psychology alone (better: on the empirical psychology of a privileged set of men, the mathematicians, see Engfer 1982, 225). He declares that logic has a solid foundation also in ontology (Wolﬀ [1740], Preliminary Discourse §89). Both pillars of this foundation of logic conspire in favoring syllogism: (1) Ontology, as it was established in the scholastic tradition, justiﬁes the dictum de omni et nullo that presides over syllogism in the ﬁrst ﬁgure (Wolﬀ [1740], §380); (2) empirical psychology ensures that the model of natural inference is the simplest syllogistic inference, that is, syllogism in the ﬁrst ﬁgure. Wolﬀ’s foundation of syllogism—and indeed of logic—on ontology and empirical psychology grants an absolute privilege to syllogisms in the ﬁrst ﬁgure. To this eﬀect, Wolﬀ maintains that ﬁgures diﬀerent from the ﬁrst (he does not admit

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the fourth ﬁgure) are not simply reducible to it but are already syllogisms in the ﬁrst ﬁgure in disguise: They are cryptic ﬁrst ﬁgure syllogisms (Wolﬀ [1740], §§382–399; see Capozzi 1982, 109–121). He also maintains that noncategorical syllogisms, consequentiae immediatae and any other kind of inference are reducible to syllogisms in the ﬁrst ﬁgure. In this way, Wolﬀ contrasts antisyllogistic conceptions, but makes no concessions to Rüdiger: The syllogisms he refers to are absolutely standard and in no way synthetic. In a letter to Leibniz, he scorns Rüdiger’s synthetic syllogistic and refers to the idea of a mathematical ratiocinatio sensualis as to one of the “paradoxa” of Rüdigerian logic (Leibniz [1860], 117). Despite his meager syllogistic, Wolﬀ oﬀers a deeply “logicist” philosophy of logic, for not only does he bring logic and mathematics together, he also considers logic prior to mathematics. This makes one wonder why he did not try a mathematical calculus of ideas. This problem is clearly related to the possibility of a heuristic. Wolﬀ knows that mathematicians use heuristic devices not reducible to syllogisms. In his German Logic he denies that “the whole algebraic calculus . . . takes place only according to syllogisms in form” (Wolﬀ [1713], IV, §24). A similar statement is to be found in his Latin Logic: “logic [i.e., logic centered on syllogism] has a notable and famous use in the art of discovery, but nevertheless it does not exhaust it” (Wolﬀ [1740], §563). Elsewhere he explains that, to discover hidden truths, it is sometimes necessary to resort to heuristic artiﬁces such as, in the a priori invention, the artiﬁces of the ars characteristica (see Arndt 1965, 1971). For, he says, this art helps separate geometric and arithmetic truths from images, so as to obtain truths from the data by means of a calculus. He grants that such an art is the most perfect science, but he believes that we only have a few examples of that art in algebra, yet none outside it (Wolﬀ [1713] IV, §22). The latter statement is revealing: To Wolﬀ the establishment of a suitable alphabet of thoughts as a prerequisite of the ars characteristica combinatoria must have appeared too great an obstacle. We know that this was a problem for Leibniz, and we know how he dealt with it. But apparently Wolﬀ was convinced that a calculus in logicis is utopian and considered it a mere desideratum. As to Leibniz’s interest in a logic of probability to be used when deliberating about political, military, medical, and juridical matters (Leibniz [1860] 1971, 110), Wolﬀ seems to doubt that a mathematical ars conjectandi could be of practical use regarding such matters (Leibniz [1860] 1971, 109). Wolﬀ’s doubt was not so strong as to make him exclude that Leibniz’s wish could ever come true, but was strong enough to make him exclude that it could come true in the foreseeable future. Nevertheless, Wolﬀ includes probability in the practical part of his logic, paying attention to the features of probable propositions, in particular to the ratio of suﬃcient and insuﬃcient reasons that make it possible to consider probability a measurable degree of certainty. In this way, he deﬁnitely abandons not only Thomasius’s obsolete treatment of probability reminiscent of the old topic but also Rüdiger’s nonmathematical analysis of probable knowledge.

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Wolﬀ made a great impact on German philosophy, and some of his doctrines were well received in Europe at large, as can be appreciated from the evident Wolﬃan imprint of some entries in the French Encyclopédie (Carboncini 1991, 188ﬀ.), where Locke and Port-Royal Logic had already found much space (Risse 1964–70, II 528). However, at ﬁrst Wolﬀ met with harsh criticism, albeit not for his logical but for his metaphysical views. In 1723, the Halle circle succeeded in convincing King Friedrich Wilhelm to banish Wolﬀ from the city because his alleged determinism, inherited from Leibniz, was a peril to religion (Wundt [1945], 234–244; Beck [1969], 258). Wolﬀ went to Marburg, where he wrote a series of works in Latin, beginning with the Latin Logic. Wolﬀ’s Latin works increased the number of his followers, so much so that a second anti-Wolﬃan oﬀensive, launched in 1734, ended in a defeat when, in 1740, Wolﬀ was readmitted to Halle with great honors. The Wolﬃan era, as concerns logic, was very positive. Despite literature that considers him an exponent of the dark ages of logic, it is diﬃcult not to give him credit for proposing a positive image of the discipline resting not only on its function as a guide in making judgments and in avoiding errors but also on the power of its inferences. But above all, it is impossible to ignore that his revaluation of syllogism diﬀered from Rüdiger’s because he used it to counteract the idea of a gap between logic and mathematics. This explains why Wolﬀ’s success promoted a revival of logic that did not simply contribute to the production of new logical textbooks—given the importance he accorded to logic in academic curricula—but spurred logical investigations. It must be stressed that those who were encouraged by Wolﬀ’s philosophy to engage in logical research did not usually follow the details of his logic. Independent authors, as well as a number of Wolﬃans, referred to old logical literature, including sixteenth- and seventeenth-century Aristotelian and scholastic treatises, and even to the works of anti-Wolﬃans, especially those of Rüdiger, undoubtedly the most distinguished of them. In this sense one must agree with Risse that if one excludes the very ﬁrst generation of Wolﬀ’s followers, it is diﬃcult to draw precise boundaries between the Wolﬃan school and its opponents (Risse 1964–70, II, 615). An example of the new post-Wolﬃan logicians is Johann Peter Reusch (1691–1758). In his fortunate Systema logicum (Reusch [1734]), though in many respects faithful to Wolﬀ, Reusch admits the inﬂuence of Aristotle, Jungius, the Port-Royal Logic, Johann Christian Lange (1669–1756), and Rüdiger (Reusch [1734], preface). As to the question of syllogistic, Reusch informs his readers about traditional doctrines and about the combinatory of syllogistic moods with a reference to Leibniz’s De Arte Combinatoria (Reusch [1734], §530). Nevertheless, he also proposes a syllogistic that, he maintains, opens the gates of the syllogistic moods (modorum cancelli) (Reusch [1734], §543), being founded on a single rule to which all syllogisms of any ﬁgure must conform: “The entire business of reasoning is done by substitution of ideas in the place of the subject or of the predicate of the fundamental proposition, that some call equation of thoughts” (Reusch [1734], §510). In other words, Reusch conceives

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of syllogisms as consisting of a single premise (propositio fundamentalis) and a conclusion obtained by assuming a new idea and substituting it for either the subject or the predicate of the propositio fundamentalis by a substitution rule. Such a substitution rule—which he means as a version of the old dictum de omni et nullo—is governed by the principle of contradiction and presupposes a network of relations among ideas. The description of the admissible idea-relations is so important for Reusch’s syllogistic, and indeed for his whole logic, that two chapters of his Systema logicum—De convenientia et diversitate idearum and De subordinatione idearum—are devoted to it. This study is clearly inﬂuenced by Rüdiger (Capozzi 1990, lxvi), but Reusch did not refer to Rüdiger as the defender of a synthetic inventive syllogistic but as the author of a syllogistic based on a deﬁnite set of idea-relations. That this was the outstanding feature of Rüdiger’s logic was clear to the historian of logic von Eberstein who, though unfavorable to Rüdigerian philosophy, in 1794 stated that Rüdiger had been the ﬁrst to determine “syllogistic ﬁgures according to the relations of concepts and not according to the position of the middle term” (von Eberstein [1794–1799], I, 112–113). No wonder the independent Wolﬃan Reusch was attracted to this approach to syllogistic so as to prefer it to Wolﬀ’s idolatry for the ﬁrst ﬁgure and to Leibniz’s combinatory of moods in the De Arte Combinatoria. In Reusch, however, there is no hint of a separation between the ratiocinatio sensualis of mathematics and the nonmathematical ratiocinatio idealis advocated by Rüdiger as an argument in favor of the inventive power of his synthetic syllogisms. This is true not only of Reusch. After Wolﬀ, logic is acknowledged as the only argumentative structure used in every ﬁeld of knowledge, from mathematics to philosophy. This is why a few logicians felt entitled to take a further step: If, according to Wolﬀ, there is no longer a gap between logic and mathematics, nothing prevents one from disregarding Wolﬀ’s restriction of logic to an outdated syllogistic. These logicians felt entitled to apply mathematical tools to ideas according to the study of idea-relations made by the adversaries of Wolﬃan logic or by independent Wolﬃans.

10. Logical Calculi in the Eighteenth Century In Jena, where Reusch was professor of logic and metaphysics since 1738, his attitude toward logic was not an exception, as can be seen in the logical work of Joachim Georg Darjes (1714–1791). But the outstanding work written in this logical context is an essay that Reusch recommended in the 1741 edition of his Systema logicum to his more specialized readers. This work is the Specimen Logicae universaliter demonstratae (Segner [1740] 1990) written by the mathematician and scientist Johann Andreas Segner (1704–1777) with the explicit aim of treating syllogistic by way of a calculus (per calculum) based on the example of algebra.

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To this end, Segner builds an axiomatic system consisting of 16 deﬁnitions, 3 postulates, and 2 axioms. The deﬁnitions introduce ideas, their relations, their arrangement in a hierarchy of genera and species, and the operations for forming ideas. Segner deﬁnes idea as a mental representation of something. If the idea is simple, its contents are obscure ideas and the simple idea is confuse for us; if the idea is composite, its contents are clear ideas and the composite idea is distinct for us. Consequently, by deﬁnition, every idea contains some idea within itself. In this way, Segner can presuppose the relation of containment (viewed from an intensional perspective) as the basic relation between two ideas. But it must be clear that Segner does not identify the content of an idea with its comprehension in the sense of the Port-Royal Logic. He simply says that given two ideas A and B, A is contained (or involved) in B if, whenever B is posited, A is also posited. The notion of containment is used to deﬁne all the relations between two ideas that are relevant for the construction of a calculus. Segner designates such relations by special algebraic symbols “−”, “=”, “>”, “ B), if A does not contain B and B contains A. IV. A is inferior to B (A < B), if A contains B and B does not contain A. V. A is coordinated to B (A × B), if A does not contain B and B does not contain A. As can be seen from this list, Segner does not have a symbol for the relation of opposition, but expresses the opposition between A and B by saying that A contains −B and B contains −A. By the expression −A Segner refers to the idea that is inﬁnitely opposite to A, and deﬁnes A as inﬁnitely opposite to −A if A contains −−A, and −−A contains A. It must be stressed that Segner does not intend an idea designated by −A as a “negative” idea opposed to a “positive” one: If by A we indicate “nontriangle,” by −A we indicate “triangle.” Ideas that can be put in a hierarchy of subordination, can also be submitted to the operations of composition and abstraction, whose possibility Segner guarantees by two special postulates. A further postulate guarantees that if A is an idea, then −A is an idea. Segner ﬁnally states two axioms that express conditions satisﬁed by some of the operations: Axiom I: If A and B are opposite ideas, then there is no idea C such that C = AB. Axiom II: If A contains B, then AB = A. In this simple system (somewhat simpliﬁed here) Segner derives a number of propositions (either problems or theorems) strictly connected to his calculus. Among the most important are the following:

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1. Given an idea A, by abstracting some of its contents, ﬁnd an idea B such that B > A. 2. If, given two ideas A and B, there is an idea AB. diﬀerent from either A or B, then A × B. 3. Given a universal idea A and its coordinate idea B, AB < A and AB < B. 4. No inﬁnite idea can be inferior or identical to a ﬁnite idea. 5. The deﬁned relations among ideas are exhaustive and reciprocally exclusive: They are not reducible one to the other. A number of theorems establish how the relation (and therefore the sign) between two ideas changes if one of them is replaced by its opposite: 6. If A = B, then −A = −B. 7. If A < B, then −A > −B. 8. If A > B, then A × −B. 9. If A × B, then −A × B; if −A × B, then either −A × B or A > B. Further theorems give an exhaustive list of valid syllogisms: 10. If A = B and C = B, then C = A. 11. If A = B and C > B, then C > A. 12. If A = B and C < B, then C < A. 13. If A = B and C × B, then C × A. 14. If A > B and C < B, then C < A. 15. If A > B and C × B, then either C < A or C × A. 16. If A > B and C > B, then C is consentient with A (i.e., is not opposite). Segner also proves a theorem that singles out all invalid syllogisms. Then he pays attention to some nonsyllogistic inferences and claims that “they shine of their own light,” whether or not we can give them syllogistic form. In particular, he proves the following inferences by composition: 17. If A = B and C = D, then AC = BD. 18. If A = B and C < D, then AC < BD. 19. If A < B and C < D, then AC < BD. Segner then proves the following theorem concerning inferences by abstraction: 20. If A is consentient with B, if C has been abstracted from A, and if D has been abstracted from B (so that C > A and D > B), then C is consentient with D.

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Thanks to a number of further deﬁnitions and three further propositions, Segner applies his system to the verbal expressions of common logic (he also pays attention to singular ideas and strictly particular propositions whose subjects bear the preﬁx “only”): “All A are B” means either A = B or A < B. “No A is B” means only A < −B. “Some A is B” means either A = B or A < B or A > B or A × B. “Some A is not B” means either A < −B or A × −B. Segner’s work proves that not all attempts to construct a logical calculus in an intensional perspective were destined to failure. Segner succeeds where Bernoulli failed (see section 7) because he never assumes that every idea has a (Port-Royalist) comprehension made of necessary attributes that cannot be modiﬁed without destroying it. Therefore, given two ideas, he simply considers three possible cases: Provided that A and B are not opposite, either A contains B, and then A < B, or A is contained in B, and then A > B, or neither idea contains the other, and then A × B. Thus Segner, unlike Bernoulli (apparently unknown to him), is under no obligation to use nouns of ordinary language as signs of ideas so as to recall their unchangeable comprehension, but uses literal symbols. When he considers the expressions of idea-relations in verbal propositions, he is under no obligation to change his intensional perspective and to consider the extensions of ideas whenever a predicate is not contained in the comprehension of a subject. We can aﬃrm a predicate B of a subject A even if they are coordinate ideas, that is, ideas whose relation is, by deﬁnition, a relation at one time of consent and of noncontainment. Some interpreters have suggested that Segner is similar to Leibniz and even a “disciple of Leibniz” (Vailati 1899, 88). Actually, Segner and Leibniz diﬀer at least inasmuch as Segner’s relation of coordination—being a relation of noncontainment—does not respond to the Leibnizian predicate-in-notion criterion. But there certainly are striking similarities. In the Specimen calculi universalis and in its Addenda, Leibniz uses an algebraic notation and an intensional perspective (see section 8). Moreover, what for Leibniz is a per se true proposition of one of the forms “ab is a,” “ab is b,” or “a is a,” has a detailed treatment in Segner. According to Segner, “A is B” is an aﬃrmation that can rest on one of the following relations between A and B, all of which produce truths: AB = A, if A < B (according to Axiom I), AB < A or AB < B if A × B (no. 3 in the list of Segner’s propositions). Equally, in his list of valid syllogisms (10–16), Segner includes Leibniz’s per se valid conclusion (“if a is b and b is c, then a is c”). As to Leibniz’s principle by which the repetition of a term is irrelevant, Segner also maintains that “the idea of the subject composed with itself cannot produce a new idea”

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(Segner [1740], 149). Also Leibniz’s praeclarum theorema, “if A is B, and D is C, then AD will be BC” (see section 8) has an elaborate equivalent in Segner’s inferences by composition. Other similarities can be found if one compares Segner’s logic with the Generales Inquisitiones, notably with respect to the calculus with negative terms (on the problems posed by the use of negative or inﬁnite predicates instead of negative propositions, and on Segner’s skilful solution, see Capozzi 1990, clx–clxiv). Segner did not know these Leibnizian doctrines, but the undeniable similarities we have stressed are not due to a miracle. They have an explanation in the fact that Segner, like Leibniz, was able to unify a variety of existing doctrines in a single system, depending on his practice of mathematics and using only an intensional approach. But that means that Segner’s logical background, albeit unsupported by knowledge of Leibniz’s relevant texts, was rich enough to oﬀer him a ﬁrm ground on which to build his calculus. In this respect, the changes that took place in German logic in the ﬁrst three decades of the eighteenth century show that Segner is representative of the logic of his time and not an inexplicable exception. But as in the case of Leibniz, this does not detract from his merits: It only emphasizes them. Let us consider one of the most interesting features of his system: the ﬁve idea-relations. Segner was not the ﬁrst to consider such relations, but depended on Rüdiger and Reusch. He was well acquainted with Reusch’s logic (the Specimen is dedicated to him) and he also knew Rüdiger’s idea-relations. In an academic dissertation of 1734, discussed by one of his students under his guidance, he expressly quotes Rüdiger’s logical work on such matters (Capozzi 1990, xcix). This does not make him less original, for it is due to him that such idea-relations are proved exhaustive and exclusive and are used as part of an adequate calculus. In this respect, Segner can be compared to the later mathematician Joseph Diez Gergonne (1771–1859). In his Essai de dialectique rationelle (Gergonne 1816–17) Gergonne considers ﬁve idea-relations using the notion of containment as basic but giving it an extensional interpretation: “the more general notions are said to contain the less general ones, which inversely are said to be contained in the former; from this the notion of relative extension of two ideas originates” (Gergonne 1816–17, 192). This extensional interpretation of the notion of containment is used by Gergonne (1816–17, 200) to classify the relations between two ideas on a par with the circles of Leonhard Euler (1707–1783) (see later in this section) and to designate each of them by a symbol. Two ideas A and B, where A is the less general idea and B is the more general one, can (1) have nothing in common, so that they stand in the relation H; (2) intersect each other, so that they stand in the relation X; (3) coincide, so that they stand in the relation I; (4) be such that A is contained in B, so that they stand in the relation C; (5) be such that that B is contained in A, so that they stand in the relation C. Like Segner, Gergonne also gives a correspondence list between the standard A, E, I, O propositions and the relations of ideas expressed by them:

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Gergonne

Propositions

Segner

I C

All A are B

A=B A” between two letters means negation (see Venn [1894], 499). Ploucquet, who takes an extensional point of view, maintains that in an aﬃrmative proposition the predicate is taken particularly. He says that this kind of particularity holds “in a comprehensive

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sense [sensu comprehensivo]” (Ploucquet 1763b, 52). Thus “All men are rational” means “All men are some rational beings,” a proposition that in his symbolism can be written as “M r,” where the letters signify the initials of the subject and the predicate. The particularity “in a comprehensive sense” of the predicate does not exclude that there could be other individuals apart from those under consideration. As for negative propositions, they are usually meant as having a universal predicate, but here too, though this may seem absurd in common language, they can have a particular predicate, as is the case with “All men are not some animals,” meaning, for instance, that they are not irrational animals. In this logical framework, Ploucquet develops a doctrine of subalternation and conversion that contrary to accepted rules, allows the conversion of particular negative propositions, for “Some A is not B” correctly converts into “No B is some A.” As to the syllogistic calculus, all one has to do is to represent the premises by Ploucquet’s symbolism and check that the premises are not both negative and that they do not contain four terms. Under these conditions, one can draw the conclusion by deleting the middle term and relating the two remaining terms, taking care that they preserve the same quantity they had in the premises. In this calculus also a syllogism in the ﬁrst ﬁgure of the form “All M are P , no S are M , then no S are P ,” which is invalid according to traditional doctrines, becomes acceptable, for it can be symbolized as follows: M p, S > M , then S > p (see Menne 1969). Another author of logical calculi is Johann Heinrich Lambert (1728–1777). This famous and eclectic scientist contributed to the study of language, metaphysics and logic, in addition to important research in the ﬁelds of optics, geometry (conic sections, perspective, theory of parallel lines, writing on the latter subject one of the basic texts in the history of non-Euclidean geometries), astronomy (comets), physics, technical applications of his theoretical works, and cosmology. Lambert’s interest in logic dates back to the ﬁfties when he wrote the so-called Six essays of an art of the signs in logic [Sechs Versuche einer Zeichenkunst in der Vernunftlehre], published only after his death (Lambert [1782–1787], I). It is rather diﬃcult to explain why Lambert decided not to publish these essays at the time that he wrote them, especially as they contain the general outline of his calculus, as well as comments on the nature of deﬁnition and on the representation of relations (see Dürr 1945). According to some interpreters (Barone 1964, 88), this decision depended on the fact that as Lambert confessed, in these writings he was attracted to the idea of discovering what “was concealed in the Leibnizian characteristic and in the ars combinatoria.” While Segner did not enter into these matters and Ploucquet refused the very idea of a universal calculus, Lambert adopted Leibniz’s ideal and, given his ignorance of the latter’s relevant texts, wanted to pursue Leibniz’s end by his own means. Now, a calculus aiming at being both formal and real presupposes an alphabet of simple elements. Therefore,

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Lambert had to postpone the publication of his early technical results until his philosophical investigation could establish such an alphabet of simple and ﬁrst concepts which, not containing composition in themselves, could not contain contradictions. Meanwhile, Lambert wrote the Neues Organon (Lambert [1764]), an important and famous book in which, according to the idea of mathesis universalis, he devoted himself to searching for the basic concepts that could help insert already acquired knowledge into a rational system and promote new discoveries. The Neues Organon consists of four parts: Dianoiology, Alethiology, Semeiotics, and Phenomenology, to which we will refer beginning with Alethiology and ending with Dianoiology. Alethiology, the doctrine of truth and error, deals with elementary concepts. Lambert gives a list of the latter, including conscience, existence, unity, duration, succession, will, solidity, extension, movement, and force. In Lambert’s view, connecting simple concepts produces truths that are not subject to change, and therefore can be considered as “eternal truths.” Eternal truths provide a foundation for all a priori sciences, in particular arithmetic, geometry, and chronometry. Semeiotics studies the relation between sign and meaning, and therefore introduces both a theory of language and the project of a characteristic. Phenomenology is the doctrine of appearance. Here Lambert, in discussing certainty and its relations to truth and error, also considers the degrees of possible certainty, and the probability of cognitions of which we have no absolute certainty. Dianoiology investigates the laws of the understanding. This part of the Neues Organon contains diagrams representing concept relations in propositions and syllogisms. Lambert represents a concept—considered in extension, that is, with respect “to all the individuals in which it appears” (Lambert 1764, Dian. §174)—as a line that can either be closed or open. He then represents the relations that two concepts entertain in the four basic propositions of categorical syllogisms: All A are B ...B A

b... a

No A is B A

a B

Some A is B b

B b ...A...

Some A is not B B

b ...A...

In the diagram representing universal aﬃrmative propositions, what counts, in addition to the length of the lines, is that A is drawn under B. The diagram representing universal negative propositions is clear. As for particular aﬃrmative propositions, the diagram shows that we only know some individuals A that are B, or at least one individual A that is B. Therefore, it remains indeterminate if also all A are B, or even all B are A. In the case of particular negative propositions the diagram shows not only that A is indeterminate but also that A is neither under B nor completely beside it, as in the case of universal negative propositions. On this basis one cannot only immediately

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make all valid conversions, but also represent all syllogisms, with the advantage of dispensing with the reduction to the ﬁrst ﬁgure (see Wolters 1980, 129–166). When the Neues Organon was published diagrams were no novelty, though Lambert introduced his diagrams before the already mentioned circular diagrams by Leonhard Euler (Euler 1768–1772): All A are B

No A is B

Some A is B

Some A is not B

B A

A

B

A

B

A

B

In fact, representations of concepts, propositions, and syllogisms by means of circles, lines, and other ﬁgures had already been devised by Johann Christoph Sturm (1661), who also introduced circular diagrams representing new syllogisms having negative terms; by Johann Christian Lange (1712, 1714); by Ploucquet (1759); and by Leibniz himself (on the history of diagrams in logic see Gardner 1983; Bernhard 2001). From what we have seen of Lambert’s and Ploucquet’s logical work, we can understand why their contemporaries were intrigued by their diﬀerent approaches to the problem of a logical calculus and wanted to assess their comparative merits. In a public debate in which Lambert and Ploucquet took part directly—reported in Bök ([1766])—Ploucquet’s Methodus calculandi was compared with Segner’s logical work, while Lambert’s diagrams in the Neues Organon underwent severe criticism. On the occasion of this debate Lambert began a correspondence with Georg Jonathan Holland (1742–1784), a pupil of Ploucquet. In a letter to Holland, Lambert criticized Ploucquet’s use of the traditional rule that nothing follows from two negative premises to exclude nonconclusive syllogism. He also criticized Ploucquet for using letters standing for the initials of substantives in syllogisms, thus showing his lack of a true symbolism (Lambert [1782] 1968, 95–96). But in an article of 1765, Lambert, though claiming that his diagrams in the Neues Organon were an example of a characteristic, acknowledged that they were only a little thing with respect to his project of a general logical calculus (Bök [1766], 153). At last, Lambert’s logical calculus was published in his Disquisitio (Lambert [1765], dated 1765 but actually printed in 1767). Here he states the aim of his calculus and lists the requisites any calculus must satisfy. As to his aim, Lambert says that he wants to ﬁnd a method for treating qualities similar to the method used in algebra for treating quantities. Just as in algebra we employ the ideas of relation, equality, proportion, and so on, so in the logical calculus we have to employ the ideas of identity, identiﬁcation, and analogy. As to the requisites a calculus must satisfy, they are the following. (1) For every operation we introduce, there must be the inverse operation, in full analogy with algebra where, when two quantities are added, it is always possible to obtain either by subtracting it from the total. (2) Given the object,

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the relations and the operations of the logical calculus, an adequate symbolism must be found. The symbols must be a perfect replacement for the things they symbolize to be safely used in their place. This means that we need a characteristic that mirrors things, a real characteristic, in which simple signs stand for simple things and are capable, once composed, to stand for composed things, so that it is also possible to proceed inversely from a composite to its simple elements. (3) Lambert also requires that we have a clear knowledge of the simple elements and the basic relations of the calculus; we must therefore know the combinatorial part of the ars characteristica combinatoria. In his opinion, the simple elements are qualities, that is, the special aﬀections of things we can consider as their attributes. Qualities are simple elements because, according to established ontological doctrines, they can be considered as “absolute” attributes, whereas other attributes, notably quantity, must be thought only “relatively” (a similar conception occurs in Leibniz’s De Arte Combinatoria). Lambert’s calculus in the Disquisitio, as it was the case with his unpublished essays of the ﬁfties, is intensional, that is, “does not concern individuals but properties” (letter to Holland 21.4.1765, Lambert [1782] 1968, 37). After trying the extensional perspective in the Neues Organon, his return to his early preference for the intension of concepts is undoubtedly due to a conscious choice. For Lambert wants to ﬁnd what is “simplest” and “ﬁrst” in concepts, but to obtain what is simpler, it is necessary to consider what is more complex, and in the case of concepts, the more complex concepts are those containing the simpler ones. Therefore, it is necessary to consider concepts as properties, as concepts containing other concepts, thus disregarding the class of individuals to which they extend. When dealing with judgments and syllogisms, Lambert’s ﬁrst aim is to establish the identity of the subject and predicate of judgments. Therefore, given the judgment “All A are B,” where A and B are not already obviously identical, Lambert establishes their identity by considering the subject as containing the predicate plus other properties. Hence the following symbolism (Lambert [1765], 461–462): All A are B

No A is B

Some A is B

Some A is not B

A = nB

A:p = B:q

mA = nB

mA:p = B:q

In the universal aﬃrmative, A = nB, n stands for the qualities which can be found in the subject A but not in the predicate B. In the universal negative, A:p = B:q, the sign “:” stands for a logical division and expresses which qualities, p and q, must be subtracted from the subject and the predicate, because neither belongs both to the subject and the predicate, so as to obtain A = B. Similarly, in the particular aﬃrmative, mA = nB, and in the particular negative, mA:p = B:q. On this basis, Lambert obtains a general formula expressing any kind of judgment: A/p = nB/q (here Lambert substitutes the

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sign of fraction for the sign “:”). From this formula one can easily derive a formula expressing any kind of syllogism: mA/p = nB/q μA/π = νC/ρ . mνC/pρ = μnB/πq To give an example of how this general formula applies to particular syllogisms, a syllogism Barbara, whose premises are B = mA and C = B, has the conclusion C = mA, whereas a syllogism Celarent, whose premises are B/q = A/p and C = νB, has the conclusion C/q = νA/p (Lambert [1782–1787], I, 102–103, 107). Despite the fact that the Disquisitio’s treatment of syllogism is very diﬀerent from that of the Neues Organon, it was disappointing for Lambert’s most competent readers. Holland sent Lambert a letter in which (beside mentioning his own tentative calculus) he observed that, however good Lambert’s calculus was, it did not achieve the declared aim to ﬁnd symbols mirroring reality. What are A, B, m, n, symbols of? Above all, which are the primitives that they are supposed to be symbols of? A deﬁnite answer, Holland concluded, could perhaps be expected from a new work Lambert had announced (Lambert [1782] 1968, 259–266). The work Holland referred to, titled Architectonic (Lambert [1771]), was published a few years later. In this treatise, which promised to give a theory of what is simple and ﬁrst in philosophical and mathematical knowledge, the author collects the results of his philosophical research going back to the mid-forties. But for all its importance as the summa of Lambert’s thought, the Architectonic provides no formal treatment, nor gives a new and complete list of simple elements that could be used as basic elements of a real characteristic, because it contains the same elements already listed in the Neues Organon. The conclusion to be drawn is that Lambert’s project shared Leibniz’s ambitions and in this respect went far beyond Segner’s and Ploucquet’s calculi, but perhaps was too ambitious and, though providing interesting details in the application of algebra to logic, can be said to be unachieved. In a sense, Lambert admitted as much in a letter (14.3.1771) to Johann Heinrich Tönnies: “should the universal characteristic belong to the same class as the philosopher’s stone or the squaring of the circle, it can at least, just as these, induce other discoveries” (Lambert [1782] 1968, 411). Ploucquet’s refusal of Leibniz’s project and Lambert’s somber admission to Tönnies may sound too pessimistic if one considers how much they and other eighteenth-century logicians—not to mention Leibniz—had progressed since Bernoulli’s failed parallelism. But especially Lambert’s assessment of universal characteristic as something similar to the squaring of the circle makes it clear that these authors believed that the construction of a satisfactory logical calculus was hindered by a possibly insurmountable obstacle: the overpowering amount of philosophical analysis to be done in the ﬁelds of metaphysics,

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semiotics, and natural language to reach a suitable alphabet of thoughts. As a matter of fact, unknown to these eighteenth-century logicians, there was an obstacle, not only on the side of philosophical analysis but also on the side of mathematics. Nineteenth-century logicians will ﬁnd out that one had to reﬂect also on the nature of mathematics and algebra, especially on their apparently exclusive link with quantity, before an algebra of logic could come to life.

11. Kant Interest for logical calculi seems to vanish at the end of the eighteenth century. We have mentioned some of the reasons behind this phenomenon, but according to a still widely received opinion this was due to the inﬂuence exerted on logic by Immanuel Kant (1724–1804). This opinion is usually justiﬁed by saying that Kant introduced confusion in logic through his notion of transcendental logic. As a matter of fact, Kant had a deﬁnite concept of logic, related to his transcendental philosophy but not to be confused with it. To evaluate Kant’s concept of logic, one must take into account his 40 years-long activity as a logic teacher, using as a textbook Georg Friedrich Meier’s Auszug aus der Vernunftlehre (Meier 1752b), a short version of the latter’s Vernunftlehre (Meier 1752a) (on Meier’s logic see Pozzo 2000). We have several texts related to this teaching activity, which constitute the socalled Kantian logic-corpus. Apart from the programs of the courses, such texts are (1) Kant’s handwritten annotations on Meier’s Auszug (the so-called logical Reﬂexionen, in Kant 1900, XVI), (2) lecture notes taken by students (Kant 1900–, XXIV; Kant 1998a, 1998b), and (3) I. Kant’s Logik, a book published in 1800 by Gottlob Benjamin Jäsche by collecting a selection of Kant’s annotations on Meier’s Auszug with Kant’s consent (Kant 1900–, IX, 1–150). These texts must be used with care and must always be compared with Kant’s published production. Nonetheless, they are essential to assess his views on logic, allowing a deeper insight into the importance of logic for Kant’s philosophy, and testifying to his knowledge of the discipline he was due to teach. As it is impossible to give details here of Kant’s treatment of logical doctrines, we will only discuss his general concept of logic.2 A comparative study of the Kantian logic-corpus shows that Kant’s concept of logic is the result of a sustained eﬀort of reﬂection lasting several years. He began as an almost orthodox Wolﬃan, founding logic on empirical psychology and ontology (Logik Blomberg, Kant 1900–, XXIV, 28). In his mature conception, however, he took the opposite view and denied that logic could be founded on either empirical psychology or ontology. To this eﬀect Kant argues that a logic founded on empirical psychology could describe human logical behavior but not prescribe laws to it. In his opinion, logical rules do not mirror what we actually do when we think, but are the standard to which our thoughts must conform if they are to have a logical form: Logic considers “not how we do think, but how we ought to think”

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(I. Kant’s Logik, Kant 1900–, IX, 14). As to the formerly accepted foundation of logic on ontology, Kant simply suppresses it, to the dismay of many of his contemporaries and later idealist philosophers. In particular, opposing Kant, Hegel proposed a new logic identical to metaphysics which, like old metaphysics, would admit that “thought (with its immanent determinations) and the true nature of things are one and the same content” (Barone 1964, 202). Thus it is rather surprising that William and Martha Kneale claim that it was Kant “with his transcendentalism who began the production of the curious mixture of metaphysics and epistemology which was presented as logic by Hegel and other Idealists of the nineteenth century” (Kneale and Kneale 1962, 355). The independence of logic from ontology and empirical psychology raises the problem of the origin and justiﬁcation of logic. Kant gives an indirect answer to the problem of the origin of logic by way of a comparison of logic with grammar. Logic and grammar—he maintains—are similar in as much as we learn to think and to speak without previous knowledge of grammatical and logical rules, and only at a later stage we become conscious of having implicitly used them. Nonetheless logic and grammar diﬀer because, as soon as we become aware of grammatical rules, we easily see that they are empirical, contingent, and subject to variations. On the contrary, once we become conscious of the logical structure of our thought, we cannot fail to appreciate that without that structure we could not have been thinking at all. Therefore logic precedes and regulates any rational thinking and is necessary in the sense that we cannot consider it contingent and variable. Kant concludes that logic “is abstracted [abstrahirt] from empirical use, but is not derived [derivirt]” from it (Reﬂexion 1612, Kant 1900–, XVI, 36) so that it can be considered a scientia scientiﬁca, whereas grammar is only a scientia empirica (Logik Busolt, Kant 1900–, XXIV, 609). This is important because the logic considered by Kant is not a natural logic that could be investigated by psychology, but is an “artiﬁcial logic.” This being the origin of logic, its justiﬁcation can be reduced to the fact that, according to Kant, logical principles such as the law of contradiction are accepted without proof: “All rules that are logically provable in general are in need of a ground [Grund] from which they are derived. Many propositions (e.g. that of contradiction) cannot be proved at all, neither a priori nor empirically” (Logik Dohna-Wundlacken, Kant 1900–, XXIV, 694). In other words, logical rules, “once known, are clear by themselves” (Reﬂexion 1602, Kant 1900–, XVI, 32). This means that logic not only is necessary, scientiﬁc, and a priori, but also is capable of justifying itself. These features make logic one of the means Kant uses in carrying through his philosophical project of explaining the possibility of experience according to his Copernican revolution. An important part of this project consists in showing that it is possible to ﬁnd all the general forms of thought—categories—without having to fall back on metaphysics or experience. Now logic (rather, one of its most important parts, i.e., the functions of judgment), which Kant has made no longer dependent on empirical

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psychology and ontology, qualiﬁes as the perfect clue to the categories. But since categories have to be completely enumerated to be employed in a complete list of principles of the understanding, which deﬁne the ﬁeld of possible experience, logic has to satisfy a further requisite: It has to be complete. Hence Kant’s well-known statement that logic “seems to all appearance to be ﬁnished and complete” (Kant 1997, B viii). Kant has been criticized for this statement, and in our opinion he lacks conclusive arguments to support it. But one must consider that a proof of the completeness of logic would have been easy if Kant had preserved the foundation of logic on empirical psychology and ontology, both ultimately guaranteed by God. It is also fair to point out that Kant envisages the possibility, for a closed system, of growing “from within,” on a par with living organisms that grow with no addition of new parts (Kant 1997, A 832/B 860). Applying this to logic, one could say that some growth in logic is possible, although within the boundaries of a systematic structure. The scientiﬁc, necessary, and self-justifying nature of logic guarantees that it has great autonomy and the maximum spectrum of application. Such prerogatives are counterbalanced by precise limitations: “Nobody can dare to judge of objects and to assert anything about them merely with logic without having drawn on antecedently well-founded information about them from outside logic” (Kant 1997, A 60/B 85). Consequently, logic is the supreme canon of truth with respect to the formal correctness of thought, but must be indiﬀerent to its contents. In this way Kant makes his concept of logic more deﬁnite. Logic, having no speciﬁc subject matter, is general. Having nothing to do with human psychology, it is pure. Concerning only the form of thought, it is merely formal. The ﬁrst consequence of this conception is that logic has to be analytic, although not in the sense that it deals with analytic judgments only. For logic is not concerned with the analytic/synthetic distinction which is left to transcendental logic: “The explanation of the possibility of synthetic judgments is a problem with which general logic has nothing to do, indeed whose name it need not even know” (Kant 1997, A 154/B 193). Logic is analytic in two senses. (1) “General logic analyzes the entire formal business of the understanding and reason into its elements, and presents these as principles of all logical assessment of our cognition” (Kant 1997, A 60/B 84). (2) Logic is analytic inasmuch as it has nothing to do with dialectic, both intended as the rhetorical art of disputation and as the part of logic dealing with probability. The most evident and better known reason for Kant’s separation of logic from dialectic is the connection between dialectic and rhetoric. A rhetorical dialectic is an art for deceiving adversaries in a dispute and for gaining consent not only disregarding truth but also purporting to produce the semblance [Schein] or illusion of truth. Kant condemns this kind of “logic” as unworthy of a philosopher (see I. Kant’s Logik, Kant 1900–, IX, 17). As to the association of dialectic with probability, it goes back to the distinction made by Aristotelian logical treatises between analytic, that is, the part of logic dealing with truth and certainty, and dialectic, that is, the part of logic dealing with

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what is probable, according to Boethius’s translation of the Greek éndoxos with the Latin probabile. This distinction was adopted by many eighteenthcentury logicians, notably by Meier who divides logic into analytica or “logic of completely certain erudite cognition,” and dialectica or logica probabilium, deﬁned as “logic of probable erudite cognition” (Meier 1752b, §6, in Kant 1900–, XVI, 72). Like many philosophers (including Leibniz), till the early seventies Kant hoped that a general logic of the uncertain could be found. Such a logic, although diﬀerent from the Aristotelian and humanist doctrines of probability and attentive to the late seventeenth-century results in the ﬁeld of probability calculus, was intended to be capable of also dealing with qualitative matters concerning justice, politics, and so on. Later on, Kant completely changed his view. He considered probability as a measurable degree of certainty—in this he agreed with Wolﬀ—which “can be expressed like a fraction, where the denominator is the number of all possible cases, the numerator is the number of actual cases” (Logik Pölitz, Kant 1900–, XXIV, 507). This view restricts probability (Wahrscheinlichkeit, probabilitas) to matters that can be subjected to a numerical calculus (games of chance and statistically based events such as mortality indexes), and excludes the possibility of an instrumental art for weighing, rather than numbering, heterogeneous reasons pro and contra a given qualitative question. Against this alleged art Kant objects that it concerns the notion of “verisimilitude” (Scheinbarkeit, verisimilitudo) rather than probability. In his view, if such an art, under the name of dialectic, belonged to logic, the latter would no longer be a canon of truth but would become an instrument for producing an illusion of truth by assigning an alleged probability—in fact a mere verisimilitude—even to questions that are beyond possible experience, such as the existence of the soul. Hence Kant’s claim that only probability restricted to matters that can be subjected to a numerical calculus is worthy of this name and, because it is contiguous to truth and certainty, belongs to the analytic part of logic and need not be dealt with in a special part of logic called dialectic (Kant 1900–, A 293/B 349). Kant’s position is drastic: Logic and dialectic must part and go separate ways. The second consequence of Kant’s view that logic is a mere formal canon of truth is that the content of logic must be limited to the doctrine of elements: concepts, judgments, and inferences. Therefore, logic must not trespass into the domains of anthropology and psychology, nor give advice for the use of logic in the ﬁelds of the natural sciences or of practical life. This means that Kant breaks away from one of the main trends of European logic, which had tried to give new life to the discipline by stressing its usefulness either as a guide for judging, or as a kind of methodology for empirical research, or as a medicine against errors, or as an epistemological exercise. In particular, Kant breaks away from Locke’s view of logic, despite the fact that he had formerly praised it, for he maintains that the study of the origin of concepts “does not belong to logic, but rather to metaphysics” (Logik Dohna-Wundlacken, Kant 1900–, XXIV, 701).

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The fact that Kant separates logic from epistemology does not mean that the texts of the Kantian logic-corpus do not contain epistemological parts. On the contrary, these texts make very interesting reading on matters such as opinion, belief, knowledge, hypotheses, probability, and so on. But these matters are no longer intended as belonging to pure logic because, to deal with them, one must take into account the content of knowledge and the human cognitive constitution, including sensibility, or at least the form of sensibility, as well as practical aspects of human action, such as the interest we have for accepting something as true. If Kant had written a logic handbook himself, he probably would have treated such matters at length, in addition to other interesting questions, such as the doctrine of logical essence, in a doctrine of method. The third consequence of Kant’s view of logic is that it is only a canon for checking the correctness of our thoughts but is incapable of invention. Kant’s sharp distinction between logic and mathematics contributes to this view. He agrees with Wolﬀ that there is a single logic to be complied with by mathematicians and nonmathematicians alike, but logic is insuﬃcient to explain why mathematics is ampliative. According to Kant, mathematics is the science that constructs a priori its concepts, that is, exhibits a priori the intuitions corresponding to them. Thus, mathematics relies also on the form of sensible intuition, so that it has content and can be inventive with respect to it. This applies not only to arithmetic and geometry but also to algebra, which is inventive because it refers (albeit mediately) to a priori intuitions. Therefore, Kant rejects the view of those who “believe that logic is a heuristic (art of discovery) that is an organ of new knowledge, with which one makes new discoveries, thus e.g. algebra is heuristic; but logic cannot be a heuristic, since it abstracts from any content of knowledge” (Logik Hechsel, Kant 1998b, 279 = ms. 9). These statements are not borne out of ignorance. Kant knew the outlines of Leibniz’s ars characteristica combinatoria, on whose utopian nature he commented in an essay of 1755 (Nova dilucidatio, Kant, 1900–, I, 390) in terms that seem to anticipate analogous statements by Ploucquet and Lambert. Moreover, his logic-corpus, as well as his works and correspondence, provide evidence that (1) he was well acquainted with the combinatorial calculus of syllogistic moods; (2) he used Euler’s (whom he quotes) circular diagrams to designate concepts, judgments, and syllogisms; (3) he knew the linear diagrams of Lambert, with whom he corresponded; (4) he probably had some knowledge of Segner’s and Ploucquet’s works; and (5) he actively promoted the diﬀusion of Lambert’s posthumous works containing the latter’s algebraic calculus. But all this did not shake his conviction that an algebraic symbolism of idea-relations and the use of letters instead of words are not by themselves a means to invention. If we consider the development of logic from humanism onward, we see that one of the basic motivations of logical research in the whole period was the demand to make logic inventive. The humanist theories of inventio, Bacon’s studies on induction, Descartes’s theory of problem solving, the ars inveniendi and a large part of Leibniz’s ars characteristica combinatoria, and so on,

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can be viewed in this perspective. Kant objected that this kind of research, while claiming to be purely formal, was meant to deal with the content of knowledge. The condition (and cost) of his objection to an inventive logic was the separation of logic from mathematics, but in this way he achieved his aim of separating pure logic from metaphysics and psychology, as well as from any transcendent foundation. This aspect of Kant’s concept of logic reappears in the philosophy of logic of some later logicians. Thus, despite substantial diﬀerences, Frege’s concept of logic seems indebted to Kant’s in several respects, such as the idea that the only logic that really counts is scientiﬁc logic, rather than some natural logic; the contention that a scientiﬁc or artiﬁcial logic provides necessary and universal rules; the condemnation of any intrusion of psychology into logic by the argument that logic is normative on a par with moral laws; the idea that logic is used for justifying knowledge rather than for acquiring new knowledge. But even Venn, who claims that Kant had “a disastrous eﬀect on logical method” (Venn [1894], xxxv) begins his own system of logic by stating: “Psychological questions need not concern us here; and still less those which are Metaphysical” (Venn [1894], xxxix). Perhaps it would have been more diﬃcult for him to make such a statement if Kant had not already made that very same statement.

Notes 1. Reference to secondary literature devoted to Leibniz’s notion of complete concept could span over many pages. We will limit ourselves to the seminal papers by Mondadori (1973) and Fitch (1979), to the discussion included in Mates (1986), and—for two recent accounts based on diﬀerent interpretations—to Zalta (2000) and Lenzen (2003). Mondadori’s and Fitch’s papers are included, together with other relevant contributions, in Woolhouse (1993). 2. The body of literature on Kant is enormous, and also literature on Kantian logic is very extensive, ranging from the relation between general and transcendental logic to the doctrines of concepts, judgments, and inferences, not to mention topics such as the relations between logic and language and mathematics. We will mention only Shamoon (1981), Capozzi (1987), Pozzo (1989), Brandt (1991), Reich (1992), Wolﬀ (1995), Capozzi (2002), and Capozzi (forthcoming) (the latter containing an extensive bibliography). A precious research tool is provided by an impressive Kantian lexicon, still in progress, many of whose volumes are devoted to Kant’s logic-corpus (Hinske 1986–).

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Camporeale, Salvatore. 1986. Lorenzo Valla, “Repastinatio liber primus”: retorica e linguaggio. In Lorenzo Valla e l’umanesimo italiano, eds. O. Besomi and M. Regoliosi, 217–239. Padova: Antenore. Capozzi, Mirella. 1982. Sillogismi e proposizioni singolari: due aspetti della critica di Wolﬀ a Leibniz. In La grammatica del pensiero: Logica, linguaggio e conoscenza nell’età dell’Illuminismo, eds. D. Buzzetti and M. Ferriani, 103–150. Bologna: Il Mulino. Capozzi, Mirella. 1987. Kant on Logic, Language and Thought. In Speculative Grammar, Universal Grammar and Philosophical Analysis of Language, eds. D. Buzzetti and M. Ferriani, 97–147. Amsterdam–Philadelphia: Benjamins. Capozzi, Mirella. 1990. Editor’s introduction to Segner [1740], xi–clxxii. Bologna: CLUEB. Capozzi, Mirella. 1994. Algebra e logica in Jakob Bernoulli. Atti del Congresso “Logica e ﬁlosoﬁa della scienza: problemi e prospettive” (Lucca 7–10 gennaio 1993), 55–74. Pisa: ETS. Capozzi, Mirella. 2002. Kant e la logica, vol. I. Napoli: Bibliopolis. Capozzi, Mirella. Forthcoming. Kant e la logica, vol. II. Napoli: Bibliopolis. Carboncini, Sonia. 1991. Transzendentale Wahrheit und Traum. Christian Wolﬀs Antwort auf die Herausforderung durch den Cartesianischen Zweifel. Stuttgart-Bad Cannstatt: Frommann-Holzboog. Cassirer, Ernst. 1922. Das Erkenntnisproblem in der Philosophie und Wissenschaft der neuern Zeit, 3rd ed., vol. II. Berlin: B. Cassirer. Chomsky, Noam. 1966. Cartesian Linguistics. A Chapter in the History of Rationalist Thought. New York–London: Harper and Row. Couturat, Louis. 1901. La Logique de Leibnitz d’après des documents inédits. Paris: Alcan. Daston, Lorraine. 1988. Classical Probability in the Enlightenment. Princeton: Princeton University Press. De Angelis, Enrico. 1964. Il metodo geometrico nella ﬁlosoﬁa del Seicento. Pisa: Istituto di Filosoﬁa. Di Liso, Saverio. 2000. Domingo de Soto. Dalla logica alla scienza. Bari: Levante. Dominicy, Marc. 1984. La naissance de la grammaire moderne. Language, logique et philosophie à Port-Royal. Liège: Mardaga. D’Ors, Angel. 1981. Las “Summulae” de Domingo de Soto. Pamplona: Universidad de Navarra (tesi doctoral). Doyle, John P. 1987–1988. Suarez on Beings of Reason and Truth. Vivarium 25, 47–75 and 26, 51–72. Doyle, John P. 1995. Another God, Chimerae, Goat-Staggs, and Man-Lions: a Seventeenth-Century Debate about Impossible Objects. Review of Metaphysics 48, 771–808. Doyle, John P. 2001. The Conimbricenses: Some Questions on Signs. Milwaukee: Marquette University Press. Dürr, Karl. 1945. Die Logistik Johann Heinrich Lamberts. In Festschrift zum 60. Geburtstag von Prof. Dr. Andreas Speiser, 47–65. Zürich: Orell Füßli. Engfer, Hans-Jürgen. 1982. Philosophie als Analysis. Studien zur Entwicklung philosophischer Analysiskonzeptionen unter dem Einﬂuss mathematischer Methodenmodelle im 17. und fruhen 18. Jahrhunderts. Stuttgart-Bad Cannstatt: Fromman-Holzboog.

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Feingold, Mordechai, Joseph S. Freedman, and Wolfgang Rother, eds. 2001. The Inﬂuence of Petrus Ramus. Basel: Schwabe. Fitch, Gregory. 1979. Analyticity and Necessity in Leibniz. Journal of the History of Philosophy, 17/1, 29–42. Reprint in Woolhouse (1993). Friedman, Russell L., and Lauge O. Nielsen. 2003. The Medieval Heritage in Early Modern Metaphysics and Modal Theory, 1400–1700. Dordrecht: Kluwer. Gardner, Martin. 1983. Logic Machines and Diagrams, 2nd ed. Brighton: Harvester Press. Garin, Eugenio. 1960. La cultura ﬁorentina nella seconda metà del ‘300 e i ‘barbari britanni’. In Rassegna della letteratura italiana, s. VII, 64, 181–195. Gaukroger, Stephen. 1989. Cartesian Logic. An Essay on Descartes’s Conception of Inference. Oxford: Clarendon Press. Giard, Luce. 1985. La production logique de l’Angleterre au XVIe siècle. Les Ètudes Philosophiques 3, 303–324. Gilbert, Neal W. 1960. Renaissance Concepts of Method. New York: Columbia University Press. Gilson, Étienne. 1913. Index scolastico-cartésien. Paris: Felix Alcan. Green-Pedersen, Niels Jorgen. 1984. The Tradition of the Topics in the Middle Ages: The Commentaries on Aristotle’s and Boethius’ Topics. München–Wien: Philosophia Verlag. Hacking, Ian. 1975a. Why Does Language Matter to Philosophy?. Cambridge: Cambridge University Press. Hacking, Ian. 1975b. The Emergence of Probability. Cambridge: Cambridge University Press. Hamilton, William. 1860–1869. Lectures on Metaphysics and Logic, 2nd ed., vol. II, appendix, eds. H. L. Mansel and J. Veitch. Edinburgh–London: Blackwood. Harnack, Adolf von. [1900] 1970. Geschichte der Königliche Preussischen Akademie der Wissenschaften zu Berlin, 3 vols. Berlin. Reprint Hildesheim: Olms. Hatﬁeld, Gary. 1997. The Workings of the Intellect: Mind and Psychology. In Logic and the Workings of the Mind: The Logic of Ideas and Faculty Psychology in Early Modern Philosophy, ed. Patricia A. Easton, 21–45. Atascadero, Calif.: Ridgeview. Hickman, Larry. 1980. Modern Theories of Higher Level Predicates. Second Intentions in the Neuzeit. Munchen: Philosophia. Hinske, Norbert. 1986–. Kant-Index. Stuttgart-Bad Cannstatt: Frommann-Holzboog. Howell, Wilbur Samuel. 1971. Eighteenth Century British Logic and Rhetoric. Princeton, N.J.: Princeton University Press. Ishiguro, Hidé. 1991. Leibniz’s Philosophy of Logic and Language. Cambridge: Cambridge University Press (2nd ed., 1st ed. 1972). Jardine, Lisa. 1982. Humanism and the Teaching of Logic. In N. Kretzmann, A. Kenny, and J. Pinborg (1982), 797–807. Jardine, Lisa. 1988. Humanistic Logic. In C. B. Schmitt, Q. Skinner, and E. Kessler (eds.) 1988, 173–198. Kneale, William, and Martha Kneale. [1962] 1975. The Development of Logic. Oxford: Clarendon Press. Kretzmann, Norman, Anthony Kenny, and Jan Pinborg, eds. 1982. The Cambridge History of Later Medieval Philosophy. Cambridge: Cambridge University Press. Lenzen, Wolfgang. 1986. “Non est” non est “est non”—Zu Leibnizens Theorie der Negation. Studia Leibnitiana 18, 1–37. Lenzen, Wolfgang. 1990. Das System der Leibnizschen Logik. Berlin: de Gruyter.

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Lenzen, Wolfgang. 2000. Guilielmi Pacidii Non plus ultra—oder: Eine Rekonstruktion des Leibnizschen Plus-Minus-Kalküls. Philosophiegeschichte und logische Analyse 3, 71–118. Lenzen, Wolfgang. 2003. Grundfragen des logischen Kalküls—Eine Art Rezension von F. Schupp (Hrg.), G. W. Leibniz, Die Grundlagen des logischen Kalküls. History and Philosophy of Logic 24, 141–162. Lenzen, Wolfgang. 2004. Calculus Universalis. Studien zur Logik von G. W. Leibniz. Paderborn: Mentis. Mack, Peter. 1993. Renaissance Argument. Valla and Agricola in the Traditions of Rhetoric and Dialectic. Leiden–New York–Köln: Brill. Mahoney, Michael S. 1980. The Beginnings of Algebraic Thought in the Seventeenth Century. In Descartes. Philosophy, Mathematics and Physics, ed. Stephen Gaukroger, 141–155. Brighton: Harvester Press. Maierù, Alfonso. 1993. La dialettica. In Lo spazio letterario del medioevo latino. 1—La produzione del testo II, eds. G. Cavallo, C. Leonardi, E. Menestò, 273–294. Rome: Salerno. Mates, Benson. 1986. The Philosophy of Leibniz: Metaphysics and Language. Oxford: Oxford University Press. Menne, Albert. 1969. Zur Logik von Gottfried Ploucquet. Akten des XIV Internationalen Kongresses für Philosophie, 45–49. Wien: Herder. Mondadori, Fabrizio. 1973. Reference, essentialism, and modality in Leibniz’s metaphysics. Studia Leibnitiana V, 74–101 (reprint in Woolhouse 1993). Mugnai, Massimo. 1992. Leibniz’ Theory of Relations. Studia Leibnitiana Supplementa XXVIII. Stuttgart: Steiner. Muñoz Delgado, Vicente. 1970. La obra lógica de los españoles en París (1500–1525). Estudios 26, 209–280. Murdoch, John. 1974. Philosophy and the Enterprise of Science in the Later Middle Ages. In The Interaction between Science and Philosophy, ed. Y. Elkana, 51–74. Atlantic Highlands, N.J.: Humanities Press. Nuchelmans, Gabriel. 1980. Late-Scholastic and Humanist Theories of the Proposition, Amsterdam–Oxford–New York: North-Holland. Nuchelmans, Gabriel. 1983. Judgment and Proposition. From Descartes to Kant. Amsterdam–Oxford–New York: North-Holland. Nuchelmans, Gabriel. 1991. Dilemmatic Arguments: Towards a History of Their Logic and Rhetoric. Amsterdam: North Holland. Ong, Walter J. 1958. Ramus, Method and the Decay of Dialogue: From the Art of Discourse to the Art of Reason. Cambridge, Mass.: Harvard University Press. Pariente, Jean-Claude. 1985. L’analyse du langage à Port-Royal. Six études logicogrammaticales. Paris: Minuit. Pérez-Ilzarbe, Paloma. 1999. El signiﬁcado de las proposiciones. Jerónimo Pardo y las teorías medievales de la proposición. Pamplona: Eunsa. Perreiah, Alan R. 1982. Humanist Critiques of Scholastic Dialectic. Sixteenth Century Journal 13, 3–22. Pozzo, Riccardo. 1989. Kant und das Problem einer Einleitung in die Logik. Ein Beitrag zur Rekonstruktion der historischen Hintergründe von Kants Logik-Kolleg. Frankfurt–Bern–New York–Paris: Peter Lang. Pozzo, Riccardo. 2000. Georg Friedrich Meiers “Vernunftlehre.” Eine historischsystematische Untersuchung. Stuttgart-Bad Cannstatt: Frommann-Holzboog.

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Reich, Klaus. 1992. The Completeness of Kant’s Table of Judgments, trans. J. Kneller and M. Losonsky of Die Vollständigkeit der kantischen Urteilstafel, Berlin (2nd ed.) 1948 Hamburg 1986 (3rd ed.). Stanford, Calif.: Stanford University Press. Risse, Wilhelm. 1964–70. Die Logik der Neuzeit, 2 vols. Stuttgart-Bad Cannstatt: Frommann-Holzboog. Risse, Wilhelm. 1965. Bibliographia Logica: Verzeichnis der Druckschriften zur Logik mit Angabe ihrer Fundorte, vol. I: 1472–1800. Hildesheim: Olms. Roncaglia, Gino. 1988. Modality in Leibniz’ Essays on Logical Calculus of April 1679. Studia Leibnitiana 20/1, 43–62. Roncaglia, Gino. 1996. Palaestra rationis. Discussioni su natura della copula e modalità nella ﬁlosoﬁa ‘scolastica’ tedesca del XVII secolo. Firenze: Olschki. Roncaglia, Gino. 1998. Sull’evoluzione della logica di Melantone. Medioevo 24, 235– 265. Roncaglia, Gino. 2003. Modal Logic in Germany at the Beginning of the Seventeenth Century: Christoph Scheibler’s Opus Logicum. In R. L. Friedman and L. O. Nielsen 2003, 253–308. Russell, Bertrand. 1900. A Critical Exposition of the Philosophy of Leibniz. Cambridge: Cambridge University Press. Schepers, Heinrich. 1959. Andreas Rüdigers Methodologie und ihre Voraussetzungen, (Kant-Studien, Ergänzungsheft 78). Köln: Kölner Universitäts-Verlag. Schmitt, Charles B., Quentin Skinner, and Eckhard Kessler, eds. 1988. The Cambridge History of Renaissance Philosophy. Cambridge: Cambridge University Press. Schupp, Franz. 2000. Introduction, trans. and commentary to G. W. Leibniz. Die Grundlagen des logischen Kalküls. Hamburg: Meiner. Shamoon, Alan. 1981. Kant’s Logic, Ph.D. dissertation, Columbia University (1979). Ann Arbor: University Microﬁlms International. Swoyer, Chris. 1995. Leibniz on Intension and Extension. Nous 29/1, 96–114. Thom, Paul. 1981. The Syllogism. München: Philosophia Verlag. Tonelli, Giorgio. 1959. Der Streit über die mathematische Methode in der Philosophie in der ersten Hälfte des 18. Jahrhunderts und die Entstehung von Kants Schrift über die “Deutlichkeit.” Archiv für Begriﬀsgeschichte 9, 37–66. Vailati, Giovanni. 1899. La logique mathématique et sa nouvelle phase de développement dans les écrits de M. J. Peano. Revue de Métaphysique et de Morale 7, 86–102. Vasoli, Cesare. 1968. La dialettica e la retorica dell’umanesimo. Milano: Feltrinelli. Vasoli, Cesare. 1974, Intorno al Petrarca e ai logici “moderni.” Miscellanea Medievalia IX, Antiqui und Moderni, Berlin: De Gruyter, 142–154. Venn, John. [1894] 1971. Symbolic Logic. Reprint of the 2nd ed. New York: Chelsea Publ. Co. Waswo, Richard. 1999. Theories of Language. In The Cambridge History of Literary Criticism III—The Renaissance, ed. G. P. Norton, 25–35. Cambridge: Cambridge University Press. Wolﬀ, Michael. 1995. Die Vollständigkeit der kantischen Urteilstafel. Mit einem Essay über Freges Begriﬀsschrift. Frankfurt: Klostermann. Wollgast, Siegfried. 1988a. Ehrenfried Walther von Tschirnhaus und die deutsche Frühaufklärung. Berlin: Akademie-Verlag. Wollgast, Siegfried. 1988b (2nd ed.). Philosophie in Deutschland zwischen Reformation und Aufklärung 1550–1650. Berlin: Akademie-Verlag.

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4

The Mathematical Origins of Nineteenth-Century Algebra of Logic Volker Peckhaus

1. Introduction Most nineteenth-century scholars would have agreed to the opinion that philosophers are responsible for research on logic. On the other hand, the history of late nineteenth-century logic clearly indicates a very dynamic development instigated not by philosophers but by mathematicians. A central outcome of this development was the emergence of what has been called the “new logic,” “mathematical logic,” “symbolic logic,” or, from 1904 on, “logistics.”1 This new logic came from Great Britain, and was created by mathematicians in the second half of the nineteenth century, ﬁnally becoming a mathematical subdiscipline in the early twentieth century. Charles L. Dodgson, better known under his pen name Lewis Carroll (1832– 1898), published two well-known books on logic, The Game of Logic of 1887 and Symbolic Logic of 1896, of which a fourth edition appeared already in 1897. These books were written “to be of real service to the young, and to be taken up, in High Schools and in private families, as a valuable addition of their stock of healthful mental recreations” (Carroll 1896, xiv). They were meant “to popularize this fascinating subject,” as Carroll wrote in the preface of the fourth edition of Symbolic Logic (ibid.). But astonishingly enough, in both books there is no deﬁnition of the term “logic.” Given the broad scope of these books, the title “Symbolic Logic” of the second book should at least have been explained. The text is based (but elaborated and enlarged) on my paper “Nineteenth Century Logic between Philosophy and Mathematics” (Peckhaus 1999).

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Maybe the idea of symbolic logic was so widely spread at the end of the nineteenth century in Great Britain that Carroll regarded a deﬁnition as simply unnecessary. Some further observations support this thesis. They concern a remarkable interest by the general public in symbolic logic, after the death of the creator of the algebra of logic, George Boole, in 1864. Recalling some standard nineteenth-century deﬁnitions of logic as, for example, the art and science of reasoning (Whately) or the doctrine giving the normative rules of correct reasoning (Herbart), it should not be forgotten that mathematical or symbolic logic was not set up from nothing. It arose from the old philosophical collective discipline logic. It is therefore obvious to assume that there was some relationship between the philosophical and the mathematical side of the development of logic, but standard presentations of the history of logic ignore this putative relationship; they sometimes even deny that there has been any development of philosophical logic at all, and that philosophical logic could therefore justly be ignored. Take for instance William and Martha Kneale’s program in their eminent The Development of Logic. They wrote (1962, iii): “But our primary purpose has been to record the ﬁrst appearances of these ideas which seem to us most important in the logic of our own day,” and these are the ideas leading to mathematical logic. Another example is J. M. Bocheński’s assessment of “modern classical logic,” which he dated between the sixteenth and the nineteenth century. This period was for him noncreative. It can therefore justly be ignored in a problem history of logic (1956, 14). According to Bocheński, classical logic was only a decadent form of this science, a dead period in its development (ibid., 20). Authors advocating such opinions adhere to the predominant views of present-day logic, that is, actual systems of mathematical or symbolic logic. As a consequence, they are not able to give reasons for the ﬁnal divorce between philosophical and mathematical logic, because they ignore the seed from which mathematical logic has emerged. Following Bocheński’s view, Carl B. Boyer presented for instance the following periodization of the development of logic (Boyer 1968, 633): “The history of logic may be divided, with some slight degree of oversimpliﬁcation, into three stages: (1) Greek logic, (2) scholastic logic, and (3) mathematical logic.” Note Boyer’s “slight degree of oversimpliﬁcation” which enabled him to skip 400 years of logical development and ignore the fact that Kant’s transcendental logic, Hegel’s metaphysics, and Mill’s inductive logic were called “logic,” as well. This restriction of scope had a further consequence: The history of logic is written as if it had been the nineteenth-century mathematicians’ main motive for doing logic to create and develop a new scientiﬁc discipline as such, namely mathematical logic, dealing above all with problems arising in this discipline and solving these problems with the ﬁnal aim of attaining a coherent theory. But what, if logic was only a means to an end, a tool for solving nonlogical

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problems? If this is considered, such nonlogical problems have to be taken note of. One can assume that at least the initial motives of mathematicians working in logic were going beyond creating a new or further developing the traditional theory of logic. Under the presupposition that a mathematician is usually not really interested in devoting his professional work to the development of a philosophical subdiscipline, one can assume that theses motives have to be sought in the mathematician’s own subject, namely in foundational, that is, philosophical problems of mathematics. Today historians have recognized that the emergence of the new logic was no isolated process. Its creation and development ran parallel to and was closely intertwined with the creation and development of modern abstract mathematics which emancipated itself from the traditional deﬁnition as a science which deals with quantities and geometrical forms and is therefore responsible for imaginabilia, that is, intuitive objects. The imaginabilia are distinguished from intelligibilia, that is, logical objects which have their origin in reason alone. These historians recognized that the history of the development of modern logic can only be told within the history of the development of mathematics because the new logic is not conceivable without the new mathematics. In recent research on the history of logic, this intimate relation between logic and mathematics, especially its connection to foundational studies in mathematics, has been taken into consideration. One may mention the present author’s Logik, Mathesis universalis und allgemeine Wissenschaft (Peckhaus 1997) dealing with the philosophical and mathematical contexts of the development of nineteenth-century algebra of logic as at least partially unconscious realizations of the Leibnizian program of a universal mathematics, José Ferreirós’s history of set theory in which the deep relations between the history of abstract mathematics and that of modern logic (Ferreirós 1999) are unfolded, and the masterpiece of this new direction, The Search for Mathematical Roots, 1870–1940 (2000a) by Ivor Grattan-Guinness, who imbedded the whole bunch of diﬀerent directions in logic into the development of foundational interests within mathematics. William Ewald’s “Source Book in the Foundations of Mathematics” (Ewald 1996) considers logical inﬂuences at least in passing, whereas the Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, edited by Ivor Grattan-Guinness (1994), devotes an entire part to “Logic, Set Theories and the Foundation of Mathematics” (vol. 1, pt. 5). In the following, the complex conditions for the emergence of nineteenthcentury symbolic logic will be discussed. The main scope will be on the mathematical motives leading to the interest in logic; the philosophical context will be dealt with only in passing. The main object of study will be the algebra of logic in its British and German versions. Special emphasis will be laid on the systems of George Boole (1815–1864) and above all of his German follower Ernst Schröder (1841–1902).

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2. Boole’s Algebra of Logic 2.1. Philosophical Context The development of the new logic started in 1847, completely independent of earlier anticipations, for example, those by the German rationalistic universal genius Gottfried Wilhelm Leibniz (1646–1716) and his followers (see Peckhaus 1994a, 1997, ch. 5). In that year the British mathematician Boole published his pamphlet The Mathematical Analysis of Logic (1847).2 Boole mentioned (1847, 1) that it was the struggle for priority concerning the quantiﬁcation of the predicate between the Edinburgh philosopher William Hamilton (1788– 1856) and the London mathematician Augustus De Morgan (1806–1871) that encouraged this study. Hence, he referred to a startling philosophical discussion which indicated a vivid interest in formal logic in Great Britain. This interest was, however, a new interest, just 20 years old. One can even say that neglect of formal logic could be regarded as a characteristic feature of British philosophy up to 1826 when Richard Whately (1787–1863) published his Elements of Logic.3 In his preface Whately added an extensive report on the languishing research and education in formal logic in England. He complained (1826, xv) that only very few students of the University of Oxford became good logicians and that by far the greater part pass through the University without knowing any thing of all of it; I do not mean that they have not learned by rote a string of technical terms; but that they understand absolutely nothing whatever of the principles of the Science. Thomas Lindsay, the translator of Friedrich Ueberweg’s important System der Logik und Geschichte der logischen Lehren (1857, English translation), was very critical of the scientiﬁc qualities of Whately’s book, but he nevertheless emphasized its outstanding contribution for the renaissance of formal logic in Great Britain (Lindsay 1871, 557): Before the appearance of this work, the study of the science had fallen into universal neglect. It was scarcely taught in the universities, and there was hardly a text-book of any value whatever to be put into the hands of the students. One year after the publication of Whately’s book, George Bentham’s An Outline of a New System of Logic appeared (1827) which was intended as a commentary to Whately. Bentham’s book was critically discussed by William Hamilton in a review article published in the Edinburgh Review (Hamilton 1833). With the help of this review, Hamilton founded his reputation as the “ﬁrst logical name in Britain, it may be in the world.”4 Hamilton propagated a revival of the Aristotelian scholastic formal logic without, however, one-sidedly preferring the syllogism. His logical conception was focused on a revision of

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the standard forms by quantifying the predicates of judgments.5 He arrived at eight standard forms (Hamilton 1859–1866, vol. 4, 1866, 287): 1. A “All A is all B”

toto-total.

2. A “All A is some B”

toto-partial.

3. I “Some A is all B”

parti-total.

4. I “Some A is some B”

parti-partial.

5. E “Any A is not any B”

toto-total.

6. E “Any A is not some B”

toto-partial.

7. O “Some A is not any B”

parti-total.

8. O “Some A is not some B” parti-partial. Hamilton’s unconsidered transition from the collective “all” to the distributive “any” has already been criticized by William and Martha Kneale (1962, 353). Hamilton used a geometrical symbolism using wedges for illustrating the eﬀects of this modiﬁcation.6 The controversy about priority arose when De Morgan, in a lecture “On the Structure of the Syllogism” (De Morgan 1846) given to the Cambridge Philosophical Society on 9 November 1846, also proposed the quantiﬁcation of the predicates.7 Neither had any priority, of course. The application of diagrammatic methods in syllogistic reasoning proposed, for example, by the eighteenth-century mathematicians and philosophers Leonard Euler, Gottfried Ploucquet, and Johann Heinrich Lambert, presupposed a quantiﬁcation of the predicate.8 The German psychologistic logician Friedrich Eduard Beneke (1798–1854) suggested to quantify the predicate in his books on logic published in 1839 and 1842, the latter of which he sent to Hamilton (see Peckhaus 1997, 191–193). In the context of this presentation, it is irrelevant to give a ﬁnal solution of the priority question. It is, however, important that a dispute of this extent arose at all. It indicates that there was a new interest in research on formal logic. This interest represented only one side of the eﬀects released by Whately’s book. Another line of research stood in the direct tradition of Humean empiricism and the philosophy of inductive sciences: the inductive logic of John Stuart Mill (1806–1873), Alexander Bain (1818–1903), and others. Boole’s logic was in clear opposition to inductive logic. It was Boole’s follower William Stanley Jevons (1835–1882; see Jevons 1877–1878) who made this opposition explicit. As mentioned earlier, Boole referred to the controversy between Hamilton and De Morgan, but this inﬂuence should not be overemphasized. In his main work on the Laws of Thought (1854), Boole went back to the logic of Aristotle by quoting from the Greek original. This can be interpreted as indicating that the inﬂuence of the contemporary philosophical discussion was not as important as his own words might suggest. In writing a book on logic he was

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doing philosophy, and it was thus a matter of course that he related his results to the philosophical discussion of his time. This does not mean, of course, that his thoughts were mainly inﬂuenced by this discussion. In any case, Boole’s early algebra of logic kept a close connection to traditional logic, in the formal part of which the theory of syllogism represented its core.9 Traditional logic not only provided the topics to be dealt with by the “Calculus of Deductive Reasoning,”10 it also served as a yardstick for evaluating the power and the reliability of the new logic. Even in the unpublished manuscripts of a sequel of the Laws of Thought titled “The Philosophy of Logic,” he discussed Aristotelian logic at length (see Boole 1997, 133–136), criticizing, however, that it is more a mnemonic art than a science of reasoning.11

2.2. The Mathematical Context in Great Britain Of greater importance than the philosophical discussion on logic in Great Britain were mathematical inﬂuences. Most of the new logicians can be related to the so-called Cambridge Network (Cannon 1978, 29–71), that is, a movement that aimed at reforming British science and mathematics which started at Cambridge. One of the roots of this movement was the foundation of the Analytical Society in 1812 (see Enros 1983) by Charles Babbage (1791–1871), George Peacock (1791–1858), and John Herschel (1792–1871). Joan L. Richards called this act a “convenient starting date for the nineteenth-century chapter of British mathematical development” (Richards 1988, 13). One of the ﬁrst achievements of the Analytical Society was a revision of the Cambridge Tripos by adopting the Leibnizian notation for the calculus and abandoning the customary Newtonian theory of ﬂuxions: “the principles of pure D-ism in opposition to the Dot-age of the University” as Babbage wrote in his memoirs (Babbage 1864, 29). It may be assumed that this successful movement triggered oﬀ by a change in notation might have stimulated a new or at least revived interest in operating with symbols. This new research on the calculus had parallels in innovative approaches to algebra which were motivated by the reception of Laplacian analysis.12 In the ﬁrst place, the development of symbolic algebra has to be mentioned. It was codiﬁed by George Peacock in his Treatise on Algebra (1830) and further propagated in his famous report for the British Association for the Advancement of Science (Peacock 1834, especially 198– 207). Peacock started by drawing a distinction between arithmetical and symbolic algebra, which was, however, still based on the common restrictive understanding of arithmetic as the doctrine of quantity. A generalization of Peacock’s concept can be seen in Duncan F. Gregory’s (1813–1844) “calculus of operations.”13 Gregory was most interested in operations with symbols. He deﬁned symbolic algebra as “the science which treats of the combination of operations deﬁned not by their nature, that is by what they are or what they do, but by the laws of combinations to which they are subject” (1840, 208). In his much praised paper “On a General Method in Analysis” (1844), Boole made the calculus of operations the basic methodological tool for analysis.

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However, in following Gregory, he went further, proposing more applications. He cited Gregory, who wrote that a symbol is deﬁned algebraically “when its laws of combination are given; and that a symbol represents a given operation when the laws of combination of the latter are the same as those of the former” (Gregory 1842, 153–154). It is possible that a symbol for an arbitrary operation can be applied to the same operation (ibid., 154). It is thus necessary to distinguish between arithmetical algebra and symbolic algebra, which has to take into account symbolic but nonarithmetical ﬁelds of application. As an example, Gregory mentioned the symbols a and +a. They are isomorphic in arithmetic, but in geometry they need to be interpreted diﬀerently. a can refer to a point marked by a line, whereas the combination of the signs + and a additionally expresses the direction of the line. Therefore symbolic algebra has to distinguish between the symbols a and +a. Gregory deplored the fact that the unequivocity of notation did not prevail as a result of the persistence of mathematical practice. Clear notation was only advantageous, and Gregory thought that our minds would be “more free from prejudice, if we never used in the general science symbols to which deﬁnite meanings had been appropriated in the particular science” (ibid., 158). Boole adopted this criticism almost word for word. In his Mathematical Analysis of Logic he claimed that the reception of symbolic algebra and its principles was delayed by the fact that in most interpretations of mathematical symbols the idea of quantity was involved. He felt that these connotations of quantitative relationships were the result of the context of the emergence of mathematical symbolism, and not of a universal principle of mathematics (Boole 1847, 3–4). Boole read the principle of the permanence of equivalent forms as a principle of independence from interpretation in an “algebra of symbols.” To obtain further aﬃrmation, he tried to free the principle from the idea of quantity by applying the algebra of symbols to another ﬁeld, the ﬁeld of logic. As far as logic is concerned this implied that only the principles of a “true Calculus” should be presupposed. This calculus is characterized as a “method resting upon the employment of Symbols, whose laws of combination are known and general, and whose results admit of a consistent interpretation” (ibid., 4). He stressed (ibid.): It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical Analysis, regardless that in its objects and in its instruments it must at present stand alone. Boole expressed logical propositions in symbols whose laws of combination are based on the mental acts represented by them. Thus he attempted to establish a psychological foundation of logic, mediated, however, by language.14 The central mental act in Boole’s early logic is the act of election used for building classes. Man is able to separate objects from an arbitrary collection which belong to given classes to distinguish them from others. The symbolic representation of these mental operations follows certain laws of combination that

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are similar to those of symbolic algebra. Logical theorems can thus be proven like mathematical theorems. Boole’s opinion has of course consequences for the place of logic in philosophy: “On the principle of a true classiﬁcation, we ought no longer to associate Logic and Metaphysics, but Logic and Mathematics” (ibid., 13). Although Boole’s logical considerations became increasingly philosophical with time, aiming at the psychological and epistemological foundations of logic itself, his initial interest was not to reform logic but to reform mathematics. He wanted to establish an abstract view on mathematical operations without regard to the objects of these operations. When claiming “a place among the acknowledged forms of Mathematical Analysis” (1847, 4) for the calculus of logic, he didn’t simply want to include logic in traditional mathematics. The superordinate discipline was a new mathematics. This is expressed in Boole’s writing: “It is not of the essence of mathematics to be conversant with the ideas of number and quantity” (1854, 12).

2.3. Boole’s Logical System Boole’s early logical system is based on mental operations, namely, acts of selecting individuals from classes. In his notation 1 symbolizes the Universe, comprehending “every conceivable class of objects whether existing or not” (1847, 15). Capital letters stand for all members of a certain class. The small letters are introduced as follows (ibid., 15): The symbol x operating upon any subject comprehending individuals or classes, shall be supposed to select from that subject all the Xs which it contains. In like manner the symbol y, operating upon any subject, shall be supposed to select from it all individuals of the class Y which are comprised in it and so on. Take A as the class of animals, then x might signify the selection of all sheep from these animals, which then can be regarded as a new class X from which we select further objects, and so on. This might be illustrated by the following example: animals A ↓ x sheep

sheep X ↓ y horned

horned sheep Y ↓ z black

black horned sheep Z

This process represents a successive selection which leads to individuals being common to the classes A, X, Y , and Z. xyz stands for animals that are sheep, horned, and black. It can be regarded as the logical product of some common marks or common aspects relevant for the selection. In his major work, An Investigation of the Laws of Thought of 1854, Boole gave up this distinction

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between capital and small letters, thereby getting rid of the complicated consequences of this stipulation. If the symbol 1 denotes the universe, and if the class X is determined by the symbol x, it is consequent that the class not-X has to be denoted by the symbol 1 − x, which forms the supplement to x, thus x(1 − x) = 0. 0 symbolizes nothing or the empty class. Now one can consider Boole’s interpretation of the universal-aﬃrmative judgment. The universal-aﬃrmative judgment “All Xs are Y s” is expressed by the equation xy = x or, by simple arithmetical transformation, x(1 − y) = 0 (p. 22): “As all the Xs which exist are found in the class Y , it is obvious that to select out of the Universe all Y s, and from these to select all Xs, is the same as to select at once from the Universe all Xs.” The universal-negative judgment “No Xs are Y s” asserts that there are no terms common in the classes X and Y . All individuals common would be represented by xy, but they form the empty class. The particular-aﬃrmative judgment “Some Xs are Y s” says that there are some terms common to both classes forming the class V . They are expressed by the elective symbol v. The judgment is thus represented by v = xy. Boole furthermore considers using vx = vy with vx for “some X” and vy for “some Y ,” but observes “that this system does not express quite so much as the single equation” (pp. 22–23). The particular-negative judgment “Some Xs are not Y s” can be reached by simply replacing y in the last formula with 1 − y. Boole’s elective symbols are compatible with the traditional theory of judgment. They blocked, however, the step toward modern quantiﬁcation theory as present in the work of Gottlob Frege, but also in later systems of the algebra of logic like those of C. S. Peirce and Ernst Schröder.15 The basic relation in the Boolean calculus is equality. It is governed by three principles which are themselves derived from elective operations (see ibid., 16–18): 1. The Distributivity of Elections (16–17): it is indiﬀerent whether from of group of objects considered as a whole, we select the class X, or whether we divide the group into two parts, select the Xs from them separately, and then connect the results in one aggregate conception, in symbols: x(u + v) = xu + xv, with u + v representing the undivided group of objects, and u and v standing for its component parts. 2. The Commutativity of Elections: The order of elections is irrelevant: xy = yx. 3. The Index Law: The successive execution of the same elective act does not change the result of the election: xn = x,

for n ≥ 2.

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Boole stressed the importance of the Index Law, which is not generally valid in arithmetic (only in the arithmetic of 0 and 1) and therefore peculiar for elective symbols. It allows one to reduce complex formulas to forms more easily capable of being interpreted. In his Investigation of the Laws of Thought (1854) Boole abandoned the Index Law and replaced it by the Law of Duality (“Boole’s Law”) xx = x, or x2 = x.16 His esteem for this law becomes evident in his claim “that the axiom of the metaphysicians which is termed the principle of contradiction. . . , is a consequence of the fundamental law of thought whose expression is x2 = x” (Boole 1854, 49). Boole referred to the derivation x2 = x x − x2 = 0 x(1 − x) = 0, the last formula saying that a class and its complement have no elements in common. Boole was heavily criticized for this “curious error” (Halsted 1878, 86) of considering the Law of Contradiction a consequence of the Law of Duality, not the other way around (the derivation works, of course, also in the other direction). Boole’s revisions came along with a change in his attitude toward logic. His early logic can be seen as an application of a new mathematical method to logic, thereby showing the eﬃcacy of this method within the broad project of a universal mathematics and so serving foundational goals in mathematics. This foundational aspect diminished in later work, successively being replaced by the idea of a reform of logic. Already in the paper “The Calculus of Logic” (Boole 1848), Boole tried to show that his logical calculus is compatible with traditional philosophical logic. Reasoning is guided by the laws of thought. They are the central topic in Boole’s Investigation of the Laws of Thought, claiming that “there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted” (1854, 6), comparing thereby the laws of thought and the laws of algebra. Logic, in Boole’s understanding, was “a normative theory of the products of mental processes” (Grattan-Guinness 2000a, 51).

2.4. Symbolic Logic within the Old Paradigm: De Morgan Although created by mathematicians, the new logic was widely ignored by fellow mathematicians. Boole was respected by British mathematicians, but his ideas concerning an algebraic representation of the laws of thought received very little published reaction.17 He shared this fate with De Morgan, the second major ﬁgure of symbolic logic at that time.18 Like Boole, the British mathematician De Morgan was inﬂuenced by algebraist George Peacock’s work on symbolic algebra, which motivated him to consider the foundations of algebra in connection with logic. He distinguished,

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for example, algebra as an art associated with what he called “technical algebra” and algebra as science, that is, “logical algebra”: “Technical algebra is the art of using symbols under regulations which . . . are prescribed as the deﬁnition of symbols. Logical algebra is the science which investigates the method of giving meaning to the primary symbols, and of interpreting all subsequent results” (De Morgan 1842, 173–174, reprint p. 338). He used algebraic symbolism in logic, being mainly interested in a reform and extension of syllogistic logic, but ignoring the operational aspect of logic as calculus. He published his main results in a series of papers in the Proceedings of the Cambridge Philosophical Society between 1846 and 1862 (reprinted in De Morgan 1966) and in his book Formal Logic (1847). He has been called “the last great traditional logician” (Hailperin 2004, 346). Among his lasting achievements is the introduction of the technical term of a universe. He spoke, for example, of the “Universe of a proposition, or of a name” that may be limited in any matter expressed or understood” (De Morgan 1846/1966, 2) but continued to distinguish two kinds of the universe of a population, “being either the whole universe of thought, or a given portion of it” (De Morgan 1853/1966, 69). In the ﬁrst of the papers “On the Syllogism,” he introduced an algebraic symbolism for the syllogism, using small letters x, y, z as names contrary to those represented by capitals X, Y , Z (De Morgan 1846/1966, 3). The relations between such names as expressed in standard forms or simple propositions are symbolized as follows (ibid., 4): P )Q P.Q PQ P :Q

signiﬁes ... ... ...

Every P is Q. No P is Q. Some P s are Qs. Some P s are not Qs.

The algebraic symbols thus signify both the quantity of the concepts involved and the copula. For the names X and Y and their contraries x and y, the following equations are valid (ibid.): X)Y = X.y = y)x X:Y = Xy = y:x Y )X = Y.x = x)y

X.Y = X)y = Y )x XY = X:y = Y :x x.y = x)Y = y)X

Y :X = Y x = x:y

xy = x:Y = y:X

De Morgan used this symbolism to reconstruct the theory of syllogism. It served as representation, not as a calculus. Only after having written the 1846 paper, De Morgan found “that the whole theory of the syllogism might be deduced from the consideration of propositions in a form in which deﬁnite quantity of assertion is given both to the subject and the predicate of a proposition,” as he reported in an “Addition,” dated 27 February 1847 (De Morgan 1966, 17). He claimed to have brought

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this idea to paper before he learned of Sir William Hamilton’s quantiﬁcation of the predicate, thereby opening the priority quarrel. De Morgan focused his subsequent logical work on the theory of the copula, following “the hint given by algebra” by separating “the essential from the accidental characteristics of the copula” (1850/1966, 50). The “abstract copula” characterized only by essential features is understood as “a formal mode of joining two terms which carries no meaning, and obeys no law except such as is barely necessary to make the forms of inference follow” (ibid., 51). The abstract copula follows two “copular conditions,” (1) transitiveness X −−− Y −−− Z = X −−− Z (2) convertibility X −−− Y = Y −−− X Aﬃrmative (−−−) and negative (−−) copula are contrary to each other. Of X −−− Y and X −− Y one or the other must be (De Morgan 1850/1966, 51). De Morgan was the ﬁrst to take seriously that traditional syllogistics was incapable of dealing with relational properties like “Smith is smaller than Jones.” His ideas concerning a logic of (two-place) relations can be regarded as his most important contributions (see Merrill 1990, chs. 5–6; Grattan-Guinness 2000a, 32–34). Already in his second paper on the syllogism, he mentioned the role of the copula for expressing the relation between what is connected. He also considered the composition of relations (1850/1966, 55), that is, in modern terms, the relative product. He studied the subject of relations “as a branch of logic” in his fourth paper on the syllogism (De Morgan 1860/1966, 208). De Morgan used capital letters L, M , N for denoting relations, lowercase letters l, m, n for the respective contraries. Additionally, two periods indicate that a relation holds, only one period that the contrary relations holds. Thus, X..LY or X.lY say that X is “some one of the objects of thought which stand to Y in the relation L, or is one of the Ls of Y ” (ibid., 220). X and Y are called “subject” and “predicate,” indicating the mode in which they stand in the relation, thus in both LY.X and X.LY , Y indicates the predicate. If the predicate is itself the subject of a relation, a composition of relations results. “Thus if X..L(M Y ), if X be one of the Ls of one of the M s of Y , we may think of X as an ‘L of M ’ of Y , expressed by X..(LM )Y , or simply by X..LM Y ” (ibid., 221). De Morgan used an accent to signify universal quantity as part of the description of the relation. LM stands for an L of every M , LM X for the same relation to many (ibid.). The converse relation of L, L , is deﬁned as if X..LY , then Y..L−1 X” (ibid., 222). De Morgan then applied this symbolism to his theory of syllogism, introducing “theorem K” as basic for what he called “opponent syllogism,” which is exempliﬁed by the following mathematical syllogism (ibid., 224–225): Every deﬁcient of an external is a coinadequate: external and coinadequate have partient and complement for their contraries,

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and deﬁcient has exient for its converse: hence every exient of a complement is a patient; which is one of the opponent syllogisms of that ﬁrst given. Theorem K says (ibid., 224) that if a compound relation be contained in another relation, by the nature of the relations and not by casualty of the predicate, the same may be said when either component is converted, and the contrary of the other component and of the component change places. One of the examples is that “if, be Z what it may, every L of M of Z be an N of Z, say LM ))N , then L−1 n))m, and nM −1 ))l” (ibid.). The problematic nature of De Morgan’s symbolism becomes obvious in his notation for complex terms. The conjunctive “P and Q” is expressed by P Q, the disjunctive (taken in the inclusive sense) by P, Q. Using this notation he formulated the laws named after him (that can, however, be found already in the work of William of Ockham): “The contrary of P Q is p, q; that of P, Q, is pq” (1847, 118). The equivalent in modern notation is ¬(p ∨ q) = ¬p ∧ ¬q, and ¬(p ∧ q) = ¬p ∨ ¬q, or in the quantiﬁcational version ¬∃x ax = ∀x ¬ax and ¬∀x ax = ∃x ¬ax.

2.5. Reception of the New Logic In 1864, Samuel Neil, the early chronicler of British mid-nineteenth-century logic, expressed his thoughts about the reasons for this negligible reception: “De Morgan is esteemed crotchety, and perhaps formalizes too much. Boole demands high mathematic culture to follow and to proﬁt from” (1864, 161). One should add that the ones who had this culture were usually not interested in logic. The situation changed after Boole’s death in 1864. In the following comments only some ideas concerning the reasons for this new interest are hinted at. In particular the roles of William Stanley Jevons and Alexander Bain are considered. These examples show that a broader reception of symbolic logic commenced only when its relevance for the philosophical discussion of the time came to the fore. 2.5.1. William Stanley Jevons A broader international reception of Boole’s logic began when Jevons (1835– 1882) made it the starting point for his inﬂuential Principles of Science (Jevons 1874). He used his own version of the Boolean calculus introduced in his Pure Logic (Jevons 1864). Among his revisions were the introduction of a simple symbolic representation of negation and the deﬁnition of logical addition as inclusive “or,” thereby creating Boolean algebra (see Hailperin 1981). He also changed the philosophy of symbolism (1864, 5):

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The forms of my system may, in fact, be reached by divesting his [Boole’s] of a mathematical dress, which, to say the least, is not essential to it. The system being restored to its proper simplicity, it may be inferred, not that Logic is a part of Mathematics, as is almost implied in Professor Boole’s writings, but that the Mathematics are rather derivatives of Logic. All the interesting analogies or samenesses of logical and mathematical reasoning which may be pointed out, are surely reversed by making Logic dependent on Mathematics. Jevons’s interesting considerations on the relationship between mathematics and logic representing an early logicistic attitude will not be discussed here. Similar ideas can be found not only in Gottlob Frege’s work, but also in that of Rudolf Hermann Lotze (1817–1881) and Schröder. Most important in the present context is the fact that Jevons abandoned mathematical symbolism in logic, an attitude that was later taken up by John Venn (1834–1923) in his Symbolic Logic (Venn 1894). Jevons attempted to free logic from the semblance of being a special mathematical discipline. He used the symbolic notation only as a means of expressing general truths. Logic became a tool for studying science, a new language providing symbols and structures. The change in notation brought the new logic closer to the philosophical discourse of the time. The reconciliation was supported by the fact that Jevons formulated his Principles of Science as a rejoinder to John Stuart Mill’s (1806–1873) System of Logic of 1843, at that time the dominating work on logic and the philosophy of science in Great Britain. Although Mill had called his logic A System of Logic Ratiocinative and Inductive, the deductive parts played only a minor role, used only to show that all inferences, all proofs, and the discovery of truths consisted of inductions and their interpretations. Mill claimed to have shown “that all our knowledge, not intuitive, comes to us exclusively from that source” (Mill 1843, bk. II, ch. I, §1). Mill concluded that the question as to what induction is, is the most important question of the science of logic, “the question which includes all others.” As a result the logic of induction covers by far the largest part of this work, a subject that we would today regard as belonging to the philosophy of science. Jevons deﬁned induction as a simple inverse application of deduction. He began a direct argument with Mill in a series of papers titled “Mill’s Philosophy Tested” (1877/78). This argument proved that symbolic logic could be of importance not only for mathematics, but also for philosophy. Another eﬀect of the attention caused by Jevons was that British algebra of logic was able to cross the Channel. In 1877, Louis Liard (1846–1917), at that time professor at the Faculté de lettres at Bordeaux and a friend of Jevons, published two papers on the logical systems of Jevons and Boole (Liard 1877a, 1877b). In 1878 he added a booklet titled Les logiciens anglais contemporaines (Liard 1878), which had ﬁve editions until 1907 and was translated into German (Liard 1880). Although Hermann Ulrici (1806–1884)

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had published a ﬁrst German review of Boole’s Laws of Thought as early as 1855 (Ulrici 1855, see Peckhaus 1995), the knowledge of British symbolic logic was conveyed primarily by Alois Riehl (1844–1924), then professor at the University of Graz in Austria. He published a widely read paper, “Die englische Logik der Gegenwart” (“English contemporary logic,” Riehl 1877), which reported mainly Jevons’s logic and utilized it in a current German controversy on the possibility of scientiﬁc philosophy. 2.5.2. Alexander Bain Surprisingly good support for the reception of Boole’s algebra of logic came from the philosophical opposition, namely from the Scottish philosopher Bain (1818–1903) who was an adherent of Mill’s logical theory. Bain’s Logic, ﬁrst published in 1870, had two parts, the ﬁrst on deduction and the second on induction. He made explicit that “Mr Mill’s view of the relation of Deduction and Induction is fully adopted” (1870, I, iii). Obviously he shared the “general conviction that the utility of the purely Formal Logic is but small; and that the rules of Induction should be exempliﬁed even in the most limited course of logical discipline” (ibid., v). The minor role of deduction showed up in Bain’s deﬁnition “Deduction is the application or extension of Induction to new cases” (40). Despite his reservations about deduction, Bain’s Logic became important for the reception of symbolic logic because of a chapter of 30 pages titled “Recent Additions to the Syllogism.” In this chapter the contributions of Hamilton, De Morgan, and Boole were introduced. One can assume that many more people became acquainted with Boole’s algebra of logic through Bain’s report than through Boole’s own writings. One example is Hugh MacColl (1837–1909), the pioneer of the calculus of propositions (statements) and of modal logic.19 He created his ideas independently of Boole, eventually realizing the existence of the Boolean calculus by means of Bain’s report. Even in the early parts of his series of papers “The Calculus of Equivalent Statements,” he quoted from Bain’s presentation when discussing Boole’s logic (MacColl 1877/78). In 1875 Bain’s logic was translated into French, in 1878 into Polish. Tadeusz Batóg and Roman Murawski (1996) have shown that it was Bain’s presentation which motivated the ﬁrst Polish algebraist of logic, Stanisław Piątkiewicz (1848–?) to begin his research on symbolic logic.

3. Schröder’s Algebra of Logic 3.1. Philosophical Background The philosophical discussion on logic after Hegel’s death in Germany was still determined by a Kantian inﬂuence.20 In the preface to the second edition of his Kritik der reinen Vernunft of 1787, Immanuel Kant (1723–1804) had written that logic had followed the safe course of a science since earliest times.

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For Kant, this was evident because of the fact that logic had been prohibited from taking any step backward from the time of Aristotle. But he regarded it as curious that logic hadn’t taken a step forward either (B VIII). Thus, logic seemed to be closed and complete. Formal logic, in Kant’s terminology the analytical part of general logic, did not play a prominent role in Kant’s system of transcendental philosophy. In any case, it was a negative touchstone of truth, as he stressed (B 84). Georg Wilhelm Friedrich Hegel (1770–1831) went further in denying any relevance of formal logic for philosophy (Hegel 1812/13, I, Introduction, XV–XVII). Referring to Kant, he maintained that from the fact that logic hadn’t changed since Aristotle one should infer that it needs to be completely rebuilt (ibid., XV). Hegel created a variant of logic as the foundational science of his philosophical system, deﬁning it as “the science of the pure idea, i.e., the idea in the abstract element of reasoning” (1830, 27). Hegelian logic thus coincides with metaphysics (ibid., 34). This was the situation when after Hegel’s death philosophical discussion on formal logic started again in Germany. This discussion on logic reform stood under the label of “the logical question,” a term created by the neo-Aristotelian Adolf Trendelenburg (1802–1872). In 1842 he published a paper titled “Zur Geschichte von Hegel’s Logik und dialektischer Methode” with the subtitle “Die logische Frage in Hegel’s Systeme.” But what is the logical question according to Trendelenburg? He formulated this question explicitly toward the end of his article: “Is Hegel’s dialectical method of pure reasoning a scientiﬁc procedure?” (1842, 414). In answering this question in the negative, he provided the occasion of rethinking the status of formal logic within a theory of human knowledge without, however, proposing a return to the old (scholastic) formal logic. The term “the logical question” was subsequently used in a less speciﬁc way. Georg Leonard Rabus, the early chronicler of the discussion on logic reform, wrote, for example, that the logical question emerged from doubts concerning the justiﬁcation of formal logic (1880, 1). Although this discussion was clearly connected to formal logic, the socalled reform did not concern formal logic. The reason was provided by the neo-Kantian Wilhelm Windelband who wrote in a brilliant survey on nineteenth-century (philosophical) logic (1904, 164): It is in the nature of things that in this enterprize [i.e., the reform of logic] the lower degree of fruitfulness and developability power was on the side of formal logic. Reﬂection on the rules of the correct progress of thinking, the technique of correct thinking, had indeed been brought to perfection by former philosophy, presupposing a naive world view. What Aristotle had created in a stroke of genius, was decorated with the ﬁnest ﬁligree work in Antiquity and the Middle Ages: an art of proving and disproving which culminated in a theory of reasoning, and after this constructing the doctrines of judgements and concepts. Once one has accepted the foundations, the safely assembled building cannot be shaken:

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it can only be reﬁned here and there and perhaps adapted to new scientiﬁc requirements. Windelband was very critical of English mathematical logic. Its quantiﬁcation of the predicate allows the correct presentation of extensions in judgments, but it “drops hopelessly” the vivid sense of all judgments, which tend to claim or deny a material relationship between subject or predicate. It is “a logic of the conference table,” which cannot be used in the vivid life of science, a “logical sport” which has its merits only in exercising the ﬁnal acumen (ibid., 166–167). The philosophical reform eﬀorts concerned primarily two areas: 1. the problem of a foundation of logic itself. It was dealt with by using psychological and physiological means, thereby leading to new discussion on the question of priority between logic and psychology, and to various forms of psychologism and anti-psychologism (see Rath 1994, Kusch 1995). 2. The problem of the applicability of logic which led to an increased interest in the methodological part of traditional logic. The reform of applied logic attempted to bring philosophy in touch with the stormy development of mathematics and sciences in that time. Both reform procedures had a destructive eﬀect on the shape of logic and philosophy. The struggle with psychologism led to the departure of psychology (especially in its new, experimental form) from the body of philosophy at the beginning of the twentieth century. Psychology became a new, autonomous scientiﬁc discipline. The debate on methodology resulted in the creation of the philosophy of science being ﬁnally separated from the body of logic. The philosopher’s ignorance of the development of formal logic caused a third departure: Part of formal logic was taken from the domain of the competence of philosophy and incorporated into mathematics where it was instrumentalized for foundational tasks. This was the philosophical background of the emergence of symbolic logic in Germany and especially the logical work of the German mathematician Schröder.

3.2. The Mathematical Context in Germany 3.2.1. Logic and Formal Algebra The examination of the British situation in mathematics at the time when the new logic emerged has shown that the creators of the new logic were basically interested in a reform of mathematics by establishing an abstract view of mathematics which focused not on mathematical objects like quantities but on symbolic operations with arbitrary objects. The reform of logic was only secondary. These results can be transferred to the situation in Germany without any problem.

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Schröder was the most important representative of the German algebra of logic.21 He was regarded as having completed the Boolean period in logic (see Bocheński 1956, 314). In his ﬁrst pamphlet on logic, Der Operationskreis des Logikkalkuls (1877), he presented a critical revision of Boole’s logic of classes, stressing the idea of the duality between logical addition and logical multiplication introduced by Jevons in 1864. In 1890, Schröder started the large project of his monumental Vorlesungen über die Algebra der Logik, which remained unﬁnished, although it increased to three volumes with four parts, of which one appeared only posthumously (1890, 1891, 1895, 1905). Contemporaries regarded the ﬁrst volume alone as having completed the algebra of logic (see Wernicke 1891, 196). Nevertheless, Schröder’s logical theory kept, like the one of Boole, close contact to the traditional shape of logic. The introduction of the Vorlesungen is full of references to that time’s philosophical discussion on logic. Schröder even referred to the psychologistic discussion on the foundation of logic, and never really freed his logical theory from the traditional division of logic into the theories of concept, judgment, and inference. Schröder’s opinion concerning the question as to what end logic is to be studied (see Peckhaus 1991, 1994b, 2004a) can be drawn from an autobiographical note (written in the third person), published in the year before his death. It contains his own survey of his scientiﬁc aims and results. Schröder divided his scientiﬁc production into three ﬁelds: 1. A number of papers dealing with some of the current problems of his science. 2. Studies concerned with creating an “absolute algebra,” that is, a general theory of connections. Schröder stressed that these studies represent his “very own object of research” of which only little was published at that time. 3. Work on the reform and development of logic. Schröder wrote (1901) that his aim was to design logic as a calculating discipline, especially making possible an exact handling of relative concepts, and, from then on, by emancipation from the routine claims of spoken language, and also to remove any breeding ground from “cliché” in the ﬁeld of philosophy as well. This should prepare the ground for a scientiﬁc universal language that, widely diﬀering from linguistic eﬀorts like Volapük [a universal language like Esperanto, very popular in Germany at that time], looks more like a sign language than like a sound language. Schröder’s own division of his ﬁelds of research shows that he didn’t consider himself a logician: His “very own object of research” was “absolute algebra,” which was similar to modern abstract or universal algebra in respect to its basic

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problems and fundamental assumptions. What was the connection between logic and algebra in Schröder’s research? From the passages quoted one could assume that they belong to two separate ﬁelds of research, but this is not the case. They were intertwined in the framework of his heuristic idea of a general science. In his autobiographical note he stressed: The disposition for schematizing, and the aspiration to condense practice to theory advised Schröder to prepare physics by perfecting mathematics. This required deepening of mechanics and geometry, but above all of arithmetic, and subsequently he became in time aware of the necessity to reform the source of all these disciplines, logic. Schröder’s universal claim becomes obvious. His scientiﬁc eﬀorts served for providing the requirements to found physics as the science of material nature by “deepening the foundations,” to quote a famous metaphor later used by David Hilbert (1918, 407) to illustrate the objectives of his axiomatic program. Schröder regarded the formal part of logic that can be formed as a “calculating logic,” using a symbolic notation, as a model of formal algebra that is called “absolute” in its last state of development. But what is “formal algebra?” The theory of formal algebra “in the narrowest sense of the word” includes “those investigations on the laws of algebraic operations . . . that refer to nothing but general numbers in an unlimited number ﬁeld without making any presuppositions concerning its nature” (1873, 233). Formal algebra therefore prepares “studies on the most varied number systems and calculating operations that might be invented for particular purposes” (ibid.). It has to be stressed that Schröder wrote his early considerations on formal algebra and logic without any knowledge of the results of his British predecessors. His sources were the textbooks of Martin Ohm, Hermann Günther Graßmann, Hermann Hankel, and Robert Graßmann. These sources show that Schröder was a representative of the tradition of German combinatorial algebra and algebraic analysis (see Peckhaus 1997, ch. 6). 3.2.2. Combinatorial Analysis Schröder developed the programmatic foundations of absolute algebra in his textbook Lehrbuch der Arithmetik und Algebra (1873) and the school program pamphlet Über die formalen Elemente der absoluten Algebra (1874). Among the sources mentioned in the textbook, Martin Ohm’s (1792–1872) Versuch eines vollkommen consequenten Systems der Mathematik (1822) is listed. It stood in the German tradition of the algebraic and combinatorial analysis which started with the work of Carl Friedrich Hindenburg (1741–1808) and his school (see Jahnke 1990, 161–322). Ohm (see Bekemeier 1987) aimed at completing Euclid’s geometrical program for all of mathematics (Ohm 1853, V). He distinguished between number

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(or “undesignated number”) and quantity (or “designated number”) regarding the ﬁrst one as the higher concept. The features of the calculi of arithmetic, algebra, analysis, and so on are not seen as features of quantities but of operations, that is, mental activities (1853, VI–VII). This operational view can also be found in the work of Graßmann, who also stood in the Hindenburg tradition. 3.2.3. General Theory of Forms Graßmann’s Lineale Ausdehnungslehre (1844)22 was of decisive inﬂuence on Schröder, especially Graßmann’s “general theory of forms” (“allgemeine Formenlehre”) opening this pioneering study in vector algebra and vector analysis. The general theory of forms was popularized by Hankel’s Theorie der complexen Zahlensysteme (1867). Graßmann deﬁned the general theory of forms as “the series of truths that is related to all branches of mathematics in the same way, and that therefore only presupposes the general concepts of equality and diﬀerence, connection and division” (1844, 1). Equality is taken as substitutivity in every context. Graßmann chooses as general connecting sign. The result of the connection of two elements a and b is expressed by the term (a b). Using the common rules for brackets we get for three elements ((a b) c) = a b c (§2). Graßmann restricted his considerations to “simple connections,” that is, associative and commutative connections (§4). These connecting operations are synthetic. The reverse operations are called resolving or analytic connections. a b stands for the form which results in a if it is synthetically connected with b: a b b = a (§5). Graßmann introduced furthermore forms in which more than one synthetic operation occur. If the second connection is symbolized with and if there holds distributivity between the synthetic operations, then the equation (a b) c) (b c) is valid. Graßmann called the c = (a second connection a connection on a higher level (§9), a terminology that might have inﬂuenced Schröder’s later “Operationsstufen,” that is, “levels of operations.” Whereas Graßmann applied the general theory of forms in the domain of extensive quantities, especially directed lines, that is, vectors, Hankel later used it to erect on its base his system of hypercomplex numbers (Hankel 1867). If λ(a, b) is a general connection of objects a, b leading to a new object c, that is, λ(a, b) = c, there is a connection Θ which, applied to c and b leads again to a, that is, Θ(c, b) = a or Θ{λ(a, b), b} = a. Hankel called the operation θ “thetic” and its reverse λ “lytic.” The commutativity of these operations is not presupposed (ibid., 18). 3.2.4. “Wissenschaftslehre” and Logic Graßmann had already announced that his Lineale Ausdehnungslehre should be part of a comprehensive reorganization of the system of sciences. His brother,

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Robert Graßmann (1815–1901), attempted to realize this program in a couple of writings published under the series title Wissenschaftslehre oder Philosophie. In its parts on logic and mathematics he anticipated modern lattice theory. He furthermore formulated a logical calculus being in parts similar to that of Boole. His logical theory was obviously independent of the contemporary German philosophical discussion on logic, and he was also not aware of his British precursors.23 Graßmann wrote about the aims of his logic or theory of reasoning (“Denklehre”) that it should teach us strictly scientiﬁc reasoning which is equally valid for all men of any people, any language, equally proving and rigorous. It has therefore to relieve itself from the barriers of a certain language and to treat the forms of reasoning, becoming, thus, a theory of forms or mathematics. Graßmann tried to realize this program in his Formenlehre oder Mathematik, published in six brochures consisting of an introduction (1872a), a general part on “Grösenlehre” (1872b) understood as “science of tying quantities,” and the special parts “Begriﬀslehre oder Logik” (theory of concepts or logic), “Bindelehre oder Combinationslehre” (theory of binding or combinatorics), “Zahlenlehre oder Arithmetik” (theory of numbers or arithmetic), and “Ausenlehre oder Ausdehnungslehre” (theory of the exterior or Ausdehnungslehre). In the general theory of quantities Graßmann introduced the letters a, b, c, . . . as syntactical signs for arbitrary quantities. The letter e represents special quantities: elements, or in Graßmann’s strange terminology “Stifte” (pins), that is, quantities which cannot be derived from other quantities by tying. Besides brackets, which indicate the order of the tying operation, he introduces the equality sign =, the inequality sign Z , and a general sign for a tie ◦. Among special ties he investigates joining or addition (“Fügung oder Addition”) (“+”) and weaving or multiplication (“Webung oder Multiplikation”) (“·”). These ties can occur either as interior ties, if e ◦ e = e, or as exterior exterior ties, if e ◦ e Z e. The special parts of the theory of quantities are distinguished with the help of the combinatorically possible results of tying a pin to itself. The ﬁrst part, “the most simple and, at the same time, the most interior,” as Graßmann called it, is the theory of concepts or logic in which interior joining e + e = e and inner weaving ee = e hold. In the theory of binding or combinatorics interior joining e + e = e and exterior weaving ee Z e hold; in the theory of numbers or arithmetic exterior joining e + e Z e and interior weaving ee = e hold, or 1 × 1 = 1 and 1 × e = e. Finally, in the theory of the exterior or Ausdehnungslehre, the “most complicated and most exterior” part of the theory of forms, exterior joining e + e Z e and exterior weaving ee Z e hold (1872a, 12–13). Graßmann thus formulated Boole’s Law of Duality using his interior weaving ee = e, but he went beyond Boole in allowing interior joining e + e = e, so coming close to Jevons’s system of 1864.

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In the theory of concepts or logic, Graßmann started with interpreting the syntactical elements, which had already been introduced in a general way. Now, everything that can be a deﬁnite object of reasoning is called “quantity.” In this new interpretation, pins are initially set quantities not being derived from other quantities by tying. Equality is interpreted as substitutivity without value change, inequality as impossibility of such a substitution. Joining is read as “and,” standing for adjunction or the logical “or.” Weaving is read as “times,” that is, conjunction or the logical “and.” Graßmann introduced the signs < and > to express sub- and superordination of concepts. The sign expresses that a concept equals or that it is subordinated another concept. This is exactly the sense of Schröder’s later basic connecting relation of subsumption or inclusion. In the theory of concepts, Graßmann expressed this relation in a shorter way with the help of the angle sign ∠. The sign T stands for the All or the totality, the sum of all pins. The following laws hold: a + T = T and aT = a. 0 is interpreted as “the lowest concept, which is subordinate to all concepts.” Its laws are a + 0 = a and a · 0 = 0. Finally Graßmann introduced the “not” (“Nicht”) or negation as complement with the laws a + a = T and a · a = 0.

3.3. Schröder’s Algebra of Logic 3.3.1. Schröder’s Way to Logic In his work on the formal elements of absolute algebra (1874) Schröder investigated operations in a manifold, called domain of numbers (“Zahlengebiet”). “Number” is, however, used as a general concept. Examples for numbers are “proper names, concepts, judgments, algorithms, numbers [of arithmetic], symbols for quantities and operations, points, systems of points, or any geometrical object, quantities of substances, etc.” (Schröder 1874, 3). Logic is, thus, a possible interpretation of the structure dealt with in absolute algebra. Schröder assumed that there are operations with the help of which two objects from a given manifold can be connected to yield a third that also belongs to that manifold (ibid., 4). He chooses from the set of possible operations the noncommutative “symbolic multiplication” c = a . b = ab with two inverse operations measuring (“Messung”)

b . (a : b) = a, a . b = a. and division (“Teilung”) b Schröder called a direct operation together with its inverses “level of operations” (“Operationsstufe”). And again Schröder realized that “the logical addition of concepts (or individuals)” follows the laws of multiplication of real numbers. But there is still another association with logic. In his Lehrbuch, Schröder √ speculated about the relation between an “ambiguous expression” like a

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and its possible values. He determined ﬁve logical relations, introducing his subsumption relations. Be A an expression that can have diﬀerent values a, a , a, . . . . Then the following relations hold (Schröder 1873, 27–29): ⎧ a ⎪ ⎪ ⎪ ⎨ a Superordination A a . ⎪ ⎪ ⎪ .. ⎩ . √ Examples: metal silver; 9 −3. ⎫ a ⎪ ⎪ ⎪ a ⎬ A. Subordination a ⎪ ⎪ ⎪ .. ⎭ . √ Examples: gold metal; 3 9.

Coordination a

a a ··· .

Examples: gold silver [in respect√to the general concept “metal”] or 3 [in respect to the general concept 9 ].

−3

Equality A = B means that the concepts A and B are identical in intension and extension. Correlation A(=)B means that the concepts A and B agree in at least one value. Schröder recognized that if he would now introduce negation, he would have created a complete terminology that allows one to express all relations between concepts (in respect to their extension) with short formulas which can harmonically be embedded into the schema of the apparatus of the mathematical sign language (ibid., 29). Schröder wrote his logical considerations of the introduction of the Lehrbuch without having seen any work of logic in which symbolic methods had been applied. It was while completing a later sheet of his book that he came across Robert Graßmann’s Formenlehre oder Mathematik (1872a). He felt urged to insert a comprehensive footnote running over three pages for hinting at this book (Schröder 1873, note, pp. 145–147). There he reported that Graßmann used the sign + for the “collective comprehension,” “really regarding it as an addition—one could say a ‘logical’ addition—that has besides the features of common (numerical) addition the basic feature a + a = a.” He wrote that he was most interested in the role the author had assigned to multiplication regarded as the product of two concepts which unite the marks being common to both concepts.

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In the Programmschrift of 1874, Schröder also gave credit to Robert Graßmann, but mentioned that he had recently found out that the laws of the logical operations had already been developed before Graßmann “in a classical work” by Boole (Schröder 1874, 7). 3.3.2. Logic as a Model of Absolute Algebra In 1877 Schröder published his Operationskreis des Logikkalkuls, in which he developed the logic of Boole’s Laws of Thought stressing the duality of the logical operations of addition and multiplication.24 An “Operationskreis” (circle of operations) is constituted by more than one direct operation together with their inverses. The “logical calculus” is the set of formulas which can be produced in this circle of operations. Schröder called it a characteristic mark of “mathematical logic or the logical calculus” that these derivations and inferences can be done in form of calculations, namely, in the ﬁrst part of logic as calculation with concepts leading to statements about the objects themselves, that is, categorical judgments, or, in Boole’s terminology, “primary propositions.” In its second part the logical calculus deals with statements about judgments as in conditional sentences, hypothetical or disjunctive judgments, or Boole’s secondary propositions. In this booklet Schröder simpliﬁed Boole’s calculus, stressing, as mentioned, the duality between logical addition and logical multiplication and, thus, the algebraic identity of the structures of these operations. Schöder developed his logic in a systematic way in the Vorlesungen über die Algebra der Logik (1890–1905) designing it as a means for solving logical problems (see Peckhaus 1998, 21–28). Again he separated logic from its structure. The structures are developed and interpreted in several ﬁelds, beginning from the most general ﬁeld of “domains” (“Gebiete”) of manifolds of arbitrary distinct elements, then classes (with and without negation), and ﬁnally proprositions (vol. 2, 1891). The basic operation in the calculi of domains and classes is subsumption, that is, identity or inclusion. Schröder presupposes a, and transitivity “If a b and at the same two principles, reﬂexivity a c, then a c.” Then he deﬁnes “identical zero” (“nothing”) and time b “identical one” (“all”), “identical multiplication” and “identical addition,” and ﬁnally negation. In the sections dealing with statements without negation, he proves one direction of the distributivity law for logical addition and logical multiplication, but shows that the other side cannot be proved; he rather shows its independence by formulating a model in which it does not hold, the “logical calculus with groups, e.g. functional equations, algorithms or calculi.” He thereby found the ﬁrst example of a nondistributive lattice.25 Schröder devoted the second volume of the Vorlesungen to the calculus of propositions. The step from the calculus of classes to the calculus of propositions is taken with the help of an alteration of the basic interpretation of the formulas used. Whereas the calculus of classes was bound to a spatial interpretation especially in terms of the part–whole relation, Schröder used in the calculus of

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propositions a temporal interpretation taking up an idea of Boole from his Laws of Thought (1854, 164–165). This may be illustrated regarding subsumption as the basic connecting relation. In the calculus of classes, a b means that the class a is part of or equal to the class b. In the calculus of propositions, this formula may be interpreted in the following way (Schröder 1891, §28, p. 13):

In the time during which a is true is completely contained in the time during which b is true, i.e., whenever . . . a is valid b is valid as well. In short, we will often say: “If a is valid, then b is valid,” “a entails b” . . . , “from a follows b.” Schröder then introduces symbols, the “sign of products” ,

two new logical and the “sign of sums” . He uses x to express that propositions referring

to a domain x are valid for any domain x in the basic manifold 1, and x to say that the proposition is not necessarily valid for all, but for a certain domain x, or for several certain domains x of our manifold 1, that is, for at least one x (Schröder 1891, 26–27).

§29, For Schröder the use of and in logic is perfectly analogous to arithmetic. The existential quantiﬁer and the universal quantiﬁer are therefore interpreted as possibly indeﬁnite logical addition or disjunction and logical multiplication or conjunction respectively.

This is expressed by the following deﬁnition, which also shows the duality of and (Schröder 1891, §30, 35). λ=n

λ=n aλ = a1 + a2 + a3 + · · · + an−1 + an aλ = a1 a2 a3 · · · an−1 an .

λ=1

λ=1

With this Schröder had all requirements at hand for modern quantiﬁcation theory, which he took, however, not from Frege but from the conceptions as developed by Charles S. Peirce (1839–1914) and his school, especially by Oscar Howard Mitchell (1851–1889).26 3.3.3. Logic of Relatives Schröder devoted the third volume of the Vorlesungen to the “Algebra and Logic of Relatives,” of which only a ﬁrst part dealing with the algebra of relatives could be published (Schröder 1895). The algebra and logic of relatives should serve as an organon for absolute algebra in the sense of pasigraphy, or general script, that could be used to describe most diﬀerent objects as models of algebraic structures. Schröder never claimed any priority for this part of his logic, but always conceded that it was an elaboration of Charles S. Peirce’s work on relatives (see Schröder 1905, XXIV). He illustrated the power of this new tool by applying it to several mathematical topics, such as open problems of G. Cantor’s set theory (e.g., Schröder 1898), thereby proving (not entirely correctly) Cantor’s proposition about the equivalence of sets (“Schröder-Bernstein Theorem”). In translating Richard

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Dedekind’s theory of chains into the language of the algebra of relatives, he even proclaimed the “ﬁnal goal: to come to a strictly logical deﬁnition of the relative concept ‘number of—’ [‘Anzahl von—’] from which all propositions referring to this concept can be deduced purely deductively” (Schröder 1895, 349–350). So Schröder’s system comes close, at least in its objectives, to Frege’s logicism, although it is commonly regarded as an antipode. 3.3.4. The Ideas of Peirce Although Schröder found his way to an algebraic approach to logic independently of Boole, he devoted his early work to a discussion and extension of the Boolean calculus. Main reference point of his mature Vorlesungen, however, was the logical work of the American “polymath” (Grattan-Guinness 2004, 545) Charles S. Peirce. Peirce contributed a great wealth of ideas to modern logic. He approached logic to its full range, interested not only in symbolic logic but also in a reform of traditional syllogistics and applications in the philosophy of science.27 In one of his ﬁrst papers on logic, Peirce improved Boole’s algebra of logic by introducing the inclusive disjunction as Jevons did before him (see Peirce 1868). He introduced “inclusion” as basic logical operator, in an algebraic spirit both for inclusion between classes and implication between propositions (Peirce 1870, WCSP 2, 360). It was later taken up by Ernst Schröder as “subsumption” . Among the ﬁve “icons” for nonrelative logic, “Peirce’s law” {(x y) x} x (see Peirce 1885, WCSP 5, 173) is outstanding. It produces an axiom system for classical propositional logic when being added to an axiom system for intuitionistic logic (see Beth 1962, 18, 128). In the paper “A Boolian Algebra with One Constant” (WCSP 4, 218–221), written around 1880, but not published before 1933, Peirce suggested replacing all logical connectors by only one interpreted as “neither P nor Q,” thereby anticipating the NOR operator, which was independently rediscovered by H. M. Sheﬀer in 1913 (Sheﬀer 1913). In his paper of 1870, Peirce took the ﬁrst step for developing a logic of relatives, thereby elaborating the ideas of De Morgan. He distinguished absolute terms, such as horse, tree, or man, from terms “whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation” (WCSP 2, 365). He discussed simple relative terms, that is, two-place relatives, and conjugate terms, that is, three- or four-place relatives like “giver of — to —” or “buyer of — for — from —” (ibid.). In his 1880 paper “On the Algebra of Logic,” he took up the topic, now speaking of singular reference for nonrelative terms and of dual and plural relatives for two- and more-place relatives. The most elaborated form of his algebra of relatives can be found in his 1885 paper, where he combined it with the theory of quantiﬁcation, the foundation of which had been formulated entirely independently of Frege by Oscar Howard Mitchell in Peirce’s Johns Hopkins logic circle (Mitchell 1883). Whereas Mitchell had developed a system limited

& & &

&

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to a theory of quantiﬁed propositional functions with two prenex quantiﬁers, Peirce developed quantiﬁers as operators on propositional functions over speciﬁc domains.28 In his 1885 paper, Peirce gave credit to Mitchell in the following way (WCSP 5, 178): All attempts to introduce this distinction [of some and all] into the Boolian algebra were more or less complete failures until Mr. Mitchell showed how it was to be eﬀected. His method really consists in making the whole expression of the propositions consist of two parts, a pure Boolian expression referring to an individual and a Quantifying part saying what individual this is. Peirce now used an index notation to express relatives. In the ﬁrst-order part of his logic (ﬁrst-intentional logic), xi yj signiﬁes that x is true of the individual i while y is true of the individual j. The quantiﬁers Σ and Π are used in analogy to their arithmetical meaning. Σi xi means that x is true of some one of the individuals denoted by i, Πi xi means that x is true of all these individuals. Applied to a ordinary language example: Let lij denote that i is a lover of j, and bij that i is a benfactor of j. Then Πi Σj lij bij means that everything is at once a lover and a benefactor of something (WCSP 5, 180). Peirce added considerations on second-intentional logic, that is, secondorder logic (ibid., 185–190) and many valued logic (ibid., 166). In later work he used furthermore “existential graphs” for a graphical representation of quantiﬁcational logic (see CP 4.293–584) which inspired several modern systems for graphical representations of logic (see, e.g., Sowa 1993, 1997). Peirce’s logical considerations were integral part of his triadic category system with ﬁrstness (possibility), secondness (existence), and thirdness (law), his semiotics, and his triadic theory of reasoning with deduction, induction, and abduction (see Hilpinen 2004, 622–628, 644–653). Most of Peirce’s path-breaking thoughts remained unpublished during his lifetime. What he was able to publish, however, excited his contemporary logicians. The best example is Schröder, whose Vorlesungen were deeply inﬂuenced by Peirce, even more, long passages read as critical comments on Peirce’s papers, especially on the seminal papers “On the Algebra of Logic” (Peirce 1880, 1885). In an intermediate word separating the halfs of volume two of the Vorlesungen Schröder wrote that after the completion of the ﬁrst half of volume two in June 1891 he had hoped to publish the second half with the logic of relatives in the autumn of the same year, but (Schröder 1905, XXIV): It is true, seldom in my life an estimation of mine failed to the same extent as then, when I judged the extension and the seriousness of the gaps in my manuscript. This was due to the fact that the only writing that seemed to be useful, Mr. Peirce’s paper on relatives [Peirce 1885], that became indeed the main basis of my volume three, has only a size of 18 pages in print (that could be printed

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on half the number of my pages), and that I thought, that I could get away with a largely reproducing report. I became aware of the enormous signiﬁcance of this paper when I worked at it in detail.

4. Conclusions Like the British tradition, but independent of it, the German algebra of logic was connected to new trends in algebra. It diﬀered from its British counterpart in its combinatorial approach. In both traditions, algebra of logic was invented within the enterprise to reform basic notions of mathematics which led to the emergence of structural abstract mathematics. The algebraists wanted to design algebra as “pan-mathematics,” that is, as a general discipline embracing all mathematical disciplines as special cases. The independent attempts in Great Britain and Germany were combined when Schröder learned about the existence of Boole’s logic in late 1873, early 1874. Finally he enriched the Boolean class logic by adopting Peirce’s theory of quantiﬁcation and adding a logic of relatives according to the model of Peirce and De Morgan. The main interest of the new logicians was to use logic for mathematical and scientiﬁc purposes, and it was only in a second step, but nevertheless an indispensable consequence of the attempted applications, that the reform of logic came into the view. What has been said of the representatives of the algebra of logic also holds for the proponents of competing logical systems such as Gottlob Frege or Giuseppe Peano. They wanted to use logic in their quest for mathematical rigor, something questioned by the stormy development in mathematics. For quite a while, the algebra of logic remained the ﬁrst choice for logical research. Authors like Alfred North Whitehead (1841–1947), and even David Hilbert and his collaborators in the early foundational program (see Peckhaus 1994c) built on this direction of logic, whereas Frege’s mathematical logic was widely ignored. The situation changed only after the publication of Whitehead’s and B. Russell’s Principia Mathematica (1910–1913). But even then important work was done in the algebraic tradition as the contributions of Clarence Irving Lewis (1883–1964), Leopold Löwenheim (1878–1957), Thoralf Skolem (1887–1963), and Alfred Tarski (1901–1983) prove.

Notes 1. Independently of each other, Gregorius Itelson, André Lalande, and Louis Couturat suggested at the 2nd Congress of Philosophy at Geneva in 1904 to use the name “logistic” for, as Itelson said, the modern kind of traditional formal logic. The name should replace designations like “symbolic,” “algorithmic,” “mathematical logic,” and “algebra of logic,” which were used synonymously up to then (see Couturat 1904, 1042). 2. For a book-length biography, see MacHale (1985). See also contemporary obituaries and biographies like Harley (1866), Neil (1865), both reprinted. For a

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comprehensive presentation of Boole’s logic in the context of British mathematics, see Grattan-Guinness (2000a). 3. Whately (1826). Risse (1973) lists 9 editions up to 1848 and 28 further printings to 1908. Van Evra (1984, 2) mentions 64 printings in the United States to 1913. 4. This opinion can be found in a letter of De Morgan’s to Spalding of 26 June 1857 (quoted in Heath 1966, xii) which was, however, not sent. Boole lists Hamilton among the “two greatest authorities in logic, modern and ancient” (1847, 81). The other authority is Aristotle. This reverence to Hamilton might not be without irony because of Hamilton’s disregard of mathematics. 5. See Hamilton 1859–1866, vol. 4 (1866), 287. 6. See his list of symbols in “Logical Notation” in Hamilton 1859–1866, vol. 4 (1866), 469–486. 7. For the priority struggle, see Heath 1966. 8. For diagrammatic methods in logic, see Gardner (1958), Bernhard (2000). 9. See the section “On Expression and Interpretation” in Boole (1847), 20–25, in which Boole gives his reading of the traditional theory of judgment. The section is followed by an application of his notation to the theory of conversion (ibid., 26–30) and of syllogism (ibid., 31–47). 10. This is the subtitle of Boole’s Mathematical Analysis of Logic (1847). 11. For the inﬂuence of Aristotelian logic on Boole’s philosophy of logic, see Nambiar (2000). 12. On the mathematical background of Boole’s Mathematical Analysis of Logic, see Laita (1977), Panteki (2000). 13. On Gregory with focus on his contributions to the foundations of the calculus see Allaire and Bradley (2002). 14. On Boole’s “psychologism,” see Bornet (1997) and Vasallo (2000). 15. For the development of quantiﬁcation theory in the algebra of logic, see Brady (2000). 16. The reason was that already the factorization of x3 = x leads to uninterpretable expressions. On Boole’s Laws of Thought see Van Evra (1977); on the diﬀerences between Boole’s earlier and later logical theory see Grattan-Guinness (2000b). 17. On initial reactions see Grattan-Guinness (2000a), 54–59. 18. For a discussion of De Morgan’s logic see Grattan-Guinness (2000a), 25–37; Merrill (1990); Sánchez Valencia (2004), 408–410, 487–515. 19. On MacColl and his logic see Astroh and Read (1998). 20. See for the following chs. 3 and 4 of Peckhaus (1997), and Vilkko (2002). 21. On Schröder’s biography, see his autobiographical note, Schröder (1901), which became the base of Eugen Lüroth’s widely spread obituary, Lüroth (1903). See also Peckhaus (1997), 234–238; and Peckhaus (2004a). 22. On the various aspects of H. G. Graßmann’s work, see Schubring (1996); Lewis (2004). 23. On Robert Graßmann’s logic and his anticipations of lattice theory see Mehrtens (1979); Peckhaus (1997), 248–250. On the relation zwischen Schröder and the Graßmann brothers see Peckhaus (1996). 24. On Schröder’s algebra of logic see Peckhaus (2004a); Sánchez Valencia (2004), 477–487; Brady (2000). 25. See Schröder (1890), 280. On Peirce’s claim to have proved the second form as well (Peirce 1880, 33) see Houser (1991). On Schröder’s proof see Peckhaus (1994a), 359–374; Mehrtens (1979), 51–56.

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26. See Mitchell (1883), Peirce (1885). On the development of modern quantiﬁcation theory in the algebra of logic see Brady (2000); Peckhaus (2004b). For Mitchell’s biography, see Dipert (1994). 27. For recent work on Peirce’s Logic, see Houser, Van Evra, and Roberts (1997); Brady (2000); Grattan-Guinness (2000a), 140–156; Hilpinen (2004). 28. Brady (2000), 6; see Peirce (1883). For Peirce’s interpretation of Mitchell see also Haaparanta (1993), 112–116.

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5

Gottlob Frege and the Interplay between Logic and Mathematics Christian Thiel

Gottlob Frege (1848–1925) has been called the greatest logician since Aristotle, but it is a brute fact that he failed to gain inﬂuence on the mathematical community of his time (although he was not ignored, as some have claimed), and that the depth and pioneering character of his work was—paradoxically— acknowledged only after the collapse of his logicist program due to the Zermelo– Russell antinomy in 1902. Because of this lack of inﬂuence in his time, a leading historian of logic and mathematics has gone so far as to deny Frege a place in the development of mathematical logic. Other historiographers of science, however, are convinced that the history of visible eﬀects of great ideas on science and scientiﬁc communities should be complemented by the recognition even of solitary insights ineﬀective at their time, because the intellectual status of such insights or discoveries will yield most valuable (and otherwise unobtainable) information about the structure and quality of the community that made them possible by providing, as it were, the native soil for their development. Knowledge of this kind is not historically useless. The neglect of Frege by the contemporaneous scientiﬁc community has two very diﬀerent reasons. First, there is little doubt that Frege maneuvered himself out of the mainstream of foundational research (or rather, never succeeded in joining this mainstream) by his insistence on using his newly developed “Begriﬀsschrift,” a logical notation the sophistication and analytical power of which the experts of the nineteenth century (as, in fact, most of those of the twentieth and the early twenty-ﬁrst centuries) failed to recognize. And second, the double disadvantage of working in the no-man’s-land between formal logic and mathematics, and of teaching at the then relatively unimportant small university of Jena gave Frege a low status in the academic world. 196

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The superﬁciality of reception is manifest, for example, in Georg Cantor’s (1885) review of Grundlagen (1884) where Cantor criticizes in condescending manner an allegedly Fregean deﬁnition of (whole) number, whereas the deﬁnition actually found in Grundlagen is quite diﬀerent and would have been worthy of a more careful study. Frege’s correction in his “Erwiderung” (1885) (which he had to publish as a—presumably paid—advertisement) went practically unnoticed. Similarly, already Ernst Schröder in his (1880) review of Frege’s Begriﬀsschrift (1879) had overlooked Frege’s revolutionary technique of quantiﬁcation, claiming (incorrectly) that its eﬀects could have been achieved in a much easier way by Boolean methods. If Frege has been regarded as the founder of modern mathematical logic, this characterization refers to his creation of classical quantiﬁcational logic in his Begriﬀsschrift of 1879 without any predecessor. As to Frege’s motivation, one can only surmise that he felt the urgent need for a logically water tight clariﬁcation of fundamental concepts of analysis like convergence, continuity, uniform continuity, and so on, the precise deﬁnition of which requires nested quantiﬁcation. The mathematical output of the Begriﬀsschrift approach rested on Frege’s replacement of the traditional analysis of elementary propositions into subject and predicate by the general analysis of a proposition into (in our case, propositional) function and argument(s), and its utilization for the expression of the generality of a statement (and of existence statements) by the employment of bound variables and quantiﬁers. For the antecedent part, classical propositional logic, Frege gave a consistent and complete (although not independent) axiom system in terms of negation and conditional, pointing out that other, equivalent axioms and also other connectives could be used, and he managed to get along with the rule of detachment and (not yet suﬃciently precise) substitution rules. In quantiﬁcational logic, he restricted himself to universal quantiﬁcation (which, together with negation, allows the expression of existential statements), and introduced the decisive concepts of the variability domain of a quantiﬁer and the scope of a quantiﬁer and of the quantiﬁed variable. The new devices enabled Frege to precisely deﬁne, for the ﬁrst time, one-one relations, a logical successor and predecessor relation, and a logical heredity relation, in such a way that the arithmetical successor and heredity relations are covered as special cases, and mathematical induction can be formulated, and turns out to have a purely logical foundation. Frege’s Foundations of Arithmetic (Die Grundlagen der Arithmetik, 1884) added, after an elaborate criticism of earlier and contemporary views on the concept of number and on arithmetical statements, a “purely logical” (today dubbed “logicist”) notion of whole number by deﬁning the number n as the extension of the concept “equinumerous to the concept Fn ,” where Fn is a model concept with exactly n objects falling under it, and of a purely logical nature guaranteed by starting with F0 = ¬x = x and constructing Fn+1 recursively from Fn . Frege’s attainment of this notion is somewhat curious because immediately before that he had described and analyzed an attempt at deﬁning number by abstraction directly from equinumerous concepts, but had repudi-

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ated this attempt because of diﬃculties that he considered insurmountable, so that he decided on the explicit deﬁnition just given. Grundlagen also introduced important logical distinctions like that of ﬁrst-order and second-order concepts, with existence and number predicates as examples of the latter. When Frege published Function und Begriﬀ in 1891 and volume 1 of his monumental Grundgesetze der Arithmetik in 1893, he had already realized that extensions of concepts, naively regarded as unproblematic in the explicit deﬁnition of number, must be introduced by an abstraction principle, too. As extensions of concepts have been a main topic of traditional logic at least since the Logic of Port Royal (1662), Frege’s treatment of abstraction in Grundlagen and in Grundgesetze centered around his discovery of the invariance property of statements about “abstract objects,” the logicist deﬁnition of number, and the general abstraction principle (exempliﬁed in Grundgesetze by Frege’s fundamental law V, vide infra) are legitimate and indeed indispensable topics of the history of formal logic. By contrast, the so-called context principle (“The meaning of a word must be asked for in the context of a proposition, not in isolation,” Grundlagen, p. X) and the dichotomy of sense and reference developed in Über Sinn und Bedeutung (1892), often regarded as his most important contribution to philosophy by drawing guidelines for semantics and for a general theory of meaning, have only a negligible role in the history of logic. However, the latter distinction is put to use by Frege in explaining the informative or cognitive value even of judgments that are derived from and therefore based on purely logical premises (as, e.g., according to the logicist thesis, all nongeometrical mathematical theorems), and is of considerable interest for the philosophy of mathematics. To derive the fundamental theorems of arithmetic precisely, that is, within a calculus incorporating strict formation rules for “well-formed formulas” and rules for the logical derivation of conclusions from premises, Frege had to revise and to augment his Begriﬀsschrift. The typically ambiguous “quantiﬁcation axiom” (Begriﬀsschrift, pp. 51 and 62) is now neatly split into a ﬁrst-order and a second-order version (Grundgesetze I, p. 61), but the most momentous change consists in the introduction of new terms of the general form “ Φ(ε),” considered to be names of a new kind of objects called courses-of-values or value-ranges (Wertverläufe) the identity condition for which is given by an abstraction principle accepted by Frege as his fundamental law V, the ﬁfth axiom of his new axiom system:

Φ(ε) = Ψ(α) ⇔ (x)(Φ(x) = Ψ(x)), where Φ(x) and Ψ(x) are functions in Frege’s general sense and the right side of the equivalence expresses the coincidence of their values for every argument, a state of aﬀairs suggesting the identity of the “courses” (or graphs) of the functions in the case of mathematical functions, and thereby the terminology of “courses-of-values.” Frege decided to regard true propositions as names of the “truth value” TRUE and false propositions as names of the “truth value”

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FALSE, respectively, and reconstructed the traditional concepts as one-place functions, the function value of which is one of the two truth values for every argument chosen. So the courses-of-values of such functions are nothing else but the traditional extensions of concepts, or mathematically spoken, the sets or classes determined by the associated function as their deﬁning condition. If concepts are taken as the functions in Frege’s fundamental law V, we get for this special case (in modern notation), { x | F (x) } = { x | G(x) } ⇔ (x)(F (x) ↔ G(x)). In this way, sets have obviously been integrated into the system of Grundgesetze, and since Frege (linking up with the traditional logic of concepts and their extensions) considers abstraction a purely logical operation, set theory becomes (or remains) a proper part of logic. The derivation of arithmetical theorems from the revised and enlarged axiom system of Grundgesetze keeps well within the limits of logic, and in this sense the present set-theoretical foundation of mathematics preserves the intentions and the spirit of Fregean logicism. It was mentioned in the beginning that Frege’s Grundgesetze system foundered at Zermelo’s and Russell’s antinomy, as shown in the appendix of volume 2 of Grundgesetze as well as in Russell’s The Principles of Mathematics, both published in 1903. Though Russell proposed to avoid the antinomy by his type theories, Frege suggested a repair of the axiom system by modifying his fundamental law V; it was shown only much later that this attempt, which has been called “Frege’s way out,” also leads to an impasse by allowing the derivation of other, more complicated antinomies. It is remarkable that the discovery and analysis of Zermelo’s and Russell’s antinomy was made possible only by the extraordinary precision, explicitness, and cogency of Frege’s Grundgesetze system, which in spite of its inconsistency remained a paradigm of a well-designed logical system well into the twentieth century. Among the little-known but precious parts of Grundgesetze, §§90 ﬀ. deserve to be highlighted because of their clear analysis of the nature and the necessary properties of an elementary proof theory and metalogic (“Die formale Arithmetik und die Begriﬀsschrift als Spiele”: Grundgesetze II, p. IX). Attention should also be given to hitherto neglected parts like Frege’s derivation of theorem χ in the appendix to Grundgesetze, where a diagonal argument is used to exhibit a fundamental inconsistency in the (traditional) notion of the extension of a concept (see Thiel 2003). Even the origin of the antinomy has not been located unequivocally up to now. According to the received view in current Frege literature, fundamental law V is responsible for the equivalent of Russell’s antinomy in Grundgesetze. This diagnosis, however, seems a bit rash. It is true that the derivation makes use of fundamental law V, but a careful analysis of it has to inspect not only the logical form of that law but also the structure of the formulae which replace the schematic letters of fundamental law V in every inference that has an instance of it as a premise. Thiel (1975) has tried to show that Frege’s formation rules for function names (which include rules for forming function

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names by the creation of empty places in complex object names) may be too liberal by allowing impredicative function names, and that names of that kind are essentially involved in the derivation of the Zermelo–Russell antinomy in Grundgesetze. A decision on this claim and the questions it raises is still open. A large part of Fregean studies in the past 50 years has been devoted to the investigation of problems that are peculiar to Frege’s systems, without visible impact on the development of the mainstream of mathematical logic invoked in the second paragraph of this chapter. Topics of this kind have been skipped here in spite of their intrinsic interest (as, e.g., the “Julius Caesar problem,” the permutation theorem, and the identiﬁcation thesis of Grundgesetze §10, Frege’s miscarried attempt at a referential completeness proof—which would have implied the consistency of the Grundgesetze system—and last but not least “Hume’s principle” and “Frege’s theorem”). A great thinker’s legacy consists not only in far-reaching insights and eﬃcient methods, it also comprises challenging problems, the solutions of which may sometimes occupy whole generations. Frege, by proving his theorem χ without recourse to Wertverläufe, exhibited an inconsistency (or at least an incoherence) in the traditional notion of the extension of a concept. He prompted our awareness of a situation the future analyses of which will hopefully not only deepen our systematic control of the interplay of concepts and their extensions but also improve our understanding of the historical development of the notion of “extension of a concept” and its historiographical assessment.

Primary Texts (Frege) 1879 Begriﬀsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a.S.: Louis Nebert; repr. in: G. Frege, Begriﬀsschrift und andere Aufsätze, ed. Ignacio Angelelli, Hildesheim: Georg Olms, 1964; Engl. Begriﬀsschrift, a formula language, modeled upon that of arithmetic, for pure thought (transl. Stefan Bauer-Mengelberg), in: Jean van Heijenoort (ed.), From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931 (Cambridge, Mass.: Harvard University Press, 1967), 1–82; Conceptual Notation. A Formula Language of Pure Thought, Modelled upon the Formula Language of Arithmetic, in: G. Frege, Conceptual Notation and Related Articles (trans. and ed. Terrell Ward Bynum, Oxford: Clarendon Press, 1972), 101–203. 1884 Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriﬀ der Zahl, Breslau: Wilhelm Koebner; crit. ed. Christian Thiel, Hamburg: Felix Meiner, 1986; Engl. in: G. Frege, The Foundations of Arithmetic. A logico-mathematical enquiry into the concept of number (bilingual ed., trans. J. L. Austin), Oxford: Basil Blackwell, 1950, rev. 2 1959). 1885 Erwiderung [to Cantor 1885, vide infra], Deutsche Litteraturzeitung 6, no. 28 (11 July 1885), Sp. 1030; Engl. “Reply to Cantor’s Review of Grundlagen der Arithmetik,” in: G. Frege, Collected Papers on Mathematics, Logic, and Philosophy, ed. Brian McGuinness (Basil Blackwell: 1984), 122.

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1891 Function und Begriﬀ. Vortrag gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft für Medicin und Naturwissenschaft. Jena: Hermann Pohle; Engl. in Beaney (1997), The Frege Reader (vide infra), 130–148. 1892 Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik 100: 25–50; Engl. in The Frege Reader (vide infra), 151–171. 1893 Grundgesetze der Arithmetik. Begriﬀsschriftlich abgeleitet, I. Band, Jena: Hermann Pohle. 1903 Grundgesetze der Arithmetik. Begriﬀsschriftlich abgeleitet, II. Band, Jena: Hermann Pohle.

Secondary Literature Beaney, Michael. 1997. The Frege Reader. Oxford/Malden, MA: Blackwell. Cantor, Georg. 1885. [Review of] G. Frege, Die Grundlagen der Arithmetik. . . . Deutsche Litteraturzeitung 6, no. 20 (16 May 1885): 728–729. Russell, Bertrand. The Principles of Mathematics. Vol. I. Cambridge: Cambridge University Press, 1903; London: George Allen & Unwin, 1937 [“Vol. I” omitted]. Schröder, Ernst. 1880. [Review of] G. Frege, Begriﬀsschrift. . . . Zeitschrift für Mathematik und Physik 25 (1880), Historisch-literarische Abtheilung, 81–94; Engl. Review of Frege’s Conceptual Notation. . . , in: G. Frege, Conceptual Notation and Related Articles (vide supra), 218–232. Thiel, Christian. 1975. Zur Inkonsistenz der Fregeschen Mengenlehre. In: idem (ed.), Frege und die moderne Grundlagenforschung. Symposium, gehalten in Bad Homburg im Dezember 1973. Meisenheim am Glan: Anton Hain, 134–159. Thiel, Christian. 2003. The extension of the concept abolished? Reﬂexions on a Fregean dilemma. In Philosophy and Logic. In Search of the Polish Tradition. Essays in Honour of Jan Woleński on the Occasion of his 60th Birthday, eds. Jaakko Hintikka, Tadeusz Czarnecki, Katarzyna Kijania-Placek, Tomasz Placek, and Artur Rojszczak (†). Dordrecht/Boston/London: Kluwer; Synthese Library, vol. 323), 269–273.

Selected Further Readings Angelelli, Ignacio. 1967. Studies on Gottlob Frege and Traditional Philosophy, Dordrecht: D. Reidel. Demopoulos, William, ed. 1995. Frege’s Philosophy of Mathematics. Cambridge, Mass./London: Harvard University Press. Dummett, Michael. 1973. 2 1981. Frege: Philosophy of Language. London: Duckworth. Dummett, Michael. 1981. The Interpretation of Frege’s Philosophy. Cambridge, Mass.: Harvard University Press. Dummett, Michael. 1991. Frege: Philosophy of Mathematics. London: Duckworth. Frege, Gottlob. 1969. Nachgelassene Schriften. Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach (eds.). Hamburg: Felix Meiner, revised and enlarged 2 1983; Engl. Posthumous Writings (trans. Peter Long/Roger White), Oxford: Basil Blackwell, 1979. Haaparanta, Leila and Jaakko Hintikka, eds. 1986. Frege Synthesized. Essays on the Philosophical and Foundational Work of Gottlob Frege. Dordrecht: D. Reidel.

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Kreiser, Lothar. 2001. Gottlob Frege. Leben–Werk–Zeit. Hamburg: Felix Meiner. Schirn, Matthias, ed. 1996. Frege: Importance and Legacy. Berlin/New York: Walter de Gruyter. Sluga, Hans D. 1980. Gottlob Frege. London/Boston/Henley: Routledge & Kegan Paul. Weiner, Joan. 2004. Frege Explained. From Arithmetic to Analytic Philosophy. Chicago/La Salle, Ill.: Open Court.

6

The Logic Question During the First Half of the Nineteenth Century Risto Vilkko

Immanuel Kant wrote, in the preface to the second edition of his Kritik der reinen Vernunft, that since Aristotle it [logic] has not required to retrace a single step, unless, indeed, we care to count as improvements the removal of certain needless subtleties or the clearer exposition of its recognized teaching, features which concern the elegance rather than the certainty of the science. It is remarkable also that to the present day this logic has not been able to advance a single step, and is thus to all appearance a closed and completed body of doctrine. (KrV, B VIII) Kant’s division of logic into its general and transcendental aspects served, during the early nineteenth century, as the basis for the removal of philosophers of logic into, roughly speaking, two opposing camps of the Herbartian formal logicians and the Hegelian idealist metaphysicians. Also it can be assumed that Kant’s disbelief in the possibilities of logic to develop any further from its alleged Aristotelian perfection discouraged many philosophers from trying to improve the logic proper and led most of them, instead, to studying the “applications” of logic, that is, the ﬁelds of study that are nowadays referred to as epistemology, psychology, methodology, and the philosophy of science. However, not all logicians of the early and mid-nineteenth century took Kant’s conception for granted. Herbart saw a promise of further development in Drobisch’s Neue Darstellung der Logik (Herbart 1836, 1267f.). Beneke wrote a few years later that even though Kant’s conception may have felt more or less credible during the 1780s, “since then, the situation has greatly changed” 203

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(Beneke 1842, 1). According to him, “logic has lost its unchangeable character. It has adopted a variety of such aspects of the possibility of which the old logicians, including Kant himself, had no idea” (ibid.). Boole wanted to remark, in 1854, that “syllogism, conversion, &c., are not the ultimate processes of Logic. It will be shown . . . that they are founded upon, and are resolvable into, ulterior and more simple processes which constitute the real elements of method in Logic” (Boole 1854, 10). De Morgan had the courage to write in 1860 that in the ﬁeld of logic “innovations have been listened to in a spirit which seems to admit that Kant’s dictum about the perfection of the Aristotelian logic may possibly be false” (De Morgan 1860, 247). After Hegel’s death in 1831, there arose in the academic circles of Germany a lively discussion concerning the makings of logic both as a philosophical discipline and as a formal and fundamental theory of science which might clarify not only the logical but also the metaphysical foundations of science. In fact, this was perhaps the most popular theme in the philosophical exchange of thoughts in Germany during the mid-nineteenth century. The most characteristic slogans in the discussion were “the logic question” and “reform of logic.” These slogans did not have very speciﬁc meanings. They were used rather loosely to refer to various competing eﬀorts to reform logic. In 1880 Leonhard Rabus (1835–1916) characterized the logic question in his book on nineteenth-century German contributions in the ﬁeld of logic as circling around the fundamental problems of the possibility and justiﬁcation of logic (Rabus 1880, 157; see Vilkko 2002). According to another late nineteenth-century German philosopher, Friedrich Harms (1819–1880), reform of logic could be sought from logic as (1) an organon of sciences, (2) a critique of sciences, or (3) a philosophical science (Harms 1874, 124). As an organon, or as a discipline of the methods of sciences, logic is the science of the forms of thought. As a critique, or as a theory of the necessary preconditions of knowledge and knowing, logic considers such questions as: How are objects given to cognition? What are the basic principles of knowing? What justiﬁes these principles? And how valid are these principles? (Harms 1881, 137). Bacon’s “inductive” reform covered logic as an organon, whereas Locke and Kant treated logic as a critique (ibid., 150). As far as Harms was concerned, this was, however, not enough. Harms wanted to stress that these two aspects of logic must be taken simply as two diﬀerent sides of the one and the same logic. In his view, the most important aspect in this reform concentrated on the form of logic as a philosophical science. He argued as follows: Since logic is a philosophical science due to its content, it must be a philosophical science also due to its form, because the content and the form of a science must coincide. Purely formal logic is, however, originally an empirical science and thus only an instrument for philosophy. When logic is reformed as a philosophical science, it is also reformed as an organon as well as a criterion of correct and consistent thinking (ibid., 121–125, 130). Hermann Ulrici (1806–1884) deﬁned the logic question as “the question about the place, the context, and the working of logic” (Ulrici 1869/70, 1). He began his most important contribution to this debate, titled “Zur logischen

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Frage” (1869/70), by stating that his conception of logic is incompatible not only with that of Hegel but also with every other attempt that denies the purely formal character of logic and tries to identify logic with metaphysics, epistemology, and/or theory of science. According to him, logic “deserves the name of a fundamental science; and it is clearly impossible for such logic to be at the same time also metaphysics and a theory of science” (ibid., 8). In other words, the logic question sprung from a genuine doubt about the justiﬁcation of the formal foundations of logic as the normative foundation of all scientiﬁc activity. On the one hand, most of the participants of the debate opposed Hegel’s attempts to unite logic and metaphysics—on the other, reform was sought to overcome the old scholastic-Aristotelian formal logic. The discussion can thus be characterized as a battle on two fronts. In any case, the need to reform was stimulated by the developments in the ﬁeld of philosophy. As Volker Peckhaus has put it: the reform endeavors that were released through this discussion scarcely considered the formal logic itself, but rather its psychological foundations and its use in theories of science that strove to seize the positive and formal sciences of that time. (Peckhaus 1997, 12) The very slogan “logic question” was used for the ﬁrst time by Adolf Trendelenburg (1802–1872). His writings provoked anew an awareness of the problematic philosophical position of formal logic. What is more, it was from his initiative that the reform discussion of logic really started around the turn of the 1840s. In 1842, he asked in his essay “Zur Geschichte von Hegel’s Logik und dialektischer Methode” whether Hegel’s dialectical method of pure thought should be treated as a scientiﬁc one. His own answer to this question was negative (Trendelenburg 1842, 414). However, of more importance was his criticism of both Herbartian formal logic and Hegelian dialectical logic in his two-volume Logische Untersuchungen (Trendelenburg 1840, I, 4–99). Before going into the details of Trendelenburg’s criticism, let us take a closer look at Herbart’s and Hegel’s conceptions of the nature and the task of logic.

1. Herbart’s Theory of the Structures of Thought Johann Friedrich Herbart (1776–1841) deﬁned philosophy as cultivation and arranging of conceptual material that is given in sense-experience (Herbart 1813, 38f.). His basic division of the ﬁeld of philosophy was the that time usual tripartite one: logic, metaphysics, and aesthetics (the most important part of which was ethics). The task of logic was to take care of the ﬁrst and the foremost duty of philosophy, that is, of conceptual clarity. The task of metaphysics was to justify concepts as objects of thought by analyzing and resolving conceptual contradictions that originate from thought itself. The third constituent, aesthetics, complemented the objects of thought by an analysis of values (Ueberweg 1923, 156f.).

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Herbart remained in many ways faithful to Kant’s conceptions. In his logic writings he himself sketched only the very basics and trusted, when it came to more advanced issues, the textbooks of, for example, Wilhelm Krug and Jakob Fries. Logic meant for him a regulative science which merely establishes the ways of handling concepts as such and lays down the law of contradiction as their highest standard. In his supplement “Hauptpuncte der Logik” to the second edition of his Hauptpuncte der Metaphysik Herbart wrote: Indeed logic is concerned with representations but not with the practice of representing. Hence, it is neither concerned with the mode and the manner of how we get to a representation, nor with the conditions of mind that are given thereby, but only with what becomes represented. (Herbart 1808, 217) In his works, Herbart gives logic such deﬁnitions as, for example, “a general science of understanding” (Herbart 1813, 67) and “a theory of the structures of thought” (Herbart 1808, 222). Logic meant for him a fundamental science that occupies itself ﬁrst of all with separating, classifying, and combining concepts as such; thereafter with making and analyzing judgments; and ﬁnally with revealing the modes of inference. The task of logic was to develop the formal consequences from the given premises (Herbart 1808, 218; 1831, 204). The fundamental point of diﬀerence between Hegel’s and Herbart’s conceptions of logic dealt with the relation between logic and metaphysics. Whereas the former drew an identity between logic and metaphysics, the latter wanted to keep the two strictly separated from each other. Herbart also insisted that for the beneﬁt of pure logic, it is necessary to avoid all psychological considerations. In the second chapter of his Lehrbuch zur Einleitung in der Philosophie, Herbart summarizes his logic conception in ﬁve theses: 1. Logic provides us with the most general regulations of separating, classifying, and combining concepts. 2. Logic presupposes concepts as known and does not distress itself with their speciﬁc contents. 3. Therefore logic is not really an instrument of such an investigation that aims at ﬁnding something novel. It rather gives instructions for revealing what we already know. 4. Nevertheless logic also points out the primary conditions of investigation in general and takes care of the important duty of paying attention to the possibility of committing errors. 5. The term “applied logic” refers to a combination of logic and psychology which, however, falls out as defective on issues where psychology does not already lead the way. (Herbart 1813, 41f.) Even though it is fully justiﬁed to characterize Herbart as an advocate of formal logic and to see his philosophy of logic as an oﬀspring of Kant’s

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empirical realism, it can be asked why the logicians of the early nineteenth century did not seem to be very interested in making an eﬀort toward further development of formal logic. One reasonable answer is based on the fact that was already pointed out in the beginning of this chapter. That is, because during the early nineteenth century logic sprung straight from Kant’s view, according to which the inherited scholastic-Aristotelian logic was to be seen as closed and complete. The only thing there seemed to be left for logicians to deal with was its applications, such as epistemology, methodology, and the philosophy of science.

2. Drobisch’s Formal Philosophy Around the mid-nineteenth century, the perhaps most eminent opponent to the idealist identiﬁcation of logic and metaphysics was the mathematician, astronomer, and philosopher Moritz Wilhelm Drobisch (1802–1896). He was one of the most important and insightful thinkers in the Herbartian school. In the ﬁrst paragraph of the introduction to his Neue Darstellung der Logik (1836), Drobisch introduced philosophy as “the general science” (Drobisch 1836, 1). According to him, it was not the task of philosophy to investigate the auxiliary apparatus of subjective cognition. That was the task of special sciences. For him, like for Herbart, philosophy meant working with purely conceptual material and trying to reach understanding of concepts in themselves (ibid., 2). In Neue Darstellung der Logik, Drobisch focused on what he called “formal philosophy,” that is, logic. He introduced logic as the doctrine of the conditions of correct and consistent thinking (ibid., 5). In his view, logic must not be understood as a description of human thinking. It must not be considered as a descriptive natural history of thinking, but rather as a normative discipline of thought in general or as a “Code of Laws of Thought” (ibid., 6). In this connection Drobisch referred to Kant’s description of logic not as a descriptive but as a demonstrative a priori science, the function of which is to take care of the necessary laws of thought, that is, as the science of the adequate use of understanding and of reason in general (see KrV, B IX, XXII). Drobisch regretted the fact that the law-giving character of logic had been spoiled by the Kantian school. Therefore he saw it necessary to once again impress on philosophers the importance of this normative aspect of logic (Drobisch 1836, 7). Drobisch was concerned about the purity of logic. Already in the preface to the ﬁrst edition of Neue Darstellung der Logik he made it clear that in what follows, logic will be understood as an independent and autonomous formal foundation of all scientiﬁc activity. He wrote that “logic is, in fact, nothing but pure formalism. It is not meant to be, and must not be, anything else” (ibid., VI). Moreover, he did not consider logic as a branch of mathematics (ibid., VIII–X). At the end of the ﬁrst edition of Neue Darstellung der Logik, there is an exceptional and incisive logico-mathematical appendix (ibid., 127–167). In this

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appendix Drobisch concentrates on the problematic connection between logic and mathematics, and introduces his algebraic construction of the simplest forms of judgment and derivation of inferences founded thereupon (ibid., 131– 136). In eﬀect, he develops an extensional algebraic calculus of classes and elementary judgments. Drobisch’s calculation apparatus follows Aristotelian theory of syllogisms. To give just a couple of examples, the classical modi BARBARA and DARAPTI are presented in the following way (ibid., 134f.): BARBARA

M =p S = m (< M = p ) S = p (where p < p)

DARAPTI

M =p M =s s=p

From today’s perspective, Drobisch’s calculus appears as one of the most interesting chapters of Neue Darstellung der Logik. It has been valued as an improvement in comparison with the intensional systems of his famous predecessors Ploucquet, Lambert, Gergonne, and Jacob Bernoulli (Thiel 1982, 763). Unfortunately, Drobisch removed his calculus from the subsequent editions of Neue Darstellung der Logik, which became one of the leading German textbooks of logic during the nineteenth century. He also executed a number of other modiﬁcations to the later editions of his book. In the preface to the second edition, he even admitted that the changes that had been carried out were so extensive that one could almost speak of two altogether diﬀerent books (Drobisch 1851, III). Trendelenburg’s hard criticism was undoubtedly an important catalyst for these changes. But was it the decisive one? Drobisch’s uncompromising attitude toward Trendelenburg in the preface to the second edition suggest that perhaps he just felt that after all his calculus was not quite ripe to be published.

3. Hegel’s Dialectical Logic The ﬁrst one of the two most important sources of Georg Wilhelm Friedrich Hegel’s (1770–1831) dialectical logic is his monumental Wissenschaft der Logik (1812/16). This much debated and thoroughly interpreted work was Hegel’s attempt to provide a comprehensive philosophical synthesis of his union of logic, metaphysics, and epistemology. The second source of Hegel’s logic is Encyclopädie der philosophischen Wissenschaften im Grundrisse, of which Hegel prepared and published three diﬀerent versions—the ﬁrst one in 1817 and the two others in 1827 and 1830. Of particular interest here is the ﬁrst part of the book: “Die Logik, die Wissenschaft der Idee an und für sich” (§§19–244). In the following we pay attention only to the third edition, which was published the year before Hegel’s death. It can be regarded as Hegel’s philosophical

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testament. Reading it this way requires, however, keeping in mind that it does not provide the reader with a thoroughly elaborated and fully developed system but only with a sketch of the foundations of one. And when considering the relations between Wissenschaft der Logik and the Encyclopädie, it is worth knowing that Hegel kept on working intensely with the former one until his death (Nicolin and Pöggeler 1959, xxix, xxxviif.). It is no surprise that Hegel’s and Herbart’s thought with regard to logic did not meet, because the very foundations and the aims of the two diﬀered from each other as greatly as day and night. The following quotation from the introduction to Wissenschaft der Logik gives the reader a hint of the distance between Hegel’s understanding of the term “logic” and that of the Herbartians: logic is to be understood as the system of pure reason, as the realm of pure thought. This realm is truth as it is without veil and in its own absolute nature. It can therefore be said that this content [of pure science] is the exposition of God as he is in his eternal essence before the creation of nature and a ﬁnite mind. (Hegel 1812, 31) Hegel built his dialectical logic on the trivet of Kant’s transcendental logic, Schelling’s identity between the real and the ideal, and Fichte’s Wissenschaftslehre. The resulting theory was designed to serve the needs of his own monumental philosophical system, which divides into the three main constituents of Science of Logic, Philosophy of Nature, and Philosophy of Spirit. Logic was regarded as the foundational science of the system. Hegel’s works provide the reader with the most comprehensive theory of metaphysical philosophy of logic. In his philosophy there is no way of separating logic and metaphysics from one another. In the Encyclopädie Hegel states that “logic therefore coincides with Metaphysics, the science of things set and held in thoughts—thoughts accredited able to express the essential reality of things” (Hegel 1830, 58). If Hegel’s philosophy in toto is a science about the real world of change and development, understood as the collective self-education of humanity about itself, then logic is the construction of the history of thinking. In the Encyclopädie Hegel deﬁnes logic as “the science of the pure Idea; pure, that is, because the Idea is in the abstract medium of Thinking” (ibid., 53). His logic does not consider the categories merely as forms of subjective thinking. They are also seen as the forms of objective Being itself. The Absolute or the Reason—which is the ultimate subject matter of Hegel’s philosophy—is a union of Thinking and Being, and it is the task of logic to develop this unity. Accordingly Hegel divided his logic into two parts: (1) the objective logic concerned with the Being, and (2) the subjective logic concerned with the Thinking. In Wissenschaft der Logik Hegel attacked ﬁercely the “dull and spiritless” (Hegel 1812, 34) attempts of formal logicians to elaborate logic as the most general deductive science of thinking. According to him, the deduction of the “so-called rules and laws, chieﬂy of inference is not much better . . . than a childish game of ﬁtting together the pieces of a colored picture puzzle”

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(ibid.). This “thinking” which formal logic strove to govern constituted for Hegel the mere form of cognition. According to him, formal logic abstracted from all content of cognition and the material constituents of knowledge are, consequently, totally independent of it. Thus, formal logic could provide only “the formal conditions of genuine cognition and cannot in its own self contain any real truth, nor even be the pathway to real truth because just that which is essential in truth, its content, lies outside logic” (ibid., 24). The two most important concepts in Hegel’s critique of the traditional scholastic-Aristotelian conception of logic are “formal” and “abstract.” His accusation of traditional logic as being merely formal thinking is based on a conception according to which logical form and content should correspond to each other. This requirement has several consequences. It explains why logic embraces for him, in addition to the problem of classiﬁcation of propositions and inferences, also the study of the categories on which these classiﬁcations are based. It implies, for example, that logic also deals with the distinctions between diﬀerent levels of knowledge correlative to the various aspects of reality given in the categories. It also has implications for Hegel’s conceptions of judgment and truth (Kakkuri 1983, 41). Hegel’s conclusion of the history of logic until the early nineteenth century was as follows: Before the dead bones of logic can be quickened by spirit, and so become possessed of a substantial, signiﬁcant content, its method must be that which alone can enable it to be pure science. In the present state of logic one can scarcely recognize even a trace of scientiﬁc method. It has roughly the form of an empirical science. (Hegel 1812, 34f.) This leads us to the central topic of Wissenschaft der Logik, that is, the problem of appropriate philosophical method. Hegel’s starting point was the assumption that philosophy had not yet found a method of its own, but merely borrowed bits and pieces from the methodologies of various sciences and, in particular, “regarded with envy the systematic structure of mathematics” (ibid., 35). The connection between logic and philosophy is inextricable because “the exposition of what alone can be the true method of philosophical science falls within the treatment of logic itself; for the method is the consciousness of the form of the inner self-movement of the content of logic” (ibid.). From Kant’s remark of elementary logic having neither lost nor gained any ground since the time of Aristotle, Hegel drew a very radical conclusion. He held the same opinion as Kant in stating that logic had not undergone any positive changes in more than 2000 years. But judging by the logic-compendia of his time the few traceable changes appeared to him as consisting “mainly in omissions” (ibid., 33). Therefore he concluded that it is “necessary to make a completely fresh start with this science” (ibid., 6). Hegel’s logic dealt not only with the traditional Aristotelian laws of thought or Kant’s logic of the understanding but also with metaphysical issues. He

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begun the introduction to the ﬁrst book of Wissenschaft der Logik by regretting the fact that “what is commonly understood by logic is considered without any reference whatever to metaphysical signiﬁcance” (ibid., 28). His at ﬁrst relative and later absolute identiﬁcation of logic and metaphysics is the fundamental point of diﬀerence between Hegel’s philosophy of logic and that of the Herbartian school. There is no undisputable clear-cut answer to the question “What kind of logic is Hegel’s logic?” It is certainly not, for example, a doctrine of the laws of formally impeccable inference. Yvon Gauthier (1984) has suggested an answer that amounts to saying that Hegelian logic is a transcendental logic, which in turn would be the study of the a priori structures of logical thought. According to him, what Hegel calls objective logic is nothing less than metaphysics in the traditional sense, and therefore it is justiﬁed to consider Hegelian logic as transcendental-metaphysical. For Hegel, transcendental-speculative logic, which deals with the most general features of thought, reaches even further than it does for Kant (ibid., 303f.). Whatever the truth, in any case Hegel’s program was one of the most inﬂuential eﬀorts to reform logic during the nineteenth century.

4. Trendelenburg’s Logical Investigations Friedrich Adolf Trendelenburg (1802–1872) was not concerned with logic as mere doctrine of the laws of correct inference. The ﬁrst two chapters of his greatest work, the two-volume Logische Untersuchungen (1840) can well be considered to discuss philosophy of logic in today’s sense of the saying, but the rest of the book—the essence of Trendelenburg’s logical investigations— is perhaps best characterized as fundamental epistemology with a strong metaphysical ﬂavor. Trendelenburg’s intention was to solve what he considered to be the ultimate task of philosophy, namely, the apparent correspondence between “the external reality of Being and the internal reality of Thinking” (Trendelenburg 1840, I, 110). In eﬀect, his Logische Untersuchungen was an attempt ﬁrst to show the defects of both Herbartian and Hegelian logic and then to supplement and reformulate them to achieve a formal and fundamental theory of science and metaphysics.

4.1. Critique of Formal Logic When talking about formal logicians, Trendelenburg meant those philosophers who attempted to explain the pure forms of thought without paying attention to the contents of thought. This tradition rested, according to Trendelenburg, on a strict distinction between thoughts and their objects, that is, between Thinking and Being. Furthermore, because in Trendelenburg’s view these socalled formal logicians took truth as simple correspondence between thoughts and their objects, they also accepted silently the presupposition of harmony

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between forms of thought and things in themselves. In particular Trendelenburg wanted to criticize those philosophers who subscribed to Herbartian conception of logic (Trendelenburg 1840, I, 4–6). Formal theories of logic had traditionally begun with a theory of concepts. Accordingly, Trendelenburg begun his criticism with some critical remarks on traditional theories of concepts and speciﬁed the target of his criticism: “In particular we consider two ingenious and consistent presentations of formal logic, the famous works of A. D. Ch. Twesten [1825] and the University of Leipzig Professor Moritz Wilhelm Drobisch [1836]” (ibid., 7). He was not content with taking concepts as given and understanding them merely as sub- and superordinate combinations of properties. He criticized this view as much too naive for uncovering the secrets of the foundations of human thought. In his view, the traditional subordination of concepts was based on nothing but simple operations of adding and subtracting properties. According to him, every attempt to ﬁnd the essence of thought with the help of such basic operations as these—or with any such alternatives as multiplication and division—remains always futile. Every theory that rested on such theory of concepts became thus “more than dubious” (ibid., 8). Hence, according to Trendelenburg, the whole ediﬁce of formal logic was built on sand. However, for the sake of argument, he assumed that there is nothing wrong with formal theories of concepts and turned to examine the fundamental principle of classical formal logic, that is, the law of identity and contradiction: “A is A and A is not not-A.” Formal logicians had traditionally believed that in the ﬁnal analysis everything else in logic derives from this principle. Trendelenburg wanted to ﬁnd out if this belief really was tenable. Even though he admitted that the principle seemed unassailable at the ﬁrst sight, he wanted the reader to pay closer attention to the latter part of it: A given concept A stands in contradiction with its negation and is logically equivalent with its double negation. According to Trendelenburg, this “blindly accepted” (ibid., 11) interpretation was insuﬃcient for explaining the nuances with regard to contents of concepts. It reduced all of the various conceptual contrasts to the pure formal logical contradiction. Trendelenburg wanted to criticize this inﬂexibility. In his opinion, every purely formal deﬁnition for identity and negation fails to explain them properly (ibid., 11–14). It may be diﬃcult to understand why Trendelenburg wanted to make such an issue about formal logic not paying attention to the contents of judgments if one does not keep in mind that his logical investigations was an attempt to elect the best parts of metaphysics and logic and to reformulate them as a general, formal, and fundamental theory of science. In the introduction to Logische Untersuchungen, he wrote that “the range of these investigations must run through the sphere of logic questions and reach for insight on the whole ﬁeld of science” (ibid., 3). After having scrutinized both formal logic and dialectical method, Trendelenburg announced that the rest of his book will be committed to answering, with regard to the objective foundations of logic, the question about the possibility of knowledge (ibid., 100f.). His project

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was more ambitious than just explicating the laws of correct and consistent deductions. Trendelenburg also probed into diﬀerent types of inference and asked if it was possible to derive all the forms of inference from what the formal logicians regarded as the basic premises of formal logic, that is, the principle of identity and the idea of concepts as combinations of properties. He found nothing to complain about the classical forms of deductive inference. However, the problematic cases were logic of induction and inferences based on analogy. “What a great shame it is,” he wrote, “if there is no ability to understand the logic of induction and analogy expressed by science; and in case it is generality and not necessity that follows, then the principle of identity and contradiction is not the [basic] principle of logic” (ibid., 18). Indeed, according to Trendelenburg, inadequate understanding of logic of induction was one of the most alarming shortcomings of early nineteenth-century formal logic. Trendelenburg dedicated the end of the ﬁrst chapter to his favorite subject, Aristotelian philosophy. He had noticed that formal logicians often appealed to Aristotle and willingly called themselves Aristotelians. Trendelenburg, however, had found a number of reasons why they should not be regarded Aristotelian. According to him (ibid., 18–21): 1. Aristotle did not propose that the forms of thought should be understood purely in themselves; 2. Understanding concepts simply as given combinations of properties does not correspond to Aristotle’s reﬁned theory of concepts; 3. The nineteenth-century formulation for the principle of identity and contradiction, “A is A and A is not not-A,” diﬀers signiﬁcantly from Aristotle’s original formulation: “The same attribute cannot at the same time belong and not belong to the same subject” (Met. 1005b 18–20); 4. Aristotle did not regard aﬃrmation and negation as purely logical forms; 5. Aristotle considered modal judgments of necessity and possibility as rooted in the nature of things; 6. Aristotle did not postulate syllogisms as merely formal relations between judgments. This was roughly what Trendelenburg left in the hands of the public for deciding whether formal logic could be taken seriously with regard to the logic question.

4.2. Critique of Metaphysical Logic If it was the most serious defect of Herbartian formal logic to strictly separate Thinking from Being and to concentrate only on the former one, then Hegelian metaphysical logic was guilty of exactly the opposite. According to Hegel, Thinking and Being could not be separated from each other. In his system

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knowledge of Being derives straight from Thinking in itself. Trendelenburg, however, could not see how this could be possible. Shortly, in his opinion the biggest fault with Hegel’s conception of logic was the attempt to completely and unjustly neglect the decisive intermediate role of the Aristotelian concept of motion in obtaining knowledge (Harms 1881, 236). Trendelenburg’s criticism of Hegel’s dialectical method ranges over a broad ﬁeld of philosophical topics. In the following we shall, however, concentrate only on those aspects and arguments that can be regarded as belonging to the ﬁeld of philosophy of logic. Trendelenburg summarized the basic situation with dialectical method in the following way: The dialectical method strives for the greatest possible. It wants to develop and create the pure Idea as if in a divine intellect—solely out of itself. Content and form are supposed to arise simultaneously. Because the pure Idea brings forth only what lies deep in itself, it must create such a world where nothing exists in itself and every thought is a genuine part of the totality. . . . We must, however, resign [from the dialectical method] at once. The means are too frail for executing the plan of such a titanic project. (Trendelenburg 1840, I, 94) Trendelenburg’s ﬁrst argument concerned the alleged presuppositionless of Hegel’s logic. According to Hegel, pure Thought needed no support from perception or sense experience. The pure Idea was the stone foundation of his logic and vice versa. Therefore, according to Hegel, logic was both quite easy and extremely hard: From diﬀerent points of view, Logic is either the hardest or the easiest of sciences. Logic is hard, because it has to deal neither with perceptions nor, like geometry, with abstract representations of the senses, but with the pure abstractions; and because it demands a force and facility of withdrawing into pure thought, of keeping ﬁrm hold on it, and of moving in such an element. Logic is easy, because its facts are nothing but our own thought and its familiar forms or terms: and these are the acme of simplicity, the ABC of everything else. (Hegel 1830, §19) Trendelenburg could not accept this view. In his opinion Hegel’s “pure Thought” did not deserve its name because it could not escape from tacitly presupposing the fundamental principle of all knowledge, that is, the Aristotelian idea of motion. According to Trendelenburg, even the most elementary dialectical steps were impossible without support from this concealed principle: “Wherever we turn to, motion remains the presupposed vehicle of the dialectically breeding Thought. . . . This spatial motion is hereupon the ﬁrst assumption of this presuppositionless logic” (Trendelenburg 1840, I, 24–29; see also Petersen 1913, 156). Trendelenburg’s second argument against the dialectical method concerned the two seemingly logical relations of negation and identity. However, Trende-

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lenburg wanted to point out that with a closer look it can be seen that it is not the logical negation that works in Hegel’s system: What is the nature of this dialectical negation? It can have a twofold character. Either it is understood in a pure logical way, so that it simply denies what the ﬁrst concept aﬃrms without replacing it with something new, or it can be understood in a real way, so that the aﬃrmative concept is denied by a new aﬃrmative concept, in what way both of the two must be replaced with each other. We call the ﬁrst instance logical negation, and the second one real opposition. (Trendelenburg 1840, I, 31) Is it now possible for the logical negation, Trendelenburg asked further, to stipulate such progress that from a given denial a new positive concept arises which exclusively unites in itself both the aﬃrmation and the negation? According to Trendelenburg’s deﬁnition for logical negation, this was totally out of question. In other words, it would be a mistake to treat the dialectical negation as logical contradiction. Hence, it must be regarded as a real opposition. However, if it is a real opposition, then it is unattainable from the logical point of view and Hegel’s dialectic is not the dialectic of pure Thought. Hence the one who takes a closer look on the so-called negations of Hegel’s dialectical logic shall in most cases discover ambiguities (ibid., 30–45). According to the rules of Hegel’s dialectical logic, identity creates a new concept of a higher level out of a given concept and its opposite. This dialectical product is the truth of its “ingredients.” Hence, dialectical identity appears to be a real unit, even though it is, in the ﬁnal analysis, only a kind of shallow similarity of abstraction. Trendelenburg could not see how it could be possible for two distinct concepts to mutate into a third, new one. He wrote that “dialectical identity oﬀers more than it has” (ibid., 55). If the dialectical identity was supposed to be some kind of an impetus of the concrete reality, then it surely could not be an identity of abstraction. According to Trendelenburg, there is an obvious contradiction between the origin of the dialectical concept of identity and its alleged eﬀect (ibid., 45–56). Trendelenburg’s third point of criticism concerned Hegel’s conception of immediacy. In Aristotle’s philosophy, Trendelenburg clariﬁed, every such element of thought is immediate that does not reduce to any other element, for example, the basic elements of representation in general or certain particulars sensed in such a manner that nothing whatsoever comes in between the sensuous representation and its object. In the nineteenth-century philosophy it was, according to Trendelenburg, more customary to use the term “immediate” in the latter sense of the word. Since the whole dialectic was in the ﬁnal analysis nothing but a chain of mediation, immediacy was out of question in this sense. However, in Hegel’s system, the concept of immediacy is prominent everywhere in the dialectical process of mediation. Now it seemed to be, in Trendelenburg’s opinion, that in this context immediacy can only mean selfsubsistence, that is, Being-for-self. Hegel himself expressed this quite clearly

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in his Encyclopädie: “Being-for-self, as reference to itself, is immediacy, and as reference of the negative to itself, is a self-subsistent, the One. This unit, being without distinction in itself, thus excludes the other from itself” (Hegel 1830, §96). Later in the same volume he exclaims that “the immediate judgment is the judgment of deﬁnite Being. The subject is invested with a universality as its predicate, which is immediate, and therefore a sensible quality” (Hegel 1830, §172; emphasis added). In Trendelenburg’s opinion, this explanation left no room for misunderstanding: In Hegel’s logic the term “immediate” refers to something foreign to his system, that is, to something sensuous. And above we just saw how in Hegel’s dialectic the function of immediacy was by no means supposed to lead the way from pure thoughts to something sensuous. Thus, Hegel’s use of the term “immediate” remains ambiguous (Trendelenburg 1840, I, 56–59). Trendelenburg closed the ﬁrst chapter of Logische Untersuchungen by estimating whether it is right to regard the Herbartian projects of formal logic as latest extensions to the Aristotelian tradition of logic. Accordingly, at the end of the second chapter, he paid attention to dialectical method having sought for its original from Plato’s Parmenides dialogue (ibid., 89). The latter part of the Parmenides dialogue (137c–166c), where Socrates and Parmenides discuss the intertwined concepts of the one and many and the problematic relations between parts and wholes, has been interpreted in many diﬀerent ways since time immemorial. Hegel recognized a resemblance between Parmenides’s holistic concept of One and his own Absolute. Admittedly there are certain similarities. However, there are also other ways of understanding the passage. Another possibility is to read Parmenides simply as Plato’s reply to his critics. According to a number of scholars, the safest principle of interpretation is to excavate the hints that Plato himself gives to the reader for understanding the dialogue. Trendelenburg subscribed to this strategy. One of these hints is also the heart of Trendelenburg’s last argument against Hegel: In the beginning of the latter part of the dialogue Parmenides suggests that Socrates could use some training in the art of dialectic so that he might be more successful in searching for solutions to Parmenides’s philosophical dilemmas (135c–136a). Thus, the arguments and proofs of the latter part can be regarded as merely heuristic. It can be read as just an evaluation of various juxtaposed philosophical arguments and theses, some of which reappear in other dialogues by Plato—others being mere formal supplements to Parmenides’s arguments. Trendelenburg also held the opinion that it is hard to ﬁnd a credible uniform philosophical doctrine hidden beneath Parmenides’s lesson to Socrates (Trendelenburg 1840, I, 89). The task of Hegel’s dialectic was to show how a closed system could seize the whole reality. The outcome, however, failed to convince Trendelenburg. It seemed to him evident that the role of perception is silently assumed everywhere in Hegel’s theory and that the concepts of the pure Thought are, in the ﬁnal analysis, nothing but diluted representations. “Intuition is,” he wrote in the closing pages to the second chapter of Logische Untersuchungen, “vital for

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human thought and it starves to death if it must try to live on its own entrails” (ibid., 96). At the end of the whole work, Trendelenburg glanced back to his work and wrote: As we have seen, formal logic is essential but not suﬃcient for accomplishing the logical task. Hegel’s dialectic, in its turn, gives a promise of more—as a matter of fact of the greatest that can be imagined—but falls out as impossible. (Trendelenburg 1840, II, 363) As the discussion on the reform of logic moved on, there appeared certain general points of agreement concerning the basic nature and the task of logic. It became common to accept that the possible reform of logic must go hand in hand with the reform of philosophy. The Kantian appreciation of mathematics against its Hegelian devaluation became rehabilitated even though the question about the relationship between logic and mathematics remained diﬃcult. On the one hand formal logic became almost resistant to philosophical criticism, but on the other hand it lost at least part of its prestige as the foremost constituent of philosophy proper as it gradually was transformed into a subdiscipline of mathematics.

5. Herbartian and Hegelian Reactions to the Criticism Trendelenburg’s Logische Untersuchungen had a devastating impact in both the Herbartian and the Hegelian camps. Academic public expected the leading representatives from both sides to formulate and present counterarguments. This they also did. The leading Herbartian philosophers were, however, a little slower in defending themselves than their Hegelian colleagues. Even though Trendelenburg explicitly aimed his censure at Drobisch and August Twesten (1789–1876), apparently most of those who took part in the evaluation discussion of Logische Untersuchungen read the ﬁrst chapter of the book as censure of Herbartian logic and metaphysics. Drobisch wrote and published several articles in defense of Herbart. Twesten did not reply on Logische Untersuchungen. It took more than 10 years before Drobisch was ready to step forth with counterarguments. The ﬁrst set of his answers was published in 1851 in the preface to the second edition of his Neue Darstellung der Logik (1851). The second one came out the year after, in the form of a journal article (Drobisch 1852). Before these two contributions, only Hermann Kern had dared to defend Herbartian philosophy against Trendelenburg’s authority. Kern published an essay for justifying Herbartian metaphysics nine years after Logische Untersuchungen (Kern 1849). In addition to Drobisch and Kern, Ludwig Strümpell appears to be the only eminent Herbartian who had the courage to defend Herbartian philosophy in public against Trendelenburg with his essay “Einige Worte über Herbart’s Metaphysik in Rücksicht auf die Beurtheilung derselben durch Herrn Professor Trendelenburg” (1855). Even

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Trendelenburg himself was astonished to ﬁnd out how long it took for his opponents to prepare any answers to his criticism (Trendelenburg 1855, 317). In the second edition of Neue Darstellung der Logik (1851), Drobisch still subscribed to formal logic. In general, Drobisch accused Trendelenburg of making formal logic appear as if it was philosophically much less sophisticated than it really was. In particular he emphasized that Trendelenburg’s statement about formal logic totally separating thoughts from their objects is incorrect: Formal logic does not presuppose pure thought and does not attempt to analyze or explain the forms of thought in abstracto. . . . Formal logic does not recognize forms without content. It only recognizes such forms that are independent of particular contents which they might fulﬁl. Contents, which they cannot completely do without, remain thus indeterminate and accidental. (Drobisch 1851, IV) Drobisch also still held that there is no insurmountable dividing wall between Aristotelian logic and formal logic (ibid., III–XIV). A year later Drobisch admitted, in a journal article “Ueber einige Einwürfe Trendelenburg’s gegen Herbart’sche Metaphysik” (1852), that it might have been a good idea to reply on Logische Untersuchungen a little sooner. However, he thought that it still was not too late to break that silence. This article was, above all, an act in defense of Herbartian metaphysics. When it comes to Trendelenburg’s arguments against Herbartian philosophy of logic, this time Drobisch only referred brieﬂy to the preface to the second edition of his Neue Darstellung der Logik (ibid., 11–12) Evidently Trendelenburg had expected more vivid reactions to the “twoedged critique” of his Logische Untersuchungen. This is at least what the critique itself (Trendelenburg 1840, 4–99), his reply to Drobisch (Trendelenburg 1855), and its extension (Trendelenburg 1867) suggest. Perhaps he had even planned the ﬁrst two chapters of Logische Untersuchungen rather as an opening of a polemic than as a coup de grâce. At least Hegelian philosophers reacted a little faster. Hegelians were naturally very sore with Trendelenburg’s criticism. Diﬀerences between their published reactions were largely due to diﬀerent personal temperaments. For instance, the leading Hegelian of the 1840s, Karl Rosenkranz (1805–1879), could not quite understand how “a man, who is so throughly familiar with Aristotle’s philosophy, [could] have sunk so deep that he denies νοησιζ τηζ νοησεως from νους” (Rosenkranz 1844, xviif.). There were, however, other Hegelians who did not manage to keep themselves as dispassionate as Rosenkranz. Karl Michelet characterized Trendelenburg’s philosophy as “jumble” (Michelet 1861, 126), and Arnold Ruge wrote, in Deutsche Jahrbücher für Wissenschaft und Kunst, that those who are dull enough to be unable to recognize the progress Hegel has stimulated have no scientiﬁc importance—and even less do they possess positive political credibility. Their work is still-born,

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a matter of deepest ignorance and complete lack of strength. Those who cannot digest Hegel, cannot digest either the heroes of the German Spirit: Luther, Leibniz and Kant! (cited in Petersen 1913, 158) Trendelenburg’s victory over both of his opponents seems to have been undisputable. In 1859 Rosenkranz conﬁrmed, in his Wissenschaft der logischen Idee (1858/59), that as a consequence of Trendelenburg’s censure in his Logische Untersuchungen the whole discussion around Hegelian dialectic had come to durable stagnation—the advance of Hegelian philosophy had ceased. Some 30 years after Logische Untersuchungen, Hermann Bonitz wrote, in his memorial essay to Trendelenburg, that “in any case it is true that now, after three decades, the substantial inﬂuence of Hegelian philosophy has been conﬁned to a very modest group of faithful adherents and that Trendelenburg has had a considerable eﬀect on this change with his criticism” (Bonitz 1872, 23). Forty years after Logische Untersuchungen, Friedrich Harms valued Trendelenburg’s contribution to the nineteenth-century philosophy of logic as easily the most signiﬁcant. Either neglecting or not knowing what for instance Gottlob Frege and Ernst Schröder had recently accomplished, he wrote that Trendelenburg’s Logische Untersuchungen is the latest signiﬁcant attempt to reform logic. . . . we are living in a time of fragmentary eﬀorts to reform logic. These attempts do not have any accurate continuity. They attempt to remodel logic from greatly varying starting-points and with greatly varying results. The future will make the best out of what Lotze, Ulrici, Ueberweg, Chr. Sigwart, and others have accomplished. (Harms 1881, 238)

References Aristotle. 1928. Metaphysica. The Works of Aristotle. Translated into English, vol. 8, ed. W. D. Ross. Oxford: Clarendon Press. Beneke, Friedrich E. 1842. System der Logik als Kunstlehre des Denkens. Berlin: Dümmler. Bonitz, Hermann. 1872. Zur Erinnerung an Friedrich Adolf Trendelenburg. Abhandlungen der königlichen Akademie der Wissenschaften zu Berlin aus dem Jahre 1872. Berlin: Buchdruckerei der königlichen Akademie der Wissenschaften, 1–39. Boole, George. 1854. An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities. London: Walton & Maberly; Cambridge: Macmillan. De Morgan, Augustus. [1860] 1966. Logic. In De Morgan, On the Syllogism and Other Logical Writings, ed. Peter Heath, 247–270. Reprint, London: Routledge & Kegan Paul. Drobisch, Moritz W. 1836. Neue Darstellung der Logik nach ihren einfachsten Verhältnissen. Nebst einem logisch-mathematischen Anhange. Leipzig: Voß. Drobisch, Moritz W. 1851. Neue Darstellung der Logik nach ihren einfachsten Verhältnissen, mit Rücksicht auf Mathematik und Naturwissenschaft, Zweite, völlig umgearbeitete Auﬂage. Leipzig: Voß.

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Drobisch, Moritz W. 1852. Ueber einige Einwürfe Trendelenburg’s gegen Herbart’sche Metaphysik. Zeitschrift für Philosophie und philosophische Kritik (Neue Folge) 21: 11–41. Gauthier, Yvon. 1984. Hegel’s Logic from a Logical Point of View. In Logic, Methodology and Philosophy of Science VI. Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science, Hanover, 1979. Studies in Logic and Foundations of Mathematics, no. 104, eds. L. Jonathan Cohen, Jerzy Łoś, Helmut Pfeiﬀer, and Klaus-Peter Podewski, 303–310. Amsterdam, New York, Oxford: North-Holland. Harms, Friedrich. 1874. Die Reform der Logik (aus den Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin). Berlin: Vogt. Harms, Friedrich. 1881. Philosophie in ihrer Geschichte, Zweiter Theil, Geschichte der Logik. Berlin: Hofmann. Hegel, Georg W. [1812] 1923. Wissenschaft der Logik. Erster Band: Die objective Logik. Ed. Georg Lasson. Reprint, Leipzig: Meiner. Hegel, Georg W. [1830] 1911. Encyclopädie der philosophischen Wissenschaften im Grundrisse. Zum Gebrauch seiner Vorlesungen. Dritte Ausgabe. Ed. Georg Lasson. Reprint, Leipzig: Meiner. Herbart, Johann F. [1808] 1887. Hauptpuncte der Logik. In the 2nd edition of Hauptpuncte der Metaphysik, ed. Karl Kehrbach, 217–226. Reprint, Langensalza: Beyer & Söhne. Herbart, Johann F. [1813] 1891. Lehrbuch zur Einleitung in die Philosophie, ed. Karl Kehrbach. Reprint, Langensalza: Beyer & Söhne. Herbart, Johann F. [1831] 1897. Kurze Encyklopädie der Philosophie aus praktischen Gesichtspuncten entworfen, ed. Karl Kehrbach. Reprint, Langensalza: Beyer & Söhne. Herbart, Johann F. 1836. Rezension von Neue Darstellung der Logik nach ihren einfachsten Verhältnissen. Nebst einem logisch-mathematischen Anhange. Von M. W. Drobisch. Göttingische gelehrte Anzeigen 10: 1267–1274. Kakkuri, Marja-Liisa. 1983. Abstract and concrete: Hegel’s logic as logic of intensions. Ajatus 39: 40–106. Kant, Immanuel. [1787] 1904. Kritik der reinen Vernunft. Zweite Auﬂage, ed. by Königlich Preußische Akademie der Wissenschaften. Reprint, Berlin: Reimer. Kern, Hermann. 1849. Ein Beitrag zur Rechtfertigung der herbartschen Metaphysik. Coburg: Gymnasium in Coburg. Michelet, Karl L. 1861. Die dialektische Methode und der Empirismus. In Sachen Trendelenburgs gegen Hegel. Der Gedanke 1: 111–126, 187–201. Nicolin, Friedhelm and Otto Pöggeler. 1959. Zur Einführung. In Hegel, Enzyklopädie der philosophischen Wissenschaften im Grundrisse, IX–LII. Hamburg: Meiner. Peckhaus, Volker. 1997. Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert. Berlin and New York: Akademie Verlag. Petersen, Peter. 1913. Die Philosophie Friedrich Adolf Trendelenburgs. Ein Beitrag zur Geschichte des Aristoteles im 19. Jahrhundert. Hamburg: Boysen. Plato. 1953. Parmenides. In The Works of Plato, Translated into English with Analyses and Introductions by B. Jowett, vol. 2, 669–718. Oxford: Clarendon Press. Rabus, Georg L. 1880. Die neuesten Bestrebungen auf dem Gebiete der Logik bei den Deutschen und Die logische Frage. Erlangen: Deichert.

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Rosenkranz, Karl. 1844. Georg Wilhelm Friedrich Hegel’s Leben. Supplement zu Hegel’s Werken. Berlin: Duncker und Humblot. Rosenkranz, Karl. 1858/59. Wissenschaft der logischen Idee. Königsberg: Bornträger. Strümpell, Ludwig. 1855. Einige Worte über Herbart’s Metaphysik in Rücksicht auf die Beurtheilung derselben durch Herrn Professor Trendelenburg. Zeitschrift für Philosophie und philosophische Kritik (Neue Folge) 27: 1–34, 161–192. Thiel, Christian. 1982. From Leibniz to Frege: Mathematical Logic between 1679 and 1879. In Logic, Methodology and Philosophy of Science VI. Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science, Hanover, 1979. Studies in Logic and Foundations of Mathematics, no. 104, eds. L. Jonathan Cohen, Jerzy Łoś, Helmut Pfeiﬀer, and Klaus-Peter Podewski, 755–770. Amsterdam, New York, Oxford: North-Holland. Trendelenburg, Adolf. 1840. Logische Untersuchungen. Berlin: Bethge. Trendelenburg, Adolf. 1842. Zur Geschichte von Hegel’s Logik und dialektischer Methode. Die logische Frage in Hegel’s Systeme. Eine Auﬀoderung [sic] zu ihrer wissenschaftlichen Erledigung. Neue Jenaische Allgemeine Literatur-Zeitung 1(97): 405–408; 1(98): 409–412; 1(99): 413–414. Trendelenburg, Adolf. [1853] 1855. Ueber Herbart’s Metaphysik und eine neue Auﬀassung Derselben. In Historische Beiträge zur Philosophie, Zweiter Band, vermischte Abhandlungen, 313–351. Berlin: Bethge. Trendelenburg, Adolf. 1867. Ueber Herbart’s Metaphysik und neue Auﬀassung Derselben, Zweiter Artikel. In Historische Beiträge zur Philosophie, Dritter Band, vermischte Abhandlungen, 63–96. Berlin: Bethge. Twesten, August. 1825. Die Logik, insbesondere der Analytik. Schleswig: Königlichen Taubstummen-Institut. Ueberweg, Friedrich. 1923. Grundriss der Geschichte der Philosophie, Vierter Theil, Die deutsche Philosophie des neunzehnten Jahrhunderts und der Gegenwart. Berlin: Mittler und Sohn. Ulrici, Hermann. 1869/70. Zur logischen Frage. (Mit Beziehung auf die Schriften von A. Trendelenburg, L. George, Kuno Fischer und F. Ueberweg.) I. Formale oder materiale Logik? Verhältniß der Logik zur Metaphysik. Zeitschrift für Philosophie und philosophische Kritik (Neue Folge) 55: 1–63; II. Die logischen Gesetze, ibid.: 185–237; III. Die Kategorieen, 56: 1–46; IV. Begriﬀ, Urtheil, Schluß, ibid.: 193–250. Vilkko, Risto. 2002. A Hundred Years of Logical Investigations. Reform Eﬀorts of Logic in Germany, 1781–1879. Paderborn: Mentis.

7

The Relations between Logic and Philosophy, 1874–1931 Leila Haaparanta

One who seeks to discuss the relations between logic and philosophy in the nineteenth century and the early twentieth century has to pay special attention to his or her use of the term “logic.” In the context of nineteenth-century and early twentieth-century philosophy, that term occasionally refers to similar activities to those we now call logic. In those days, logic could mean what we nowadays tend to call logic proper, that is, working with formal systems that resemble those of mathematics. However, it could also mean activities that we would now wish to label as “epistemology,” “philosophy of science,” “philosophy of language,” or “philosophy of logic.” Therefore, it may sound strange to promise to discuss the relations between nineteenth-century and early twentieth-century logic and philosophy. It is more to the point to claim that this chapter gives a survey of the ﬁeld of philosophy where (1) the philosophical foundations of modern logic were discussed and (2) where such themes of logic were discussed that were on the borderline between logic and other branches of the philosophical enterprise, such as metaphysics and epistemology. What will be excluded in this chapter are the formal developments on the borderline between logic and mathematics, hence, contributions made by such logicians as Augustus De Morgan (1806–1871), George Boole (1815–1864), Ernst Schröder (1841–1902), and Giuseppe Peano (1858–1932), for example (see chapters 4 and 9 in this volume). Gottlob Frege (1848–1925) and Charles Peirce (1839–1914) are included, since their work in logic is closely related to and also strongly motivated by their philosophical views and interests. In addition, this chapter pays attention to a few philosophers to whom logic amounted to traditional Aristotelian logic and to those who commented on the nature of logic from a philosophical perspective without making any signiﬁcant contribution to the development of formal logic. 222

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The choice of the years 1874 and 1931 has its reasons. In 1874 Franz Brentano (1838–1917) held his inaugural lecture “Über die Gründe der Entmutigung auf philosophischem Gebiete” in Vienna. The lecture showed the way to philosophers who made a sharp distinction between subjective psychological acts studied by the empirical sciences and the objective contents of those acts represented by means of logic (Brentano 1929, 96). Basically the same distinction had already been made by Bernard Bolzano (1781–1848) in his Wissenschaftslehre (1837). In 1931, which is the last year that is taken into account in this chapter, a volume of Erkenntnis was published which contained Rudolf Carnap’s (1891–1970) criticism of Martin Heidegger (1889–1976) titled “Überwindung der Metaphysik durch logische Analyse der Sprache” and Arendt Heyting’s (1898–1980) “Die intuitionistische Grundlegung der Mathematik,” which was inspired by Edmund Husserl’s (1859–1938) and Heidegger’s thoughts. Those articles were important in view of the division of philosophical schools in the twentieth century. This chapter is far from being the whole story of the relations between logic and philosophy 1874–1931. Instead, it consists of a number of themes and opens up a few perspectives on the period. There is slight emphasis on German philosophy. The chapter focuses on Frege, Husserl, and Peirce. Frege and Peirce are chosen because of their central role in the development of modern logic. Husserl is chosen because he wrote a great deal on the philosophical problems related to the logical enterprise. If we use the labels of our time, we would say that Husserl was one of the most important philosophers of logic of his own time.

1. The Historical Setting, 1874–1931 Even if Kant thought that no signiﬁcant changes are possible in logic, his own transcendental logic raised several new themes that we could now call philosophy of logic. Transcendental logic was philosophy of certain logical categories, especially of their metaphysical limits and epistemological import. After Kant, the role of those categories was discussed in various ways. There were philosophers such as Johann Gottlieb Fichte (1764–1814), Friedrich Wilhelm Schelling (1775–1854), and G. W. F. Hegel (1770–1831) who anchored logical categories to the world, who argued that logical categories are categories of being, of what there is (see chapter 6). In the second half of the century, the situation changed. Philosophers started to debate on the relation between logic and psychology. That debate increased interest in the epistemological questions related to logic, but it also brought about a new formulation of metaphysical or ontological problems. The basic question was no longer what the most general structure of the world is. Instead, philosophers pondered on whether there was a speciﬁc abstract realm that had thoughts as its denizens and that logic could represent. In his Lehrbuch der Logik (1920), Theodor Ziehen listed and characterized various groups of nineteenth-century logic (Ziehen 1920, 155–216). The main

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opposition in the last decades of the century was that between psychologists and antipsychologists. In the ﬁrst half of the century, there were a number of psychologists such as Friedrich Beneke (1798–1854), Otto Friedrich Gruppe (1804–1876), William Whewell (1794–1866), and August Comte (1798–1857). In Germany several schools arose in the late nineteenth century that attacked psychologism, such as neo-Kantians like Hermann Cohen (1842–1918) and Paul Natorp (1854–1924) and logicists like Frege and Husserl. Husserl’s early views are usually considered psychologistic. Among logicists there were philosophers whom Ziehen called value-theoretical logicists, such as Wilhelm Windelband (1848–1915) and Heinrich Rickert (1863–1936), and moderate logicists like Hermann Lotze (1817–1881) and Gustav Teichmüller (1832–1888). Moderate or weak psychologists included several thinkers, for example, Christoph Sigwart (1830–1904), Wilhelm Wundt (1832–1920), Benno Erdmann (1851–1922), and Theodor Lipps (1851–1914).1 The contents of these doctrines will be clariﬁed later in this chapter. The opposition between logic and metaphysics became important in a new way in the beginning of the twentieth century, when Carnap raised criticism against Heidegger’s views. Researchers who have tried to trace the origin of the distinction between the analytic and the phenomenological, more generally, the Continental tradition, have paid attention to the debate between Carnap and Heidegger concerning the relation between logic and metaphysics. Various interpretations can be proposed concerning the core of Carnap’s criticism. It is not clear how Heidegger would have defended his view or attacked Carnap’s position. Michael Friedman (1996, 2000), among others, has studied the theme by taking the historical context into account. He has argued that the roots of Carnap’s thought were in the neo-Kantianism of the Marburg school, while Heidegger’s philosophy ensued from the Southwest school. According to Friedman, that diﬀerence largely explains the fact that Carnap emphasized the role of logic, while Heidegger stressed the centrality of questions concerning human beings and their values. There are various ways of making the distinction between the analytic and the phenomenological tradition. Several criteria have been suggested, such as their attitudes toward the history of philosophy, toward their own history, toward science, and toward the idea of scientiﬁc philosophy; their views on what are the central problems of philosophy, the objects of philosophical research, and the methods of philosophy; and their attitudes toward the ideal of clarity in philosophy. It has been suggested that views on the relation between logic and metaphysics are an important criterion if we wish to divide philosophers into the two camps. The diﬀerent criteria turn out to be problematic in closer scrutiny.2 One who seeks to locate the diﬀerences between the two traditions cannot ignore the fact that there were at least two ideas that early phenomenology, especially Husserl’s thought, and most of early analytic philosophy shared. First, there was the idea of pure philosophy, which presupposed a belief in the sharp distinction between knowledge a priori and knowledge a posteriori. That

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belief had one of its origins in late nineteenth-century antipsychologism. Second, there was the belief in the method of analysis as the method of philosophy. That method, though in diﬀerent versions, was used by Frege, the “godfather” of analytic philosophy, and Husserl, the pioneer of phenomenology. These two common features were intertwined in various ways.3 Carnap’s and Heidegger’s debate had its background in lively discussion concerning the philosophy of logic and mathematics that was going on particularly in German philosophy at the end of the nineteenth century and at the beginning of the twentieth century. Hermann Lotze (1817–1881), who was professor at the University of Göttingen, inﬂuenced a number of those who took part in the discussion. On Lotze’s view, objectivity is not the same as that actuality (Wirklichkeit) which belongs to concrete beings. Lotze also regarded abstract objects like thoughts and values as objective in the sense that they are valid. Frege was one of Lotze’s students, and so was Bruno Bauch, Frege’s colleague in Jena, a neo-Kantian philosopher and the founder of the society for German idealism. Frege also belonged to that society. Heinrich Rickert, who was professor in Freiburg, was also inﬂuenced by Lotze’s philosophy. Carnap was a student of Bauch’s, while Heidegger was a student of Rickert’s. Rickert and Windelband were central ﬁgures of the neo-Kantianism of Southwest Germany.4 Frege’s philosophical environment was not the Southwest school but rather the Marburg school. Frege was most likely to receive his concept of truth value from Windelband, who was one of the so-called value-theoretical logicists; they were philosophers who thought that besides the moral values the realm of values includes the truth values studied by logic.5 Like Frege, Husserl criticized psychologists and held the view that logic and mathematics study abstract objects, such as numbers and thoughts, that is, the structure of thoughts and the inferential relations between thoughts. At the beginning of the twentieth century, Husserl was professor in Göttingen, until he moved to Freiburg in 1916, to follow Rickert in the professorship. Husserl’s follower in Freiburg was Heidegger. Frege was a logicist in two meanings, Husserl in one meaning of the word. A competing doctrine, namely formalism, was represented by David Hilbert (1862–1943), who was Husserl’s colleague and friend in Göttingen. In the philosophy of mathematics, logicism meant two things; on the one hand, it was the view that arithmetic or even the whole of mathematics can be reduced to logic; on the other hand, it was the view that numbers are abstract objects that are independent of the human mind. On this latter view, mathematical knowledge has to do with these very objects. Frege’s logicist program had to do with arithmetic, and it included deﬁning the concept of number by means the concepts of “extension of a concept” and “equinumerous.” Frege took extensions of concepts to be logical objects.6 Husserl’s studies in the foundations of logic and mathematics were closely connected to the rise of phenomenology. Logicism in the latter meaning was a natural starting point of Husserl’s phenomenology; namely, if the objects

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of logic and mathematics have their origins in consciousness, even if not in empirical consciousness studied by psychology, as Husserl argued, one has to ﬁnd out how to make the distinction between empirical and non-empirical consciousness. If psychology is interested in empirical consciousness, and if philosophy, including the philosophy that studies the relations between the subject and logic and mathematics, is interested in pure consciousness, how are these consciousnesses distinguished from each other?7 Are we dealing with ontologically two diﬀerent consciousnesses, or are we talking about two diﬀerent points of view to the same consciousness? If the latter holds, what do we mean by saying that in the last analysis it is the same consciousness we are talking about, and if it is one and the same, how can we justify the claim that it is one and the same consciousness? Several diﬃcult ontological and epistemological questions arise. Therefore, it is not surprising that Husserl moved from studies in the philosophical foundations of logic to studies of consciousness. Husserl asked how logic as science is possible. He wished to justify the ﬁeld of knowledge called logic, but it often seems that he also wished to justify a certain logic, namely, classical logic, by studying its origin in consciousness. There has been a debate on whether Husserl wished to take a position on the correctness of logical systems, and if he did, whether he was a conservative or a revisionist in logic. One has raised the question whether Husserl would have suggested giving up the law of excluded middle or any other law of logic, if we cannot ﬁnd philosophical justiﬁcation for those laws. Phenomenologists often emphasize the incommensurability of philosophy and the sciences. We could think that Husserl’s philosophical studies and logicians’ debate on the acceptability of various logics are incommensurable. Dieter Lohmar has, for his part, sought to show that Husserl was a moderate revisionist. In his view, Husserl thought that it is possible that we cannot ﬁnd justiﬁcation for all laws of classical logic.8 Radical revisionists in logic were Oskar Becker (1889–1964) and Heyting, who sought to change logic on the basis of Husserl’s and Heidegger’s thought. They tried to develop intuitionistic logic by using Husserl’s concepts of meaning intention and meaning fulﬁllment or disappointment.9 As noted, one of Heyting’s papers, inspired by Husserl and Heidegger, came out in the same volume of Erkenntnis where Carnap had his criticism of Heidegger, the criticism where Carnap sought to show by means of logic that Heidegger’s sentences are meaningless. In this chapter, I proceed as follows. First, I consider logic as a category theory, hence, as a doctrine that is interested in the categories of thought and being. I discuss those views in which logic is understood as the ideal language that mirrors reality in the right way. Husserl’s formal ontology is related to these doctrines. This consideration brings us to Heidegger’s view of metaphysics. I then take up the ideas of one world and of the plurality of worlds and consider the theories of modalities. The doctrine of three realms was much discussed in the late nineteenth century. I will consider the acknowledgment of the third realm and the reasons for such an acknowledgment. This question

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is tied to the debate on psychologism and antipsychologism and the problem of the objectivity of the realm of thoughts that logic speaks about. The important point made by antipsychologists was epistemological rather than ontological. That point brings us to the question of the possibility of logical knowledge and the various ways of answering the question given by Kant, Frege, and Husserl. I then continue with epistemological considerations and discuss Frege’s idea that in logical inference no intuitive gaps are allowed. If a logical theorem is justiﬁed, there is no reference to intuition in the inferential chain. In addition, philosophers raised the question concerning the justiﬁcation of traditional logical laws and a speciﬁc logical language. Husserl was one of those who raised such questions. That theme also brings us to intuitionistic logic and to the relations between logic and experience. Finally, Frege’s and Peirce’s methodologies of logic are discussed, and Frege’s semantic views are presented.

2. The Relations between Logic, Metaphysics, and Ontology In nineteenth-century logic and philosophy, logic was often understood contentually or materially (inhaltlich). The idea that logic has content received various meanings. (1) Logic was regarded as contentual in the sense that it was assumed to speak about the objects of the world. Kant’s transcendental logic was contentual in this sense in a peculiar way; it showed us the form of the phenomenal world. Hegelian logic was contentual, because it sought to mirror the historical development of reality. (2) Logic was taken to be contentual in the sense of being transcendental, that is, being a picture of the a priori conditions of all thought. (3) Logic was thought to have content in the sense that it was assumed to speak about the objects of the abstract realm, that is, to convey thoughts, which were considered objective. (4) Logic was thought to have content in the sense that it was assumed to mirror the structure of the psychological realm. Philosophers who regarded logical categories as categories of being or as categories of objects of knowledge and experience represented the ﬁrst or the second position. Leibniz and Kant belonged to that tradition of logic, even if their views otherwise diﬀered radically. Frege also thought that an ideal language can be discovered that is the correct mirror of the universe. However, Frege is also famous for his writings about objective thoughts and of his view of logic as a representative of the realm of abstract objects. Hence, besides being a mirror of all that there is, for Frege logic was a mirror of a speciﬁc realm; the problem for interpreters has been whether Frege considered that realm in the framework of epistemology only, or whether he regarded its objects as having an ontological status. This doctrine, whether in its epistemological version or both its epistemological and its ontological version, was in opposition with the fourth doctrine listed, which was called psychologism.

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2.1. The Leibnizian Starting Point: Logic as the Mirror of Reality Leibniz was the most prominent of the pre-Fregean thinkers who maintained that the terms of our natural language do not correspond to the things of the world in a proper way and that we should therefore construct a new language which mirrors correctly the whole universe. One important characteristic of modern logic was that unlike traditional logic, it proposed a new language— mathesis universalis, lingua characteristica, begriﬀsschrift, or whatever it was called by various authors. Modern logicians, primarily Frege, wished to establish a new language that mirrors the world and replaced the grammatical subjectpredicate analysis of sentences by the argument-function analysis. Therefore, the term “linguistic turn,” as applied to Frege, may lead us astray, if we do not remember that Frege also turned away from language. That is, unlike traditional logicians, he paid little attention to grammatical concepts like those of subject and predicate in his logical studies. In addition to the dream of ideal language, there was the idea of calculus strongly emphasized by Boole and his followers. It meant the eﬀort to formulate the rules of logical inference explicitly by presenting logical and non-logical vocabulary, formation rules, and transformation rules. Boole stated as follows: We might justly assign it as the deﬁnitive character of a true Calculus, that it is a method resting upon the employment of symbols, whose laws of combination are known and general, and whose results admit of a consistent interpretation. (Boole 1965, 4) The nineteenth century saw a breakthrough of the two ideas, even if emphases varied among logicians. Frege stressed that he did not want to put forward, in Leibniz’s terms, only a calculus ratiocinator, by which he primarily meant the rules of logical inference. He argued that his conceptual notation was to be a lingua characterica, which was the term that he used for Leibniz’s lingua characteristica. That is, his notation was to be a proper language which speaks about all that there is.10 Frege raised criticism against Boole, because in his view Boole merely focused on developing a Leibnizian calculus in his logical works. However, this was not exactly what Boole himself thought of his project, because he included the idea of logic as a mental or philosophical language in his philosophical remarks on logic (Boole 1958, 11, and Boole 1965, 5). It has been argued in the literature that since diﬀerent logicians emphasized diﬀerent sides of the Leibnizian project, they ﬁnally came to advocate conﬂicting views of the basic nature of logic. It has been claimed that Boole, Peirce, and Schröder, for example, were inclined to stress the importance of developing a calculus, whereas Frege and the early Russell were among those who laid more emphasis on the idea of logic as a universal language. The systematic consequences of the two views have been studied by a number of authors, especially Jean van Heijenoort (1967), Warren D. Goldfarb (1979), and Jaakko Hintikka (1979, 1981a, 1981b). According to these studies, those

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who stressed the idea of logic as language thought that language speaks about one single world. This was the position to which Frege was committed. He thought that there is one single domain of discourse for all quantiﬁers, as he assumed that any object can be the value of an individual variable and any function must be deﬁned for all objects. This is what was stated by his principle of completeness (Grundsatz der Vollständigkeit) (GGA II, §§56–65). On the other hand, those who supported the view that logic is a calculus were ready to give various interpretations or models for their formal systems. This appears to have been Boole’s standpoint. Boole wrote: Every system of interpretation which does not aﬀect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. (Boole 1965, 3) However, it is not clear how this passage ought to be interpreted. It is noteworthy that Boole’s statement is not far from what Frege thought. Frege also wished to construct such a language as can be applied to various ﬁelds like arithmetic and geometry (BS, 1964, “Vorwort,” XII). However, ﬁelds of application are not what is meant by the distinction between the one-world and the many-world view. Moreover, Frege wanted to develop both a language and a calculus; if he wanted to develop them as they are understood by contemporary scholars, he could not consistently support both of the implications stressed by those scholars, that is, he could not preach for the one-world view and for the plurality of worlds at the same time. The twentieth-century perspective has also given more content to the two views. It has been claimed that those who support the idea of logic as language tend to think that they cannot step beyond the limits of language and that this prevents them from developing a proper semantic theory for their language. On the other hand, it has been argued that those who endorse the view of logic as calculus are inclined to think that it is possible to look at a formal system, as it were, from the outside and develop a semantic theory for it. For example, even if Frege introduced his doctrine of senses (Sinne) and references (Bedeutungen), which is a semantic doctrine, he did not believe that he could propose a proper semantic theory for a formal or a natural language. He repeatedly pointed out that he can only give suggestions and clues concerning his basic semantic concepts and the semantic properties of his conceptual notation.11 Frege made the distinction between language and calculus on the basis of his interpretation of Leibniz’s project, but he was not conscious of all the implications of the two views of logic which have been detected in the literature. Hence, there are at least three diﬀerent (though closely connected) stories to be told, as far as the ideas of a universal language and calculus are concerned. There is the story of the content which Leibniz gave to his idea, the story Frege and Boole told about their projects, and the story told from the

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twentieth-century perspective that tries to capture the far-reaching systematic implications of the two extreme positions. In Frege’s hands, the dream of a universal language was tied to the task of philosophy. Otherwise Frege did not write much about the task of philosophy. In the beginning of his Begriﬀsschrift (1879) he writes that if one task of philosophy is to free the human mind from the power of word by revealing the mistakes that are often almost unavoidably caused by the use of language, then his conceptual notation, which has been constructed for this purpose, will be a useful tool for a philosopher (Frege, BS, 1964, XII–XIII). Frege often complains that natural language leads us astray. However, he nowhere states that it would be the only task of philosophy to clarify language. There is one story to be told concerning the relations between Kant and Frege which illuminates Frege’s position among the opponents and the supporters of metaphysics. In the preface of his Begriﬀsschrift, Frege states that he tries to realize Leibniz’s idea of lingua characterica. The term was most likely to come from the Leibniz edition by J. E. Erdmann from the years 1839 and 1840, as the word characterica is used there instead of the word characteristica used by Leibniz (see Haaparanta 1985, 102–117). Adolf Trendelenburg also used the same word in his writing “Über Leibnizens Entwurf einer allgemeinen Charakteristik” (1867). According to Trendelenburg, philosophers ought to construct a Leibnizian universal language, Begriﬀsschrift, by taking Kant’s theory of knowledge into account. In his view, Kant’s contribution was that he distinguished the conceptual and the empirical component of thought and stressed the importance of studying the conceptual component. Trendelenburg also tells us about Ludwig Benedict Trede, who in his article “Vorschläge zu einer nothwendigen Sprachlehre” in 1811 tried to create a universal language by following Leibniz and Kant. Frege also called his language conceptual notation, which he, it is true, took to be a less successful name for it. He also used the expression “the formula language of pure thought” in the subtitle of his book Begriﬀsschrift and the expression “the intuitive representation of the forms of thought” in his article “Über die wissenschaftliche Berechtigung einer Begriﬀsschrift” (1882) (Frege, BS, 1964, 113–114). The above-mentioned connections have been noticed and also stressed by a few scholars several years ago (see Sluga 1980; Haaparanta 1985). Even if there were no similarities whatsoever between Trede’s notation and Frege’s language, we can say that by his reference to Trendelenburg Frege told us something about the philosophical background of his conceptual notation. On the basis of what has been said, we may argue that Frege’s conceptual notation was itself a philosophical position taking. It was not in favor of psychologistic transcendentalism, according to which the necessary conceptual conditions which make knowledge and experience possible are typical of the human mind. Nor was it in favor of transcendental idealism, if we think that a transcendental idealist is one who acknowledges a transcendental subject. We can say, however, that Frege was a transcendentalist in a very weak sense; he tried to write down the forms of thought, which Kant would have called

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the necessary conditions of knowledge and experience. It is, of course, obvious that Frege’s conceptual notation was not a codiﬁcation of those forms that we ﬁnd in Kant’s table of categories. The Vienna Circle gave a special treatment to the new logic that Frege had developed. The manifest of the Vienna Circle was directed against metaphysics, and the same spirit can be found in many other writings of the members of the circle. In the manifest, the new logic was described as a neutral system of formulas, a symbolism which is free from the slag of the historical languages, as a tool by means of which it is possible to show that the statements made by metaphysicians and theologians are pseudo-statements, that they express feeling of life, which would be properly expressed by art. The Vienna Circle regarded the close relation with traditional languages as the main problem of metaphysics. They also blamed metaphysics for assuming that thought can know itself without empirical material; that kind of knowing was sought by transcendental philosophy. The Vienna Circle declared that it is not possible to develop metaphysics from “pure thought” (Der Wiener Kreis 1973, 308). They believed that logical analysis overcomes not only scholastic metaphysics but also Kantian and modern apriorism. That position taken by the Vienna Circle meant the rejection of synthetic judgments a priori and hence the rejection of transcendental knowledge. Hence, if we draw a line from Kant to Frege and further to the Vienna Circle, there is a crucial change in how the relations between being and the pure forms are understood. It is as early as in his Allgemeine Erkenntnistheorie (1918) that Schlick raised the question of whether there are any pure forms of thought and answered that thought with its judgments and concepts does not press any form on reality (Schlick 1918, 304–305). For Schlick, that means the repudiation of Kant’s philosophy (ibid., 306). In his article “Die Wende der Philosophie” (1930) he argued that the greatest change is due to a new insight concerning the nature of the logical, which was made by Frege, Russell, and particularly Wittgenstein. According to that new understanding, the pure form is merely the form of an expression, but that form cannot be presented (Schlick 1938, 33–34). It is true that Frege did not present the system of signs called conceptual notation, if presenting it had meant giving a semantic theory for the system in a metalanguage. If Frege thought that forms of thought are proper objects of knowledge, that knowledge was for him a kind of immediate recognition. Recognition of the correct forms, the result of which is the creation of conceptual notation, can be called immediate intellectual seeing or intuition. In his late writings in 1924 and 1925, Frege stressed that we see correctly, if natural language does not disturb our intellectual seeing. Moreover, when Frege discussed certain important features of his language, such as the distinctions between the diﬀerent meanings of “is,” which are existence, predication, identity, and class inclusion, he gave lengthy arguments for the distinctions. One of the most central reasons he put forth was that his new language takes care of the diﬀerence between individuals and concepts, which is missing both in Aristotelian logic and in Boole’s logic, and that the

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diﬀerence is mirrored by the distinction between identity and predication as well as by the distinction between predication and class inclusion. For example, to preserve the distinction between objects and concepts, Frege considered it necessary to realize that the “is” of identity diﬀers from the “is” of predication and, moreover, that this distinction reﬂects how things really are (“Über Begriﬀ und Gegenstand,” 1892, KS, 168). Moreover, the motivation for denying that existence is a ﬁrst-order predicate came from Kant’s thought. Frege also gave a positive contribution by trying to tell what existence is, namely, that it is a second-order concept. We can say that Frege had not only a view of the word “being” but also a view of the forms of being, which are forms of thought, and those forms were meant to be codiﬁed as his ideal language.12 There was a well-known controversy between logic and metaphysics in the early days of the analytic tradition and the phenomenological movement, to which I already referred. The Vienna Circle declared in 1929 that the new logic, the ideal language developed by Frege, Russell, and Whitehead, frees philosophy from considering the true nature of reality. It was believed that by means of the new formula language, it was possible to show that metaphysical statements are meaningless. It was not thought that the very ideal language would have a metaphysical content. For a logical empiricist, Heidegger’s philosophy was an example of the meaninglessness of metaphysics. In 1931 Carnap published his article “Überwindung der Metaphysik durch logische Analyse der Sprache,” in which he studied Heidegger’s sentences and stated that the sentences of a metaphysician cannot be combined with the ways in which logic and science proceed. In his Was ist Metaphysik? (1929) as well as in the afterwords of its later editions, Heidegger discussed the criticism that had been raised against the way he used the word “nichts.” According to Heidegger, nothing is the origin of negation, not the other way round. His message was that logic has its origin in the being of Dasein (Heidegger 1992, 37) and philosophy can never be measured by means of the standards of the idea of science (ibid., 41). Hence, for Heidegger the origin of the logical concept of being was the being of Dasein. There thus seemed to be a sharp contrast between Heidegger, who spoke about the meaning of being and a linguistic philosopher who spoke about the diﬀerent meanings of the word “is.” It was Frege who distinguished the diﬀerent meanings of “is” in his conceptual notation, and therefore it may seem that Frege was clearly among those who wished to limit the talk about being to the word “is.” This is not the case, as Frege was not an opponent of metaphysics. It is more to the point to say that Frege’s thought lay somewhere between the philosophy of the Vienna Circle and Heidegger’s fundamental ontology. The view of philosophy held by the Vienna Circle was characterized by the fact that philosophy was taken to be an art of using a tool and the good tool was Frege’s, Russell’s, and Whitehead’s formula language. However, the language lost the metaphysical content that it had for Frege. The pure forms were interpreted as the forms of a system of signs; the system of signs was no

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more “an intuitive representation of the forms of thought,” as Frege wrote. In its manifest the Vienna Circle declared that there are no depths in science but there is surface everywhere (Der Wiener Kreis 1973, 306). In that sense, the circle also wanted philosophy to be like science. Both Frege and Heidegger were interested in the philosophical basis of logic, Frege mainly in the epistemological basis and Heidegger in the origin of logic in the being of Dasein. Both thought that there is something under the “surface.” The Vienna Circle thought that philosophy is activity; that was especially emphasized by Schlick in “The Future of Philosophy” (1931). Schlick referred to Wittgenstein, for whom philosophy was not a theory but a certain kind of activity, that is, of clarifying meanings and writing formulas which do the job of clariﬁcation (Schlick 1938, 132). It is true, the incentive for that kind of philosophizing was given by Frege, but it would be far from the truth to argue that Frege held that view. Edmund Husserl touched on the relations between logic and being in several connections, for example, when he distinguished between formal ontology and material ontologies in his Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie I (1913). It was already in the ﬁrst volume of the Logische Untersuchungen (1900) that he divided logic into two parts according to the two tasks that he believed logic to have. One of the tasks was to give the formal categories of meaning, whereas the other was to put forward the formal categories of objects. Husserl listed the basic concepts of pure logic or analytical categories both in the Logische Untersuchungen I (A 244–245/B 243–244) and in the Ideen I (Husserl 1950b, 26–32). His categories of meaning include such concepts as belong to the essence of the proposition or apophansis, such as subject and predicate, conjunctive, disjunctive, and hypothetical connections, that is, what we would call logical connectives, and the concepts of concept, proposition, and truth. In addition to the categories of meaning, he gave a list of pure formal objective categories, such as object, property, relation, state of aﬀairs, identity, whole and part, number, and genus and species. Husserl called these categories of objects substrate-categories. In the Ideen I, Husserl states that “formal ontology contains the forms of all ontologies . . . and prescribes for material ontologies a formal structure common to them all” (Husserl 1950b, 27; Kersten’s translation, 21), and then goes on with treating formal ontology and pure logic as synonymous terms. He also claims that pure truths of meaning can be converted into pure truths of objects (ibid., 28). In the Logische Untersuchungen, Husserl pays attention to the distinction between formal and empirical (or material) concepts, as well as to the distinction between formal or analytic propositions and laws and material propositions and laws. He states that concepts like something, one, object, quality, relation, association, plurality, number, order, ordinal number, whole, part, magnitude, and so on, have a basically diﬀerent character from concepts like house, tree, color, tone, space, sensation, feeling, and so on, which for their part express genuine contents. It is not clear how we should make the

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distinction between form and content. What is clear, however, is that Husserl and his contemporaries took the very distinction between form and matter, or form and content, to be essential to logical studies, whether logicians were interested in concepts or in inferences in which the concepts were used.

2.2. The Absence of the Metaphysics of Modalities Our contemporary modal logic is usually considered as an extension of the two-valued predicate calculus that was developed in the late nineteenth century. However, the roots of our modal theory reach far back to Aristotelian logic, which regarded modal logic as a legitimate branch of logical studies. Interest in modal notions is a new phenomenon among logicians only when it is considered in the framework of the developments of those late nineteenth-century logicians who are honored as the pioneers of modern logic. In the beginning of the twentieth century, logicians were not willing to discuss modal concepts. They were mainly inspired by the extensionalist program which was preached by Frege, among others, and codiﬁed in the Principia Mathematica. Modal notions seemed to escape all treatments that are interested only in references. Later in the twentieth century, logicians proposed axiomatic systems for modal logic, which, however, ﬁrst avoided all systematic semantic considerations. Since the late 1950s, they introduced and developed interpretations for the axioms of modal systems. These interpretations are useful for clarifying which systems of axioms most naturally correspond to our intuitions concerning modal notions and their relations. Leibniz is an important ﬁgure behind our contemporary modal logic. It is also known that he is an important ﬁgure behind Frege’s logical work. Nonetheless, given that Frege set out to realize what Leibniz had dreamt of, it is surprising that he was reluctant to develop modal logic in the early twentieth century. Even if he started from Leibniz’s program in arguing that we must construct a proper language that represents the world, he was not true to Leibniz’s view that there could be alternative worlds to which our ideal language would be related. Frege’s conceptual notation was meant to represent only one world. As already noted, this doctrine of Frege’s is most clearly visible in his requirement that all of the predicates of the language must be deﬁned for all objects. Frege’s formula language was thus meant to speak about all that there is, and its quantiﬁers were meant to range over all individuals. Modal logic, as we understand it nowadays, was thus blocked out in the very beginning. Frege gave another reason for his unwillingness to discuss the concepts of necessity and possibility within the limits of his logic. The reason was that those concepts do not concern logic at all but that they have to do with the nature of the grounds of our judgments (BS, §4). For Frege, logic is interested in the objective realm of thoughts.13 Frege regarded the act of judging as a psychological phenomenon, which belongs to the realm of our private minds.14 Hence, even if Frege severely criticized all eﬀorts to reduce logical laws to

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psychological laws, he restricted modal notions to the realm of psychology, thus agreeing with psychologists. He did not believe that thoughts are necessary or possible as such, but he insisted that they are necessary or possible for our private minds. Like psychologists, he connected modal concepts with the concept of certainty and took them to modify our acts of thinking, which are units of the subjective realm. Signs for modal concepts did not play any role in his ideal language (for Frege’s views in more detail, see chapter 12 in this volume).

3. The Relations between Logic, Epistemology, and Psychology 3.1. Logical Psychologism In the late nineteenth century, the question of what logic mirrors, if it mirrors something, was mostly discussed in a way that was determined by the debate on the relation between logic and psychology. Contemporary naturalism has its roots in late nineteenth-century psychologism. The word “naturalism” was also used in the late nineteenth century. Like contemporary naturalism, late nineteenth-century naturalism and its version called psychologism had various contents(see Haaparanta 1995, 1999b; Kusch 1995). In her book Philosophy of Logics (1978) Susan Haack distinguishes between strong and weak logical psychologism. According to the strong view, logic describes our thought and may also tell us how we ought to think (Haack 1978, 238). In his book Husserl and Frege (1982), J. N. Mohanty describes strong logical psychologism as a doctrine according to which logic is a branch of psychology, the laws of logic describe actual human thought, and psychological study is therefore both suﬃcient and necessary for studying the foundations of logic (Mohanty 1982, 20). In Haack’s terminology, weak logical psychologism is the view that logic determines how we ought to think (Haack 1978, 38). Mohanty, for his part, characterizes the weak version as a thesis that it is necessary but not suﬃcient to study human thinking processes if we want to clarify the theoretical foundations of logic (Mohanty 1982, 20). Many logicians who are regarded as antipsychologists (Frege, for example) might accept what Haack calls weak logical psychologism. However, they would not say that determining the norms of thought would be the only or the basic task of logic (GGA I, “Einleitung,” XV). Logical psychologism had two diﬀerent roots in nineteenth-century philosophy. First, there was an interpretation of transcendentalism which regarded the transcendental conditions of experience as the conditions determined by the mental structure of the human race. Second, there was the tradition of empiricism, which attempted to base all knowledge on experience. German, French, and British logical psychologism in the nineteenth century was so complicated a doctrine that many ways of classifying it are possible. It could

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be a doctrine concerning the basic concepts of logic, the basic laws of traditional logic, or the nature of logical inference. If logical psychologism was understood as a theory that primarily concerns the conceptual tools of logic, it either claimed that such concepts as unity, plurality, negation, and possibility are structural features of the human mind, or it argued that those concepts are abstracted from sense perception. The latter position is linked with the empiricist tradition of the modern times. The former position followed if one interpreted transcendentalism by saying that the transcendental conditions of experience are determined by the structure of our factual minds. If logical psychologism was a doctrine concerning such laws as the law of excluded middle or the law of noncontradiction, it either maintained that those laws are structural features of the human mind or claimed that those laws have their origin in sense perception. There were a number of philosophers who stressed that the laws of logic have an empirical basis in sense perception, but who did not call themselves psychologists. J. S. Mill, for example, did not want to take that label (Mill 1906, Book II). If a psychologist claimed that the basic laws of logic represent the constant and innate structures of the human mind, he regarded the laws of logic as factual in the sense that he took them to be research objects of the science called psychology. Hence, both the empiricist and the transcendentalist version of psychologism were epistemological theories that tried to reveal the natural origin of logic and to justify certain logical concepts, logical laws, and logically valid inferences by means of the revealed origin. The foregoing classiﬁcation contained two basic forms of psychologism. Husserl also hinted at a similar division, when he distinguished between empirical and transcendental psychology as two diﬀerent bases of psychologism (LU I, A 123/B 123). One of the versions abstracts such laws as the law of noncontradiction from the objects of experience, whereas the other version pushes the structure of the mental realm into the objects of experience. In his Philosophie als strenge Wissenschaft (1910–1911) and in his lecture notes “Logik als Theorie der Erkenntnis” (1910–1911) Husserl characterized naturalism in various ways.15 He stated that naturalism is a phenomenon consequent on the discovery of nature, which is to say, nature considered as a unity of spatiotemporal being subject to exact laws of nature (PsW, 79). He also remarked that psychology is concerned with “empirical consciousness,” with consciousness from the empirical point of view, whereas phenomenology is concerned with “pure consciousness,” which is consciousness from the phenomenological point of view (PsW, 91). For Husserl, the phenomenological point of view was the philosophical point of view. Moreover, he continued that any psychologistic theory “naturalizes” pure consciousness (PsW, 92). Naturalizing pure consciousness amounts to identifying it with empirical consciousness. If the realm of pure consciousness had been the realm of norms for Husserl, his criticism would have been that naturalism deduces norms from facts. However, the core of the distinction between pure and empirical consciousness was not at that point.

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In his lectures on logic as the theory of knowledge (1910–1911), Husserl distinguished between laws of logic, laws of natural sciences, and norms created by human beings. The distinction corresponds to that made by Frege in his preface to the ﬁrst volume of the Grundgesetze. Husserl admitted that there are major reasons which speak in favor of logical psychologism. However, he explained by means of an analogy that logical psychologism is not true. In his “Göttinger Vorlesungen über Urteilstheorie” in the summer term of 1905 Husserl talked about the analogy between geometry and logic. There he points out that it is common to draw a false analogy; the psychologistic view is that the art of logical reasoning is related to psychology as geodesy is related to geometry or as technical physics is related to theoretical physics (19b). In his lectures in 1910 and 1911, Husserl explained what he thought is the right analogy (20b). Just as geodesy is related to ideal geometry, normative logic is related to logic as a theoretical discipline. Moreover, just as behind geodesy there is a natural science or several of them, likewise behind normative logic there is psychology. In Husserl’s view, the norms of logic are inferred from the facts of pure or theoretical logic, not from the facts given by psychology; the facts given by pure logic have to do with the structures of propositions and with the inferential relations between propositions. Husserl also stated in his lecture notes that naturalistic philosophy is characterized by the fact that it acknowledges only one ﬁeld of possible knowledge, which is nature (17a). Moreover, he stated that naturalism recognizes only one method of giving foundations for knowledge; it argues that all knowledge is based on experience (17b). In Husserl’s view, the essential diﬀerence between naturalism and antinaturalism was that naturalism does not acknowledge the ideal realm. Husserl characterized the ideal realm as eternal, self-identical, timeless, spaceless, unmovable, and unchangeable; he did not state that it is something that is expressed by normative propositions. He also remarked that there is no mysticism in such a view (28a, 28b). As we will see in the next section, in his later writings he expressed his view in constructivist terms and stressed the diﬀerence between two attitudes more than the diﬀerence between the two realms.

3.2. Antipsychologism and the Doctrine of the Third Realm In the passages quoted, Husserl acknowledged what is called “the third realm” by Frege. The doctrine of the three realms can be found in Lotze. According to Lotze, the being of abstract objects is not like the being of concrete objects. Instead, abstract objects are valid, geltend. Lotze took it to be important to distinguish between what is valid and what is (was gilt and was ist) (Lotze 1874, 16 and 507). Frege presented a doctrine of three realms, by means of which he expressed his view on the being and the being known of logical categories and of thoughts that are constituted by those categories. In the ﬁrst volume of his Grundgesetze der Arithmetik (1893) and in his article “Der Gedanke” (1918) he made a

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distinction between the subjective realm of ideas (Vorstellungen), the objective realm of actual (wirklich) objects, and the realm of objects that do not act on our senses but are objective, that is, the realm of such abstract objects as numbers, truth values, and thoughts (GGA I, XVIII–XXIV; KS, 353). His conceptual notation, which he called the formula language of pure thought, was meant to mirror parts of the third realm, as it was meant to present the structure of thoughts and the inferential relations between thoughts. It is usually assumed that Frege’s acknowledgment of the third realm was a Platonic doctrine. Some interpreters have challenged the received view, but others, most notably Tyler Burge (1992), have given strong arguments for the view that Frege held a Platonic ontology; Burge also emphasizes that Frege did not seek to defend his position, except for showing problems in competing views, and that he did not make any eﬀort whatsoever to develop a sophisticated version of his ontology. In spite of Burge’s carefully documented argumentation, other interpretations remain serious candidates. When Frege discussed his third realm in his “Der Gedanke,” he remarked that he must use metaphorical language. In other words, such expressions as “the content of consciousness” and “grasping the thought” must not be understood literally (KS 359, n. 6). As Frege expressed his worry about the fact that natural language leads us astray as early as in the preface of his Begriﬀsschrift, the interpretation that Frege did not take numbers or thoughts to have being in the proper sense of the word “being” is at least worth considering. Frege did think that the objects of the third realm are objective, hence, independent of subjective minds. That is not yet an ontological position. On Frege’s view, thoughts and their constitutive logical categories are denizens of the third realm, but their being is not like the being of the denizens of the objective and actual realm. Thomas Seebohm has argued that Frege presented a transcendental argument to the eﬀect that the existence of mathematical objects and logical categories is a necessary condition of the meaningfulness of mathematical and logical practice (Seebohm 1989, 348). If that argument holds, Frege’s acknowledgment of the third realm would have ensued from his epistemological views. Husserl argued that we must acknowledge an ideal realm of abstract objects to avoid psychologism. He pointed out that there is an unbridgeable diﬀerence between the sciences of the real and the sciences of the ideal, as the former are empirical, while the latter are a priori. Husserl realized that if we acknowledge the ideal realm, we must face an epistemological problem concerning our access to this realm. Most of Husserl’s logical studies after his Logische Untersuchungen are an eﬀort to answer this question by means of phenomenology. In his last logical works, titled Formale und transzendentale Logik (1929) and Erfahrung und Urteil (1939), which was published posthumously, he sought to show that we have an access to the denizens of the ideal realm, because we have set the structure of transcendental consciousness to those denizens, hence, we have maker’s knowledge of that realm.

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Husserl did not think that we could be mistaken about what the correct logical categories are. His problem was how to give a justiﬁcation for what he regarded as our true beliefs concerning those categories. In his sixth logical investigation (LU II, 1901, 1921) Husserl studies the components of meaning which determine the form of a proposition and calls them categorial meaningforms (Bedeutungsformen). In his view, those forms are expressed in natural language in several ways, for example, by deﬁnite and indeﬁnite articles, numerals, and by such words as “some,” “many,” “few,” “is,” “and,” “if—then,” and “every” (LU II, A 601/B2 129; LU II, A 611/B2 139). Husserl asked what the origin of logical forms is, when nothing in the realm of real objects seems to correspond to them (LU II, A 611/B2 139). He took it to be a problem how the logical words originally get their meaningfulness, hence, what kind of activity of a subject is required so that the logical words become meaningful. In his last works on logic, Husserl sought to show that that activity is precisely the activity of transcendental consciousness. Kant interpreted logical categories as the pure concepts of understanding, which correspond to certain types of judgments and which give form to the objects of experience. In his Begriﬀsschrift, Frege, for his part, introduced eight signs as the basic signs of his formula language of pure thought; those signs expressed the basic logical categories and made it possible for Frege to present most types of judgments listed in Kant’s table. As was noted, in Frege’s doctrine of the three realms the logical categories were regarded as constitutive for the denizens of the third realm called thoughts.

3.3. On the Possibility of Logical Knowledge 3.3.1. What Is Logical Knowledge? Kant is famous for his eﬀort to answer the so-called transcendental questions, such as “How is pure mathematics possible?”, “How is pure natural science possible?”, and “How is metaphysics as science possible?” This type of questions have two readings. One either wants to know whether x is possible and wishes to have a justiﬁcation for its possibility, or one assumes that x is possible and tries to ﬁnd out the conditions of its possibility.16 If one raises the question concerning the possibility of logical knowledge, one may think of two questions, ﬁrst, whether logical knowledge is possible at all, and second, if it is, under what conditions it is possible. This section is a short study of a few late nineteenth-century and early twentieth-century logicians’ and philosophers’ views of that possibility. Frege’s and Husserl’s views will again be in focus. By logical knowledge, one may mean knowledge which is reached by means of logical inference, hence, knowledge based only on logical truths. For example, knowing that p & q → p

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would be an example of logical knowledge. By logical knowledge, one may also mean knowledge concerning the basic concepts of logic, the logical forms of propositions, the basic laws of logic, or the rules of logical inference. For example, statements like Existence is a logical concept. The logical form of the sentence “Man is an animal” is “∀x(F (x) → G(x))”. The law of noncontradiction holds. would express logical knowledge in the intended meanings. In his Grundlagen der Arithmetik (1884, §3) Frege stated that the distinctions between a priori and a posteriori, synthetic and analytic, concern not the content of the judgment but the justiﬁcation for making the judgment. He excluded the naturalistic interpretation of his claim and stated that by his distinctions he intends to refer to the ultimate ground on which rests the justiﬁcation for holding a proposition to be true. He continued that the problem becomes that of ﬁnding the proof of the proposition. By his characterizations of analytic and synthetic truths and truths a priori and a posteriori, he expressed the view that the justiﬁcation of analytic truths a priori comes from general logical laws and deﬁnitions. In his view, logical laws neither need nor admit of justiﬁcation. However, the question remains what Frege would have named as the source of knowledge if he had thought that we can know the structure of the ideal logical language in the proper sense of knowing. Did he think that we know that existence is a logical concept? If he thought that way, what would he have labeled as the source of knowledge, hence, what would have been a justiﬁcation for such a claim? As already stated in section 3.2, Husserl studied the components of meaning which determine the form of a proposition and called them categorial meaningforms (Bedeutungsformen). He took it to be a problem how the logical words originally get their meaningfulness, hence, what kind of activity of a subject is required so that the logical words become meaningful. In Husserl’s thought, the questions of origin were linked with the questions of justiﬁcation. 3.3.2. Can We Have Logical Knowledge? Emil Lask on Kant’s View Kant thought that categories, hence, logical concepts, have their origin in the logical forms of propositions. However, he took the list of the logical forms of propositions for granted. If Kant’s transcendental deduction was a justiﬁcation of certain logical concepts, the idea of that justiﬁcation was to show the role of those concepts in cognition and experience; it was to show how the pure concepts of understanding contribute to making objects of knowledge possible and how they are linked with the forms of intuition. Expressed in contemporary terminology, Kant sought to give us the epistemological foundation of logic by showing how the pure forms of thought

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are applied to sensuous experience. Moreover, by that project Kant also tried to show us how logical knowledge in general is possible. He argued that logical concepts are hidden in objects of experience, they have their origin in those objects, and we can have knowledge of them precisely via their link to what is given in intuition. Emil Lask, a student of Rickert, whose thought inﬂuenced Heidegger’s early philosophy, praised Kant’s Copernican revolution in his work on logic and the doctrine of categories. According to his writing published in 1911, Kant had shown that certain questions concerning objects belong to logic, hence not to metaphysics (LP, 31).17 However, Lask argued that Kant’s critique of knowledge could not touch on the questions concerning logic or the logical forms of objects in a proper manner (LP, 260–262). In his view, that followed because Kant was committed to a two-world doctrine in which a distinction was made between the world of sensory objects and the transcendent world. Lask argued that as Kant neither regarded logic as sensory nor took it to be metaphysical, he made it homeless (heimatlos; LP, 263). We may disagree on Lask’s two-world interpretation of Kant’s thought. However, it is interesting to ﬁnd out how Lask solved the problem concerning the homelessness of logic which he thought to have found in Kant’s philosophy. His starting point was to give up the two-world doctrine and replace it by an epistemological doctrine concerning the concept of objectivity. That doctrine came from Lotze. As was noted, according to Lotze the being of abstract objects is not like the being of concrete objects. Lotze took them to be valid, and he considered it to be important to distinguish between what is valid and what is (Lotze 1874, 16 and 507). Lask supported that kind of division between two worlds (LP, 6), but he did not consider it an ontological distinction. Instead, for him that was a distinction between two diﬀerent attitudes or points of view, which we can take toward our sensory experience (LP, 48–49, and 88–91). Lask thought that the logical attitude considers psychological, physical, and cultural beings in a way that diﬀers from the attitude of everyday experience and scientiﬁc activity; it is interested in what is valid for those beings. On Frege’s View In his “Über die wissenschaftliche Berechtigung einer Begriﬀsschrift” (1882) Frege writes: “a perspicuous representation of the forms of thought (eine anschauliche Darstellung der Denkformen) has . . . signiﬁcance extending beyond mathematics. May philosophers, then, give some attention to the matter!” (BS, 1964, 114). The forms of thought Frege talked about were not meant to be the forms which the human mind happens to have. However, Frege thought that we (or he) can have an access to those forms and they can be written down as a language, as a conceptual notation (begriﬀsschrift), as he thought to have done in his book titled Begriﬀsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. These forms were not tied to human psychology, but they were pure, hence, not naturalistically characterizable forms. If Frege thought that logical forms can be known in the proper sense of knowing, he must have meant by “knowing” some kind of immediate

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recognition of the presence of the forms of thought. That recognition, the result of which is Frege’s conceptual notation, could be called intuition, in the sense of immediate intellectual seeing. How is this intellectual seeing possible at all? In view of Frege’s conceptual notation, which is meant to be a genuine language that speaks about the world and carries ﬁxed meanings, intellectual seeing presupposes grasping the correct structure of thoughts. Frege did not think that meanings could be given syntactically, hence, for him, giving meanings to logical constants did not amount to giving inferential rules, say, rules of introduction and elimination. For him, meanings of logical function names were found by means of grasping thoughts and by analyzing them. Still, Frege assumed that meanings are present in the syntax of the conceptual notation and there is no way of giving a semantic theory for that notation. He thought that our knowledge concerning the structure of the ideal logical language, hence, the basic logical concepts and the logical forms of propositions, and concerning the basic laws of logic carries its own justiﬁcation, which has to do with “immediate seeing,” which is not disturbed by sensory data. However, even if Frege did not seek to present any theory of logical knowledge, that did not mean that his ideal language would not have been motivated by epistemological considerations.18 On Husserl’s View Husserl’s doctrine of categorial perception in the Logische Untersuchungen was meant to be a solution to the problem concerning the origin of logical knowledge. Husserl introduced a new concept of perception that was not sensuous perception. In his view, categorial meanings are originally related to objects of sense perception but in a peculiar manner; logical forms are in the objects of sensuous acts but hidden in them as it were. In categorial perception, which was the term Husserl used, the subject sees the sensuous object diﬀerently; he or she perceives the object via logical forms, hence, the object is for him or her in these forms (LU II, A 615/B2 143). Sensuous objects are objects of sensuous acts, whereas ideal objects are objects which arise in that kind of “seeing diﬀerently” (LU II, A 617/B2 146). In Husserl’s view, such acts as the act of conjunction, disjunction, and generalization need sensuous acts which are their foundation (LU II, A 618/B2 146). In the later edition of the sixth logical investigation in 1921, Husserl remarked that these acts that are not founding acts are in relation to what appears in the sensuous founding acts (LU II, B2 146).19 Lask and Husserl thus shared the idea that the philosophical nature of logic must be studied by studying the attitudes or the points of view which the subject of knowledge has to the objects of knowledge. Hence, in Husserl’s view the origin of logical concepts is in sense perception; logical forms can become ideal objects studied by the science called logic, because there is a subject who sees the objects of perception in an explicating manner (LU II, A 623–625/B2 151–153). Logical concepts, like the concepts of whole and part, are possibilities in objects which become articulated in categorial acts (LU II, A 627/B2 155). Logical forms are not in themselves

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but the subject makes them exist. Husserl thus found the origin of logical concepts in the activity of the subject. For Husserl, the forms of thought have been set into sensuous objects and as objective they can be known by us. To know them is to construct new categorial objects, and that constructing is categorial intuition. On this reading of Husserl’s text, logical knowledge is possible because it is knowledge concerning our own constructions. Frege thought that we cannot take distance from logical categories, we can only write them down when we see them correctly; we cannot present an epistemological theory for them. Nonetheless, Frege gave us several epistemological arguments which aimed at supporting his choice of a certain kind of logical language. Unlike Frege, Husserl thought that logical categories and even logical laws need and can be given justiﬁcation, which means giving an epistemological theory for logic.20 A somewhat surprising conclusion can be drawn if we pay attention to the connection between the views of the possibility of logical knowledge just discussed and of the nature of philosophy. The philosophers of the Vienna Circle thought that Frege’s, Russell’s, and Whitehead’s logic was a neutral system of formulas and a useful tool for clarifying thoughts, hence, philosophy was for them a certain kind of activity, namely, the activity of clarifying thoughts by means of the new tool. They did not suggest that philosophers ought to present theories of anything, not even of logical knowledge. Later, it has been typical of the analytic tradition to put forward philosophical, formal, and semiformal theories of various kinds, including theories of logic, or logics, and of natural language. From that perspective, Husserl’s way of thinking of the possibility of logical knowledge and his search for a theory of that kind of knowledge is a more natural background for the analytic tradition than Frege’s approach.

4. Discovery, Justiﬁcation, and Intuition 4.1. The Rejection of Intuition The problem of justiﬁcation became a central theme in the philosophy of logic during the ﬁrst decades of the twentieth century. The role of intuition as a justiﬁer was discussed by logicians and philosophers. From what has been said, it seems that Husserl opposed reference to intuition in cases where Frege was ready to rely on intuitive knowledge. If we argue that way, we suggest that Husserl’s demand for justiﬁcation goes further than that of Frege’s. It is true Husserl thought that even if propositions that are taken to be basic in a formal system are not in need of justiﬁcation in terms of logical inference, they need another kind of justiﬁcation, namely, a philosophical justiﬁcation. Unlike Frege, Husserl thought that logical categories and logical laws need and can be given philosophical justiﬁcation in the sense of giving a philosophical or an epistemological theory for logic. In section 4.2.1, I trace Husserl’s view back to its Kantian origins.

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In a postumous work published in 1991, J. Alberto Coﬀa considers the semantic tradition from Bolzano to Carnap, hence from the early nineteenth century until the early twentieth century. According to Coﬀa, the semantic tradition reacts against Kant’s philosophy. He claims that that tradition tried to get rid of all references to intuition, which it took to be Kant’s great problem. Coﬀa points out that the semantic tradition can be deﬁned by means of its problem, its enemy, its goal, and its strategy. According to Coﬀa, its problem was a priori, its enemy was pure intuition, on which Kant relied when he studied the possibility of mathematics, its aim was to develop a concept of a priori in which pure intuition played no role, and its strategy was to base that theory on the development of semantics (Coﬀa 1991, 22). Coﬀa also argues that in geometry it is particularly necessary to refer to constructions which are seen immediately but that even calculus, which was the strongest branch of eighteenth-century mathematics, had the same practice (ibid., 23). Coﬀa remarks that by the end of the nineteenth century Bolzano, Helmholz, Frege, Dedekind, and many others helped settle that Kant was not right, that concepts without intuition were not empty (ibid., 140). The pioneers of logic at the end of the nineteenth century stressed that in the ﬁeld of logic one is not allowed to refer to intuition; each inferential step must be written down. Particularly, Frege’s conceptual notation was meant to be a tool by means of which each step in the process of inference can be written down exactly without any resort to intuition. However, even if Frege and Peirce, among others and maybe most prominently, were creating a new logic and Frege tried to carry out a program which aimed at reducing arithmetic to logic, they did not, and they did not even want to, get rid of intuition altogether. Of course the very concept of intuition was problematic for them. If we look at the pages of Frege’s Begriﬀsschrift, we notice that he appeals to what we would nowadays call our pattern recognition abilities both in his analysis of sentences and in his ways of presenting inferences. Peirce laid even more emphasis on the role of intuition. For example, in 1898 he praised Kant for understanding the role of constructions or diagrams in mathematical inference. He wrote that mathematical inference proceeds by means of observation and experiment and that the necessary nature of this inference is merely caused by the fact that a mathematician observes and tests a diagram which is his own creation (“The Logic of Mathematics in Relation to Education,” 1898, CP, 3.560). In 1896, Peirce noted that logic has to do with observing facts concerning mental constructions (NE 4, 267). He very often stressed the value of ﬁgures in inference and states in 1902 that all knowledge has its origin in observation (NE 4, 47–48). Of course these pioneers of modern logic did not assume that a logician is able to see all the consequences of given premises, and they did not give a logician a permission to refer to seeing the conclusion of given premises immediately. Nevertheless, they thought that when taking the shortest steps in an inferential process, a logician does something that can be naturally called perceiving or seeing.

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4.2. Husserl’s Problem of Justiﬁcation and Frege’s and Peirce’s Discoveries Even if the idea of axiomatic science in logic is not discussed in this section, the methods of logical discovery and justiﬁcation deserve attention. Frege claimed that all great scientiﬁc improvements of modern times have their origin in the improvement of method. He wrote: I would console myself on this point with the realization that a development of method, too, furthers science. Bacon, after all, thought it better to invent a means by which everything could easily be discovered than to discover particular truths, and all great steps of scientiﬁc progress in recent times have had their origin in an improvement of method. (Frege, BS, 1964, XI; Frege 1972, 105) The method Frege proposed for science was his begriﬀsschrift, the new logic, but there was even a deeper truth in his statement. A better method was also needed if one wished to improve logic. In his paper “Explanation of Curiosity the First” (1908) Peirce described Euclid’s procedure in proving theorems. Euclid ﬁrst presented his theorem in general terms and then translated it into singular terms. Peirce paid attention to the fact that the generality of the statement was not lost by that move. The next step was construction, which was followed by demonstration. Finally, the ergo-sentence repeated the original general proposition. Peirce laid much emphasis on the distinction between corollarial and theorematic reasoning in geometry. He took an argument to be corollarial if no auxiliary construction was needed. For Peirce, construction was “the principal theoric step” of the demonstration (CP, 4.616). Peirce also stressed that it is the observation of diagrams that is essential to all reasoning and that even if no auxiliary constructions are made, there is always the step from a general to a singular statement in deductive reasoning; that means introducing a kind of diagram to reasoning. Peirce’s methodological interests are well known. For example, in 1882 he stated in his “Introductory Lecture on the Study of Logic”: “This is the age of methods; and the university which is to be the exponent of the living condition of the human mind, must be the university of methods” (W 4, 379). Moreover, in his “Introductory Lecture on Logic” (1883) he made an interesting remark on methodology. He wrote: But modern logicians generally, particularly in Germany, do not regard Logic as an art but as a science. They do not conceive the logician as occupied in the study of methods of research, but only as describing what they call the normative laws of thought, or the essential maxims of all thinking. Now I have not a high respect for the Germans as logicians. I think them very unclear and obtuse. But I must admit that there is much to be said in favor of distinguishing

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Logic from Methodology. . . . Let us say then that Logic is not the art of method but the science which analyzes method. (W 4, 509–510) As Peirce thus regarded logic as science, it is no surprise that he was also interested in the methodological commitments and choices of the one who works in the science of logic. There is an interesting history of method from Kant to Frege and Husserl. What was especially important is that all the way the task is twofold. On the one hand, Kant considered transcendental forms, that is, logical concepts, to be our method or tools for reaching the phenomenal world, as he considered them to be our tools for constructing that world. On the other hand, he regarded it as necessary to have a proper method, which is transcendental analysis, for knowing those very tools. Frege’s task was also twofold. Frege set out to ﬁnd a new method for science, which would be his begriﬀsschrift, but he also needed a new method of discovering that very method. In Husserl’s philosophy the method of ﬁnding the method came to be a method of knowing the ideal world. That happened because Husserl considered the logical tools to have being as structures of that world. According to Husserl, logic tries to claim something about the structure of the realm of ideal objects, which is strange for us in the sense that it is independent of our subjective mental realms. Husserl’s question brings us back to the question of method, as Husserl assumed that we have knowledge of the ideal realm only if there is a reliable route from our subjective minds to the objective realm, that is, only if we have proper tools for reaching that realm. Therefore, for him the foundational task was to know and describe these very tools. 4.2.1. Husserl and the Justiﬁcation of Logic Husserl’s question “How is logic as science possible?” amounted to the question “What were the methods of discovery and justiﬁcation that justiﬁed modern logic as science?” Husserl also proposed this problem for those who are interested in the foundations of logic. He compared the activities of a practicing artist with those of a scientist. He argued that both of them are in an equally bad shape if we think of how conscious they are of the principles of their creation or their evaluation. Husserl even claimed that mathematics has no special position on this issue. A mathematician is often unable to inform us of his steps of discovery or to give us a proper theoretical evaluation, that is, a justiﬁcation, of his results (LU I, A 9–10/B 9–10). Husserl proposed that all discovery and testing rest on regularities of form and that regularities of form also make the theory of science, that is, logic, possible (LU I, A 22/B 22). Husserl’s thought lends itself easily to the framework of the philosophical tradition introduced by Kant. His main works in the ﬁeld of logic bear Kantian labels in their very titles. His trilogy of logic consisted of the book titled Logische Untersuchungen I–II (1900–1901), the ﬁrst volume of which he calls Prolegomena zur reinen Logik, Formale und transzendentale Logik (1929), and Erfahrung und Urteil (1939), which was posthumously completed and

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published by Ludwig Landgrebe. Even if Husserl attached his philosophy to Kantian themes, he was convinced that he had to raise heavy criticism against Kant’s ideas. He blamed Kant for having failed to achieve a “pure” theory of knowledge, which would be free from all naturalistic elements, such as psychological and anthropological assumptions. No more was he satisﬁed with neo-Kantians’ developments, which he called transcendental psychology (LU I, A 92–97/B 92–97).21 He admitted, though, that Kant’s philosophy also had features that go beyond psychologism (LU I, A 94/B 94, note). In his early writings, Husserl seemed to speak in favor of psychologism, for example, in his book Philosophie der Arithmetik (1891), which Frege, the devoted antipsychologist, heavily attacked in 1894 (“Rezension von: E. Husserl, Philosophie der Arithmetik, Erster Band, Leipzig, 1891,” KS, 179–192). Some scholars, for example Mohanty (1982), have disputed that Husserl was a psychologist in the sense that Frege gave to the term. Mohanty stresses that Frege’s criticism led Husserl to revise some parts of his theory of number and it may have made him pay more attention to distinguishing between act, content, and object. However, Mohanty points out that it could not lead Husserl to reject such a version of psychologism which Frege attacked simply because Husserl never subscribed to that version (Mohanty 1982, 22–26). However that may be, it was at the very end of the nineteenth century that Husserl clearly joined the antipsychologistic camp, which his Logische Untersuchungen testiﬁed. It may be noted that in that work he also pointed out that he does not want to reject everything that he has done in his Philosophie der Arithmetik (LU II, B1 283, note). In the Logische Untersuchungen, the main starting points for Husserl were Bolzano, Lotze, and Brentano, to whom Husserl paid homage in those two logical works (LU I, A 219–227/B 219–227, and LU II, A 344–350/B1 364–370). Bolzano (1837) had introduced Sätze an sich and Vorstellungen an sich, which he regarded as neither existing in space and time nor depending on our mental acts (Bolzano 1929, §19). Hence, Bolzano distinguished the proposition itself from our thinking of it and acknowledged a speciﬁc realm of ideal objects, for which he did not admit proper existence, though. As was noted, Lotze, for his part, considered being and validity to be two senses of actuality (Wirklichkeit) and distinguished between the being of concrete things and the validity of abstract objects. For him, validity was a way of being independent of subjective mental acts (Lotze 1874, 507). Even if Brentano was not a defender of abstract entities, he distinguished between mental acts and their objects, which have intentional inexistence in those acts but need not have any real existence (Brentano 1924, 124–125). Husserl was inﬂuenced by Brentano already from the middle of the 1880s, when he was Brentano’s student in Vienna.21 As was noted in section 3.2, Husserl approved of those ideas and made a distinction between the real and the ideal. He stated: There is an essential, quite unbridgeable diﬀerence between sciences of the ideal and sciences of the real. The former are a priori, the latter

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empirical. The former set forth ideal general laws grounded with intuitive certainty in certain general concepts; the latter establish real general laws, relating to a sphere of fact, with probabilities into which we have insight. (LU I, A 178/B 178; LI I, 185) Husserl observed that once the distinction between the ideal and the real realm is acknowledged, we quite naturally come to realize one crucial problem. This problem constituted an important part of Husserl’s criticism against Kant. In 1929 Husserl maintained that because Kant did not make the distinction between the ideal and the real, he failed to ask one important question. Because Kant did not assume any world of ideal objects of thought, he could not ask how we can have an access to these objects (FTL, 233–235). In the Formale und transzendentale Logik, Husserl was explicit in stressing the importance of Kant’s theories concerning the Humean problem, which include his doctrine of transcendental synthesis and of transcendental abilities in general. Husserl praised Kant’s questions concerning our knowledge and its presuppositions. However, he blamed Kant for not asking transcendental questions about formal logic (FTL, 228–230). Kant took Aristotelian logic to be a complete system, which needs no major corrections. All we can do for what he called general logic was to make it more elegant; the proper task of that logic, which is to expose and prove the formal rules of all thought, had already been accomplished, in Kant’s view (KRV, B viii–ix). Kant asked how pure mathematics is possible, how pure natural science is possible, and how metaphysics as natural disposition and as science is possible (KRV, B 20–22), but he did not ask how logic as science is possible. Husserl believed that if Kant had distinguished between the ideal and the real realm, it would have occurred to him to ask such an epistemological question. Husserl concluded that both Hume and Kant realized the transcendental problem of the constitution of what he called the real realm. He thought that they failed to see the corresponding problem concerning the constitution of the ideal objects, such as the judgments and the categories which belong to the sphere of reason and which logic is interested in. In other words, Kant did not make his analytic a priori a problem (FTL, 229–230). Husserl’s question in his logical works can thus be formulated in three ways: (1) How can we have knowledge of the realm of ideal objects? (2) How can we rely on what logic claims? (3) How can we justify the analytic truths a priori? These formulations have close connections. The ideal realm consists of abstract objects like numbers and thoughts, and it is precisely logic that tries to say something about the structure of thoughts and about the inferential links between thoughts. Therefore, because Kant did not ask how we can know anything about the ideal realm, he did not ask how logic as science is possible, either. Moreover, since logical laws are analytic a priori, Husserl asked how we can rely on the analytic a priori claims which logic oﬀers to us.22 Husserl thus blamed Kant for not being able to ask how we can have knowledge of the ideal realm. We could certainly defend Kant by the argument

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that because he did not postulate any such problematic realm as the realm of abstract objects, he did not need to face such epistemological problems as Husserl. We may also say that even if Kant did not ask Husserl’s question, his Kritik der reinen Vernunft served as an answer to that question. However, the point in Husserl’s argument might be construed as the statement that we cannot know anything that is not made objective, hence that the proper deﬁnition of the concept of knowledge implies that the object of knowledge is distinguished from the knowing subject. On this construal of his argument, Husserl required that logical concepts and laws are something that can be known in the proper sense of knowledge. If this is what he meant, the point of his criticism was that Kant did not set the conceptual tools of logic outside consciousness to study those tools.23 Husserl thus asked the question which Kant did not ask and tried to do what Kant did not do, namely, lay the epistemological foundations of logic. But what was actually the philosophical incentive of the question concerning how logic as science is possible? From Galilei and Descartes to Kant, philosophers had sought for a ﬁrm foundation for modern natural science, for mathematics and even for metaphysics. If we believe that the history of logic can be reconstructed as a Kuhnian science, hence, that the question of foundations arises in logic when the received framework is threatened, we quite naturally see the nineteenth century as a revolutionary period in logic. Aristotelian logic was losing ground in those days, and new formal developments arose. What this period needed, then, was an epistemological justiﬁcation for either the old logic or for those new suggestions. Hence, on this construal, Husserl’s question was necessitated by the new developments of logic in the nineteenth century. Husserl remarked: “how could such a logic [scientiﬁc logic] become possible while the themes belonging to it originally remained confused?” (FTL, 158; Husserl 1969, 178). The foundational crisis was not the most perspicuous reason for the question concerning the possibility of logic as science. The question arose as a natural consequence of the various confrontations within logic and philosophy of logic in the nineteenth century. As we saw in Husserl’s case, it arose from a philosophical position that postulated a speciﬁc realm of abstract objects like thoughts which logic speaks about. If that kind of realm is assumed and acknowledged, it is quite natural to ask how we can have knowledge of it, that is, how we can rely on logic which is supposed to speak about it. But why does anyone want to assume such an objective realm? I already suggested one answer that had to do with the proper concept of knowledge. Other guesses can also be made. Husserl’s argumentation suggests that historically the objectivity of the ﬁeld of interest of logic was probably necessitated by a proper criticism against a psychological or anthropological interpretation of Kant’s transcendentalism, which was represented by such logicians as Jakob Fries and Benno Erdmann, for example. Fries thought that logical concepts must be understood as the ways in which the human species organizes experience, and the logical laws must be construed as anthropological laws.24

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On Husserl’s judgment, the philosophy of logic of his own day was strongly anthropologistic; he even argued that it was rare to ﬁnd a thinker who would be free from the inﬂuence of that doctrine (LU I, A 116/B 116). In addition to the empiricist tradition, psychologism in logic had a natural connection with Kant’s transcendentalism, for the transcendental structure of human thought was easily construed as a psychological structure, which is typical of the human race. If one wanted to save transcendental logic from that kind of reading, one had better regard the transcendental structure as the structure of some objective realm.25 4.2.2. The Role of Judgments in Frege’s and Peirce’s Logical Discoveries Frege and Peirce discovered quantiﬁcation theory independently of each other. They both introduced a new formula language in which arguments or indices were distinguished from functions or relative terms. In his paper “Über den Zweck der Begriﬀsschrift” (1883) Frege remarked: In fact, it is one of the most important diﬀerences between my way of thinking and the Boolean way—and indeed I can add the Aristotelian way—that I do not proceed from concepts but from judgements. (BS 1964, 101) That Frege opposed Aristotle and Boole has been noticed by all interpreters, but it was about 30 years ago that Frege’s way of thinking was taken under more extensive historical consideration. Interpreters such as Hans Sluga (1980) linked Frege’s view with Kant’s idea that judgments have priority over their constitutive concepts. Kant was also one of Peirce’s philosophical heroes. Murray Murphey (1961) already noted that Peirce’s logical discovery brought him closer to Kant, as Peirce distinguished between indices and relative terms, hence, as it were, wrote down Kant’s distinction between intuitions and concepts. In his paper “Booles rechnende Logik und die Begriﬀsschrift” (1880/81), Frege clariﬁed the diﬀerence between his conceptual notation and Boolean logic. He stated that the real diﬀerence is that in logic he avoids a division into two parts, of which the ﬁrst is dedicated to the relation of concepts, that is, to primary propositions, and the second to the relation of judgments, that is, to secondary propositions, by construing judgments as prior to concept formation (ibid., 14 and 52). He continued that unlike Boole, he reduces his primary propositions to the secondary ones, which comes up in that he construes the subordination of two concepts as a hypothetical judgment (ibid., 17–18). This result came out when Frege broke up the judgment which contained subordination, which is a relation between two concepts. Before Frege was able to do this, he had to realize the distinction between individuals and concepts. This is what he also emphasized. In the article he remarked that his view does justice to that distinction. In Frege’s view, the problem with Boole’s notation lay in that Boole’s letters never meant individuals but always extensions of concepts. The distinction between individuals and concepts, or

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more generally functions, hence, between proper names and function names, was a crucial part in Frege’s discovery. It seems on the basis of Frege’s remarks in the Grundlagen that even if Frege criticized Kant’s concept of intuition, he viewed the distinction between intuitions and concepts as a precursor of his own distinction (GLA, §27, n.). The same methodological change from the Boolean method to the analysis of judgments was essential to Peirce’s discovery. I already mentioned that Kant was an important ﬁgure behind Peirce’s philosophy. As Murphey remarked, it was the manner in which Kant discovered his categories that interested Peirce most of all (Murphey 1961, 33). In the 1870s, Peirce discovered his logic of relatives, which was inspired by De Morgan’s ideas and Boole’s algebra of logic. Peirce’s articles titled “The Logic of Relatives” (1883) and “On the Algebra of Logic: A Contribution to the Philosophy of Notation” (1885) contained the ﬁrst presentation of his quantiﬁcation theory, which he himself called his general algebra of logic and which, as he wrote, he developed on the basis of O. H. Mitchell’s, his student’s, ideas (CP, 3.363 and 3.393). The ﬁrst important change from Boole’s logical algebra was that Peirce added indices to relations. Indices referred directly to individuals. Second, he introduced the quantiﬁers “some” and “every.” When he introduced his two improvements of logic, Peirce referred to Mitchell’s article “On a New Algebra of Logic” (1883). He expressed his indebtedness to Mitchell regarding both indices and quantiﬁers. However, when he described Mitchell’s way of using indices, he deviated from what Mitchell said. Peirce interpreted Mitchell’s formula “F1 ” as “the proposition F is true of every object in the universe” and formula “Fu ” as “the proposition F is true of some object in the universe.” For Mitchell, the symbol F was any logical polynomial involving class terms and their negatives. He did not take it to be a proposition, but rather called it a predicate or a description of every or some part of the universe (Mitchell 1883, 75 and 96). Moreover, Peirce used the concept of individual, which Mitchell did not use. Otherwise, it is true that Mitchell had both indices and quantiﬁers, as Peirce declared. Mitchell supported the view that objects of thought, in which logic is interested, are either class terms or propositions, but that every proposition expresses a relation among class terms (Mitchell 1883, 73). Because Mitchell thought that, basically, every proposition expresses a relation among class terms, he relied on the Boolean method, which started from concepts and came up with propositions by combining concepts. It is precisely this way of thinking which Frege attacked, as we noted. Hence, even if Mitchell did suggest indices and quantiﬁers, the new logical language cannot be encountered in his treatment. Peirce’s contribution was to take propositions as the starting point of analysis and generate a distinction between relative terms and the names of individuals. In his article “On a New List of Categories” (1867), which was meant to improve Kant’s doctrine of categories, Peirce relied on the subject-predicate form of propositions and assumed that in the aggregate of a subject and a predicate the subject represented what he calls substance (CP, 1.547 and 1.548).

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For Peirce, substance was the present in general, hence not an individual. It is not until the 1880s that individuals in the sense of Kant’s intuitions appeared in Peirce’s logical notation. These observations suggest that between Peirce’s “New List” (1867) and his discovery of the new notation (1883, 1885) there was a methodological change, which contributed to his logical discovery. Hence, the decisive insight both for Frege and Peirce was that a judgment is not an aggregate of terms that represent concepts or classes but that its elements have diﬀerent kinds of roles in their contexts. Two of those basic roles are that of representing relations and that of denoting individuals.

5. Origins of Twentieth-Century Semantics: Frege’s Distinction between Sinne and Bedeutungen Even if Frege did not have any semantic theory, he expressed views of semantic concepts and had considerations in his works that can be called semantic. For Frege, the Sinne, senses, of sentences are thoughts and the Bedeutungen, references, of sentences are truth values, the True and the False. Sentences are compounded out of proper names, which refer to objects, and function names, which refer to functions. The Sinne of function names are simply parts of thoughts.26 But what are the Sinne expressed by proper names? In “Über Sinn und Bedeutung” (1892), Frege remarked that the sense of a proper name is a way the object to which this expression refers is presented, or a way of “looking at” this object. Furthermore, he stated that the sense expressed by a proper name belongs to the object to which the proper name refers. In other words, for Frege, senses were not primarily senses of names but senses of references. Hence, it is more advisable to speak about senses expressed by names than senses of names. Frege also gave examples of senses, like “the Evening Star” and “the Morning Star” as senses of Venus, and “the teacher of Alexander the Great” and “the pupil of Plato” as senses of Aristotle (“Über Sinn und Bedeutung,” KS, 144). Nonetheless, Frege admitted that we speak meaningfully about entities which do not exist. In his view, a sentence lacks only a truth value—but not a sense—if it contains a name that has no reference (“Über Sinn und Bedeutung,” KS, 148). Russell adopted a critical standpoint against this idea, according to which an expression can have a sense although it lacked a reference. In his article “On Denoting” (1905) he argued that a sentence like “The present King of France is bald” should be construed as the sentence “One and only one being has the property of being the present King of France, and that being is bald.” The property of being the present king of France does not belong to any being, and therefore the sentence is false. Moreover, according to Russell, the sentence “The present King of France is not bald” is false if it means that there is an entity which is now king of France and is not bald. Russell, however, suggested another analysis for the latter sentence which says that it

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is false that there is an entity that is now king of France and is bald. On this interpretation, the sentence turns out to be true (Russell 1956, 53). Frege regarded it as possible for an object to be given to us in a number of diﬀerent ways. He observed that it is common in our natural language that one single proper name expresses many of those senses which belong to an object. For Frege, to each way in which an object is presented there corresponds a special sense of the sentence that contains the name of that object. The diﬀerent thoughts that we get from the same sentence have the same truth value. In Frege’s view, we must sometimes stipulate that for every proper name there is just one associated manner of presentation of the object denoted by the proper name (“Der Gedanke,” 1918, KS, 350). However, he believed that diﬀerent names for the same object are unavoidable, because one can be led to the object in a variety of ways (“Über den Begriﬀ der Zahl,” 1891/92, NS, 95). For Frege, our knowledge of an object determines what sense, or what senses, the name of the object expresses to us. One sense or a number of senses provides us only with one-sided knowledge (einseitige Erkenntnis) of an object. Frege argued: “Complete knowledge [allseitige Erkenntnis] of the reference would require us to be able to say immediately whether any given sense belongs to it. To such knowledge we never attain” (“Über Sinn und Bedeutung,” KS, 144; Frege 1952, 58).27 On the basis of Frege’s hints, we may conclude that his concept of Sinn is thoroughly cognitive. Many of his formulations suggest that Sinne are complexes of individual properties of objects, hence, something knowable. If this interpretation of the concept of Sinn were correct, it would have been Frege’s view that we know an object completely only if we know all its properties, which is not possible for a ﬁnite human being. It would also follow that according to Frege, each object could in principle have an inﬁnite number of names which would correspond to the modes of presentation of the object. Frege did not hold the position that knowing some arbitrary property or complex of properties of an object constitutes knowing the object completely since, for him, a necessary condition for knowing an object would be knowing all the properties of that object. Nevertheless, on the suggested interpretation he thought that in a weaker sense we know an object precisely by knowing some properties of that object. It is true Frege’s weaker sense of knowing an object is not free from problems, either, even if it is more natural than the stronger sense. This is because Frege does not explain which properties of an object one must know to know the object. In Frege’s view, we are not able to speak about the senses of proper names as senses, for if we start speaking about them, they turn into objects, which, again, have their own senses. But what are these objects in case we speak about the senses expressed by proper names? Frege said that senses can be named (“Über Sinn und Bedeutung,” KS, 144–145) and proposed such examples as “the teacher of Alexander the Great” and “the pupil of Plato.” But if senses were complexes of the properties that belong to objects, as suggested, their names ought to be such as “being the teacher of Alexander the Great” or

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“being the pupil of Plato.” Frege’s examples suggest that when we name a sense of an object, we do not name any new object which would be a complex of individual properties of that object, but we name the original object in a new way. Hence, it follows from these examples that we do not succeed in naming a sense of an object as any new object, after all. Instead, we only name the object itself as considered under the description with which the sense provides us. There has been much discussion on what Frege’s motivation for adopting the distinction between senses and references might have been. When he introduced the distinction, he primarily referred to identity statements. It seems as if the distinction between Sinn and Bedeutung had, above all, been meant to give an adequate account of the symbol of identity, which Frege wanted to preserve in his language. By making the distinction between Sinn and Bedeutung, he sought to give a natural reading for identity statements. When introducing the concepts of sense and reference, Frege tried to solve the problems that what we now call intensional contexts caused for what we now call his idea of extensional language. The principle of functionality, which we may call the principle of compositionality in the case of references, is the core of that idea.28 Everything worked well according to what we would call truth tables when Frege constructed complex sentences out of simple sentences by means of conditionality (BS, §5). The trouble for Frege was caused by what became later called intensional contexts. Frege tried to deal with those contexts by introducing the concepts of indirect sense and indirect reference, the latter being the same as the normal sense of an expression. Frege claimed that in certain indirect contexts our words automatically switch their references to what normally are their senses. In a letter to Russell, he even recognized the need for using special signs for words in indirect speech (BW, 236). For example, in the complex sentences “A believes that a is P ” and “A believes that b is P ,” “that a is P ” and “that b is P ” name two diﬀerent thoughts, since “a” and “b” have diﬀerent senses. Let us assume that a and b have the same normal reference. Given that the truth value of the complex sentence is considered to be the value of a function whose arguments are the references of the components of the sentence, it does no harm to what we call the principle of functionality even if the complex sentences have diﬀerent truth values. Since the arguments of the function diﬀer from each other, that is, because a and b have diﬀerent indirect references, the references of the complex expressions may quite well be diﬀerent, and the principle of functionality is thus saved. Frege’s theory of Sinn and Bedeutung was not only a solution oﬀered to the problems that indirect contexts caused to the idea of extensional language, but it was also a direct consequence of his idea of a universal language. As noted, Frege’s begriﬀsschrift, conceptual notation, was meant to be a realization of Leibniz’s great idea. Leibniz thought that the terms of our natural language do not correspond to the things of the world in a proper way, and therefore we ought to construct a new language which mirrors correctly the whole universe.29 He dreamed of a language that speaks about the actual world in the sense of mirroring the individual concepts instantiated in this world. Frege’s

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world diﬀered from that of Leibniz in the sense that for him the actual world was the only world. For Frege, Sinne were something that we cannot avoid when we try to reach the world by means of our language. Frege’s belief in the inescapability of Sinne can thus be considered a special form of the Kantian belief that we must always consider objects through our conceptual systems. In “Ausführungen über Sinn und Bedeutung” (1982–1985) he remarked: “Thus it is via a sense and only via a sense that a proper name is related to an object” (NS, 135; Frege 1979, 124). Hence, the distinction between senses and references was something that Frege would have accepted in any case because of his belief in the role of conceptual machinery in reaching the world. That observation brings us back to where we started, namely, to how Frege understood the nature of his conceptual notation.30

Notes I have used extracts from my article “Analysis as the Method of Logical Discovery: Some Remarks on Frege and Husserl,” Synthese 77 (1988), 73–97, with the kind permission of Springer Science+Business Media. The chapter also contains passages from my article “Existence and Propositional Attitudes: A Fregean Analysis,” Logical Analysis and History of Philosophy 4 (2001), 75–86, which appear here with the kind permission of Mentis, and from my article “Finnish Studies in Phenomenology and Phenomenological Studies in Finland,” in Leila Haaparanta and Ilkka Niiniluoto (eds.), Analytic Philosophy in Finland, Poznań Studies in the Philosophy of the Sciences and the Humanities, vol. 80 (Rodopi, Amsterdam, 2003), 491–509, which appear here with the kind permission of Rodopi. I have used the manuscripts “Göttinger Vorlesungen über Urteilstheorie” (1905) and “Logik als Theorie der Erkenntnis” (1910–1911) with the kind permission of the Husserl Archives at the University of Leuven. 1. For the debate between psychologists and antipsychologists, see, for example, Kusch (1995). 2. See Friedman (1996, 2000) and Haaparanta (1999a, 2003). 3. See Beaney (2002) and Haaparanta (2007). 4. See Haaparanta (1985, 1999a) and Friedman (1996, 2000). 5. See Gabriel (1986). Cf. Ziehen (1920), 132–240. 6. See, for example, Haaparanta (1985) and Mancosu (1998). Also see Detlefsen (1992) and chapters 9 and 14 in this volume. 7. See Haaparanta (1988, 1999b). 8. See Lohmar (2002a, 2002b). 9. See Becker (1927) and Heyting (1930a, 1930b, 1931). 10. See, for example, Leibniz (1961a), 84 and 192, and Leibniz (1961b), 29, 152, and 283. See, for example, Frege, “Booles rechnende Logik und die Begriﬀsscrift” (1880/1881), NS, 9–52, “Über den Zweck der Begriﬀsschrift” (1883), BS (1964), 98, “Über die Begriﬀsschrift des Herrn Peano und meine eigene” (1896), KS, 227, GGA II, §§56–65, and “Anmerkungen Freges zu: Philip E. B. Jourdain, The development of the theories of mathematical logic and the principles of mathematics” (1912), KS, 341. For the terminological diﬀerence between Leibniz and Frege, see Haaparanta (1985), 11, and its references.

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11. The idea that Jaakko Hintikka (1979, 1981a, 1981b) has labeled as the idea of the ineﬀability of semantics and to which Hugly (1973) has also paid attention in Frege’s logic is visible at various points in Frege’s writings. For example, see Frege’s remarks on senses in “Über Sinn und Bedeutung” (1892), KS, 144–145, on functions in “Über Begriﬀ und Gegenstand” (1892), KS, 170, on the concept of identity in “Rezension von: E. G. Husserl, Philosophie der Arithmetik I” (1894), KS, 184, and on the concept of truth in “Der Gedanke” (1918), KS, 344. Also see his informal explanations of the semantics of his conceptual notation, “Darlegung der Begriﬀsschrift,” in GGA I. See Haaparanta (1985), 33, 41–43, 61–62, and 66. 12. See, for example, “Dialog mit Pünjer über Existenz” (before 1884), in NS, GLA, §53, “Über Begriﬀ und Gegenstand,” (1891), KS, 173, and Frege’s letter to Hilbert 6.1.1900, BW, 75. See also Haaparanta (1985). 13. See Frege’s “Vorwort” to GGA I. Also see his article “Der Gedanke” (1918), KS, 342–362. 14. See “Der Gedanke,” KS, 351, where Frege discusses the realm of representations (Vorstellungen). In Frege’s view, representations like sense impressions and feelings need someone who has them. Obviously, acknowledging the truth of a thought, that is, judging, needs someone who acknowledges. Frege’s terminology thus suggests that he takes the acts of judging to belong to the realm of our minds. 15. References are to the manuscripts “Göttinger Vorlesungen über Urteilstheorie” (1905) and “Logik als Theorie der Erkenntnis” (1910–1911). 16. See, for example, Kemp Smith (1962), 43–45. 17. See Haaparanta and Korhonen (1996), 40–41. See Crowell (1992) and Friedman (1996), 58–59. 18. See Haaparanta (1985). 19. See Haaparanta and Korhonen (1996), 42. 20. See Haaparanta (1988). 21. See Husserl’s biography in Schuhmann (1977). Even if the doctrine of “propositions in themselves” was popular among a number of Husserl’s predecessors and contemporaries, Husserl’s view can also be interpreted as ensuing from certain internal motives of his philosophy. This kind of reading is suggested by Cooper-Wiele (1989), who emphasizes the role of the idea of a totalizing act in Husserl’s thought. See Cooper-Wiele (1989), 11 and 90–108. 22. For Husserl’s concept of formal or analytical law, see LU II, A 246–251/B1 252–256. For Husserl’s discussion concerning the relationship between logical laws and the analytic a priori, see, for example, Husserl (1950b), 28. 23. The same problem had also been tackled by Hegel from an opposite point of view. In Hegel’s view, Kant’s problem was that his critical philosophy tried to study the faculty of knowledge before the act of knowing. Hegel argued that other tools can be studied before they are used, but the use and study of logical tools is one and the same process (Hegel, 1970, §10 and §41, Zusatz 1). 24. See Fries (1819), 8. Also see Fries (1827), 4. For Erdmann’s psychologistic interpretation of transcendentalism, see Erdmann (1923), 472–477. For Frege’s criticism of Erdmann, see GGA I, “Vorwort,” xv–xvi. 25. Kusch (1995) has studied the sociological aspects of the debate on psychologism. My presentation is restricted to those aspects that are internal to the philosophical discussion. For Husserl’s criticism of psychologism, also see Willard (1984), 143–166. 26. See Haaparanta (1985). Also see chapter 13 in this volume.

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27. The word “complete” is not a good translation for allseitig, but it is in any case not so misleading as the word “comprehensive” chosen by Geach and Black. A better expression would, perhaps, be “knowledge from every angle.” 28. See Frege’s argumentation in “Über Sinn und Bedeutung.” 29. See note 10. See also Leibniz (1969), sec. 8. 30. There are a great number of studies in late nineteenth-century and early twentieth-century philosophy, especially Frege and Husserl, that one could recommend for further reading, for example, Beaney (1996), Bilezki and Matar (1998), Dummett (1993), Floyd and Shieh (2001), Glock (1999), Hill (1991), Hill and Rosado Haddock (2000), Kreiser (2001), Macbeth (2005), Mendelsohn (2005), Reck (2002), Schumann (1977), Tieszen (1989, 2004), Tragesser (1977), and Weiner (2004).

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Lohmar, Dieter. 2002a. Elements of a Phenomenological Justiﬁcation of Logical Principles, including an Appendix with Mathematical Doubts concerning some Proofs of Cantor on the Transﬁniteness of the Set of Real Numbers. Philosophia Mathematica 10: 227–250. Lohmar, Dieter. 2002b. The Transition of the Principle of Excluded Middle from a Principle of Logic to an Axiom: Husserl’s Hesitant Revisionism in Logic. In New Yearbook of Phenomenology and Phenomenological Philosophy, ed. Burt Hopkins and Steven Crowell, 53–68. Madison: University of Wisconsin Press. Lotze, Hermann. 1874. System der Philosophie, Erster Teil: Drei Bücher der Logik. Leipzig: Verlag von G. Hirzel. Macbeth, Danielle. 2005. Frege’s Logic. Cambridge, Mass.: Harvard University Press. Mancosu, Paolo, ed. 1998. From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. New York: Oxford University Press. Mendelsohn, Richard L. 2005. The Philosophy of Gottlob Frege. Cambridge: Cambridge University Press. Mill, John Stuart. [1843] 1906. A System of Logic: Ratiocinative and Inductive. New York: Longmans, Green. Mitchell, O. H. 1883. On a New Algebra of Logic. In Studies in Logic by Members of the Johns Hopkins University, ed. Charles Peirce, 72–106. Boston: Little, Brown. Mohanty, J. N. 1982. Husserl and Frege. Bloomington: Indiana University Press. Murphey, Murray G. 1961. The Development of Peirce’s Philosophy. Cambridge, Mass.: Harvard University Press. Peirce, Charles Sanders. 1931–1935. Collected Papers of Charles Sanders Peirce (CP), vols. I–VI, ed. Charles Hartshorne and Paul Weiss. Cambridge, Mass.: Harvard University Press. Peirce, Charles Sanders. 1958. Collected Papers of Charles Sanders Peirce (CP), vols. VII–VII, ed. Arthur Burke. Cambridge, Mass.: Harvard University Press. Peirce, Charles Sanders. 1976. The New Elements of Mathematics by Charles S. Peirce (NE), vols. 1–4, ed. Carolyn Eisele. The Hague: Mouton. Peirce, Charles Sanders. 1984. Writings of Charles S. Peirce: A Chronological Edition (W), vol. 4, ed. E. Moore et al. Bloomington: Indiana University Press. Reck, Erich H., ed. 2002. From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy. New York: Oxford University Press. Russell, Bertrand. 1956. Logic and Knowledge: Essays 1901–1950, ed. Robert Charles Marsh. London and New York: Routledge. Schlick, Moritz. 1918. Allgemeine Erkenntnislehre. Wien: Verlag von Julius Springer. Schlick, Moritz. 1938. Gesammelte Aufsätze 1926–1936, Wien: Gerold. Schlick, Moritz. 1986. Die Probleme der Philosophie in ihrem Zusammenhang, Vorlesung aus dem Wintersemester 1933/34, hrsg. von H. Mulder, A. J. Kox, und R. Hegselmann. Frankfurt am Main: Suhrkamp. Schuhmann, Karl. 1977. Husserl-Chronik. Denk- und Lebensweg Edmund Husserls. The Hague: Martinus Nijhoﬀ. Seebohm, Thomas. 1989. Transcendental Phenomenology. In Husserl’s Phenomenology: A Textbook, ed. J. N. Mohanty and W. R. McKenna, 345–385. Washington, D.C.: Center for Advanced Research in Phenomenology & University Press of America. Sluga, Hans D. 1980. Gottlob Frege. London: Routledge. Tieszen, Richard. 1989. Mathematical Intuition: Phenomenology and Mathematical Knowledge. Dordrecht: Kluwer.

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Tieszen, Richard. 2004. Husserl’s Logic. In Handbook of the History of Logic, vol. 3: From Leibniz to Frege, ed. Dov M. Gabbay and John Woods, 207–321. Amsterdam: Elsevier. Tragesser, Robert S. 1977. Phenomenology and Logic. Ithaca, N.Y.: Cornell University Press. Trendelenburg, Adolf. 1867. Über Leibnizens Entwurf einer allgemeinen Charakteristik. In Adolf Trendelenburg, Historische Beiträge zur Philosophie, Dritter Band: Vermischte Abhandlungen, 1–47. Berlin: Verlag von G. Bethge. Weiner, Joan. 2004. Frege Explained: From Arithmetic to Analytic Philosophy. La Salle, Ill.: Open Court. Willard, Dallas. 1984. Logic and the Objectivity of Knowledge: A Study in Husserl’s Early Philosophy. Athens: Ohio University Press. Der Wiener Kreis. [1929] 1973. The Vienna Circle of the Scientiﬁc Conception of the World (Wissenschaftliche Weltauﬀassung, Der Wiener Kreis). In Otto Neurath: Empiricism and Sociology, ed. Martha Neurath and Robert S. Cohen, 301–318. Dordrecht: Reidel. Ziehen, Theodor. 1920. Lehrbuch der Logik auf positivistischer Grundlage mit Berücksichtigung der Geschichte der Logik. Bonn: A. Marcus & E. Webers Verlag.

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A Century of Judgment and Inference, 1837–1936: Some Strands in the Development of Logic Göran Sundholm Dedicated to Per Martin-Löf on the occasion of his 60th birthday. “O judgement! thou art ﬂed to brutish beasts, And men have lost their reason.” —Julius Caesar, Shakespeare My oﬃce in the present chapter is to tell how, within a century, the notions of judgment and inference were driven out of logical theory and replaced by propositions and (logical) consequence. Systematic considerations guide the treatment. My history is unashamedly Whiggish: A current position will be shown as the outcome, or even culmination, of a historical development. No apology is oﬀered, nor, in my opinion, is one needed. Philosophy in general, and the philosophy of logic in particular, treats of conceptual architecture. The logical ediﬁce is an old one and its supporting concepts have a venerable pedigree. Many parts of the building are buried in the past. Thus, the study of conceptual architecture has to be aided by conceptual archaeology. In the light The present chapter is based on lectures that I have given to second-year philosophy students at Leyden since 1990, and also draws on my inaugural lecture (1988). Per MartinLöf’s (1983) Siena lectures were an important source of inspiration, as were innumerable subsequent conversations with him on the history and philosophy of logic. In recent years conversations with my colleagues Maria S. van der Schaar and E. P. Bos have also been helpful. I am also indebted to Dr. Björn Jespersen and Dr. van der Schaar for valuable comments on the penultimate draft and to Dott.ssa Arianna Betti for much appreciated help with word processing. The material has been treated in invited lecture-courses, at the ESSLI Summerschool in Saarbrücken 1991, and at the universities of Siena 1992, Campinas and Rio de Janeiro 1993, Turku 1998, and Amsterdam 1999, as well as in a complete semester-course at Stockholm 1994. I am indebted to hosts and participants alike. My Cracow 1999 LMPS 11 lecture, now published as (2002), brieﬂy tells the inference half of the tale. Translations into English are in general my own.

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of the many changes that logic underwent during my chosen period, this may even seem rather apposite. Within the philosophy of logic, to understand what a present-day position is, it is often essential to understand how it became what it is. Furthermore, the systematic philosophical underpinning of presentday logic is not ﬁxed; the balance is not ready to be drawn up. Accordingly, a survey of the historical development that led to the various options is a required aid for an informed choice among contemporary alternatives. In one essential respect, though, mine diﬀers from a Whig history. The ﬁnal outcome is not necessarily seen as an improvement on earlier but now largely abandoned views. My own preferences go in the direction of anti-realism, but a deliberate attempt has been made to adopt a neutral stance when describing the various positions. The heroes and villains of my plot, in rough chronological order, are John of St. Thomas, Bernard Bolzano, Franz Brentano, Gottlob Frege, the Ludwig Wittgenstein of the Tractatus, Arend Heyting, and Gerhard Gentzen. Minor roles will be played by Immanuel Kant, Johann Gottlieb Fichte, David Hilbert, Bertrand Russell and G. E. Moore, Harold Joachim, and L. E. J. Brouwer. The treatment will not be exhaustive. In particular, many eminent logicians will not be treated, even though they do belong to the period under consideration, for the simple reason that their contributions did not touch the systematic theme that uniﬁes my exposition. The criteria for inclusion and exposition are based also on systematic considerations. It is my conviction, with respect to our present stage of logical knowledge, both systematic and historical, that this deserves preference above a mere recording of chronological facts. The systematic framework in which such facts are ﬁtted confers coherence and memorability on the unfolding tale. Such a procedure is not without its dangers. They have been faced with great lucidity by Jonathan Barnes: On the one hand, no discussion of the ancient theories will have any value unless it is conducted in moderately precise and rigorous terms; and on the other, a rigorous and precise terminology was unknown in the ancient world. If I insist on precision I shall be guilty of anachronism. If I stick to the ancient formulations, I shall be guilty of incoherence. I prefer anachronism.1 Barnes’s point is well taken and applies with equal force to the nineteenth century. Taking my cue from him, if methodological demands force me into anachronism, I would rather be coherent than (chronologically) right. The (Oxford English) dictionary explains logic as the art and science of valid reasoning. In my chosen century, the central notion of logic is that of judgment. Its form and function in inference will play a crucial role in the sequel. Changes in the conception of judgment and, concomitantly, of inference, are central here. Other topics, such as the position of the law of the excluded third, its function as a criterion for signiﬁcance, and its relation to the knowability of truth, also serve to structure the chapter. The (un)deﬁnability of truth, as well as the nature of the formal calculus used (if any), will also so serve.

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1. L’ancien Régime: The Logic That Was to Be Overturned The preface to early editions of Quine’s Methods of Logic opened with the terse observation that “Logic is an old subject, and since 1879 it has been a great one.”2 One would be hard put not to agree with the ﬁrst part of Quine’s quip, but a number of us have taken issue with the second. Surely, logic was great also prior to 1879, the year in which Frege published his Begriﬀsschrift. George Boolos and Hilary Putnam have respectively dated the inauguration of logical greatness to 1847 and 1854 on the strength of the appearance of George Boole’s logical works.3 Contrary to the received Massachusetts wisdom of Harvard and MIT, it seems obvious to me that the year 1837 deserves pride of place within the history of logic as the proper counterpart to 1879.4 To grasp the substance and magnitude of the logical revolution, we have to consider in outline the kind of logic that was superseded. To a large extent it was nothing but a latter-day version of traditional logic, with the typical methodological accretions that became common after the Port Royal Logic.5 We do well to remember that traditionally logic was conceived of as more wideranging than what is today the case. As a matter of fact, the “sweet Analytics of Aristotle”—Prior and Posterior—are not addressed to the same problematic. The Analytica Priora is devoted to the theory of consequence, that is, an answer is oﬀered to the question: What follows from what? The Analytica Posteriora, on the other hand, treats of the theory of demonstration, where the crucial question is: How does one obtain further knowledge from known premises? Present-day logic restricts itself to the theory of consequence and relegates the theory of demonstration to epistemology. In the nineteenth century, on the other hand, these epistemological concerns constituted a part of logical theory. At the beginning of my chosen period, the traditional patrimony is still very much in charge. The following familiar square oﬀers a convenient starting point for (my description of) the successive revolutions in logic: The Traditional Structure of Logic: Operation of the Intellect

(Mental) Product

(External) Sign

1

Simple Apprehension

Concept, Idea, (Mental) Term

(Written/spoken) Term

2

Judging, Composition/Division of two terms

Judgment, (Mental) Proposition: S is P .

Assertion, (Written/spoken) Proposition

3

Reasoning, Inferring

(Mental) Inference

(Written/spoken) Inference, Reasoning

The diagram6 employs a conceptual order of priority from left to right, from acts, via products, to signs: Acts of various kinds have mental products that

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(may) have (outward) linguistic signs, be they spoken or written. But for this “horizontal” order of conceptual priority there is also a “vertical” order of priority among the (act-)kinds, that is, the operations of the mind. The proper subject matter of logic is reasoning, that is, the third operation of the mind. Nevertheless, the two other operations have to be included in the domain of logic, since inferences are built from judgments. Judgments, in turn, are formed through the composition, or division, of two concepts (“terms”). In logic, the conceptual order starts with terms, and proceeds via judgment to inference. The traditional diagram exhibits a characteristic tripartite act

object ←→ sign

structure. Indeed, Johann Gottlieb Fichte went so far as to claim that essentially there are only two philosophical positions with respect to its epistemological components act object.7 Either you give the object through the act, in which case—with Fichte—you are an idealist, or you direct the act toward the prior object, in which case you are a dogmatist.8 Under this act/object structure, concepts are objects of acts of grasping (“apprehending”), and similarly the judgments made (“mental propositions”) are products of the acts of judging. With respect to the third operation, though, the traditional position is not consequent. To sort this out we note a basic ambiguity in the term inference. On the one hand, inference may be taken in the sense of an inference pattern (German Schlussweise). Such a pattern, or mode, of inference can be given by means of a schema I: J1 J2 . . . Jk , J where I deliberately have allowed more than the customary two premises of traditional syllogisms. The mode I of inference corresponds to a rule of inference according to which you have the right to make, that is, to know, the judgment J, provided that you have already made, that is, provided that you already know, the judgments J1 , J2 , . . . , Jk . On the other hand, inference can also pertain to an act of inference, say, for instance, one made according to the mode I. Such an act has, or perhaps better, proceeds according to, the structure | | | J1 J2 . . . J3 9 . J The product of an act of inference, though, is not an act of inference—the act, clearly, does not have itself as product—nor is it the mode of inference I, according to which the act was carried out; on the contrary, it is the judgment made J. The traditional diagram is accordingly in error when it puts (mental)

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inference in the product place of the act of inference (reasoning). What could such a “mental inference” be? No suitable entity seems available for service in the role. The inference mode is not an act of inference, nor a product of such an act; it is a blueprint, or manual, for inference acts that have products. An inference act is a mediate act of judgment, in which one judgment, the conclusion, is known on the basis of certain other judgments, the premises, being known. Thus, an act of inference is a particular kind of judging, whence its (mental) product is a judgment made. Already Kant famously reversed one of the above orders of priority, namely, that between rows 1 and 2. Concepts are no longer held to be prior to judgments: “We can reduce all actions of reason to judgements, so that reason generally can be regarded as a capacity for judgement.”10 This reversal, in one form or other, we shall encounter in most of the thinkers here considered. Also other paradigm shifts in philosophy can be accounted for in terms of the traditional diagram. The most original contribution of twentieth-century philosophy, namely, the abolition of the primacy of the inner mental life that was eﬀected by Wittgenstein,11 can be seen as nothing but a reversal of the priorities between the second and third columns. The outward sign is no longer conceptually posterior to the inner product.

2. Speech Act Intermezzo: A Uniﬁed Linguistic Account for Some Nineteenth-Century Changes Traditionally, the linguistic counterpart to the mental judgment made is the assertion. This term, in common with other English -ion words, exhibits a process/product ambiguity.12 It may concern the act of asserting (judging) or the product of such an act, the assertion (judgment) made. The appropriate linguistic tool for assertion is the declarative sentence. In general, when S is a declarative sentence the question Is it true that S? may legitimately be put. An assertion that snow is white is readily eﬀected by means of a single utterance of the declarative sentence Snow is white.13 By convention, in the absence of counterindications that it should not be so held, a single utterance of a declarative is an assertion. For instance, the declarative S is not used assertorically in Consider the example: S. or He claimed that S, but I don’t know whether it really constitutes so.14

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Not every use of a declarative sentence is assertoric, but assertoric uses can be recognized as such since the counterquestions: How do you know that S is true? What are your grounds?15 are a legitimate response to an assertoric use of S. The content of the assertion eﬀected by means of an assertoric utterance of the declarative Snow is white, that is, the assertion that snow is white, is given by means of a nominalized that clause, that snow is white. In general, a single utterance of this clause alone will not serve to eﬀect an assertion that snow is white.16 To get back to a declarative, a single utterance of which will so serve, one must either append is true or preﬁx it is true to the clause in question. Then we obtain, respectively, that snow is white is true and it is true that snow is white, single utterances, either of which do suﬃce for asserting that snow is white. Note that the ﬁrst of these two formulations admits of the preﬁx the content. It then yields a yet fuller but still equivalent formulation of the judgment made: The content that snow is white is true. The second formulation, though, resists the corresponding interpolation, which results in ungrammatical nonsense: It is true the content that snow is white. These considerations suggest that judgable content A is true is the proper form of judgment, when one prefers a unary form of judgment that makes explicit the content judged in the judgment made.17 The content in question will be given by a that-clause formed from a declarative S. The judgment made in or by the act of judging that is made public through the assertoric utterance of the declarative S accordingly, takes the form that S is true.18

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It must be stressed, though, that this path to this unary, content-explicit form of judgment is manifestly not language independent, because it draws extensively on linguistic considerations, albeit very simple ones. As such, it would be rejected out of hand by most major ﬁgures considered here, even though the unary form of judgment itself is accepted. Diﬀerent analyses oﬀered by various logicians reach the same result, but diﬀerent routes are taken. Nevertheless the speech act theory route to the unary form of judgment constitutes as good an example as any of the characteristic—twentieth-century— linguistic turn in philosophy that was inaugurated by Frege (1891): ontological and epistemological questions are now answered (while recast in linguistic form) via a detour through language.19 However, drawing on the traditional conceptual link between judgment and assertion, namely, that between mental object and exterior sign, the above exposition, in spite of its anachronistic (twentieth-century) ﬂavor, explains why the (nineteenth-century) unary form of judgment has to take the form it has.

3. Revolution: Bolzano’s Annus Mirabilis I postulated that 1837 was a crucial year for logic, no reason being given. However, in this year the four hefty tomes of Bernard Bolzano’s Wissenschaftslehre made their weighty appearance.20 This event constitutes the greatest revolution in logical theory since Aristotle, even though the Wissenschaftslehre fell stillborn from the press, as far as near-time inﬂuence is concerned, owing to clerical and political censorship. Indeed, in the preface to the second edition of his main work (the ﬁrst edition of which appeared in the year of Bolzano’s birth) a very distinguished professor of philosophy could still write: “Since Aristotle, [Logic] has not had to retreat a single step. Also remarkable is that it has not been able to take a single step forward, and thus to all appearance is closed and perfect,”21 which state of aﬀairs continued until the coming of the second nineteenth-century revolution in logic. Within logical theory, 1879, the year of Quine’s choosing, is the counterpart to the second revolutionary year 1848. Traditional logic was ﬁrst and foremost a term logic, rather than a propositional logic. In spite of the medieval scholastic achievements concerning the theory of consequentiae, and the insights of the—much earlier—Stoic logic, the syllogism, in one version or other, still ruled supreme, which circumstance renders Kant’s opinion considerably less farfetched than it might seem today. For instance, his own conception of logic as set out in the Jäsche Logik (whether it be truly Kantian or not) is cast entirely in the customary traditional mold.22 Bolzano’s revolution with respect to the traditional picture is threefold. First, the middle (“product”) column of the traditional schema is objectiﬁed. The mental links are severed, and thus, in particular, the traditional notions mental term (concept, idea) and mental proposition (judgment) are turned into their ideal, or Platonist, counterparts idea-in-itself (Vorstellung an sich)

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and proposition-in-itself (Satz an sich).23 Second, the pivotal middle square of the diagram is altered: The judgment made no longer takes the traditional (S is P ) form. Logic is no longer term logic. Instead Bolzano uses the propositional, unary form of judgment that was canvassed above, with his Sätze an sich taking the role of judgable contents: The Satz an sich S is true.24 Third, Bolzano bases his logical theory, not on inference (from judgments known to judgment made), but on (logical) consequence between propositions.25 Judgment is dethroned and its content now holds pride of place in logical theory. Needless to say, Bolzano, a priest steeped in the tradition, does not jettison everything traditional: A Satz an sich, that is, the judgable content, rather than the judgment made, has (or can brought to) the canonical form V1 has V2 , which is very close to the Aristotelian form S is P . Instead of the Aristotelian judgment Man is mortal we ﬁnd the Bolzanian content Man has mortality. The precise reasons for this shift from the concrete mortal to the abstractum mortality need not detain us here; in essence, Bolzano takes the Aristotelian form of judgment and turns it into a form of content, where the contents are objectiﬁed denizens of the ideal—Platonic—third realm.26 Bolzano’s key notion is that of proposition-in-itself: The idea-in-itself is explained as a part of a proposition-in-itself that is not a proposition-in-itself. Bolzano’s logical objectivism is a Platonism: As already noted, his crucial an sich notions are all ideal. We are not told very much about what ideal means here. Instead, his manner of proceeding is that of a via negativa: a list of nonapplicable attributes is oﬀered. Thus, the ideal realm is characterized as atemporal, aspatial, inert, nonlinguistic, nonmental, unchangeable, nongenerated. . . . Furthermore, the propositions-in-themselves serve in various logical roles, in particular as contents of mental acts and declarative sentences.27 However, not only propositions and their parts are ideal an sich notions: The truth of a true proposition-in-itself is truth-in-itself. Bolzano’s explanation of truth is an interesting one. According to him, all propositions have or can be brought to the logical form (V1 has V2 ), and so truth only has to be explained

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for propositions of this form: The proposition (V1 has V2 ) is true if and only if V1 really (German wirklich) has V2 , for instance, the proposition-in-itself that snow has whiteness is true if and only if snow really has whiteness.28 This, virtually “disquotationalist,” rendering is compatible with currently fashionable “minimalist” positions concerning truth. Bolzano, however, was on the road toward a more substantial notion of truth when he noted that the following proportion holds concerning truth and a certain kind of existence, namely, that of instantiation (German Gegenständlichkeit), that is, the higherorder property of an idea-in-itself of being instantiated: the similarity between this relation among propositions and . . . that among ideas is obvious. Namely, what holds, concerning ideas, for the circumstance whether indeed a certain object falls under them or not, holds, concerning propositions, for the circumstance whether truth pertains to them or not.29 In the form of a proportion: proposition-in-itself idea-in-itself = . truth instantiation Thus, what it is for a proposition-in-itself to be true is what it is for an idea-in-itself to have something falling under it. In other words, applied to my (snow-bound) stock example: the proposition-in-itself that snow is white is true (or has truth, in the terminology preferred by Bolzano) precisely when the idea-initself the whiteness of snow has nonemptiness, that is, when some entity falls under the whiteness of snow.30 Bolzano here anticipates something of considerable importance for the analysis of truth, and we shall have occasion to return to his comparison in the sequel. Bolzano’s apparatus for logical analysis, comprising propositions, ideas, and instantiation, is highly versatile.31 Thus, for instance, as Leibniz knew, the four categorical Aristotelian judgments are readily cast in the required form. For instance, an E judgment, No V1 are V2 , is rendered the Idea (in-itself) of a V1 that is V2 does not have existence (Gegenständlichkeit).32

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The notion of truth in itself for propositions-in-themselves is bivalent: For every proposition-in-itself A, A has truth or A has falsity, also in-itself. The an sich character of the truth of true propositions-in-themselves is one of the pillars on which Bolzano’s logical realism rests.33 Another is the reduction of epistemological matters to the Platonist an sich notions. The ﬁrst instance of this reduction concerns judgment: a judgment of the novel form, that is, proposition-in-itself A is true is correct (richtig) if A really is a truth-in-itself.34 This reduces the epistemic notion of the correctness for judgments to the Platonist an sich notion of truth for propositional contents. Here Bolzano pays a price—in my opinion too high a price—for his iron-hard realism in logic and epistemology. Under the Bolzano reduction, a blind judgment, a mere guess, without any trace of justiﬁcation, is a piece of knowledge (an Erkenntnis).35 The only thing that matters is the an sich truth, whether knowable or not, of the proposition-in-itself that serves as content of the judgment in question.36 Thus, for instance, according to Bolzano, if, independently of any counting, it happens to hit bull’s eye, my unfounded claim that the City Hall at Leyden has 1234 window panes, is simply a piece of knowledge. In this I, for one, cannot follow him. Bolzano deserves high praise for his lucid and uncompromising realism. Also antirealists proﬁt from reading him: His version of realism is one of the very best on oﬀer.37 Admitting blind judgments as pieces of knowledge, however, is not just realism but realism run rampant. Bolzano’s transformation of the third and ﬁnal notion in the traditional picture, namely, that of inference, makes an unmistakably modern impression. The changed form of judgment transforms the inference schema I into I : A1 is true A2 is true . . . Ak is true . C is true An inference according to I is valid if the proposition-in-itself C is a logical consequence of the propositions-in-themselves A1 , A2 , . . . , Ak . Such a logische Ableitbarkeit—Bolzano’s terminology—holds between the A’s and C when each uniform variation V of all nonlogical ideas that makes all the A’s true also makes C true.38 In other—more modern—words, C is a logical consequence of A1 , A2 , . . . , Ak when the proposition-in-itself (A1 & A2 & · · · & Ak ) ⊃ C is not just true but logically true, that is, true under all uniform variations of its nonlogical parts.39 The notion of an Ableitbarkeit provides yet another Bolzano reduction of an epistemic notion to Platonist an sich notions. In the same fashion that

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Bolzano reduced the (epistemic) correctness (Richtigkeit) of the judgment made to the an sich truth of its an sich content, the validity of an inference is also reduced to, or in this case perhaps better, replaced by something on the level of the Platonist contents of judgments. Indeed, the fourth chapter, §§223–268, of the Wissenschaftslehre bears the title “Von den Schlüssen,” but deals with Ableitbarkeiten among propositions-in-themselves, rather than with judgments that are made on the basis of certain other judgments already having been made.40 Thus the inference is valid or not, irrespective of whether it transmits knowledge from premise judgments to the conclusion judgment, solely depending on the an sich truth-behavior of the propositions-in-themselves that serve as contents of the judgments in question, under all variations with respect to suitable in-themselves parts of the relevant propositions. Bolzano’s position is accordingly threatened not just by the phenomenon of blind knowledge. Under his account also inference can be blindly valid, irrespective of whether it preserves knowability from premise(s) to conclusion. Logical consequence (logische Ableitbarkeit) is a relation that may obtain between any propositions whatsoever, be they true or false. Bolzano also studies another consequence relation among propositions, but now restricted to the ﬁeld of truths-in-themselves only, that he calls Abfolge (grounding). The theory of Bolzano’s grounding relation is diﬃcult and as yet not very well explored; it can be seen as yet another reduction of epistemic notions to Platonist ones. Consider the inference I : A is true . B is true When I is valid, that is, preserves knowledge from premise to conclusion, and the premise is known, the judgment A is true serves to ground the judgment B is true. Then a certain relation obtains between the propositions A and B that serve as contents of the judgments in question. Abfolge can be seen as a “propositionalization” Abf (A, B) of that relation: The relation of grounding, which holds in the ﬁrst instance between pieces of knowledge, that is, between judgments known, is turned into a propositional relation (“connective”) between propositions, that is, contents of judgments. Every truth has a grounding tree that is partially ordered according to the Abfolge relation.41 It can be seen as an ideal proof that shows why the true proposition is true, somewhat along the lines of Aristotelian demonstrations διοτι.42 In the light of Bolzano’s innovations and ensuing reductions, it is important to distinguish between the holding of a consequence, that is, the preservation of truth from antecedent propositions to consequent proposition, and the validity of an inference ﬁgure, that is, the preservation of knowability from premise judgments to conclusion judgment.43 This insight is lost to modern philosophy of logic that largely accepts the Bolzano reduction to such an extent that (validity of) inference and (logical holding of) consequence are identiﬁed.

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4. Revisionism; the “Novel” Contributions of Brentano Franz Brentano, in lectures given at the Universities of Würzburg and Vienna, from the early 1870s onward, proposed another revision of traditional doctrine. Because of his distaste for all Platonist notions in logic, such as Bolzano’s proposition-in-itself, Brentano rejected the single unary form of judgment that ascribes truth to a Platonist content.44 Instead, he canvassed the use of two unary forms of judgments, namely, α IS (exists), in symbols α+, and α IS NOT (does not exist), in symbols α−, where α is a (general) concept. Brentano, however, was not the ﬁrst to note this. Already Bolzano explicitly considered these forms, under the respective guises of α has Non-Emptiness (Gegenständlichkeit) and α has Emptiness, and determined their most important properties. In particular, we already noted, Bolzano knew that the four Aristotelian categorical judgments can be dealt with using these two forms.45 Credibility might not be stretched to the point of credulity if we surmise that this anticipation provides one of the reasons for Brentano’s staggering lack of generosity toward the Great Bohemian: When . . . I drew attention to Bolzano, this . . . in no way, was intended to recommend Bolzano as a teacher and leader to the young people. What they could learn from him, I dare say, they could learn better from me. . . . And . . . as I myself never took a single thesis from Bolzano, so I was never able to convince my pupils that they would ﬁnd there a true enrichment of their philosophical knowledge.46 Under the circumstances, “methinks the learned Gentleman doth protest too much!” However, it is not unlikely that also Bolzano’s logical objectivism disqualiﬁed him as a “teacher and leader” in the eyes of Brentano, who distrusted all kinds of logical Platonism. Of more lasting value than Brentano’s employment—and alleged rediscovery—of the Leibniz–Bolzano reductions are his views on the blind judgment.47 These have profound consequences for his formulation of the traditional laws of thought, such as noncontradiction and excluded third, as well as for the relation between truth and evidence. Young man Brentano construed evidence as “experience of truth” (German Erlebnis der Wahrheit—Husserl’s terminology), whence the order of dependence goes from truth to evidence.48

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Later, under the pressure from the phenomenon of blind judgment, he reversed this order of priority and held that truth (correctness, German Richtigkeit) should be seen as possibility for evident judgment: Truth pertains to the judgments of he who judges rightly, that is, to the judgements of him who judges what someone would judge who judged with evidence; that is, he who asserts what would be asserted also by someone judging with evidence.49 Similarly he is led to a negative formulation of the law of excluded middle: It is impossible that someone, who rejects something that is wrongly accepted by someone else, rejects it wrongly, as well as that someone who accepts something, that is wrongly accepted by someone, accepts it wrongly, presupposed . . . that both judge with the same mode of representation and with the same mode of judgement.50 From an antirealist point of view, Brentano is certainly on the right track; he refrains from asserting that a content must be either true or false, in entire independence of whether it is known to be so. His formulation, though, is not entirely correct. Brentano, the great crusader against the blind judgment, here forgets to take it into account. Of course, it is possible that the object A is wrongly accepted by P1 , as well as wrongly rejected by P2 , namely, when P1 and P2 both judge blindly, that is, without evidence. On the other hand, the corresponding formulation of Noncontradiction is correct: It is impossible that someone rightly rejects what is rightly accepted by someone else.

5. Functions Triumphant: Frege’s Account of Judgment and Inference Frege, pace Quine, is generally held to have inaugurated the revolution in logic. From the present perspective though, his contribution is remarkably slender. Logical objectivism, with its novel unary judgment, is present wholesale already in Bolzano, where it is cast in a more perspicuous form. Frege, furthermore, does not treat of logical consequence among propositions, or Thoughts, as he called them. For better or worse, Bolzano, with his insistence on replacing inference with the notion of consequence, makes a much more modern impression than Frege, whose traditional views on inference have come in for much criticism. We must not forget, however, that Frege was a mathematician and from the outset his aims were those of a mathematician rather than of a philosopher. His contributions to my topic are all subservient to the aim of providing a secure foundation for mathematical analysis, very much in the style of traditional Aristotelian foundationalism: One seeks a small number of primitive concepts, and basic truths concerning those primitives, in terms of which, at least a very sizable part and preferably all, of mathematics can be formulated, while its truths can be derived by means of primitive inference steps, where the basic

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axioms and primitive inference steps are made evident from the concepts they contain.51 In the Begriﬀschrift booklet from 1879 (what turns out to be) a preliminary version of the formal language is given and the basic notions explained. In the Grundlagen der Arithmetik from 1884 the program of securing the mathematical theorems by means of reducing the mathematical axioms to logical theorems is spelled out informally. However, Frege was aware of the fact that he had only made plausible the reduction of arithmetic to logic, since, possibly at the instigation of the Brentanist Carl Stumpf, the Grundlagen development was informal and not carried out in the begriﬀsschrift. Thus Frege could not guarantee that his demonstration were really gap-free. The means of demonstration, whether logical or arithmetical, were not explicitly listed. Accordingly, his inferences have not been made evident solely from the concepts employed in them, and so the arithmetical ediﬁce remains shaky. The (considerable) changes in the begriﬀsschrift that were put into eﬀect around 1890 served to make the formal execution of the logicist program feasible; unfortunately, the project failed owing to the emergence of the Zermelo–Russell paradox in Frege’s system. Thus, when compared to Bolzano, Frege’s most important contribution is his begriﬀsschrift.52 By creating this formal language, Frege provides a partial realization of the Leibnizian calculus ratiocinator project. That an inference step, or axiom, is valid depends on contentual aspects pertaining to the notions from which the step, or axiom, in question has been built.53 However, once such a step has been explicitly formulated and validated in terms of contents, it is mechanically recognizable as such. No further contentual, “intuitive” considerations are required to determine whether the inference in question is valid; being of the appropriate syntactic form suﬃces and that form is mechanically, or “blindly,” recognizable. As far as the theoretical framework is concerned, Frege’s one step over and beyond Bolzano is minute but with enormous consequences. In both early and mature formulations of his theory of judgment, Bolzano’s unary form of judgment is retained: The circumstance that S is a fact and a judgment is not the mere grasping of a Thought, but the acknowledgment of its truth.54 Frege, however, by training and profession was a mathematician. His teaching activity was mainly devoted to analytical geometry. Through his mentor Ernst Abbe, one of Riemann’s few students, he also gained access to the latest developments in the then emerging function theory, that is, that branch of mathematical analysis that deals with analytic functions in the complex plane. His logical revolution draws heavily on the notion of function: Instead of Bolzano’s clumsy form of content “A has b”, Frege carves up his contents using the versatile form P (a),

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that is, function P applied to argument a. Frege’s logic is mathematized from the outset. It is especially well suited for coping with Weierstraß’s rigorous treatment of analysis; indeed, the notation could have been (and probably was) invented for the very purpose.55 The familiar concepts of pointwise continuity, and its reﬁnement into uniform continuity, illustrate this: (∀x ∈ I)(∀ε > 0)(∃δ > 0)(∀y ∈ I)(|x − y| < δ ⊃ |f (x) − f (y)| < ε) and (∀ε > 0)(∃δ > 0)(∀x ∈ I)(∀y ∈ I)(|x − y| < δ ⊃ |f (x) − f (y)| < ε). These succinct formulations show how admirably the Fregean quantiﬁer is geared to expressing distinctions involving multiple generality.56 A verbal, natural language treatment would be much harder to take in. Frege’s function-theoretic conception of logic imposed an interesting bifurcation on his views on truth. Mathematicians speak of the value of a function for a certain argument. For instance, 2 + 2 is the value of the function x + 2 for the argument 2. In the ﬁrst instance, the plus-two function takes numbers into numbers, but owing to Frege’s doctrine of universality, it has to be extended into one deﬁned for all objects. One then makes use of what Quine has called a “don’t care” argument, for instance, r+2 if r is a number; ξ + 2 =def the Moon otherwise. Adopting the same perspectives also at the level of sentences, from the complete sentence Caesar conquered Gaul, we get the function ξ conquered Gaul, which must also be deﬁned for all objects, including me, the Moon, and Louis XIV, as well as the number of those grains of sand at Syracuse beach that were not counted by Archimedes when writing the Sandreckoner, and plutonium, an element unknown at the time of Frege. A value of the conquering Gaul function will have to be something close to a judgable content, or Thought. It will not, however, be a judgable content, because it is not invariant under diﬀerent descriptions of the argument. Frege’s by now notorious example concerning the planet Venus makes this clear: Venus = Venus, The Morning Star = Venus, and the Evening Star = Venus are three values of the function ξ = Venus.

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Since in all three cases the argument—Venus = The Morning Star = The Evening Star—is the same, the value has to be the same. The Thought expressed is diﬀerent in all three cases.57 Accordingly, the value of the function for the argument Venus (under any description) is not a Thought. Instead Frege avails himself of certain ideal objects, the True and the False, that are known as “truth values,” and serve as appropriate function-values. By the truth value of a sentence, Frege understands the circumstance that it is true or that it is false.58 Thus, the common function-value in the three cases above is the truth value the True. Sentences are then seen as truth value names.59 In his elucidation of the revised begriﬀsschrift Frege lays down, for each regular sentence, under what condition it is a name of the True. The sentence then expresses, or has as its sense (Sinn), the Thought that this truth condition is fulﬁlled.60 Frege’s theory of meaning is a bipartite mediation theory, very much along the lines of early medieval theories of signiﬁcation: The sign expresses its sense that refers to an entity (called Bedeutung by Frege). In Frege’s theory a number of themes are dealt with that were touched on in the section 2. Frege deemed it necessary to include in his begriﬀsschrift a speciﬁc symbol, that makes explicit the assertoric force that the Kundgabe of a judgment made carries. Frege’s view of inference has come in for much criticism; an inference is an “act of judgement, which is made, according to logical laws, on the basis of judgements already made.”61 On the symbolic level, this is reﬂected in the omnipresence of the judgment stroke, both on premises and conclusion, in Frege’s formal inference-ﬁgures in the Gg. Modus ponens, for instance, takes the form A⊃B A 62 . B The sign “” has changed its meaning and in the logic of today it is an ordinary (meta)mathematical predicate applicable to certain (meta)mathematical objects, namely wﬀ’s, that is, elements of a free algebra of “expressions” generated over a certain “alphabet.” When ϕ ∈ wﬀ, “ ϕ” has the meaning there exists an inductively deﬁned derivation-tree of wﬀ’s with ϕ as end formula; in particular, the Frege sign does no longer function as a force indicator, but can be negated and occur in an antecedent of an implication.63 In Frege, however, it is clear that it expresses assertoric force. Thus, both premises and conclusion of inferences are known, since, as we remarked, assertions made do contain claims to knowledge. The practice of drawing inferences from mere hypothesis, however, in particular as embodied in the works of Gerhard Gentzen, is held to refute Frege at this point.64 Frege was ﬁrmly committed to realism: Being true (Wahrsein) is something diﬀerent from being held true (Fürwahrhalten), be it by one, be it by many, be it by all, and

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can in no way be reduced to it. It would be no contradiction that something is true that is held false by all.65 This is a clear statement of one of the central roles of truth, namely, its metaphysical role. By this I understand the task of truth to hold open the possibility of making mistakes. It is a minimum requirement on any viable epistemological position that it must allow for the possibility of mistaken acts of knowledge: “What is true is independent of our acknowledgement. We can make mistakes.”66 The opposite, “Protagorean” position would make man the measure of all things and would equate truth with truth-for-us. It would constitute an epistemological nihilism, where anything goes, along the lines of moral nihilism within ethics: “If God is dead, everything is permitted.” Mistaken deeds, be they logico-epistemic or ethical, presuppose a norm. Frege avails himself of the required norm via the notion of truth for judgable contents. He then reduces the rightness (Latin rectitudo) of epistemic acts, that is, the notion that is needed, strictly speaking, to uphold metaphysical realism, to that of the correctness of the judgment made in such an act, and that correctness ﬁnally to the truth of the Thought that serves as content of the judgment made. Truth for Thoughts, ﬁnally, is bivalent: Any Thought is either true or false, come what may. Frege secures this via his doctrine of sharp concepts. Thoughts are the result of applying concepts, that is, functions from objects to truth values, to objects. Concepts have to be sharply deﬁned on all objects: The Law of Excluded Third is really the requirement that concepts be sharply delineated in another guise. An arbitrary object Δ either falls under the concept Φ, or it does not fall under it: tertium non datur.67 Thus Frege, and before him Bolzano, secures the metaphysical role of truth, namely, that of providing the notion of rightness for epistemic acts, via the bivalence of truth for judgmental contents. This is not the only way to secure the notion of rightness for acts; Brentano, for instance, rejected the notion of proposition (Thought) and instead used product-correctness as the basic, absolute notion. Wittgenstein, on the other hand, did not take propositional truth as the basic notion, the way Frege and Bolzano did, but reduced it to the ontological notion of obtaining with respect to states of aﬀairs. One can also take the notion of rightness as a primitive notion sui generis, which is my own preferred option.68 Frege throughout his career held the view that truth (for propositions) is sui generis and indeﬁnable. Since the Thought that S is the same as the Thought that it is true that S, every Thought contains (the notion of) truth and so there is no neutral ground left from which to formulate a deﬁnition: Every putative deﬁniens irreducibly contains the deﬁniendum in question. Frege’s realism, just like Bolzano’s, is a logical one: There is no attempt at a further ontological reduction of propositional truth. For Frege, a fact is nothing but

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a true Thought and correspondence theories of truth are ﬁrmly rejected.69 His ontology is very sparse: objects, functions, and that is all—no facts, no states of aﬀairs, no tropes, or what have you. Frege held the wheel at the ﬁrst bend of the linguistic turn. His only category distinction is that between saturated and unsaturated entities, and this ontological distinction draws on the linguistic distinction between expressions with and without gaps into which other expression may be ﬁtted. In spite of his thoroughgoing realism, Frege appears committed to the view that every true proposition can be known as such: The most secure demonstration is obviously the purely logical, which, abstracting from the particular character of the things, rests only on the laws on which all knowledge depends. We then divide all truths that require a justiﬁcation into two kinds, in that for the one, the demonstration can proceed purely logically, for the other has to be based on facts of experience.70 Truths are then divided into those that need justiﬁcations and those that do not; the former are split into those that have purely logical demonstrations and those whose demonstrations rest on experiential facts. Thus, in either case, it appears that if the truth is one that stands in need of justiﬁcation, then there is a demonstration. Thus all truths can be known: If it needs no justiﬁcation, it can be known from itself, whereas truths that do need justiﬁcation can be known through a demonstration, be it logical or empirical.

6. Truth Made: The Correspondence Theory Strikes Back Half a decade after Frege’s Hochleistungen, G. E. Moore inaugurated his realist apostasy from the Hegelianism of his philosophical apprenticeship by adopting something very much like Bolzano’s theory of propositions with an an sich notion of simple truth. In this he was soon followed by Bertrand Russell.71 Russell and Moore were not crystal clear (to put it mildly). The best formulation of their novel theory was oﬀered by a staunch upholder of the old order, the idealist H. H. Joachim, whose aptly titled (1906) book The Nature of Truth has a chapter Truth as a Quality of Independent Entities. His characterization of the an sich theory of truth is a powerful one: “Truth” and “Falsity,” in the only strict sense of the terms, are characteristics of “Propositions.” Every Proposition, in itself in an entire independence of mind, is true or false; and only Propositions can be true or false. The truth or falsity of a Proposition is, so to say, its ﬂavor, which we must recognize, if we recognize it at all, immediately: much as we appreciate the ﬂavor of pineapple or the taste of gorgonzola.72

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Joachim also articulated clearly the possibility of unknowable truths on the an sich reading of truth: “The independent truth will be and remain entirely in itself, unknown and unknowable.”73 In an oblique way Russell had already admitted of the possibility of unknowable truths: Now, for my part, I see no possible way of deciding whether propositions of inﬁnite complexity are possible or not; but this at least is clear, that all the propositions known to us (and it would seem, all propositions that we can know) are of ﬁnite complexity.74 In philosophy, claims that something cannot be done are dangerous and invariably tend to provoke attempts to achieve what has been denied. Frege’s view that truth is sui generis and cannot be deﬁned was challenged even before it had been published:75 After yet another decade of logico-semantical soul-searching Moore and Russell were veering toward the correspondence theory of truth.76 Both gave reductions of truth in ontological terms by means of a truth-maker 77 analysis in the form proposition A is true = there exists a truth-maker for A. In a truth-maker analysis, to each proposition there is related a suitable notion of truth-maker and also a suitable notion of existence with respect to such truth-makers. Moore chose “facts” as his truth-makers and Russell used “complexes.” For Moore, a proposition is true if it corresponds to an existing fact, and for Russell it is true if the complex to which it corresponds exists. The intricacies of their respective ontologies of facts and complexes need not detain us here; both were superseded by Wittgenstein’s Tractatus and are now merely of historical interest. The Tractatus rests on three main pillars, to wit (i) Wittgenstein’s famous picture theory of linguistic representation; (ii) the doctrine of logical atomism, according to which every proposition is a truth-function of elementary propositions; and (iii) the Saying/showing doctrine. Of these the picture theory serves to structure the work.78 In a brief attempt at an exposition, I treat the proposition (∗)

Peter is the father of John

as if it were a Tractarian elementary proposition.79 Thus, our example (∗) is an elementary proposition of the form aRb. Hence, it must (?) immediately (?) strike us as a picture and indeed even one that obviously resembles its subject matter (4.12). How can we make sense of this? On the ontological side, in the world, we have the state of aﬀairs that Peter and John stand in the father-son relation. We now have to construe the propositional sign used to express the proposition (∗) as a fact that

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serves to present this state of aﬀairs. The two structures—linguistic and ontological—have to be, in mathematical parlance, isomorphic. Language

World

“Peter” Peter “John” John Q(a, b) father-son relation Q(“Peter”, “John”) Peter and John’s standing in the father-son relation Our task to ensure isomorphism between language and world amounts to ﬁnding an appropriate Q-relation. Obviously the ﬁeld of such a relation must consist of expressions and this is the key to Wittgenstein’s solution: Q(α, β) =def the expressions α and β stand, respectively, immediately to the left and to the right of the sign-array “is the father of.”80 Hence, “that ‘Peter’ stands in a certain relation, namely the Q-relation, to ‘John’, says that Peter and John stand in the father-son relation” (3.1432). Using the Q-relation, the sentence-sign (∗) is (or can be viewed as) a fact, since the two proper names do stand in the Q-relation. This syntactic fact in turn serves to present the state of aﬀairs that Peter is the father of John. When this state of aﬀairs exists (or obtains), it is a fact, and the proposition is true. In this case the proposition is a picture of the fact. According to the picture theory, every atomic, or elementary, proposition E presents a state of aﬀairs (Sachverhalt) SE that may or may not obtain (4.21).81 Accordingly, if the presented state of aﬀairs SE obtains the elementary proposition is true and depicts (what is then) the fact SE (4.25, 2). States of aﬀairs are logically independent of each other; from the obtaining of one nothing can be concluded about the obtaining of another (2.062). A point (Wahrheitsmöglichkeit) v in logical space LS is an assignment of + (obtains) and − (does not obtain) to each state of aﬀairs (4.3); in other words, a point in logical space is a function v from states of aﬀairs to {+, −}. Thus, LS = {+, −}SV , that is, the collection of functions from the collection SV of Sachverhalte to {+, −}. A situation (Sachlage) σ in logical space is a partition of LS into two parts σ + and σ − (2.11). Points in the positive part σ + are compatible and those in the negative part σ − are incompatible with σ. A proposition A is a truth-functional combination of elementary propositions (5).82 The truth-functional composition of the proposition A determines whether A is true or false with respect to or at a point v in LS. A point v ∈ LS induces a {T(rue), F(alse)}-valuation v on truth-functional propositions in the following way: For an elementary proposition E, v(E) = T

if v(SE ) = +;

v(E) = F

if v(SE ) = −.

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Thus, an elementary proposition is true at a point v if v is compatible with the state of aﬀairs that the elementary proposition presents. For the proposition A = N(ξ), v(A) = T if v(B) = F for every proposition B in the range ξ;83 v(A) = F otherwise. Thus, a generalized (joint) negation is true only if all the negated propositions are true (6, 5.5ﬀ.). A proposition C is a logical consequence of a class Γ of propositions if for every v ∈ LS such that v(A) = T for every A ∈ Γ also v(C) = T (5.11, 512). Because every proposition is obtained through repeated applications of the N-operation to (suitably presented) ranges of propositions, the explanation determines fully whether a proposition is true or false at a point in logical space (5.501–3). The sense (Sinn) of the proposition A is a certain Sachlage σA in LS (4.021, 4.2).84 The positive part of the sense of A is given by { v ∈ LS | v(A) = T }, and similarly for the negative part, of course. The thesis of truth-functionality then ensures that the Sachlage σA , that is, the sense of A can be “computed” from the symbol A. From this epitome it should be clear that the Tractarian logical theory is a realism of the kind that was inaugurated by Bolzano.85 However, Wittgenstein carries the logical realism of Bolzano and Frege to a ﬁtting conclusion: The logical realism of Bolzano is here replaced by an ontological realism. Propositional truth, the primitive an sich notion of logical realism, is reduced one step further to a prior ontological notion, namely, the obtaining of states of aﬀairs. Neither Bolzano nor Frege ignored epistemological issues; in fact, they were of an all-encompassing importance for Frege’s logicist project. Wittgenstein, on the other hand, deliberately eschews epistemic concerns in logic, for instance, the Frege–Russell assertion sign (4.442). Also the epistemic notion of inference is eliminated in favor of logical consequence by means of the Bolzano reduction (5.132). Nevertheless, concerning the deployment of logic, Wittgenstein held that it must be possible to compute mechanically from the symbols alone whether one proposition follows from another (5.13, 6.126, 6.1262). He was wrong in this. In general, the “computation” cannot be executed, owing to its inﬁnitary character. When he wrote the Tractatus, Wittgenstein was not aware of the unsolvability of the general Entscheidungsproblem for the predicate calculus. It was discovered—by Church and Turing—only in 1936, and poses an insuperable technical obstacle for the Tractarian philosophy of logic and language. Thus, Wittgenstein’s vision that everything important concerning logic could be read oﬀ mechanically am Symbol allein was rendered illusory. Wittgenstein was certainly aware of the fact that reasoning presupposes a correctness norm, because otherwise correct (right) and correct-for-me coincide, in which case there is no possibility for mistakes anymore. However, rather than taking rightness (rectitudo) of acts as a primitive notion, he adopts an

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ontological reduction of rightness. The order of explanation runs as follows: The rightness of the act of inference is reduced to the correctness of the product of the act, that is the judgment made, or knowledge obtained, which notion, in its turn, is reduced to the truth of the content. The truth of the content, ﬁnally, is reduced to an ontological notion, namely, that of the obtaining of states of aﬀairs.86 If objectivity is guaranteed at that level, say, in the form of bivalence for states of aﬀairs—a state of aﬀairs either obtains or does not obtain—it can be exported back to other levels, whence the possibility for mistakes is held open. In diagram form: The truth-maker reduction in Wittgenstein’s TRACTATUS (4) (2) {content of object} ← act of knowledge ↓ SC obtains ↔ [{Proposition C} is true] ↑ ↑ (1) state of aﬀairs [object of the act] (3) = [asserted statement, statement known]. From an epistemological point of view, the rightness notion for acts of knowledge is the most crucial one. It is enough to uphold the diﬀerence between appearance and reality, and, as such, constitutes the minimum requirement on a viable epistemology.87 The need for an ultimate correctness-norm for acts of knowledge, Wittgenstein certainly knew and accepted. Whereas I prefer to take it as primitive, Wittgenstein in the Tractatus reduces the rightness of the act to the correctness of the assertion made, and that in turn to the truth of the propositional content, which, ﬁnally, is reduced to the obtaining (and nonobtaining) of the corresponding state of aﬀairs. Committed realists, when challenged, often reduce the norm of rightness one step further, from the notion of obtaining for states of aﬀairs, to “reality itself,” which accordingly has to provide for the obtaining and nonobtaining of states of aﬀairs. When this reduction is coupled with the idea that “reality itself” is the sum total of all (material) objects and the wish to treat also reality itself as a material object, conceptual confusion results. However, without being a transcendent notion, reality cannot fulﬁll its required role as norm. It certainly cannot be subject to contingent facts the way material objects are, because such facts are responsible to the norm, whence it cannot be a material object. Wittgenstein had thought harder about these issues than most and such confusion is certainly avoided in the Tractatus: “Reality is the obtaining and non-obtaining of states of aﬀairs” (2.06). On such a view, the notion of obtaining (and nonobtaining) of states of aﬀairs can (pleaonastically) be reduced to reality itself. On the other had, “reality itself” thus construed is in no way less transcendent a notion than that of the obtaining of states of aﬀairs.

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7. Constructive Proofs of Propositions, the Traditional Form of Judgment Resurfaces Moore and Russell continue, or perhaps rediscover, the realist stance in logic that had been advocated by Bolzano. Classical, bivalent logic is upheld for an sich bearers of truth—the “propositions”—either by means of a primitive sui generis notion of an sich truth (Bolzano, Frege) or by means of an ontological reduction of truth via a truth-maker analysis (Moore, Russell, Wittgenstein). One would not expect such metaphysical generosity concerning truth to come cheap. The currency in which the price has to be paid is, however, epistemological rather than metaphysical: Unknowable truths cannot be ruled out. The issue is by now a familiar one, owing to the works of Michael Dummett, who has challenged realist accounts of truth on meaning-theoretical grounds: Bivalent truth cannot serve as a key concept in an adequate theory of meaning, owing to the occurrence of propositions with undecidable truth-conditions.88 However, Dummett was not the ﬁrst to challenge unreﬂective realism. Already in the 1880s, the Berlin mathematician Leopold Kronecker and his pupils, among whom was Jules Molk, challenged the automatic use of realist logic: Deﬁnitions should be algebraic and not merely logical. It is not enough just to say: “something either is or is not.” Being and nonbeing have to be set forth with respect to the particular domain within which we operate. Only in this way do we take a step forward. If we deﬁne, for instance, an irreducible function as a function that is not reducible, that is to say, that is not decomposable into other functions of a ﬁxed kind, we do not give an algebraic deﬁnition at all, we only enunciate what is but a simple logical truth. In Algebra, for it to be rightful to give this deﬁnition, it must be preceded by the indication of a method that permits one, with the aid of ﬁnitely many rational operations, to obtain the factors of a reducible function. Such a method only confers an algebraic sense on the words reducible and irreducible.89 In other words, the following “deﬁnition” is not a permissible one: 1 if the Riemann hypothesis is true; f (x) =def 0 if the Riemann hypothesis is false. When the deﬁnition is read classically (or “logically”), the function f is constant and therefore, trivially, a computable function. However, at the moment of writing, we are unable to compute the “computable” function in question. On the “logical” view, f (14), say, is a natural number, but its numerical value cannot be ascertained. Deﬁnitions by means of undecided cases do not admit the eﬀective substitution of deﬁniens for deﬁniendum. They contravene the canon for deﬁnitions that has been with us for three centuries, ever since

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Pascal.90 This is the price that a mathematician has to pay for unrestricted use of classical logic. His language will then contain nonprimitive terms that cannot be eliminated in favor of primitive vocabulary: Accordingly, there is no guarantee that meaning has been conferred on the terms in question. The Kronecker criticism, in my opinion rightly, rules out deﬁnition of functions by means of undecidable separation of cases. Possibly a classical mathematician could live happily without these contrived functions. However, Dirichlet’s famous deﬁnition of the function that is 1 on rational real numbers and 0 on irrational real numbers provoked a change in the conception of what a function is and can hardly be dismissed for want of mathematical interest. It also proceeds by an undecided separation of cases. Many proofs in classical analysis make use of this method. For instance, the standard “bisection of intervals” proof of the Bolzano–Weierstraß theorem that every bounded inﬁnite set of real numbers has an accumulation point proceeds in exactly this fashion.91 Again, these are mathematical matters and perhaps the classical logician, rather than the classical mathematician, need not be worried. Alas, this hope turns out to be forlorn: We only have to notice that Frege’s explanation of the classical quantiﬁer is cast in the form of an undecided separation of cases for matters to become more serious. Quantiﬁer(phrase)s are function(expression)s that take (expressions for) propositional functions and yield (expressions for) propositions. Propositions, for the mature Frege, are ways of specifying truth values, and it seems advisable to make explicit also the relevant domain of quantiﬁcation.92 Accordingly, we consider a truth value valued function A[x] ∈ {The True, The False}, provided that x ∈ D. Frege then deﬁnes the universal quantiﬁer by means of the following explanation: The True, if A[a/x] = The True, provided a ∈ D; (∀x ∈ D)A[x] =def The False, otherwise. However, when the domain D is inﬁnite, unsharp, or otherwise undecidable, the separation of cases cannot be carried out and the deﬁned quantiﬁer cannot be eliminated. Uncharitably put “the classical logician literally does not know what he is talking about.” To my mind, this is the strongest way to marshal undecidability considerations against classical logic. The law of excluded middle is not the real issue.93 Already the classical rules of quantiﬁer formation are unsound: They do not guarantee that “propositions” formed accordingly actually do have content. Until 1930, content was a very live issue. Work on the foundations of mathematics was dominated by the wish to secure a foundation for the practice of mathematical analysis after the ε-δ fashion of Weierstraß that satisﬁes the following conditions:

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(Ai) a formal system is given, in a syntactically precise way, (Aii) with meaning explanations that endow the well-formed expression of its formal language with content, (Aiii) in such a way that its axioms and rules of inference are made evident, and (B) classical logic is validated. Frege’s GGA was the ﬁrst substantial attempt to meet the double desiderata of contentual formalization (A) and classical logic (B), but it foundered on the Zermelo–Russell paradox: Somewhere in Frege’s §§29–31 there is an error, since otherwise every regular expression would have a Bedeutung and every derivable expression would be a name of the True. Whitehead and Russell also failed in their attempted Principia Mathematica execution of the foundationalist program: Their meaning explanations do not suﬃce to make evident the “Axioms” of Inﬁnity, Choice, and Reducibility. Similarly, Wittgenstein’s Tractatus provides (an attempt at) a semantic superstructure for the formal languages designed by Frege and Peano (as modiﬁed by Whitehead and Russell), as does the work of Frank Ramsey (1926). By 1930, faith in the project is waning: Carnap (1931) represents logicism’s last stand. The metamathematical Hilbert program (1926) was an attempt to secure the unlimited use of classical logic, at the price of giving up content, by means of an application of positivist philosophy of science to mathematics. The use of classical logic and impredicative methods are all ﬁne as long as “the veriﬁable consequences,” that is, those theorems that do have content, actually “check out.”94 Passing content by, this means that every free-variable equation between numerical terms that is derivable using also ideal axioms without content has to be correct, when read with content. Hilbert discovered that this holds if the ideal system is consistent, that is, does not derive, say, the formula 0 = 1. In a way, this would have been an ideal approach to the foundations of mathematics for the working mathematician. The conceptual analysis required for foundational work, at which a mathematician does not necessarily excel, is replaced by a clear-cut mathematical issue, to be resolved by a (meta)mathematical proof, just like any other mathematical problem. Alas, it was too good to be true: With the appearance of Gödel (1931) all hope ended here, but the mathematical study of languages without content, which Hilbert had introduced in pursuit of a certain philosophical program, stayed on as an mathematical research program even when the philosophical position had collapsed. Shortly after 1930, the ﬁrst wave of (meta)mathematical results come in: Tarski and Lukasiewicz (1930), the already mentioned Gödel (1931), and Tarski (1933a, 1933b). Under the inﬂuence of these (meta)mathematical successes, even Carnap, the last logicist diehard, jettisons content and anything goes: Up to now, in constructing a language the procedure has usually been, ﬁrst assign a meaning to the fundamental mathematico-logical

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symbols, and then to consider what sentences and inferences are seen to be logically correct in accordance with this meaning. . . . The connection will only become clear when approached from the opposite direction: let any postulates and rules be chosen arbitrarily; then this choice, whatever it may be, will determine what meaning is to be assigned to the fundamental logical symbols.95 The weakness of the position to which Carnap converted is obvious: If anything goes, what guarantee is there that content can be assigned? After all, there had been a few attempts at securing analysis already, meaning explanations and all, that had foundered on inconsistencies in the underlying formalisms. In such a calculus, demonstrably, no content can be had. Carnap’s novel gospel is an extremely liberal one: Principle of Tolerance. It is not our business to set up prohibitions, but to arrive at conventions. . . . In logic, there are no morals. Everyone is at liberty to build up his own logic . . . as he wishes.96 A quarter of a century earlier, at the same time when, in Cambridge, Russell and Moore bit the bullet of unknowable truths, Carnapian licentiousness was rejected on the other side of the North Sea in the (1907) doctoral dissertation of a young Amsterdam mathematician who took over the torch of mathematical constructivism from Kronecker. L. E. J. (“Bertus”) Brouwer (1881–1966) claimed that language use was responsible to the mathematical deed of construction and not the other way round: In the ediﬁce of mathematical thought thus erected, language plays no part other than that of an eﬃcient, but never infallible or exact, technique for memorizing mathematical constructions, and for communicating them to others so that mathematical language by itself can never create new mathematical systems. But because of the highly logical nature of mathematical language the following question naturally presents itself. Suppose that, in mathematical language, trying to deal with an intuitionist mathematical operation, the ﬁgure of an application of one of the principles of classical logic is, for once, blindly formulated. Does this ﬁgure of language then accompany an actual languageless mathematical procedure in the actual mathematical system concerned? 97 In particular, laws of whatever theoretical logic have no validity on their own, but have to be applied in such a fashion that they do ensure proper content. A year after his thesis, Brouwer reaches the conclusion that the law of excluded middle cannot guarantee that the required deed of construction can be executed, whence it has to be rejected not as false but as unfounded.98 Thus he refrains from asserting that “A ∨ ¬A is true.”99 Also the method of proofs by means of nonconstructive dilemma that proceeds by obtaining

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the conclusion that C is true from both the assumption that A is true as well as from the opposite assumption that A is false and concludes that C is true, is rejected, as is the method of indirect (or apagogic) proof, when one assumes a negative claim, obtains a contradiction and concludes a positive claim from this contradiction. Reductio ad absurdum proofs, on the other hand, are perfectly acceptable to constructivists: In these one proves a negative claim from a positive assumption that yields a contradiction. Here a method is provided for obtaining a contradiction from an assumption that constitutes a construction for the negation. Mere formulation or postulation does not automatically confer validity on the rules in question. Formulation alone is not enough to secure preservation of content at the level of the mathematical deed of construction. In this, surprisingly enough, Brouwer resembles Frege who, at roughly the same time, severely criticized formalist accounts of mathematics for their lack of content.100 For Frege, however, it was a commonplace that the contents expressed by declaratives have to be bivalent propositions, tertium non datur. Frege hoped to secure this by making the bond between propositions and truth values a tight one: A proposition is a means of presenting a truth value. Owing to lack of eﬀectiveness in some of the chosen means of presentation, for example, quantiﬁcation with respect to an inﬁnite domain via an undecidable separation of cases, an operational want of content is the result. Accordingly, Brouwer, as well as other mathematical constructivists who insist on the constructional deed in mathematics, will have to provide for another notion of proposition than that of (a mode of presentation of) a truth value, if the formal logical calculi shall not be void of content. This Brouwer did only by precept in his mathematical work: With a lifelong love-hate relationship to language, he never took to formalization and the emerging symbolic calculi of logic.101 It was left to others, to wit Hermann Weyl, one of few ﬁrst-rate mathematicians with a sympathy for intuitionism, and Brouwer’s pupil Arend Heyting, to formulate the required notions explicitly. Brouwer’s style of exposition in his intuitionistic writings was not to everybody’s taste and Weyl, who deftly wielded a polemical pen, took over the early propaganda work, at which he excelled. From his study at Göttingen, Weyl had ﬁrsthand knowledge of Husserl’s phenomenology, and this inﬂuence can be seen in his writings around 1920.102 It was left to him, possibly drawing on work of Schlick and Pfänder, to formulate explicitly the required notion of constructive existence to be applied in a constructive truth-maker analysis: An existential proposition—for instance, “there is an even number” —is not at all a proper judgement that expresses a state of aﬀairs; existential states of aﬀairs are an empty invention of logicians. “2 is an even number”: that is a real judgement that expresses a state of aﬀairs; “there is an even number” is only a judgement-abstract that has been obtained from this judgement.103

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Here we have a novel form of judgment, namely, α exists, where α is a general concept. Its assertion condition is given by the rule a is an α , α exists whence one is entitled to assert that α exists only if one already knows an α.104 The contribution of Heyting is twofold. First, he gave an explicit formulation of the proper intuitionistic rules of logic.105 Second, he intervened decisively in a confused debate whether logic according to intuitionists would need a third truth value: true, false, and undeﬁned, thereby leading to a law of the “excluded fourth,” and so on.106 In his intervention Heyting formulated explicitly a constructivist notion of proposition that admits of a truth-maker analysis: A proposition p, for example, “Euler’s constant is rational” expresses a problem, or better still, a certain expectation (that of ﬁnding two integers a and b such that C = a/b) that may be realized or disappointed.107 Here the intuitionistic novelty is introduced: proofs of propositions, that is, judgable contents, rather than judgments. All previous proving in the history of logic and mathematics had been at the level of judgment and not at that of their contents. These proofs of propositions are not epistemic but ontological in character; inspection of the examples given by Heyting and Brouwer reveals that they are common or garden mathematical objects: functions, ordered pairs, and so on. A proposition A is given by a certain set Proof(A) of proofobjects for the proposition in question. Many alternative formulations have been oﬀered: Proposition Intention Expectation Problem Type Set Speciﬁcation

Proof

Heyting (1934)

Fulﬁllment

Heyting (1930), (1931)

Solution Object Element Program

Heyting (1930), Kolmogorov (1932) Howard (1980) Martin-Löf (1982) Martin-Löf (1982)

The explanation of the standard logical constants then take the following form: ⊥

There are no proofs for ⊥.

& When a is a proof for A and b is a proof for B, a, b is a proof for A & B. ∨

When a is a proof for A, i(a) is a proof for A ∨ B. When b is a proof for B, j(a) is a proof for A ∨ B.

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⊃ When b is a proof for B, provided x is a proof for A, λx.b is a proof for A ⊃ B. ∀ When D is a set, P is a proposition, when x ∈ D, and b is a proof for P , when x ∈ D, λx.b is a proof for (∀x ∈ D)P . ∃

When D is a set, a ∈ D, P is a proposition, when x ∈ D, and b is a proof for P [a/x], a, b is a proof for (∃x ∈ D)P .108

The constructivist truth-maker analysis then takes the form proposition A is true = Proof(A) exists, where the notion of existence is the constructive (Brouwer–)Weyl existence already explained.109 The wheel has come full circle: A judgment made that ascribes truth to a proposition is elliptic for another judgment in the fully explicit form: a is a Proof(A), which is nothing but a judgment of the traditional form: S is P .110 Transformation of the form of judgment Traditional binary form S is P

Existential unary forms Brentano ± Concept α IS (exists)

Bolzano 1837 unary form Prop. A is true

Frege 1879 Prop. P (A) is true

Russell, Moore, 1910 Truth-maker analysis Prop. A is true = The concept Truth-maker [A] exists Realist:

Constructive:

Tractatus 1921 Elementary prop. A is true = Sachverhalt SA obtains

Heyting 1930 Prop. A is true = Proof(A) exists Weyl 1921 Constructive existence

p is a Proof(A)

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8. Inference versus Consequence: How Gentzen Had It Both Ways The interpreted formal systems of Frege, Whitehead and Russell, and Heyting were all axiomatic. These systems (are meant to) have an intended interpretation in terms of the respective meaning explanations. In such systems, a formal derivation is or can be read as a proof that shows that its conclusion formula, when read according to its interpretation, does express a truth. In the modern metamathematical systems of propositional and predicate logic, on the other hand, the end-formula has no intended interpretation, but has to be true under any truth value assignment or set-theoretic interpretation, respectively. Frege, furthermore, explicitly held that one can only draw inferences from known premises. This claim has been controverted, most famously by Gentzen, who created another kind of formalism in his 1933 Göttingen dissertation.111 The derivable objects are still formulae, but may depend on assumptions, and several rules serve to discharge open assumptions. A derivation takes the general form: A1 , A2 , . . . , Ak . . . . (D) . C where A1 , . . . , Ak are the undischarged assumption on which the end-formula C depends. The rules of inference are divided into two groups of introduction and elimination rules. The conjunction introduction rule (&I), say, allows you to proceed to the conclusion A & B, given two derivations of A and B, respectively, that depend on open assumptions in the lists Γ and Δ, respectively. The derivation of A & B depends on open assumptions in the joint list Γ, Δ. The rule (&E) of conjunction elimination, on the other hand, allows you to obtain the conclusion A from the premise A & B, and also the conclusion B from the same premise, while the open assumptions remain unchanged. The rule (⊃I) of implication introduction allows one to proceed to A ⊃ B from the premise B that has been derived from assumption formulae in the list Γ, while discharging as many premises of the form A as one wants—one, many, or none. The derivation of A ⊃ B depends on assumptions in the list Γ1 , where Γ1 coincides with Γ, except possibly for some deleted occurrences of the assumption formula A. The system is convenient to work with when one actually has to ﬁnd the derivations in question. Michael Dummett put the case for Gentzen’s natural deduction as follows: Frege’s account of inference allows no place for a[n] . . . act of supposition. Gentzen later had the highly successful idea of formalizing inference so as to leave a place for the introduction of hypotheses.

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Indeed, “it can be said of Gentzen that it was he who showed how proof theory should be done.”112 However, Dummett’s comparison between Frege and Gentzen is not entirely fair, since it does not take the metamathematical paradigm shift into account. For Frege, the formal system was a tool in the epistemological analysis of mathematics: it was actually used for for proving theorems. For Gentzen, (meta)mathematician, or Beweistheoretiker, the formal system was Hilbertian, that is, an object of study, without content, about which one proves (meta)mathematical theorems, such as, for instance, his famous (1936, 1938) consistency theorem by means of ε0 -induction. For a fair comparison, the respective formal systems of Frege and Gentzen accordingly have to be placed on an equal footing: We either divest Frege systems of their content and treat them as if they were metamathematical, or we supply meaning explanations for the key notions in Gentzen’s systems, so as to endow its object “language” with content. The present chapter is devoted to the notion of judgment, and an inference is nothing but a judgment of a particular (mediate) kind. However, without content no judgment, so it is to the second of these alternatives that we have to turn. Our task is to give a reformulation, call it Gentzen, of Gentzen, at the same level of interpretation as that provided by Frege. The early stages of the conversion present no diﬃculties: It is clear that the wﬀ’s in the formal language, say, of ﬁrst-order arithmetic, can be interpreted as propositions. The syntactic terms are readily turned into numerical expressions, and the predicates < and = obviously lend themselves for interpretation as the computable numerical relations less than and identity, respectively. So far so good; with respect to elementary syntax and semantics, Frege and Gentzen march in step. The diﬃculties arise when we turn to the pragmatic dimension that is involved in Frege’s use of the turnstile as an assertion sign, that is, as an explicit force indicator. Gentzen does not use a turnstile, but if he had it would undoubtedly have been used as a Kleene–Rosser theorem predicate; Gentzen was a (meta)mathematician. Here we see a ﬁrst diﬃculty for Gentzen: Gentzen (and with him other metamathematicans) used his wﬀ’s in two roles. Wﬀ’s are fed to connectives, that is, Frege’s Gedankengefüge, to build other, more complex wﬀ’s: Accordingly, for Gentzen they are propositions. On the other hand, Gentzen also used wﬀ’s as end formulae of derivation trees: Accordingly, for Gentzen, the wﬀ’s also have to be turned into theorems, that is, assertions (judgments made) that propositions are true. Here Gentzen confronts a potentially damaging ambiguity. However, we must allow him the same leeway as that oﬀered to Frege: He can make use of the turnstile as an assertion sign, and also other force indicators, should he want to do so. The obvious option for Gentzen is to use two force indicators, one for assertion () and another for assumption (). Finally, Gentzen also has to interpret the derivation trees of Gentzen. The Gentzen derivation D will be interpreted by means of the following procedure:

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i. Append “is true” to each wﬀ that stands on its own at a node in D (rather than as part of another wﬀ); ii. preﬁx transformed wﬀ’s “A is true” of assumption formulae by the assumption sign “” and preﬁx transformed “A is true” of conclusion formulae by the assertion sign “”; iii. interpret all wﬀ’s in D as propositions. The result is the tree D :

(D )

A1 is true, .

A2 is true, .

. . . , Ak is true, . .

. C is true. We may have some hope that D will serve as a ﬂow-chart for a proof-act that yields the knowledge that proposition C is true. However, this simpleminded approach does not work: The interaction of the two kinds of force—assumption and assertion—is more involved. Consideration of an example, in which prooftheoretical experts will recognize one of Dag Prawitz’s reduction ﬁgures, makes this clear:113 [A] | (d) B | A⊃B A B This tree is dressed according to the procedure and transformed into the tree

(d )

A is true (1) | | B is true (2) A ⊃ B is true (3) A is true B is true (5)

(4)

The force apparatus is almost equal to its task: the proposition A occurs as part of an assertion (4), of an assumption (1), and as an unasserted part of an assertion (3). The notation is rich enough to distinguish these cases clearly. With respect to the proposition B matters are less fortunate, though. For the proposition B, assertion (5) and unasserted part (3) are coped with, in the same way as for the proposition A. The premise that B is true of the (⊃I) rule (2), however, is neither assumed nor asserted, and its force cannot be expressed with the two force indicators at hand. There one asserts that the proposition B is true, provided that the proposition A is true.114 One must note, though, that it is not the assertion that is hypothetical or conditional; the assertion

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is categorical, whereas the notion of truth has been made conditional. We no longer ascribe outright truth to the proposition B, but only the constrained notion . . . is true, provided that A is true. Thus we have an unconditional, categorical assertion that conditional truth pertains to the proposition B. Strictly speaking, this is a novel form of judgment. The derivation tree D above, where the assumptions A1 is true, A2 is true, . . . , Ak is true are still open, or undischarged, does not allow for the ascription of outright truth to the proposition C, but only of truth on condition that A1 is true, A2 is true, . . . , Ak is true. The general case of the weakened, conditional truth in question will then be: . . . is true (A1 is true, A2 is true, . . . , Ak is true). Accordingly nodes in derivation trees are not covered with statements of the form A is true, but with statements of the conditional form. Eﬀecting this transformation, the derivation tree D ultimately takes the form D : A1 is true (A1 is true), A2 is true (A2 is true), . . . , Ak is true (Ak is true) . . . . (D ) . C is true (A1 is true, A2 is true, . . . , Ak is true). The relevant notion of assertion is still categorical, but the truth that is asserted of various proposition may be weakened. We must distinguish between the two statements: i. proposition A ⊃ B is true, ii. B is true, provided that A is true, or its (synonymous) variant, ii . if A is true, then B is true. The statement (i) is explained classically via truth-making of atomic propositions and then inductively via the truth tables, say, and constructively in terms of an assertion-condition demanding a (canonical) proof-object, as in section 7. From the constructive point of view, an assertion of the ﬁnal statement in D , that is, (∗)

C is true (A1 is true, . . . , Ak is true),

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demands a dependent proof-object: (∗∗)

c is a proof of C, provided that x1 is a proof of A1 , . . . , xk is a proof of Ak .

Accordingly, the conditional statement (∗) represents a novel form of judgment, with the assertion condition (∗∗). This suggests how natural deduction derivations should be interpreted: They are notations for dependent proof-objects. Gentzen did not have only one format for natural deduction derivations but two. Sometimes they are considered as mere notational variants.115 In the present context their diﬀerences are signiﬁcant. In 1936 he used a sequential format for the derivations.116 The derivable objects are no longer well-formed formulae, but sequents. A sequent A1 , A2 , . . . , Ak ⇒ C lists all the open assumptions on which C depends. Derivations have no assumptions, but axioms only of the form A ⇒ A, with the Gentzen interpretation A is true, provided that A is true, indeed, something undeniably correct, albeit not very enlightening. Consideration of the tree D shows that its top formulae are axioms of this kind and that the conditional statements at the nodes in the tree are nothing but sequents in another notation. Because there are no acts of assumption, no discharge of assumptions takes place, but antecedent (assumption-)formulae can get struck out; for instance, the rule (⊃I) takes the form A, Γ ⇒ B . Γ⇒A⊃B Conjunction introduction (&I) will be Γ⇒A Δ⇒B . Γ, Δ ⇒ A & B On the Gentzen interpretation the sequent A1 , . . . , Ak ⇒ C is interpreted as C is true, on condition that A1 is true, . . . , Ak is true, and the Gentzen sequent should properly include the truth ascriptions: A1 is true, . . . , Ak is true ⇒ C is true.

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Derivations in the sequential format of natural deduction describe, or are, blueprints for proof-acts that certain propositions are conditionally true. Properly speaking, we have here a treatment of consequence relations among propositions: The consequent proposition is true when the antecedents are all true. One should also note that the statement proposition C is true is a special case of the sequent, when the number k of antecedent propositions = 0. The sequents can also be read as closed sequents (A1 , . . . , Ak ) ⇒ C.117 Just as the Gentzen sequents represent a novel form of judgment, so do these closed sequents, and their Gentzen interpretation should be the sequent (A1 , . . . , Ak ) ⇒ C holds. To have the right to assert that a closed sequent holds we must give a verifying object. This is a function f that takes proofs a1 , . . . , ak of the antecedent propositions into a proof f (a1 , . . . , ak ) of the consequent proposition C. We must distinguish between three equiassertible statements: the proposition A ⊃ B is true (demands a proof of A ⊃ B); the conditional statement (open sequent) A true ⇒ B true (demands a dependent proof b of B, provided that x is a proof of A); the closed sequent (A) ⇒ B holds (demands a function from Proof(A) into Proof(B)). The assertion condition is diﬀerent in all three cases, but one can be met only if the other two can also be met. Furthermore, one reason that these notions are not always kept apart is that all three are refutable by the same counterinstance, namely, a proof-object a of A and a dependent proof-object c for the open sequent B true ⇒ ⊥ true. With this distinction, my treatment of the sequent calculus comes to an end. At the level of assertion, there is apparently little to choose between Gentzen and Frege. It only seems fair to let Frege have the last word: The repeated Frege conditional ... C Ak .. . A1

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is commonly read as the implication (A1 ⊃ (A2 ⊃ (· · · ⊃ (Ak ⊃ C) · · · ))). However, rotating the Frege conditional 90° clockwise, while altering the notation only slightly, produces another familiar result as the late Pavel Tichý (1988, pp. 248–252) observed, namely A1 , A2 , . . . , Ak ⇒ C.118 The correspondence between the calculi of Frege and Gentzen operates even with respect to the ﬁne structure of the rules, sometimes even exhibiting a surprising(?) resemblance of terminology. Thus, the following question acquires some urgency: Did Gentzen read Frege’s Grundgesetze prior to 1933, the year in which his dissertation was composed?119 Be that as it may. Bolzano gave us a coherent theory of (logical) consequence between propositions. Frege was more right about inference from judgments made to judgment than he is given credit for. However, only in Gentzen’s sequential natural deduction do we have a theory that treats of both consequence as well as inference.

Brief Biographical Notes 1. Bernard Bolzano, 1781–1848120 A Bohemian priest of Italian origin, who held the chair of Philosophy of Religion at the Charles University in Prague from 1805 until 1820, when he was summarily dismissed, as well as barred from public teaching and preaching, for holding too liberal views concerning matters both spiritual and temporal, gave fundamental contributions to mathematical analysis (“Bolzano–Weierstraß theorem”). A wholly admirable man, he led a retiring life with friends in the Bohemian countryside, devoting himself to logical and mathematical researches. The magnitude of Bolzano’s contribution to logical theory, as well as to philosophy in general, can hardly be overestimated. Being censored, it was left unrecognised, thereby retarding logical progress by half a century. Appreciation is mounting with the growing volume of the Gesamtausgabe, and Bolzano might yet receive the credit that is so amply his due: “the greatest philosopher of the nineteenth century, bar none.”121 2. Franz Brentano, 1838–1917 Brentano belonged to a prominent German cultural family. Ordained a priest, he held, after impressive Aristotle studies, a (Catholic) extraordinary chair in Philosophy at Würzburg. Misgivings over the deﬁnition of Papal Infallibility in 1870 led him to renounce the priesthood and change his chair for one in Vienna, where his lectures aquired cult status as society happenings. The Concordat between Austria and the Vatican allegedly forbade Austrian ex-priests to marry and led him to resign also this chair and his Austrian citizenship in 1880; having taken Saxon citizenship he then married, conﬁdently expecting reappoinment. This never happened, reputedly

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at the personal instigation of the emperor; after 15 years as Privatdozent, the former Professor Brentano left Austria and settled in Florence. Blindness darkened his last decade, to which belong important brief essays on truth as well as a wide-ranging correspondence with utterly devoted pupils. Italy’s entry into World War I forced a ﬁnal move to Switzerland, where Brentano died in 1917. Brentano, an outstandingly successful lecturer and supervisor, though not devoid of dictatorial leanings, had highly able doctoral students, who bitterly disappointed him by not speaking in unison with their master’s voice. Nevertheless, his inﬂuence ranges wide, not only among devoted Brentanists, but also in two major schools of twentieth-century philosophy, to wit, the LvowWarsaw school under Twardowski, the ﬁrst to introduce analytical techniques, and Husserl’s phenomenology. 3. Gottlob Frege, 1848–1925 A German mathematician at Jena who taught (mainly) analytical geometry several hours a week, never reached the rank of Ordinarius but gave fundamental contributions to the foundations of logic and mathematics. In the Begriﬀsschrift and Die Grundlagen der Arithmetik the program of reducing arithmetic to logic, as well as the logic to which it was to be reduced, are set out with great lucidity. His attempt at a fully rigorous execution of his foundationalist program, in the Grundgesetze der Arithmetik, proved to be irredeemably ﬂawed owing to the emergence of the Zermelo–Russell paradox within the system. Three important essays from the early 1890s provide a philosophical underpinning for the Grundgesetze. Of these, Über Sinn und Bedeutung is commonly regarded as the origin of modern philosophy of language. Frege founded no school, and, for a long time was only known through and for his inﬂuence on major ﬁgures such as Russell, Carnap, and Wittgenstein. A deeply conservative man in matters cultural and political, Frege died forgotten in the Weimar Republic to which he could not relate. His contributions to logic, its philosophy, and the philosophies of mathematics and language are now recognized in their own right, and not only as an inﬂuence on others, whereby Frege rightly emerges as a major thinker of the nineteenth century. 4. Ludwig Wittgenstein, 1889–1951 The youngest son of Karl Wittgenstein, a main architect of the Industrial Revolution in Austria, as well as one the wealthiest men in Europe, was educated at the Oberrealshule in Linz, where Adolf Hitler was a fellow pupil, and subsequently at the Technische Hochschule, Berlin-Charlottenburg, and Manchester University, prior to settling at Cambridge, where his work on the foundations of logic ripened in close contact with Bertrand Russell, in relation to whom Wittgenstein went through the stages of pupil, co-worker, and implacable critic. At the outbreak of World War I, Wittgenstein volunteered for the Austrian army, and during the war he reﬁned and deepened his views on logic that were published in the aphoristic Logisch-Philosophische Abhandlung, now universally known as Tractatus (Logico-Philosophicus). After the Great War, Wittgenstein gave away his in-

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herited fortune and became a schoolteacher in the hinterlands of lower Austria. A Vienna lecture by L. E. J. Brouwer, March 1928, rekindled his interest in philosophy, and led him to return to England. Keynes wrote to his wife: “God is in England. I met him at the 5.15 train.” Elected a fellow of Trinity, and from 1939 professor in succession to G. E. Moore, Wittgenstein developed an entirely novel conception of philosophy, on which he published nothing. A stern man, more unsparing of himself than of others, Wittgenstein died in 1951, his last words being: “Tell them I have had a wonderful life.”

Notes 1. Barnes (1988, p. 48). I am indebted to my Leyden colleague Dr. J. van Ophuijsen for drawing my attention to this marvelous passage. 2. (1950), p. vii. 3. Putnam (1982), Boolos (1994). Boolos also canvasses 1858, the year in which Dedekind cut the rationals, as a candidate. 4. I have argued as much, in and out of print, since 1988 (p. 4). 5. Thus texts would comprise three main parts, or “books”: Of Terms, Of Judgement, and Of Reasoning, and possibly a ﬁnal part treating Of Method. Kant’s Jäsche Logik is a good case in point. 6. The diagram draws on a similar one in Maritain (1946, p. 6) but is reasonably standard. Maritain’s source, and also that of virtually all other neo-Thomists, is the splendid Ars Logica by John of St. Thomas. 7. Fichte (1797) (which bears the title Wissenschaftslehre). The convenient representation of the act/object distinction was introduced in Martin-Löf (1987). 8. A committed anti-antirealist, or the unbiased reader, might prefer the less pejorative realist for the other alternative. 9. Here the vertical bars above the judgments J1 , J2 , . . . , Jk represent acts that yield, respectively, the judgment in question. 10. KdrV, A69. “Wir können alle Handlungen des Verstandes auf Urteile zurückführen, so daß der Ve r s t a n d überhaupt als ein Ve r m ö g e n z u u r t e i l e n vorgestellt werden kann.” 11. And by Heidegger, or so I have been told. 12. As does judgment: act of judging versus judgment made. 13. Recent scholarly tradition associates this familiar example with Tarski (1944). It is, nevertheless, considerably older than so. We ﬁnd it in Boole (1854, p. 52), as well as in Hilbert and Ackermann (1928, p. 4) (who undoubtedly have it from Boole). The latter is cited by Tarski in Der Wahrheitsbegriﬀ. However, the ultimate source for the present-day logical obsession with arctic meteorology might well be Aristotle’s Prior Analytics, Book A, ch. iv, where we ﬁnd a discussion of things—among them snow—that admit the predication of “white.” 14. Here the letter S serves as a schematic letter for declaratives. 15. For lack of space, in what is after all an inquiry into the recent history of logic, I must here leave well-known (spät) Wittgensteinian claims to the contrary without due consideration. I simply register my conviction that they do not present an insurmountable obstacle, and that my view as given in the text is essentially correct.

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16. Sometimes a single utterance of the nominalization will, nevertheless, eﬀect the required assertion, for example, when responding to the question: “Which of the two alternatives is the true one?” 17. The apt term judgable content we owe to Frege’s Bs. His beurtheilbarer Inhalt has been variously rendered into English as (i) possible content of judgment (Geach), (ii) content that can be judged (Van Heijenoort), (iii) judicable content (Jourdain), and (iv) judgable content (Dummett). Of these, the last deserves preference over the second for the sake of brevity, while the ﬁrst is likely to cause serious confusion, owing to its pointing in the direction of modal logic. 18. This is not the full story. The right to ask for grounds, when faced with an assertion made, shows that there is an implicit claim to knowledge contained in the assertoric force with which the sentence has been uttered, and which sometimes comes to the fore, for instance, in Moore’s paradoxical assertion of It is raining but I do not believe it. Thus, I know that snow is white, which is in the performative ﬁrst person, might have been a more felicitous form to use for the assertion made by means of my assertoric utterance of the declarative “Snow is white,” were it not for the fact that it is prone to be conﬂated with the third-person use of the ﬁrst person, which, when applied to me, is synonymous with Göran Sundholm knows that snow is white. 19. This linguistic turn in philosophy was so named by Gustav Bergmann (1964, p. 177). The term gained wide currency after Richard Rorty (1967) chose it for his title. 20. Weighty in every sense of the word; its four volumes add up to a total of close to 2500 pages. 21. Kant, KdrV, B VIII (my translation). 22. The Kantian authority for Jäsche’s text is not undisputed, see Boswell (1988). 23. The English rendering of Bolzano’s Satz an sich is a matter of some delicacy. The modern, Moore-Russell notion of proposition, being an English counterpart of the Fregean Thought (German Gedanke), really is an an sich notion, and, for our purposes, essentially the same as Bolzano’s Satz an sich. Thus, proposition-initself is pleonastic: The in-itself component is already included in the proposition. Furthermore, the mental propositions and their linguistic signs, that is, written or spoken propositions, as explained, carry assertoric force, whereas Bolzano’s Sätze an sich manifestly do not, serving, as they do, in the role of judgmental content. Accordingly, it might be better to use Sentence in itself, which does not seem to carry the presumption of assertoric force. However, as Ockham and other medieval thinkers noted, the propositio mentalis, and its matching exterior signs, can be further analyzed into propositio judicationis, which does carry assertoric force, and propositio apprehensionis, which does not. Ockham has Quodlibetal Questions with congenial titles: Questio V:vi “Is an act of apprehending really distinct from an act of judging?” and Q iv:16 “Does every act of assenting presuppose an act of apprehending with respect to the same object?” So from this point of view, Bolzano’s proposition-in-itself is obtained by severing the (mental) links that tie the propositio apprehensionis to its mental origin and its linguistic signs.

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24. WL, §34. 25. Occasionally I shall permit myself to drop the “in-itself” idiom in the interest of perspicuity and readability and speak just of “propositions.” 26. Contrary to a common misapprehension, Frege, who employed the term third realm in 1918, is not its progenitor. It was in general use in neo-Kantian circles. Gottfried Gabriel’s lemma Reich, drittes in the Historisches Wörterbuch der Philosophie tells the full story. 27. In some theories, for instance, that of Frege, propositions (thoughts) are explained as the meanings of declarative sentences. This is not Bolzano’s way of proceeding, the sui generis, absolutely mind- and language-independent propositionsin-themselves, are there in their own right, so to say, and they are capable of fulﬁlling various logical oﬃces, among them that of serving as sentence meanings. 28. The reader will please note long shadows being cast forward toward Tarski (1944). Bolzano (§28) has a discussion of whether really is really necessary in the right side of this explanation. 29. WL, §154 (4). “Auch leuchtete jedem die Ähnlichkeit ein, die zwischen diesem Verhältnisse unter den Sätzen und zwischen jenem, welches . . . unter Vorstellungen . . . obwaltet. Was nähmlich bei Vorstellungen den Umstand gilt, ob ein gewisser Gegenstand durch sie in der Tat vorgestellt werde, das gilt bei Sätzen der Umstand, ob ihnen Wahrheit zukomme oder nicht.” 30. When snow is white, the idea-in-itself the whiteness of snow is instantiated and the idea the blackness of snow is not. 31. Sebestik (1992) oﬀers a beautiful précis of Bolzano’s framework. 32. WL, §138. The categorical judgments of the A and I forms and O are treated of, respectively, at §225 Anmerkung, and §171. The treatment of an O judgment (Some α is not β) follows the pattern of the I judgments: The idea-in-itself of an α that is not β has Gegenständlichkeit. 33. The falsity of false propositions-in-themselves is also an sich. 34. WL, §34. The status of my chosen form of judgment [A is true] is very delicate with respect to Bolzano’s system. On the one hand, it has to be a proposition-in-itself, since the iteration of . . . is true is the key step in Bolzano’s non-apagogic “proof” that there are inﬁnitely many true propositions-in-themselves (WL, §32). On the other hand, propositions-in-themselves are supposed to be sentence meanings, as well as the bearers of truth and falsity, as is clearly documented by the following passage (cited from Mark Textor [1996], p. 10) in Bolzano’s Von der mathematischen Lehrart (my translation): “not what grammarians call a proposition, namely the linguistic expression, but rather the sense of this expression, that must always be only one of true or false, is for me a proposition in itself or an objective proposition.” The sense (Sinn) of the declarative sentence (grammatical proposition) “Snow is white” is that snow is white. Furthermore, that-clauses are what yield grammatical declarative sentences when saturated with “is true” or “is false.” Accordingly, that-clauses seem to be the appropriate linguistic counterparts to propositions-in-themselves. But then, the ascription of truth to a proposition-in-itself is not a proposition-in-itself, since the declarative “the proposition-in-itself A is true” isn’t a that-clause, that is, does not have the required form for being (the linguistic counterpart of) a proposition-in-itself. This note attempts to answer Wolfgang Künne, who objected, after my Cracow lecture, that for Bolzano, [A is true] is just another proposition-in-itself, but not a judgment. In spite of the considerations, on balance, I am inclined to think that

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this might be an instance where I let my systematic preferences override historical subtleties, and that some injustice is done to Bolzano. 35. Franz Brentano, for whom the problem of the blind judgment became a major issue, and whose views will be considered in the sequel, propagated this apt terminology. As Per Martin-Löf has pointed out to me, the notion and term might ultimately derive back to Plato, The Republic, 506c: “opinions divorced from knowledge, are ugly things[.] The best of them are blind. Or do you think that those who hold some correct opinion without evidence diﬀer appreciably from blind men who go the right way?” 36. WL, §36. Fairness bids me to report that Bolzano was aware of a certain awkwardness in his doctrine at this point. In WL, §314, with the telling title “Are there Deﬁnite Limits to Our Capacity for Knowledge?,” he notes: For since every judgment that agrees with the truth is a piece of knowledge, even if that agreement is only accidental and had come about only by way of previous errors, it can very well be seen that the limits of our capacity for knowledge, if we were able to abide by such a broad deﬁnition, would ﬂuctuate everywhere, since mere chance and even a mistake could contribute to its enlargement. In this connection we should further note that according to Bolzano, every truth is knowable, since God knows the truth of every true proposition-in-itself, whence it can be known: ab esse ad posse valet illatio. 37. In much the same way, Bolzano proﬁted immeasurably from having Kant, the foremost idealist of the age, as his main target. That can be seen by comparing the pristine clarity of Bolzano’s work with the murkiness of early Moore and Russell 60 years later. Their realism was the result of an apostasy from and battle with a much inferior version of idealism, namely the British Hegelianism of Bradley, Green, and Bosanquet. Another example of the same phenomenon is provided by Wittgenstein, who, according to Geach (1977, vi), held Frege’s Der Gedanke in low esteem: “it attacked idealism on its weak side, whereas a worthwhile criticism of idealism would attack it just where it was strongest.” The destructive side of a philosophical position seems to gain in quality with the target it attacks. 38. Apparently Bolzano was unable to give a material criterion for what it is to be a (non)logical idea, but then so were his successors, who oﬀered virtually identical accounts of logical truth and consequence a century later. 39. The foregoing brief formulations do not do perfect justice to Bolzano on a number of scores. (i) Logical consequence (in the modern sense) is a two-place relation between antecedent and succedent propositions, whereas Ableitbarkeit, be it logical or not, is a three-place relation between antecedent proposition(s), consequent proposition(s), and idea(s) (that occur in at least one antecedent or consequent proposition), where the ideas indicate the places where the variation takes place. Logical Ableitbarkeit considers variation with respect to the collection of all nonlogical ideas that occur in the antecedent and consequent propositions. As a limiting case (possibly one rejected by Bolzano) one might consider merely material Ableitbarkeit that consists in the preservation of truth under variation with respect to no ideas, and which holds between A1 , A2 , . . . , Ak and C, when the implication (A1 & A2 & · · · & Ak ) ⊃ C

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is just true, but not necessarily logically true. (ii) With respect to the consequence (or sequent in the terminology of Hertz and Gentzen) A1 , A2 , . . . , Ak ⇒ B1 , B2 , . . . , Bm , Bolzano demands that the A’s and the B’s be compatible, that is, there must be some suitable variation that makes them all true. Furthermore (iii), modern theory holds that the multiple-succedent sequent should be read as A1 & A2 & · · · & Ak ⊃ B1 ∨ B2 ∨ · · · ∨ Bm ; the sequent is valid when this implication is a logical truth. Bolzano, however, uses another meaning for the sequent, namely, A1 & A2 & · · · & Ak ⊃ B1 & B2 & · · · & Bm ; according to Bolzano an Ableitbarkeit with many succedent propositions holds when every variation that makes all antecedent propositions true also makes every (and not just at least one) succedent proposition true. In Siebel (1996) Bolzano’s theory of Ableitbarkeit is studied in depth and related (with due consideration for signiﬁcant diﬀerences) to a number of well-known modern topics, such as Russell’s theory of propositional functions, the Quine-AjdukiewiczTarski account of logical truth and consequence, and the relevance logic of Anderson and Belnap. Nevertheless, in spite of the sometimes considerable diﬀerences, it is proper to regard Bolzano as the founder of the modern theory of (logical) consequence among propositions; he is the ﬁrst to reduce the validity of inference (from judgment to judgment) to a matching relation among propositions (-in-themselves) that serve as contents of the relevant premise and conclusion judgments, respectively. 40. In WL, part III (“Erkenntnislehre”), ch. II (“Von den Urtheilen”), §300 (“mediation of a judgement through other judgements”), Bolzano considers also inferences proper, that is, mediate acts of judgments, and not only their Platonist simulacra, namely, consequence relations (Ableitbarkeiten) among the respective judgmental contents. Lack of space prevents me from developing this theme any further. 41. A true proposition-in-itself can stand in the relation of Abfolge to more than one grounding proposition. 42. WL, §220. See Aristotle, An. Post., I:13. 43. Validity of an inference ﬁgure must be distinguished from that of validity (rightness) of an act of inference. An act of inference, that is, a mediate act of judgment, is valid (right, real, or true) if its axioms, that is, according to Frege’s GLA, §3, p. 4, characterization, judgments neither capable of nor in need of demonstration, really are correct, and the inference-ﬁgures employed therein really are valid, that is, do preserve knowability. 44. Compare, for instance, the fragments reprinted in part III and appendix 2 of Brentano (1930), with titles such as “Against so-called Judgmental Contents” and “On the Origin of the Erroneous Doctrine of the entia irrealia.” 45. Furthermore, these reductions were well known already to Leibniz, for instance in the General Inquisitions, §§146–151. Franz Schmidt provides a list of 28 (!) diﬀerent Leibnizian reductions of the four categorical judgments in Leibniz (1960, pp. 524– 529). For instance, the singular aﬃrmative judgment “Some A are B” is rendered alternatively as “AB is,” “AB is a thing,” and “AB has existence,” by reductions 17,

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22, and 27. Here Bolzano’s German hat Gegenständlichkeit is matched by Leibniz’s Latin est or est Ens. 46. Letter from Brentano to Hugo Bergmann, June 1, 1909, quoted from Bergmann (1968, pp. 307–308) (my translation). Wenn ich . . . auf Bolzano aufmerksam machte, so geschah dies, . . . , keineswegs, um den jungen Leuten Bolzano als Lehrer und Führer zu empfehlen. Was sie von ihm, das dürfte ich mich sagen, konnten sie besser von mir lernen. . . . Und wie gesagt, wie ich selbst von Bolzano nie auch nur einen einzigen Satz entnommen habe, so habe ich auch niemals meinen Schülern glaubhaft gemacht, daß sie dort eine wahre Bereicherung ihrer philosophischen Erkenntnis gewinnen würden. 47. Brentano (1889, Anm. 27, pp. 64–72) and the fragments in Brentano (1930, part IV) are important here. Also relevant is his Versuch über die Erkenntnis, that is, Brentano’s (1903) attempt at an Essay on human knowledge after the fashion of Locke and Leibniz. For instance, its ﬁrst part bears the grandiloquent title: Destroy prejudice! An appeal to the contemporary age, that it free itself, in the spirit of Bacon and Descartes, from all blind Apriori. 48. Note that this use of the term evidence is diﬀerent from its use within current analytical philosophy of science and the Anglo-Saxon common-law legal systems. (“My lord, I beg leave to enter exhibit 4 into evidence.”) There one is concerned with supporting evidence for a claim. Brentano’s use is concerned with that which is evident (known). Evidence is the quality that pertains to what is evident. 49. Brentano (1930, p. 139). 50. Brentano (1956, p. 175, §39). According to the editor, this negative formulation of the Law of Excluded Third derives from an unpublished fragment “Über unsere Axiome” from 16.2.1916. 51. Scholz (1930) remains the standard treatment of Aristotelian foundationalism. 52. Frege (1879) is a book with the title Begriﬀsschrift, whereas “begriﬀsschrift” is an English (loan-)word for the eponymous formal language developed in that work. This is not an ideal solution to the title/notion ambiguity of the German term. Using either of the two standard English renderings—ideography and concept(ual ) notation—seems a worse option, though. 53. This is how it ought to be; regarding Frege, his GGA axiom 5 concerning Werthverläufe and the use of classical second-order impredicative quantiﬁcation remain unjustiﬁed. In place of notions we could speak of terms or concepts here. Either choice runs the risk of being taken in too narrow a sense, though. Today a term is a syntactic entity only, often associated with the formation rules of ﬁrst-order predicate calculus, whereas for Frege a concept is conﬁned to a certain kind of function. 54. The ﬁrst formulation combines passages from (1879, pp. 2, 4): “Der Umstand, dass. . .” and “. . .ist eine Thatsache.” Apparently the form of judgment is (. . .ist eine Thatsache), where the blank has to be ﬁlled with an “Umstand.” Accordingly the form of the judgment made through an assertoric utterance of “Snow is white” is: The circumstance that snow is white is a fact

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In place of circumstance, Frege also allows for “Satz.” In (1918, p. 74, n. 8) he identiﬁes fact with true Thought, which yields the ﬁnal reformulation: The proposition that snow is white is true. The second formulation is taken from 1892 (p. 34, n. 7): Ein Urteil ist mir nicht das bloße Fassen eines Gedankens, sondern die Anerkennung seiner Wahrheit. 55. Frege (1880/81, pp. 36 ﬀ.) attempts to sell his begriﬀsschrift to the mathematicians by treating of the standard notions pertaining to continuity, but to no avail, alas. The mathematicians, and among them to his shame Felix Klein, did not rise to the occasion and the paper was rejected by, for instance, the Mathematische Annalen. In desperation, Frege sought refuge with the philosophers at the Zeitschrift für Philosophie und philosophische Kritik, but they proved equally cold-hearted. In spite of it being unpublished, Frege’s piece must be give full evidentiary value since it was written for publication and repeatedly submitted. Later Frege established very good relations with the Zeitschrift where some of this very best papers appeared, in 1882 and 1892. 56. That is, iterated combinations of “for all/there is” and “there is/for all.” The passage from continuity to uniform continuity provides a clear example of the shift from “∀∃” to “∃∀”. The (linear) logical notation employed here is reasonably standard, using inverted A and E for Alle (all) and Es gibt (there is). It is due to Gerhard Gentzen (1934–35), but derives in essence from Peano, via mediation through Whitehead. Frege’s own begriﬀsschrift is two-dimensional and has great versatility, as well as a strange beauty of its own. It was never able, however, to gain proselytes, and so it perished with its progenitor in the early stages of the mounting metamathematical revolution in the late 1920s. 57. Frege’s checkered struggle toward an identity criterion for propositions (his Thoughts) is long and fascinating; see Sundholm (1994c). 58. Frege (1892, p. 34): “Ich verstehe unter dem Wahrheitswerte eines Satzes den Umstand, daß er wahr ist oder daß er falsch ist.” 59. Frege’s notion of a proper name (Eigenname), following the German translation of John Stuart Mill’s System of Logic, comprises not just grammatical proper names but singular terms in general. 60. GGA, I, §32. Note that this formulation admit the equation of proposition (Thought) with truth-condition, the Thought that snow is white = the Thought expressed in “snow is white” = the truth-condition of “snow is white,” whence for a declarative sentence S: S = that S is true = the truth-condition of “S” is fulﬁlled = the proposition expressed in “S” is true. Thus also, the Thought that S = the thought that the truth-condition of “S” is fulﬁlled.

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61. Frege (1906b, II, p. 387): “eine Urteilsfällung, die auf Grund schon früher gefällter Urteile nach logischen Gesetzen vollzogen wird.” 62. Bs, §6, and GGA, I, §14. The change from Frege’s two-dimensional notation to a one-dimensional Gentzen notation is not always anodyne, but here, where the concern is assertoric force, rather than the speciﬁc contents, it seems innocent enough. 63. The ins and outs of Frege’s assertion sign are treated very well in Stepanians (1998, chs. 1–5). Concerning the origin of its use as a theorem predicate, see Kleene (1952, p. 88, p. 526). 64. Dummett (1973, p. 309, p. 435). I beg to diﬀer and will return to the issue in section 8. 65. GGA, I, preface, pp. xv–xvi: Wahrsein ist etwas anderes als Fürwahrgehalten werden, sei es von Einem, sei es von Vielen, sei es von Allen, und es ist in keiner Weise darauf zurückzuführen. Es ist kein Widerspruch, dass etwas wahr ist, was von Allen für falsch gehalten wird. This marvelous credo is embedded in a passage pp. xv–xvii that is highly germane to the realism issue. 66. Nachlass, p. 2. (The Logik of the 80s): “Was wahr ist, ist unabhängig von unser Anerkennung. Wir können irren.” It is not required that there be mistaken acts of knowledge, but only that their possibility is not ruled out. 67. GGA, II, p. 69: Das Gesetz des ausgeschlossenen Dritten ist ja eigentlich nur in anderer Form die Forderung, dass der Begriﬀ scharf begrenzt sei. Ein beliebiger Gegenstand Δ fällt entweder unter den Begriﬀ Φ, oder er fällt nich unter ihn: tertium non datur. 68. The crucial primacy of the sui generis notion of rightness was noted by Martin-Löf (1987, 1991). In the light of this, Sundholm (2004) spells out various interrelations between diﬀerent roles of truth. 69. Locus classicus: “Der Gedanke” (1918). 70. Bs, preface, p. IX: Die festeste Beweisführung ist oﬀenbar der rein logische, welche, von der besonderen Beschaﬀenheit der Dinge absehend, sich allein auf die Gesetze gründet, auf denen alle Erkenntnis beruht. Wir theilen danach alle Wahrheiten, die einer Begründung bedürfen, in zwei Arten, idem der Beweis bei den einen rein logisch vorgehen kann, bei den andern sich auf Erfahrungsthatsachen stützen muss. GLA, §§3–4, contains a further elaboration of this theme into an account of the distinctions analytic/synthetic, a priori/a posteriori. Frege’s considerations here, successively stepping from a known truth to its grounds seeking the ultimate laws of justiﬁcations, are strongly reminiscent of Bolzano’s use of his grounding trees with respect to Abfolge. 71. Moore (1898, 1902) and Russell (1903, appendix A, §477, 1904). Cartwright (1987) treats of their early theory in some depth. The notion of proposition is here essentially the same as in Bolzano, and, to some extent, Frege. The grave responsibility for mistranslating the Fregean Gedanke (Thought) into proposition rests on Moore and Russell: Moore (1898, p. 179) introduced the terminology: “We have approached

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the nature of the proposition or judgment. A proposition is composed not of words, nor yet of thoughts, but of concepts.” Russell (1903, appendix A, §477) completes the error by coupling Frege’s Gedanke with his own proposition. Through PM this misidentiﬁcation eventually became standard throughout all of modern logic. In its original sense from the tradition, a proposition was either the (mental) judgment made, or its outward announcement in language, whereas after Russell and Moore it is turned into the content of a proposition in the original sense. 72. Joachim (1906, p. 37), who apparently wrote this marvelous passage in ignorance of Bolzano, drawing only on what he could ﬁnd in Russell and Moore, for instance: [There] is no problem at all in truth or falsehood; that some propositions are true and some false, just as some roses are red and some white; that belief is a certain attitude towards propositions, which is called knowledge when they are true, error when they are false. (Russell 1904, p. 523) Note how Russell adheres to the Bolzano reduction of knowledge to the mere truth of its content. Wittgenstein (Tractatus 6.111) also took notice of this passage from Russell. 73. Joachim (1906, p. 39). Russell and Moore both responded to Joachim’s book in Mind. Moore’s response is particularly interesting: “That some facts are facts, and some truths true, which never have been, are not now, and never will be experiences at all, and which are not timelessly expressed either” (1907, p. 231). What Moore countenances here are propositions that will remain unknown at all times; that, though, does not make them unknowable. The opposite view presupposes what Lovejoy (1936) called the principle of Plenitude, namely, that all potentialities will eventually become actual. (Martin-Löf 1991 rejects the application of Plenitude to knowability: what is knowable need never be known.) Only a year later did Moore commit himself in a review of William James: “It seems to me, then, that very often we have true ideas which we cannot verify; true ideas, which in all probability no man will ever be able to verify” (Moore 1907–08, p. 103). 74. Russell (1903, p. 145). (I am indebted to Prof. Peter Hylton for drawing my attention to this passage.) Since inﬁnitely complex propositions have to be unknowable, one way of deciding the issue concerning their existence is to deny that there are unknowable propositions. Because Russell is unable to pronounce on the issue, this means that he does not want to rule out unknowable truths. Also this passage was noticed by Wittgenstein, see Tractatus 4.2211. Russell’s is the earliest position (known to me) that allows for unknowable truths. Frege, as we saw, rejects them; every truth either is knowable in itself or has a Begründung, that is, a proof. 75. The undeﬁnability of truth was claimed in print only in Der Gedanke (1918, p. 60). In Nachlass, p. 140 (Logik 1897), Frege had made the same points almost verbatim. They in turn go back in nuce to the Logik of the 80s. 76. Moore in the lecture course from 1910–11 that was published later (1953). Russell in a number of places, for example, 1912 (p. 74) and PM, p. 43. 77. The notion was explicitly formulated by Mulligan, Simons, and Smith (1984). 78. The classic Stenius (1960) remains eminently readable. Hacker (1981) oﬀers the best presentation of the theory and its diﬃculties. 79. It most certainly is not; Peter and John, assuming they are empirical subjects, are complexes (5.541–5.421) composed of thoughts (3), that is, picture-facts (2.16),

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and will be analyzed in terms of the propositions that describe the complexes in question (3.24). The transcendent subject (5.63–5.641), on the other hand, “thinks out” the sentence-senses, which constitute the method of projection to the world (3.11–3.13), whereas the empirical subject is composed of sentential signs, that is, thought-facts. 80. The deﬁnition of the Q-relation reminds one of the ways that Frege formed unsaturated expressions. It is clear, I think, that this is one of very many places where the inﬂuence of “the great works” of Frege (see the preface to the Tractatus) can be felt. 81. In the next few paragraphs I use expository devices from the metatheory of the propostional calculus to survey the logico-semantical doctrines of the Tractatus. References to the Tractatus are by thesis number. Enderton (1972) contains the relevant model theory. 82. Wittgenstein’s notion of proposition (Satz) is not that of Bolzano–Frege– Russell (Satz an sich/Gedanke/proposition). In the Tractatus a proposition is a meaningful sentence in use and the Sätze might well better be rendered sentences in English translation. 83. Here N is Wittgenstein’s generalized Sheﬀer-stroke that negates every member of the range ξ of propositions. 84. The Fregean proposition is a sense, whereas the Tractarian proposition (sentence) has sense. 85. Jan Sebestik (1990) suggests that Robert Zimmerman’s Gymnasium textbook Philosophische Propädeutik, which is replete to the point of plagiarism with material taken from Bolzano’s Wissenschaftslehre, might be the missing link between Bolzano and Wittgenstein. 86. After the metamathematical revolution around 1930, Wittgenstein’s ontological notion, obtaining of the states of aﬀairs makes the elementary proposition true, is transformed into the model-theoretic: A |= ϕ, that is, the set-theoretical structure A satisﬁes the wﬀ ϕ (Tarski and Vaught 1957). See also Sundholm (1994b). 87. The crucial epistemological role of rightness in upholding the distinction between appearance and reality was noted and stressed by Martin-Löf (1987). 88. Dummett (1976) is the locus classicus, while Dummett (1991) oﬀers a booklength treatment. The secondary literature on Dummett’s argument has reached the proportions of an avalanche. Sundholm (1986) is an early survey, and Sundholm (1994a) approaches Dummett’s position from a more severely constructivist standpoint. 89. Molk (1885, p. 8): Les déﬁnitions devront être algébraiques et non pas logiques seulement. Il ne suﬃt pas de dire: “Une chose est ou et non pas.” Il faut montrer ce ques veut dire être et ne pas être, dans le domaine particulier dans lequel nous nous mouvons. Alors, seulement nouns faisons un pas en avant. Si nous déﬁnissions, par exemple, une fonction irréductible comme une fonction qui n’est pas réductible, c’est a dire quie n’est pas décomposable en d’autres fonctions d’une nature déterminée, nous ne donnons point de déﬁnition algébraique, nous n’énonçons qu’une simple vérité logique.

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Pour qu’en Algèbre, nous soyons en droit de donner cette déﬁnition, il faut qu’elle soit précédé de l’exposé d’une méthode nous permettant d’obtenir a l’aide d’un nombre ﬁni d’operations rationelles, les facteurs d’une fonction réductible. Seule cette méthode donne aux mots réductible et irréductible. 90. See the Port-Royal Logic, Arnauld and Nicole (1662, part IV, ch. III). 91. The ∞ set D ⊆ , being bounded, is contained in a closed real interval I. Deﬁne I0 =def I =def [a0 , b0 ] Ik+1 =def [ak , ak + bk /2] Ik+1 =def [ak + bk /2, bk ]

if this left half of Ik contains ∞ many points from D; otherwise. (NB. Here we cannot decide whether a half has ∞ many points from D.)

Hence, each of the nested intervals Ik contains ∞ many points from D, and length (Ik ) → 0, when k → ∞. Thus, k Ik contains exactly one point that is the required accumulation point for D. 92. Frege does not include the set D, the domain of quantiﬁcation, since he quantiﬁes over all individual objects. 93. Martin-Löf (1983, p. 33) hints at this way of understanding Brouwer’s criticism. It was noted explicitly by Aarne Ranta (1994, p. 38). See also Sundholm (1998), where also Poincaré’s criticism of impredicability is cast in the same mold. The law of excluded middle does not only serve as a principle of reasoning. It is also used meaning-theoretically to delimit the notion of proposition. Thus, for Frege, a proposition is a method for determining one of the truth values True and False. Similarly, every proposition implies itself and something which is not a proposition implies nothing, Russell (1903, §16) notes, and goes on to use “P ⊃ P ” as an explanation of what it is for something P to be a proposition. But an assertion that P ⊃ Q is true is equivalent to is an assertion that P is false or Q is true. Thus an assertion that P is a proposition amounts to an assertion that P is false or P is true. The issue resurfaces in the Cambridge Letter R 12 from Wittgenstein to Russell, June 1913, where “ ‘aRb.v.∼aRb’ must follow directly without the use of any other premiss.” Also Cantor’s explanation of a well-deﬁned set (1882, p. 114) makes meaning-theoretical use of the law of excluded middle. 94. Appropriately enough, free-variable equations between computable terms, with only true numerical substitution instances, are called veriﬁzierbar (veriﬁable) in the canonical exposition Hilbert and Bernays (1934, p. 237). 95. Carnap (1934, p. xv). 96. Carnap (1934, pp. 51–52). 97. Brouwer (1981, p. 5). This formulation, albeit late, expresses Brouwer’s lifelong view. 98. Brouwer (1908). 99. One does not, of course, claim that A ∨ ¬A is false, that is, that ¬(A ∨ ¬A) is true, because the latter claim is refutable outright: Assume that ¬(A ∨ ¬A) is true. Assume further that A is true. Under this assumption, A ∨ ¬A is also true. Therefore, the assumption that A is true leads to a contradiction. Therefore, A is false, now only under the sole assumption that ¬(A ∨ ¬A) is true. Hence ¬A is true,

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still under the same assumption. But then, under the same assumption, also A ∨ ¬A is true. Thus the assumption that ¬(A ∨ ¬A) is true leads to the conclusion that also A ∨ ¬A is true, which is a contradiction. Therefore the assumption is wrong and ¬(A ∨ ¬A) is false. Thus, ¬¬(A ∨ ¬A) is true. 100. GGA, II, §§87–147, as well as his undigniﬁed diatribes (1906a) and (1906b), (1908), against Korselt and Thomae, respectively. The need for content in mathematical sign-languages is a theme that Frege pursued from his earliest writings; see for instance, the long Nachlass paper on Boole’s logic and his own begriﬀsschrift (1880/81) and above all (1882). 101. Even after World War II—Brouwer lectured regularly at Cambridge from 1946 to 1951—he would proclaim, apparently with a deadpan face, that “Absurdity of absurdity of absurdity is equivalent to absurdity,” rather than use the pellucid ¬¬¬A ↔ ¬A. See Brouwer (1981, p. 12). 102. Most clearly perhaps in the introduction to Weyl (1918a), but also in the treatment of logic in (1918b). 103. Weyl (1921, p. 54): Ein Existentialsatz—etwa “es gibt eine gerade Zahl”—ist überhaupt kein Urteil im eigentlichen Sinne, das einen Sachverhalt behauptet; ExistentialSachverhalte sind eine leere Erﬁndung der Logiker. “2 ist eine gerade Zahl”: das ist ein wirkliches, einem Sachverhalt Ausdruck gebendes Urteil; “es gibt eine gerade Zahl” ist nur ein aus diesem Urteil gewonnenes Urteilsabstrakt. 104. The novel form of judgment and the explicit formulation of the rule that provides its assertion-condition are both due to Per Martin-Löf (1994). 105. Heyting (1930a). 106. The debate in question is treated in Thiel (1988) and Franchella (1994). 107. Heyting (1930b, p. 958): Une proposition p, comme, par example, “la constante d’Euler est rationelle”; exprime un problème, ou mieux encore une certaine attente (celle de trouver deux entiers a et b tels que C = a/b), qui pourra être réalisée ou déçue. 108. This table is based on a streamlined formulation oﬀered by Per MartinLöf (1984), and, in each case, lays down what a canonical proof-object is for the proposition in question. For the signiﬁcance of canonical in this context, see Sundholm (1997), where a full exposition of the intuitionistic meaning explanations is oﬀered. 109. It should be stressed that these meaning explanations for the logical constants, and the ensuing truth-deﬁnition, are neutral with respect to the underlying logic; in fact the framework can be viewed as a Tarskian truth deﬁnition—another neutral account. If we allow nonconstructive existence claims, also classical logic holds under the proof-object semantics. We have to show, reasoning nonconstructively, that Proof(A ∨ ¬A) = ∅. Assume that Proof(A) = ∅. Let a ∈ Proof(A); then i(a) ∈ Proof(A ∨ ¬A). Assume that Proof(A) = ∅. Then λx.x ∈ Proof(A) → Proof(⊥) = Proof(¬A), and so j(λx.x) ∈ Proof(A ∨ ¬A). Hence, in either case, Proof(A ∨ ¬A) = ∅, so the proposition A ∨ ¬A is true. Q.E.D.

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110. In Martin-Löf’s constructive type theory (1982, 1984) the elliptic form of judgment A is true is replaced by the explicit p is a Proof(A); Martin-Löf (1983) makes clear that this constitutes a return to the traditional S is P form of judgment. 111. Published as Gentzen (1934–35). 112. Dummett (1973, p. 309, and p. 435, respectively). 113. Prawitz (1965, p. 37). Prawitz prefers the opposite order between the two premises of the (⊃I) rule, but this is of no importance for the present point. 114. Provided that, given that, on condition that, under the assumption that, under the hypothesis that. . . . Many variations in the wording are possible here. 115. For instance by Prawitz (1971, remark 1.6, p. 243), Dummett (1977, pp. 121– 122, and 1991, p. 248), as well as Sundholm (1983). 116. The sequential form of natural deduction uses both introduction rules and elimination rules. It must not be confused with the sequent calculus of Gentzen (1934–35) that uses no elimination rules but has both left and right introduction rules, on both sides of the sequent arrow. 117. Gentzen did not consider closed sequents; the exploration of their theory is due to Peter Schroeder-Heister (1981, 1984, 1987), half a century after Gentzen. 118. Kutschera (1996) and Schroeder-Heister (1999) both discuss the matter in apparent unawareness of Tichý’s explicit treatment. Tichý’s remarkable chapter 13— Inference—merits attention, as does his paper “On Inference” (1999). 119. The Übersicht (1934–35, p. 176) does mention Frege, Russell, and Hilbert as particularly important for the formalization of logical inference, but the remark does not presuppose familiarity with the details of Frege’s formalization. The introduction to Hilbert and Ackermann (1928, p. 2), which Gentzen did know, makes similar mention of the same authors. 120. Full biographies are available for a number of authors treated of in the present chapter: Frege (Kreiser 2001), Wittgenstein (McGuinness 1988; Monk 1990), Brouwer (Van Dalen 1999), and Gentzen (Menzler-Trott 2001). 121. Simons (1999, p. 115).

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Russell, Bertrand. 1904. Meinong’s theory of complexes and assumptions. Mind 13: 204–219, 336–354, 509–524. Russell, Bertrand. 1912. The Problems of Philosophy. Oxford: Oxford University Press. 1982. Scholz, Heinrich. 1930. Die Axiomatik der Alten. Blätter für deutsche Philosophie 4: 259–278. Cited from Scholz (1961), 27–45. English translation by Jonathan Barnes, Ancient Axiomatic Theory. In Articles on Aristotle 1: Science, eds. J. Barnes, M. Schoﬁeld, and R. Sorabji, 1975, 50–60. London: Duckworth. Scholz, Heinrich. 1961. Mathesis Universalis, eds. Hans Hermes, Friedrich Kambartel, and Joachim Ritter. Basel: Benno Schwabe Verlag. Schroeder-Heister, Peter. 1981. Untersuchungen zur regellogischen Deutung von Aussagenverknüpfungen. Diss., Bonn. Schroeder-Heister, Peter. 1984. A natural extension of natural deduction. Journal of Symbolic Logic 49: 1284–1300. Schroeder-Heister, Peter. 1987. Structural Frameworks with Higher-Level Rules. Habilitationsschrift. University of Konstanz, Department of Philosophy. Schroeder-Heister, Peter. 1999. Gentzen-Style Features in Frege. In Abstracts of the 11th International Congress of Logic, Methodology, and Philosophy of Science (Cracow, August 1999), 499. Cracow. Sebestik, Jan. 1990. The Archeology of the Tractatus: Bolzano and Wittgenstein. In Wittgenstein—Towards a Re-Evaluation, Proc. 14th Int. Wittgenstein Symp., Kirchberg am Wechsel, 13–20 August 1989, Vol. I, eds. Rudolf Haller and J. Brandl, 112–128. Wien: Verlag Hölder-Pichler-Tempsky. Sebestik, Jan. 1992. The Construction of Bolzano’s Logical System. In Bolzano’s Wissenschafstlehre 1837–1987. International Workshop Firenze, 16–19 settembre 1987, 163–177. Firenze: Leo S. Olschki. Siebel, Mark. 1996. Der Begriﬀ der Ableitbarkeit bei Bolzano. St Augustin: Academia Verlag. Simons, Peter. 1999. Bolzano, Brentano, and Meinong: Three Austrian Realists. In German Philosophy since Kant (Royal Institute of Philosophy Supplement 44), ed. Anthony O’Hear, 109–136. Cambridge: Cambridge University Press. Stenius, Erik. 1960. Wittgenstein’s Tractatus: A Critical Expostion of Its Main Line of Thought. Oxford: Basil Blackwell. Stepanians, Markus. 1998. Frege und Husserl über Urteilen und Denken. Paderborn: Schöningh. Sundholm, B. G. 1983. Systems of Deduction. In Handbook of Philosophical Logic I, eds. D. Gabbay and F. Guenthner, 133–188. Dordrecht: Reidel. Sundholm, B. G. 1986. Proof Theory and Meaning. In Handbook of Philosophical Logic, eds. D. Gabbay and F. Guenthner, vol. III, 471–506. Dordrecht: Reidel. Sundholm, B. G. 1988. Oordeel en Gevolgtrekking: Bedereigde Species? (Judgement and Inference. Endangered Species?). Inauguaral Lecture, Leyden University, September 9, 1988. Published in pamphlet form by the university. Sundholm, B. G. 1994a. Vestiges of Realism. In The Philosophy of Michael Dummett, eds. Brian McGuinness and G. Oliveri, 137–165. Dordrecht: Kluwer. Sundholm, B. G. 1994b. Ontologic versus Epistemologic: Some Strands in the Development of Logic, 1837–1957. In Logic and Philosophy of Science in Uppsala, eds. Dag Prawitz and Dag Westerståhl, 373–384. Dordrecht: Kluwer. Sundholm, B. G. 1994c. Proof-theoretical semantics and Fregean identity-criteria for propositions. Monist 77: 294–314.

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Sundholm, B. G. 1997. Implicit epistemic aspects of constructive logic. Journal of Logic, Language and Information 6: 191–212. Sundholm, B. G. 1998. Intuitionism and Logical Tolerance. In Alfred Tarski and the Vienna Circle (Vienna Circle Institute Yearbook), eds. Jan Wolenski and E. Köhler, 135–149. Dordrecht: Kluwer. Sundholm, B. G. 2002. A Century of Inference: 1837–1936. In Logic, Methodology and Philosophy of Science 11, Cracow 1999, eds. Jan Wolenski and K. Placek. Dordrecht: Kluwer. Sundholm, B. G. 2004. Antirealism and the Roles of Truth. In Handbook of Epistemology, eds. I. Niiniluoto, M. Sintonen, and J. Wolenski, 437–466. Dordrecht: Kluwer. Tarski, Alfred. 1933a. Pojece prawdy wjezykach deukcyjnynch, Travaux de la Société des Sciences et des Letttres de Varsovie, Classe III Sciences Mathématiques et Physiques, vol. 34. (German translation by Leopold Blaustein: Tarski 1935). Tarski, Alfred. 1933b. Einige Betrachtungen über die Begriﬀe der ω-Widerspruchsfreiheit und der ω-Vollständigkeit. Monatshefte für Mathematik und Physik 40: 97–112. Tarski, Alfred. 1935. Der Wahrheitsbegriﬀ in den formalisierten Sprachen. In Studia Philosophica I (1936), 261–405. Lemberg: Polish Philosophical Society. Oﬀprints in monograph form dated 1935. Tarski, Alfred. 1944. The semantic conception of truth and the foundations of semantics. Philosophy and Phenomenological Research 4: 341–375. Tarski, Alfred. 1956. Logic, Semantics, Metamathematics. Oxford: Clarendon Press. Tarski, Alfred, with Jan Lukasiewicz. 1930. Untersuchungen über den Aussagenkalkül. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III 23: 30–50. English translation in Tarski (1956, pp. 38–59). Tarski, Alfred, with Robert Vaught. 1957. Arithmetical extensions of relational systems. Compositio Mathematicae 13: 81–102. Textor, Mark. 1996. Bolzanos Propositionalismus. Berlin: De Gruyter. Thiel, Christian. 1988. Die Kontroverse um die intuitionistische Logik vor ihre Axiomatisierung durch Heyting im Jahre 1930. History and Philosophy of Logic 9: 67–75. Tichý, Pavel. 1988. Frege’s Foundations of Logic. Berlin: De Gruyter. Tichý, Pavel and Jindra Tichý. 1999. On Inference. In The LOGICA Yearbook 1998, Philosophical Institute, Czech Academy of Science, 73–85. Prague: FILOSOFIA Publishers. Van Dalen, Dirk. 1999. Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer, vol. 1. The Dawning Revolution. Oxford: Clarendon Press. Weyl, Hermann. 1918a. Raum–Zeit–Materie. Berlin: Julius Springer. Weyl, Hermann. 1918b. Das Kontinuum. Leipzig: Veit. Weyl, Hermann. 1921. Über die neue Grundlagenkrise der Mathematik. Mathematische Zeitschrift 10: 39–79. English translation in Mancosu (1998), pp. 86–118. Wittgenstein, Ludwig. 1921. Logisch-philosophische Abhandlung. Annalen für Naturphilosophie 14: 198–262. Reprinted with English translation in Wittgenstein (1922). Wittgenstein, Ludwig. 1922. Tractatus logico-philosophicus. London: Kegan Paul.

9

The Development of Mathematical Logic from Russell to Tarski, 1900–1935 Paolo Mancosu, Richard Zach, and Calixto Badesa

The following nine itineraries in the history of mathematical logic do not aim at a complete account of the history of mathematical logic during the period 1900–1935. For one thing, we had to limit our ambition to the technical developments without attempting a detailed discussion of issues such as what conceptions of logic were being held during the period. This also means that we have not engaged in detail with historiographical debates which are quite lively today, such as those on the universality of logic, conceptions of truth, the nature of logic itself, and so on. While of extreme interest, these themes cannot be properly dealt with in a short space, as they often require extensive exegetical work. We therefore merely point out in the text or in appropriate notes how the reader can pursue the connection between the material we treat and the secondary literature on these debates. Second, we have not treated some important developments. While we have not aimed at completeness, our hope has been that by focusing on a narrower range of topics our treatment will improve on the existing literature on the history of logic. There are excellent accounts of the history of mathematical logic available, such as, to name a few, Kneale and Kneale (1962), Dumitriu (1977), and Mangione and Bozzi (1993). We have kept the secondary literature quite present in that we also wanted to write an essay that would strike a balance between covering material that was adequately discussed in the secondary literature and presenting new lines of investigation. This explains, for instance, why the reader will ﬁnd a long and precise exposition of Löwenheim’s (1915) theorem but only a short one on Gödel’s incompleteness theorem: Whereas there is hitherto no precise presentation of the ﬁrst result, accounts of the second result abound. Finally, the treatment of the foundations of mathematics is quite restricted, and it is 318

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ancillary to the exposition of the history of mathematical logic. Thus, it is not meant to be the main focus of our exposition.1 Page references in citations are to the English translations, if available, or to the reprint edition, if listed in the bibliography. All translations are the authors’, unless an English translation is listed in the references.

1. Itinerary I. Metatheoretical Properties of Axiomatic Systems 1.1. Introduction The two most important meetings in philosophy and mathematics in 1900 took place in Paris. The First International Congress of Philosophy met in August and so did, soon after, the Second International Congress of Mathematicians. As symbolic, or mathematical, logic has traditionally been part of both mathematics and philosophy, a glimpse at the contributions in mathematical logic at these two events will give us a representative selection of the state of mathematical logic at the beginning of the twentieth century. At the International Congress of Mathematicians, Hilbert presented his famous list of problems (Hilbert 1900a), some of which became central to mathematical logic, such as the continuum problem, the consistency proof for the system of real numbers, and the decision problem for Diophantine equations (Hilbert’s tenth problem). However, despite the attendance of remarkable logicians like Schröder, Peano, and Whitehead in the audience, the only other contributions that could be classiﬁed as pertaining to mathematical logic were two talks given by Alessandro Padoa on the axiomatizations of the integers and of geometry, respectively. The third section of the International Congress of Philosophy was devoted to logic and history of the sciences (Lovett 1900–1901). Among the contributors of papers in logic we ﬁnd Russell, MacColl, Peano, Burali-Forti, Padoa, Pieri, Poretsky, Schröder, and Johnson. Of these, MacColl, Poretsky, Schröder, and Johnson read papers that belong squarely to the algebra of logic tradition. Russell read a paper on the application of the theory of relations to the problem of order and absolute position in space and time. Finally, the Italian school of Peano and his disciples—Burali-Forti, Padoa, and Pieri—contributed papers on the logical analysis of mathematics. Peano and Burali-Forti spoke on deﬁnitions, Padoa read his famous essay containing the “logical introduction to any theory whatever,” and Pieri spoke on geometry considered as a purely logical system. Although there are certainly points of contact between the ﬁrst group of logicians and the second group, already at that time it was obvious that two diﬀerent approaches to mathematical logic were at play. Whereas the algebra of logic tradition was considered to be mainly an application of mathematics to logic, the other tradition was concerned more with an analysis of mathematics by logical means. In a course given in 1908 in

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Göttingen, Zermelo captured the double meaning of mathematical logic in the period by reference to the two schools: The word “mathematical logic” can be used with two diﬀerent meanings. On the one hand one can treat logic mathematically, as it was done for instance by Schröder in his Algebra of Logic; on the other hand, one can also investigate scientiﬁcally the logical components of mathematics. (Zermelo 1908a, 1)2 The ﬁrst approach is tied to the names of Boole and Schröder, the second was represented by Frege, Peano, and Russell.3 We will begin by focusing on mathematical logic as the logical analysis of mathematical theories, but we will return later (see itinerary IV) to the other tradition.

1.2. Peano’s School on the Logical Structure of Theories We have mentioned the importance of the logical analysis of mathematics as one of the central motivating factors in the work of Peano and his school on mathematical logic. First of all, Peano was instrumental in emphasizing the importance of mathematical logic as an artiﬁcial language that would remove the ambiguities of natural language, thereby allowing a precise analysis of mathematics. In the words of Pieri, an appropriate ideographical algorithm is useful as “an instrument appropriate to guide and discipline thought, to exclude ambiguities, implicit assumptions, mental restrictions, insinuations and other shortcomings, almost inseparable from ordinary language, written as well as spoken, which are so damaging to speculative research” (Pieri 1901, 381). Moreover, he compared mathematical logic to “a microscope which is appropriate for observing the smallest diﬀerence of ideas, diﬀerences that are made imperceptible by the defects of ordinary language in the absence of some instrument that magniﬁes them” (382). It was by using this “microscope” that Peano was able, for instance, to clarify the distinction between an element and a class containing only that element and the related distinction between membership and inclusion.4 The clariﬁcation of mathematics, however, also meant accounting for what was emerging as a central ﬁeld for mathematical logic: the formal analysis of mathematical theories. The previous two decades had in fact seen much activity in the axiomatization of particular branches of mathematics, including arithmetic, algebra of logic, plane geometry, and projective geometry. This culminated in the explicit characterization of a number of formal conditions for which axiomatized mathematical theories should strive. Let us consider ﬁrst Pieri’s description of his work on the axiomatization of geometry, which had been carried out independently of Hilbert’s famous Foundations of Geometry (1899). In his presentation to the International Congress of Philosophy in 1900, Pieri emphasized that the study of geometry is following arithmetic in becoming more and more “the study of a certain order of logical relations; in freeing itself little by little from the bonds which still keep

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it tied (although weakly) to intuition, and in displaying consequently the form and quality of purely deductive, abstract and ideal science” (Pieri 1901, 368). Pieri saw in this abstraction from concrete interpretations a unifying thread running through the development of arithmetic, analysis, and geometry in the nineteenth century. This led him to a conception of geometry as a hypothetical discipline (he coined the term “hypothetico-deductive”). In fact he goes on to assert that the primitive notions of any deductive system whatsoever “must be capable of arbitrary interpretations in certain limits assigned by the primitive propositions,” subject only to the restriction that the primitive propositions must be satisﬁed by the particular interpretation. The analysis of a hypothetico-deductive system begins then with the distinction between primitive notions and primitive propositions. In the logical analysis of a hypothetico-deductive system it is important not only to distinguish the derived theorems from the basic propositions (deﬁnitions and axioms) but also to isolate the primitive notions, from which all the others are deﬁned. An ideal to strive for is that of a system whose primitive ideas are irreducible, that is, such that none of the primitive ideas can be deﬁned by means of the others through logical operations. Logic is here taken to include notions such as, among others, “individual,” “class,” “membership,” “inclusion,” “representation,” and “negation” (383). Moreover, the postulates, or axioms, of the system must be independent, that is, none of the postulates can be derived from the others. According to Pieri, there are two main advantages to proceeding in such an orderly way. First of all, keeping a distinction between primitive notions and derived notions makes it possible to compare diﬀerent hypothetico-deductive systems as to logical equivalence. Two systems turn out to be equivalent if for every primitive notion of one we can ﬁnd an explicit deﬁnition in the second one such that all primitive propositions of the ﬁrst system become theorems of the second system, and vice versa. The second advantage consists in the possibility of abstracting from the meaning of the primitive notions and thus operate symbolically on expressions which admit of diﬀerent interpretations, thereby encompassing in a general and abstract system several concrete and speciﬁc instances satisfying the relations stated by the postulates. Pieri is well known for his clever application of these methodological principles to geometrical systems (see Freguglia 1985; Marchisotto 1995). Pieri refers to Padoa’s articles for a more detailed analysis of the properties connected to axiomatic systems. Alessandro Padoa was another member of the group around Peano. Indeed, of that group, he is the only one whose name has remained attached to a speciﬁc result in mathematical logic, that is, Padoa’s method for proving indeﬁnability (see the following). The result was stated in the talks Padoa gave in 1900 at the two meetings mentioned at the outset (Padoa 1901, 1902). We will follow the “Essai d’une théorie algébrique des nombre entiers, précédé d’une introduction logique a une théorie déductive quelconque.” In the Avant-Propos (not translated in van Heijenoort 1967a) Padoa lists a number of notions that

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he considers as belonging to general logic such as class (“which corresponds to the words: terminus of the scholastics, set of the mathematicians, common noun of ordinary language”). The notion of class is not deﬁned but assumed with its informal meaning. Extensionality for classes is also assumed: “a class is completely known when one knows which individuals belong to it.” However, the notion of ordered class he considers as lying outside of general logic. Padoa then states that all symbolic deﬁnitions have the form of an equality y = b where y is the new symbol and b is a combination of symbols already known. This is illustrated with the property of being a class with one element. Disjunction and negation are given with their class interpretation. The notions “there is” and “there is not” are also claimed to be reducible to the notions already previously introduced. For instance, Padoa explains that given a class a to say “there is no a” means that the class not-a contains everything, that is, not-a = (a or not-a). Consequently, “there are a[’s]” means: not-a = (a or not-a). The notion of transformation is also taken as belonging to logic. If a and b are classes and if for any x in a, ux is in b, then u is a transformation from a into b. An obvious principle for transformations u is: if x = y then ux = uy. The converse, Padoa points out, does not follow. This much was a preliminary to the section of Padoa’s paper titled “Introduction logique a une théorie déductive quelconque.” Padoa makes a distinction between general logic and speciﬁc deductive theories. General logic is presupposed in the development of any speciﬁc deductive theory. What characterizes a speciﬁc deductive theory is its set of primitive symbols and primitive propositions. By means of these, one deﬁnes new notions and proves theorems of the system. Thus, when one speaks of indeﬁnability or unprovability, one must always keep in mind that these notions are relative to a speciﬁc system and make no sense independently of a speciﬁc system. Restating his notion of deﬁnition he also claims that deﬁnitions are eliminable and thus inessential. Just like Pieri, Padoa also speaks of systems of postulates as a pure formal system on which one can reason without being anchored to a speciﬁc interpretation, “for what is necessary to the logical development of a deductive theory is not the empirical knowledge of the properties of things, but the formal knowledge of relations between symbols” (1901, 121). It is possible, Padoa continues, that there are several, possibly inﬁnite, interpretations of the system of undeﬁned symbols which verify the system of basic propositions and thus all the theorems of a theory. He then adds: The system of undeﬁned symbols can then be regarded as the abstraction obtained from all these interpretations, and the generic theory can then be regarded as the abstraction obtained from the specialized theories that result when in the generic theory the system of undeﬁned symbols is successively replaced by each of the interpretations of this theory. Thus, by means of just one argument that proves a proposition of the generic theory we prove implicitly a proposition in each of the specialized theories. (1901, 121)5

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In contemporary model theory, we think of an interpretation as specifying a domain of individuals with relations on them satisfying the propositions of the system, by means of an appropriate function sending individual constants to objects and relation symbols to subsets of the domain (or Cartesian products of the same). It is important to remark that in Padoa’s notion of interpretation something else is going on. An interpretation of a generic system is given by a concrete set of propositions with meaning. In this sense the abstract theory captures all of the individual theories, just as the expression x + y = y + x captures all the particular expressions of the form 2 + 3 = 3 + 2, 5 + 7 = 7 + 5, and so on. Moving now to deﬁnitions, Padoa states that when we deﬁne a notion in an abstract system we give conditions which the deﬁned notion must satisfy. In each particular interpretation the deﬁned notion becomes individualized, that is, it obtains a meaning that depends on the particular interpretation. At this point Padoa states a general result about deﬁnability. Assume that we have a general deductive system in which all the basic propositions are stated by means of undeﬁned symbols: We say that the system of undeﬁned symbols is irreducible with respect to the system of unproved propositions when no symbolic deﬁnition of any undeﬁned symbol can be deduced from the system of unproved propositions, that is, when we cannot deduce from the system a relation of the form x = a, where x is one of the undeﬁned symbols and a is a sequence of other such symbols (and logical symbols). (1901, 122) How can such a result be established? Clearly one cannot adduce the failure of repeated attempts at deﬁning the symbol; for such a task, a method for demonstrating the irreducibility is required. The result is stated by Padoa as follows: To prove that the system of undeﬁned symbols is irreducible with respect to the system of unproved propositions it is necessary and suﬃcient to ﬁnd, for any undeﬁned symbol, an interpretation of the system of undeﬁned symbols that veriﬁes the system of unproved propositions and that continues to do so if we suitably change the meaning of only the symbol considered. (1901, 122)6 Padoa (1902) covers the same ground more concisely but also adds the criterion of compatibility for a set of postulates: “To prove the compatibility of a set of postulates one needs to ﬁnd an interpretation of the undeﬁned symbols which veriﬁes simultaneously all the postulates” (1902, 249). Padoa applied his criteria to showing that his axiomatization of the theory of integers satisﬁed the condition of compatibility and irreducibility for the primitive symbols and postulates. We thus see that for Padoa the study of the formal structure of an arbitrary deductive theory was seen as a task of general logic. What can be said about

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these metatheoretical results in comparison to the later developments? We have already pointed out the diﬀerent notion of interpretation which informs the treatment. Moreover, the system of logic in the background is never fully spelled out, and in any case it would be a logic containing a good amount of set-theoretic notions. For this reason, some results are taken as obvious that would actually need to be justiﬁed. For instance, Padoa claims that if an interpretation satisﬁes the postulates of an abstract theory, then the theorems obtained from the postulates are also satisﬁed in the interpretation. This is a soundness principle, which nowadays must be shown to hold for the system of derivation and the semantics speciﬁed for the system. For similar reasons the main result by Padoa on the indeﬁnability of primitive notions does not satisfy current standards of rigor. Thus, a formal proof of Padoa’s deﬁnability theorem had to wait until the works of Tarski (1934–1935) for the theory of types and Beth (1953) for ﬁrst-order logic (see van Heijenoort 1967a, 118–119, for further details).

1.3. Hilbert on Axiomatization In light of the importance of the work of Peano and his school on the foundations of geometry, it is quite surprising that Hilbert did not acknowledge their work in the Foundations of Geometry. Although it is not quite clear to what extent Hilbert was familiar with the work of the Italian school in the last decade of the nineteenth century (Toepell 1986), he certainly could not ignore their work after the 1900 International Congress in Mathematics. In many ways Hilbert’s work on axiomatization resembles the level of abstractness also emphasized by Peano, Padoa, and Pieri. The goal of Foundations of Geometry (1899) is to investigate geometry axiomatically.7 At the outset we are asked to give up the intuitive understanding of notions like point, line, or plane and consider any three system of objects and three sorts of relations between these objects (lies on, between, congruent). The axioms only state how these properties relate the objects in question. They are divided into ﬁve groups: axioms of incidence, axioms of order, axioms of congruence, axiom of parallels, and axioms of continuity. Hilbert emphasizes that an axiomatization of geometry must be complete and as simple as possible.8 He does not make explicit what he means by completeness, but the most likely interpretation of the condition is that the axiomatic system must be able to capture the extent of the ordinary body of geometry. The requirement of simplicity includes, among other things, reducing the number of axioms to a ﬁnite set and showing their independence. Another important requirement for axiomatics is showing the consistency of the axioms of the system. This was unnecessary for the old axiomatic approaches to geometry (such as Euclid’s) because one always began with the assumption that the axioms were true of some reality and thus consistency was not an issue. But in the new conception of axiomatics, the axioms do not express truths but only postulates whose consistency must be investigated.

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Hilbert shows that the basic axioms of his axiomatization are independent by displaying interpretations in which all of the axioms except one are true.9 Here we must point to a small diﬀerence with the notion of interpretation we have seen in Pieri and Padoa. Hilbert deﬁnes an interpretation by ﬁrst specifying what the set of objects consists in. Then a set of relations among the objects is speciﬁed in such a way that consistency or independence is shown. For instance, for showing the consistency of his axioms, √ he considers a domain given by the subset of algebraic numbers of the form 1 + ω 2 and then speciﬁes the relations as being sets of ordered pairs and ordered triples of the domain. The consistency of the geometrical system is thus discharged on the new arithmetical system: “From these considerations it follows that every contradiction resulting from our system of axioms must also appear in the arithmetic deﬁned above” (29). Hilbert had already applied the axiomatic approach to the arithmetic of real numbers. Just as in the case of geometry, the axiomatic approach to the real numbers is conceived in terms of “a framework of concepts to which we are led of course only by means of intuition; we can nonetheless operate with this framework without having recourse to intuition.” The consistency problem for the system of real numbers was one of the problems that Hilbert stated at the International Congress in 1900: But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a ﬁnite number of logical steps based upon them can never lead to contradictory results. (1900a, 1104) In the case of geometry, consistency is obtained by “constructing an appropriate domain of numbers such that to the geometrical axioms correspond analogous relations among the objects of this domain.” For the axioms of arithmetic, however, Hilbert required a direct proof, which he conjectured could be obtained by a modiﬁcation of the arguments already used in “the theory of irrational numbers.”10 We do not know what Hilbert had in mind, but in any case, in his new approach to the problem (1905b), Hilbert made considerable progress in conceiving how a direct proof of consistency for arithmetic might proceed. We will postpone treatment of this issue to later (see itinerary VI) and go back to specify what other metatheoretical properties of axiomatic systems were being discussed in these years. By way of introduction to the next section, something should be said here about one of the axioms, which Hilbert in his Paris lecture calls axiom of integrity and later completeness axiom. The axiom says that the (real) numbers form a system of objects which cannot be extended (Hilbert 1900b, 1094). This axiom is in eﬀect a metatheoretical statement about the possible interpretations of the axiom system.11 In the second and later editions of the Foundations of Geometry, the same axiom is also stated for points, straight lines and planes:

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(Axiom of completeness) It is not possible to add new elements to a system of points, straight lines, and planes in such a way that the system thus generalized will form a new geometry obeying all the ﬁve groups of axioms. In other words, the elements of geometry form a system which is incapable of being extended, provided that we regard the ﬁve groups of axioms as valid. (Hilbert 1902, 25) Hilbert commented that the axiom was needed to guarantee that his geometry turn out to be identical to Cartesian geometry. Awodey and Reck (2002) write, “what this last axiom does, against the background of the others, is to make the whole system of axioms categorical. . . . He does not state a theorem that establishes, even implicitly, that his axioms are categorical; he leaves it . . . without proofs” (11). The notion of categoricity was made explicit in the important work of the “postulate theorists,” to which we now turn.

1.4. Completeness and Categoricity in the Work of Veblen and Huntington A few metatheoretical notions that foreshadow later developments emerged during the early years of the twentieth century in the writings of Huntington and Veblen. Huntington and Veblen are part of a group of mathematicians known as the American postulate theorists (Scanlan 1991, 2003). Huntington was concerned with providing “complete” axiomatizations of various mathematical systems, such as the theory of absolute continuous magnitudes (positive real numbers) (1902) and the theory of the algebra of logic (1905). For instance, in 1902 he presented six postulates for the theory of absolute continuous magnitudes, which he claims to form a complete set. A complete set of postulates is characterized by the following properties: 1. The postulates are consistent; 2. They are suﬃcient; 3. They are independent (or irreducible). By consistency he means that there exists an interpretation satisfying the postulates. Condition 2 asserts that there is essentially only one such interpretation possible. Condition 3 says that none of the postulates is a “consequence” of the other ﬁve. A system satisfying the conditions (1) and (2) we would nowadays call “categorical” rather than “complete.” Indeed, the word “categoricity” was introduced in this context by Veblen in a paper on the axiomatization of geometry (1904). Veblen credits Huntington with the idea and Dewey for having suggested the word “categoricity.” The description of the property is interesting: Inasmuch as the terms point and order are undeﬁned one has a right, in thinking of the propositions, to apply the terms in

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connection with any class of objects of which the axioms are valid propositions. It is part of our purpose however to show that there is essentially only one class of which the twelve axioms are valid. In more exact language, any two classes K and K of objects that satisfy the twelve axioms are capable of a one-one correspondence such that if any three elements A, B, C of K are in the order ABC, the corresponding elements of K are also in the order ABC. Consequently any proposition which can be made in terms of points and order either is in contradiction with our axioms or is equally true of all classes that verify our axioms. The validity of any possible statement in these terms is therefore completely determined by the axioms; and so any further axiom would have to be considered redundant. [Note: Even were it not deducible from the axioms by a ﬁnite set of syllogisms] Thus, if our axioms are valid geometrical propositions, they are suﬃcient for the complete determination of Euclidean geometry. A system of axioms such as we have described is called categorical, whereas one to which it is possible to add independent axioms (and which therefore leaves more than one possibility open) is called disjunctive. (Veblen 1904, 346) A number of things are striking about the passage just quoted. First of all, we are used to deﬁne categoricity by appealing directly to the notion of isomorphism.12 What Veblen does is equivalent to specifying the notion of isomorphism for structures satisfying his 12 axioms. However, the fact that he does not make use of the word “isomorphism” is remarkable, as the expression was common currency in group theory already in the nineteenth century. The word “isomorphism” is brought to bear for the ﬁrst time in the deﬁnition of categoricity in Huntington (1906–1907). There he says that “special attention may be called to the discussion of the notion of isomorphism between two systems, and the notion of a suﬃcient, or categorical, set of postulates.” Indeed, on p. 26 (1906–1907), the notion of two systems being isomorphic with respect to addition and multiplication is introduced. We are now very close to the general notion of isomorphism between arbitrary systems satisfying the same set of axioms. The ﬁrst use of the notion of isomorphism between arbitrary systems we have been able to ﬁnd is Bôcher (1904, 128), who claims to have generalized the notion of isomorphism familiar in group theory. Weyl (1910) also gives the deﬁnition of isomorphism between systems in full generality. Second, there is a certain ambiguity between deﬁning categoricity as the property of admitting only one model (up to isomorphism) and conﬂating the notion with a consequence of it, namely, what we would now call semantical completeness.13 Veblen, however, rightly states that in the case of a categorical theory, further axioms would be redundant even if they were not deducible from the axioms by a ﬁnite number of inferences.

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Third, the distinction hinted at between what is derivable in a ﬁnite number of steps and what follows logically displays a certain awareness of the diﬀerence between a semantical notion of consequence and a syntactical notion of derivability and that the two might come apart. However, Veblen does not elaborate on the issue. Finally, later in the section Veblen claims that the notion of categoricity is also expressed by Hilbert’s axiom of completeness as well as by Huntington’s notion of suﬃciency. In this he reveals an inaccurate understanding of Hilbert’s completeness axiom and of its consequences. Baldus (1928) is devoted to showing the noncategoricity of Hilbert’s axioms for absolute geometry even when the completeness axiom is added. It is, however, true that in the presence of all the other axioms, the system of geometry presented by Hilbert is categorical (see Awodey and Reck 2002).

1.5. Truth in a Structure These developments have relevance also for the discussion of the notion of truth in a structure. In his inﬂuential paper (1986), Hodges raises several historical issues concerning the notion of truth in a structure, which can now be made more precise. Hodges is led to investigate some of the early conceptions of structure and interpretation with the aim of ﬁnding out why Tarski did not deﬁne truth in a structure in his early articles. He rightly points out that algebraists and geometers had been studying “Systeme von Dingen” (systems of objects), that is, what we would call structures or models (on the emergence of the terminology, see itinerary VIII). Thus, for instance, Huntington in (1906–1907) describes the work of the postulate theorist in algebra as being the study of all the systems of objects satisfying certain general laws: “From this point of view our work becomes, in reality, much more general than a study of the system of numbers; it is a study of any system which satisﬁes the conditions laid down in the general laws of §1.”14 Hodges then pays attention to the terminology used by mathematicians of the time to express that a structure A obeys some laws and quotes Skolem (1933) as one of the earliest occurrences where the expression “true in a structure” appears.15 However, here we should point out that the notion of a proposition being true in a system is not unusual during the period. For instance, in Weyl’s (1910) deﬁnition of isomorphism, we read that if there is an isomorphism between two systems, “there is also such a unique correlation between the propositions true with respect to one system and those true with respect to the other, and we can, without falling into error, identify the two systems outright” (Weyl 1910, 301). Moreover, although it is usual in Peano’s school and among the American postulate theorists to talk about a set of postulates being “satisﬁed” or “veriﬁed” in a system (or by an interpretation), without any further comments, sometimes we are also given a clariﬁcation that shows that they were willing to use the notion of truth in a structure. A few examples will suﬃce.

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Let us look at what might be the ﬁrst application of the method for providing proofs of independence. Peano in “Principii di geometria logicamente esposti” (1889) has two signs, 1 (for point) and c ε ab (c is a point internal to the segment ab). Then he considers three categories of entities with a relation deﬁned between them. Finally he adds: Depending on the meaning given to the undeﬁned signs 1 and c ε ab, the axioms might or might not be satisﬁed. If a certain group of axioms is veriﬁed, then all the propositions that are deduced from them will also be true, since the latter propositions are only transformations of those axioms and of those deﬁnitions. (Peano 1889, 77–78) In 1900, Pieri explains that the postulates, just like all conditional propositions are neither true nor false: they only express conditions that can sometimes be veriﬁed and sometimes not. Thus for instance, the equality (x + y)2 = x2 + 2xy + y 2 is true, if x and y are real numbers and false in the case of quaternions (giving for each hypothesis the usual meaning to +, ×, etc.). (Pieri 1901, 388–389) In 1906, Huntington: The only way to avoid this danger [of using more than is stated in the axioms] is to think of our fundamental laws, not as axiomatic propositions about numbers, but as blank forms in which the letters a, b, c, etc. may denote any objects we please and the symbols + and × any rules of combination; such a blank form will become a proposition only when a deﬁnite interpretation is given to the letters and symbols—indeed a true proposition for some interpretations and a false proposition for others. . . From this point of view our work becomes, in reality, much more general than a study of the system of numbers; it is a study of any system which satisﬁes the conditions laid down in the general laws of §1. (Huntington 1906–1907, 2–3)16 In short, it seems that the expression “a system of objects veriﬁes a certain proposition or a set of axioms” is considered to be unproblematic at the time, and it is often read as shorthand for a sentence, or a set of sentences, being true in a system. Of course, this is not to deny that in light of the philosophical discussion emerging from non-Euclidean geometries, a certain care was exercised in talking about “truth” in mathematics, but the issue is resolved exactly by the distinction between axioms and postulates. Whereas the former had been taken to be true tout court, the postulates only make a demand, which might be satisﬁed or not by particular system of objects (see also on the distinction, Huntington 1911, 171–172).

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2. Itinerary II. Bertrand Russell’s Mathematical Logic 2.1. From the Paris Congress to the Principles of Mathematics 1900–1903 At the time of the Paris congress, Russell was mainly familiar with the algebra of logic tradition. He certainly knew the works of Boole, Schröder, and Whitehead. Indeed, the earliest drafts of The Principles of Mathematics (1903; POM for short) are based on a logic of part-whole relationship that was closely related to Boole’s logical calculus. He also had already realized the importance of relations and the limitations of a subject-predicate approach to the analysis of sentences. This change was a central one in his abandonment of Hegelianism17 and also led him to the defense of absolute position in space and time against the Leibnizian thesis of the relativity of motion and position, which was the subject of his talk at the International Congress of Philosophy, held in Paris in 1900. However, he had not yet read the works of the Italian school. The encounter with Peano and his school in Paris was of momentous importance for Russell. He had been struggling with the problems of the foundation of mathematics for a number of years and thought that Peano’s system had ﬁnally shown him the way. After returning from the Paris congress, Russell familiarized himself with the publications of Peano and his school, and it became clear to him that “[Peano’s] notation aﬀorded an instrument of logical analysis such as I had been seeking for years” (Russell 1967, 218). In Russell’s autobiography, he claims that “the most important year of my intellectual life was the year 1900 and the most important event in this year was my visit to the International Congress of Philosophy in Paris” (1989, 12). One of the ﬁrst things Russell did was to extend Peano’s calculus with a worked-out theory of relations and this allowed him to develop a large part of Cantor’s work in the new system. This he pursued in his ﬁrst substantial contribution to logic (Russell 1901b, 1902b), which constitutes a bridge between the theory of relations developed by Peirce and Schröder and Peano’s formalization of mathematics. At this stage Russell thinks of relations intensionally, that is, he does not identify them with sets of pairs. The notion of relation is taken as primitive. Then the notion of the domain and co-domain of a relation, among others, are introduced. Finally, the axioms of his theory of relations state, among other things, closure properties with respect to the converse, the complement, the relative product, the union, and the intersection (of relations or classes thereof). He also deﬁnes the notion of function in terms of that of relation (however, in POM they are both taken as primitive). In this work, Russell treats natural numbers as deﬁnable, which stands in stark contrast to his previous view of number as an indeﬁnable primitive. This led him to the famous deﬁnition of “the cardinal number of a class u” as “the class of classes similar to u.” Russell arrived at it independently of Frege, whose deﬁnition was similar, but he was apparently inﬂuenced by Peano, who discussed such a deﬁnition in 1901 without, however, endorsing it. In any case, Peano’s inﬂuence

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is noticeable in Russell’s abandonment of the Boolean leanings of his previous logic in favor of Peano’s mathematical logic. Russell now accepted, except for a few changes, Peano’s symbolism. One of Peano’s advances had been a clear distinction between sentences such as “Socrates is mortal” and “All men are mortal,” which were previously conﬂated as being of the same structure. Despite the similar surface structure, the ﬁrst one indicates a membership relation between Socrates and the class of mortals, whereas the second indicates an inclusion between classes. In Peano’s symbolism we have s ε φ(x) for the ﬁrst and φ(x) ⊃x ψ(x) for the second. With this distinction Peano was able to deﬁne the relation of subsumption between two classes by means of implication. In a letter to Jourdain in 1910, Russell writes: Until I got hold of Peano, it had never struck me that Symbolic Logic would be any use for the Principles of mathematics, because I knew the Boolean stuﬀ and found it useless. It was Peano’s ε, together with the discovery that relations could be ﬁtted into his system, that led me to adopt Symbolic Logic. (Grattan-Guinness 1977, 133) What Peano had opened for Russell was the possibility of considering the mathematical concepts as deﬁnable in terms of logical concepts. In particular, an analysis in terms of membership and implication is instrumental in accounting for the generality of mathematical propositions. Russell’s logicism ﬁnds its ﬁrst formulation in a popular article written in 1901 where he claims that all the indeﬁnables and indemonstrables in pure mathematics stem from general logic: “All pure mathematics—Arithmetic, Analysis, and Geometry—is built up of the primitive ideas of logic, and its propositions are deduced from the general axioms of logic” (1901a, 367). This is the project that informed the Principles of Mathematics (1903). The construction of mathematics out of logic is carried out by ﬁrst developing arithmetic through the deﬁnition of the cardinal number of a class as the class of classes similar to it. Then the development of analysis is carried out by deﬁning real numbers as sets of rationals satisfying appropriate conditions. (For a detailed reconstruction see, among others, Vuillemin 1968, RodriguezConsuegra 1991, Landini 1998, Grattan-Guinness 2000.) The main diﬃculty in reconstructing Russell’s logic at this stage consists in the presence of logical notions mixed with linguistic and ontological categories (denotation, deﬁnition). Moreover, Russell does not present his logic by means of a formal language. After Russell ﬁnished preparing POM, he also began studying Frege with care (around June 1902). Under his inﬂuence, Russell began to notice the limitations in Peano’s treatment of symbolic logic, such as the lack of diﬀerent symbols for class union and the disjunction of propositions, or material implication and class inclusion. Moreover, he changed his symbolism for universal and existential quantiﬁcation to (x)f (x) and (Ex)f (x). He adopted from Frege the symbol for the assertion of a proposition. His letter to Frege of June 16, 1902, contained the famous paradox, which had devastating consequences for Frege’s system:

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Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. Therefore we must conclude that w is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a deﬁnable collection does not form a totality. (Russell 1902a, 125) The ﬁrst paradox does not involve the notion of class but only that of predicate. Let Imp(w) stand for “w cannot be predicated of itself,” that is, ∼w(w). Now we ask: Is Imp(Imp) true or ∼Imp(Imp)? From either one of the possibilities the opposite follows. However, what is known as Russell’s paradox is the second one oﬀered in the letter to Frege. In his work Grundgesetzte der Arithmetik (Frege 1893, 1903), Frege had developed a logicist project that aimed at reconstructing arithmetic and analysis out of general logical laws. One of the basic assumptions made by Frege (Basic Law V) implies that every propositional function has an extension, where extensions are a kind of object. In modern terms we could say that Frege’s Basic Law V implies that for any property F (x) there exists a set y = { x : F (x) }. Russell’s paradox consists in noticing that for the speciﬁc F (x) given by x ∈ / x, Frege’s principle leads to asserting the existence of the set y = {x : x ∈ / x }. Now if one asks whether y ∈ y or y ∈ / y from either one of the assumptions one derives the opposite conclusion. The consequences of Russell’s paradox for Frege’s logicism and Frege’s attempts to cope with it are well known, and we will not recount them here (see Garciadiego 1992). Frege’s proposed emendation to his Basic Law V, while consistent, turns out to be inconsistent as soon as one postulates that there are at least two objects (Quine 1955a).18 Extensive research on the development that led to Russell’s paradox has shown that Russell already obtained the essentials of his paradox in the ﬁrst half of 1901 (Garciadiego 1992; Moore 1994) while working on Cantor’s set theory. Indeed, Cantor himself already noticed that treating the cardinal numbers (resp., ordinal numbers) as a completed totality would lead to contradictions. This led him to distinguish, in letters to Dedekind, between “consistent multiplicities,” that is, classes that can be considered as completed totalities, from “inconsistent multiplicities,” that is, classes that cannot, on pain of contradiction, be considered as completed totalities. Unaware of Cantor’s distinction between consistent and inconsistent multiplicities Russell in 1901 convinced himself that Cantor had “been guilty of a very subtle fallacy” (1901a, 375). His reasoning was that the number of all things is the greatest of all cardinal numbers. However, Cantor proved that for every cardinal number there is a cardinal number strictly bigger than it. Within a few months this conundrum led to Russell’s paradox. In POM we ﬁnd, in addition to the two paradoxes we have discus

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The Development of Modern Logic

Edited by

Leila Haaparanta

2009

Oxford University Press, Inc., publishes works that further Oxford University’s objective of excellence in research, scholarship, and education. Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With oﬃces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam

Copyright © 2009 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data The development of modern logic / edited by Leila Haaparanta. p. cm. Includes bibliographical references. ISBN 978-0-19-513731-6 1. Logic—History. I. Haaparanta, Leila, 1954– BC15.D48 2008 160.9—dc22 2008016767

9 8 7 6 5 4 3 2 1 Printed in the United States of America on acid-free paper

Preface

This volume is the result of a long project. My work started sometime in the 1990s, when Professor Simo Knuuttila urged me to edit, together with a few colleagues, a volume on the history of logic from ancient times to the end of the twentieth century. Even if the project was not realized in that form, I continued with the plan and started to gather together scholars for a book project titled The Development of Modern Logic, thus making a reference to the famous book by William and Martha Kneale. Unlike that work, the new volume was meant to be written by a number of scholars almost as if it had been written by one scholar only. I decided to start with thirteenth-century logic and come up with quite recent themes up to 2000, hence, to continue the history written in The Development of Logic. My intention was to ﬁnd a balance between the chronological exposition and thematic considerations. The philosophy of modern logic was also planned to be included; indeed, at the beginning the book had the subtitle “A Philosophical Perspective,” which was deleted at the end, as the volume reached far beyond that perspective. The collection of articles is directed to philosophers, even if some chapters include a number of technical details. Therefore, when it is used as a textbook in advanced courses, for which it is also planned, those details are recommended reading to students who wish to develop their skills in mathematical logic. In 1998, we had a workshop of the project with most of the contributors present. It was a ﬁne beginning, organized by the Department of Philosophy at the University of Helsinki and by the Philosophical Society of Finland. We got ﬁnancial support from the Academy of Finland and from the Finnish Cultural Foundation, which I wish to acknowledge. I moved to the University of Tampere in the fall of 1998. Unlike logic perhaps, life sometimes turns out to be chaotic. As we were a large group, it was no surprise that various personal and professional matters inﬂuenced the process of writing and editing. Still, we

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Preface

happily completed the volume, which became even larger than was originally intended. I wish to thank the contributors, from whom I have learned a great deal during the editorial process. It has been a pleasure to cooperate with them. Renne Pesonen and Risto Vilkko kindly assisted me with the editorial work. I am very grateful to my colleagues for useful pieces of advice. There are so many who have been helpful that it is impossible to name them all. My special thanks are due to Auli Kaipainen and Jarmo Niemelä, who prepared the camera-ready text for publication. Jarmo Niemelä also assisted me with compiling the index. I wish to thank Peter Ohlin, editor at Oxford University Press, who has been extremely helpful during the process. I have beneﬁted considerably from the help of my editors, Stephanie Attia and Molly Wagener, of Oxford University Press. The ﬁnancial support given by the Academy of Finland is gratefully acknowledged. I have done the editorial work at the University of Tampere, ﬁrst at the Department of Mathematics, Statistics and Philosophy and then at the Department of History and Philosophy. Finally, I wish to express my deep gratitude to my mother and to my husband, whose support and encouragement have been invaluable. L. H.

Contents

Contributors

ix

1. Introduction 3 Leila Haaparanta 2. Late Medieval Logic 11 Tuomo Aho and Mikko Yrjönsuuri 3. Logic and Philosophy of Logic from Humanism to Kant Mirella Capozzi and Gino Roncaglia

78

4. The Mathematical Origins of Nineteenth-Century Algebra of Logic Volker Peckhaus 5. Gottlob Frege and the Interplay between Logic and Mathematics Christian Thiel 6. The Logic Question During the First Half of the Nineteenth Century Risto Vilkko 7. The Relations between Logic and Philosophy, 1874–1931 Leila Haaparanta

159 196 203

222

8. A Century of Judgment and Inference, 1837–1936: Some Strands in the Development of Logic 263 Göran Sundholm 9. The Development of Mathematical Logic from Russell to Tarski, 1900–1935 318 Paolo Mancosu, Richard Zach, and Calixto Badesa

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Contents

10. Set Theory, Model Theory, and Computability Theory Wilfrid Hodges 11. Proof Theory of Classical and Intuitionistic Logic Jan von Plato

471

499

12. Modal Logic from Kant to Possible Worlds Semantics Tapio Korte, Ari Maunu, and Tuomo Aho

516

Appendix to Chapter 12: Conditionals and Possible Worlds: On C. S. Peirce’s Conception of Conditionals and Modalities Risto Hilpinen

551

13. Logic and Semantics in the Twentieth Century 562 Gabriel Sandu and Tuomo Aho 14. The Philosophy of Alternative Logics Andrew Aberdein and Stephen Read

613

15. Philosophy of Inductive Logic: The Bayesian Perspective Sandy Zabell 16. Logic and Linguistics in the Twentieth Century Alessandro Lenci and Gabriel Sandu 17. Logic and Artiﬁcial Intelligence Richmond H. Thomason

724

775

848

18. Indian Logic 903 J. N. Mohanty, S. R. Saha, Amita Chatterjee, Tushar Kanti Sarkar, and Sibajiban Bhattacharyya Index

963

Contributors

Andrew Aberdein, Humanities and Communication, Florida Institute of Technology, [email protected]ﬁt.edu. Tuomo Aho, Department of Philosophy, University of Helsinki, [email protected]ﬁ. Calixto Badesa, Department of Logic, History and Philosophy of Science, University of Barcelona, [email protected] Sibajiban Bhattacharyya, Department of Philosophy, University of Calcutta (died in 2007). Mirella Capozzi, Department of Philosophical and Epistemological Studies, University of Rome “La Sapienza,” [email protected] Amita Chatterjee, Department of Philosophy and Center for Cognitive Science, Jadavpur University. Leila Haaparanta, Department of History and Philosophy, University of Tampere, [email protected]ﬁ. Risto Hilpinen, Department of Philosophy, University of Miami, [email protected] Wilfrid Hodges, Department of Mathematics, Queen Mary, University of London, [email protected] Tapio Korte, Department of Philosophy, University of Turku, [email protected]ﬁ. Alessandro Lenci, Department of Computational Linguistics, University of Pisa, [email protected] ix

x

Contributors

Paolo Mancosu, Department of Philosophy, University of California, Berkeley, [email protected] Ari Maunu, Department of Philosophy, University of Turku, [email protected]ﬁ. J. N. Mohanty, Department of Philosophy, Temple University, Philadelphia. Volker Peckhaus, Department of Humanities, University of Paderborn, [email protected] Stephen Read, Department of Philosophy, University of St. Andrews, [email protected] Gino Roncaglia, Department of Humanities, University of Tuscia, Viterbo, mc3430[email protected] S. R. Saha, Department of Philosophy, Jadavpur University. Gabriel Sandu, Department of Philosophy, University of Helsinki; Department of Philosophy, Sorbonne, [email protected]ﬁ. Tushar Kanti Sarkar, Department of Philosophy, Jadavpur University. Göran Sundholm, Department of Philosophy, University of Leiden, [email protected] Christian Thiel, Department of Philosophy, University of Erlangen, [email protected] Richmond H. Thomason, Department of Philosophy, University of Michigan, [email protected] Risto Vilkko, Department of Philosophy, University of Helsinki, [email protected]ﬁ. Jan von Plato, Department of Philosophy, University of Helsinki, [email protected]ﬁ. Mikko Yrjönsuuri, Department of Social Sciences and Philosophy, University of Jyväskylä, [email protected]ﬁ. Sandy Zabell, Department of Mathematics, Northwestern University, [email protected] Richard Zach, Department of Philosophy, University of Calgary, [email protected]

The Development of Modern Logic

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1

Introduction Leila Haaparanta

1. On the Concept of Logic When we state in everyday language that a person’s logic fails, we normally mean that the rules of valid reasoning, which ought to guide our thinking, are not in action for some reason. The word “logic” of our everyday language can usually be analyzed as “the collection of rules that guide correct thinking or reasoning.” That collection is assumed to be known naturally; a rational human being follows those rules in normal circumstances, even if he or she could not formulate them, that is, express them in language. When the word “logic” (in Greek logos “word,” “reason”) refers to one subﬁeld of philosophy or of mathematics, it usually means the discipline concerning valid reasoning or the science that studies that kind of reasoning. In his logical studies, Aristotle (384–322 b.c.) considered inferences, which are called syllogisms. They consisted of two premises and a conclusion, and the validity of the argument of a syllogistic form was determined by the structure of the argument. If the premises of a syllogism were true, the conclusion was also true. According to Aristotle, the basic form of a judgment is “A is B,” where “A” is a subject and “B” is a predicate. Forms of judgments include “Every A is B,” “No A is B,” “Some A is B,” and “Some A is not B.” Unlike Aristotelian logic, modern formal logic is called symbolic or mathematical, as it studies valid reasoning in artiﬁcial languages. Until the nineteenth century logic was mainly Aristotelian. Following Aristotle, the main focus was on judgments that consisted of a subject and a predicate and that included such words as “every,” “some,” and “is” in addition to letters corresponding to the subject and the predicate. The Stoics, for their part, were 3

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interested in what is nowadays called propositional logic, in which the focus is on such words as “not,” “and,” “or,” and “if–then.” It was not until the nineteenth century that symbolic logic, which had its model in mathematics, became a serious rival of Aristotelian logic. The grammatical analysis of judgments was challenged in the late nineteenth century by logicians who took the model of analysis from mathematics. The words “function” and “argument” became part of the vocabulary of logic, and predicates that expressed relations as well as quantiﬁers were included in that vocabulary. In the new logic, which was mostly developed by Gottlob Frege (1848–1925) and Charles Peirce (1939–1914) and which was codiﬁed in Principia Mathematica (1910, 1912, 1913), written by A. N. Whitehead (1861–1947) and Bertrand Russell (1872–1970), the rules of logical inference received a new treatment, as the pioneers of modern logic tried to give an exact formulation of those rules in an artiﬁcial language. Except for the collection of the rules of valid reasoning and the discipline or the science that focuses on those rules, the word “logic” means a speciﬁc language that fulﬁlls certain requirements of preciseness. It also means a ﬁeld of research that focuses on such a language or such languages. Since the seventeenth century, it has been typical of the ﬁeld called logic to construct and study a formal language or formal languages called logic or logics. The old Aristotelian logic heavily relied on natural language. Aristotle and his followers thought that natural language reﬂects the forms of logical inference and other logical relations, even the form of reality. The pioneers of modern logic sought to construct an artiﬁcial language that would be more precise than natural languages. In the twentieth century those languages called logics have been used as models of natural languages; hence, modern logic that rejected the grammatical analysis of judgments has, among other things, served as a tool in linguistic research. It is important to note that the pioneers of modern logic, such as G. W. Leibniz (1646–1716) and Frege, did not intend to present any tools of studying natural languages; they wished to construct a symbolic language that would overcome natural language as a medium of thought in being more precise and lacking ambiguities that are typical of natural language. As the views of the tasks and the aims of logic have varied in history, we may wonder whether Aristotle and the representatives of modern logic, for example, Frege, were at all interested in the same object of research and whether it is possible to talk about the same ﬁeld of research. In spite of diﬀerences, we may name a few common interests whose existence justiﬁes the talk about research called logic and the history of that ﬁeld. In each period in the history of logic, researchers called logicians have been interested in concepts or terms that are not empirical, that is, whose meanings are not, or at least not incontestably, based on sensuous experience, and that can be called logical concepts or terms. What concepts or terms have been regarded as logical has varied in the history, but interest in them unites Aristotle, William of Ockham, Immanuel Kant, and Frege as well as logicians in the twentieth and twenty-ﬁrst centuries. Other

Introduction

5

points of interest have been the so-called laws of thought, for example, the law of noncontradiction and the law of excluded middle. A third theme that unites logicians of diﬀerent times is the question of the validity of reasoning. In several chapters of the present volume, the question concerning the nature and the scope of logic is discussed in view of the period and the logicians that are introduced to the reader.

2. What Is Modern Logic? The starting point of modern logic is presented in textbooks in various ways depending on what features are regarded as the characteristics of modernity. Some say modern logic started together with modern philosophy in the late Middle Ages, while others think that it started in the seventeenth century with Leibniz’s logic. Still others argue that the beginning of modern logic was 1879, when Frege’s Begriﬀsschrift appeared. If the beginning of modern logic is dated to the seventeenth century, its pioneers include Leibniz, Bernard Bolzano (1781–1848), Augustus De Morgan (1806–1871), George Boole (1815–1864), John Venn (1834–1923), William Stanley Jevons (1835–1882), Frege, Peirce, Ernst Schröder (1841–1902), Giuseppe Peano (1858–1932), and Whitehead and Russell. Unlike many contemporary logicians, modern logicians believed that there is one and only one true logic. Leibniz was the most important of those thinkers who argued that the terms of our natural language do not correspond to the objects of the world in a proper way and that therefore we have to construct a new language, which mirrors the world correctly. Following Leibniz, modern logicians sought to construct an artiﬁcial language that would be better than natural languages. If we think that this kind of eﬀort is an important feature of modern logic, then we may say that modern logic started with Leibniz. The idea of calculus has also been an important feature of modern logic. Logic has been considered a system which consists of logical and nonlogical vocabulary, formation rules, and transformation rules; the formation rules tell us what kind of sequences of symbols are well formed, and the transformation rules are the basis on which logical reasoning is performed like calculating. Many early pioneers of modern logic relied on the grammatical subjectpredicate analysis in analyzing sentences that was also part of traditional logic, as mentioned above. It was not until Frege’s logic that this division was rejected. The division between arguments and functions thus became central in logic. Frege also stressed that it was the distinction between individuals and concepts that he wants to respect. If we stress that feature, we may say that the philosophical ideas of modern logic can be found in medieval nominalists, but that they did not become codiﬁed in formal languages until the latter half of the nineteenth century in Frege’s and Peirce’s discoveries. Those two logicians also made quantiﬁers into the basic elements of logic. As modern thinkers, many late medieval philosophers were interested in individuals, but

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The Development of Modern Logic

the distinction between an individual and a concept was not taken into account in logic until Frege’s and Peirce’s discoveries. Frege regarded his logic as an axiomatic theory. That feature can also be considered a typical feature of modern logic. As was said before, it is often thought that Frege’s Begriﬀsschrift gave birth to modern logic. In that book there were many logical discoveries, such as the theory of quantiﬁcation and the argument-function analysis. Frege’s book was both philosophical and mathematical. Later, in the ﬁrst volume of his Grundgesetze der Arithmetik (1893), Frege states that he is likely to have few readers; all those mathematicians stop reading who, when seeing the words “concept,” “relation,” and “judgment” think: “It is metaphysics, we do not read it,” and those philosophers stop reading, who, when seeing a formula, shout: “It is mathematics, we do not read it” (p. xii). Charles Peirce discovered the logic of relatives in the 1870s. That logic was inspired by Boole’s algebra of logic and De Morgan’s theory of relations. Peirce’s articles “The Logic of Relatives” (1883) and “On the Algebra of Logic: A Contribution to the Philosophy of Notation” (1885) contain the ﬁrst formulation of his theory of quantiﬁcation that he calls his general algebra of logic. Peirce’s algebra diﬀered from that of Boole’s especially in that Peirce introduced signs that refer to individuals in addition to signs that signify relations. Second, he introduced the quantiﬁers “all” and “some.” Frege only used the sign for generality and deﬁned existence by means of generality and negation. Both the logicians rejected Boole’s idea that judgments are formed by combining subjects and predicates. Frege and Peirce, who made their important discoveries independently of each other, Peirce maybe with his group of students and Frege alone, had common features. They were both philosophers and mathematicians and could combine philosophical ideas with technical novelties in their logical thought. Frege and Peirce both invented a notation for quantiﬁers and quantiﬁcation theory almost simultaneously, independently of each other. Therefore they can be regarded as the principal founders of modern logic. However, as many scholars have emphasized, most notably Jean van Heijenoort in his paper “Logic as Calculus and Logic as Language” (1967), Jaakko Hintikka in his papers “Frege’s Hidden Semantics” (1979) and “Semantics: A Revolt Against Frege” (1981), and Warren Goldfarb in his article “Logic in the Twenties: The Nature of the Quantiﬁer” (1979), the two logicians seem to be far apart philosophically. The division between the two traditions to which the logicians belong has also been emphasized by a number of authors of the present volume. The distinction between the two conceptions of logic, namely, seeing logic as language versus seeing it as calculus, has been suggested from the perspective of twentieth-century developments, but the origin of the division has been located in nineteenth-century logic. Diﬀerent interpretations of the history of logic follow depending on how the distinction is understood. According to van Heijenoort, Hintikka, and Goldfarb, those who stressed the idea of logic as language thought that logic speaks about one single world. It is certain

Introduction

7

that Frege held that position. He thought that there is one single domain of discourse for all quantiﬁers, as he assumed that any object can be the value of an individual variable and any function must be deﬁned for all objects. On the other hand, those who supported the view that logic is a calculus gave various interpretations or models for their formal systems. That was Boole’s and his followers’ standpoint. Several other features of the two traditions are mentioned in the chapters of the present volume. The volume titled Studies in the Logic of Charles Sanders Peirce (1997) introduces another pair of traditions, which are mathematical logic and algebraic logic and which are also touched upon in the present collection of articles. Ivor Grattan-Guinness states in his contribution to the volume on Peirce that the phrase “mathematical logic” was introduced by De Morgan in 1858 but that it served to distinguish logic using mathematics from “philosophical logic,” which was also a term used by De Morgan. However, in Grattan-Guinness’s terminology, De Morgan’s logic was part of the algebraic tradition; using algebraic methods in logic would be typical of what he calls algebraic logic. The most common phrase used in the nineteenth century was “the algebra of logic” or sometimes “logical algebra.” In the ﬁgure which Grattan-Guinness presents to us, Boole, De Morgan, Peirce, and Schröder belong to the tradition of algebraic logic, while Peano and Russell belong to the tradition of mathematical logic. It seems that many of those who belong to the tradition of logic as calculus belong to the tradition of algebraic logic in Grattan-Guinness’s division, and that many of those who think that logic is a language belong to what Grattan-Guinness calls the tradition of mathematical logic. Grattan-Guinness gives us a few typical features of the two traditions that he discusses. In algebraic logic, laws were stressed, while in mathematical logic axioms were emphasized. Moreover, he states that in mathematical logic, especially in the logicist version represented by Russell, logic was held to contain all mathematics, while in algebraic logic it was maintained that logic had some relationship with mathematics. In Grattan-Guinness’s view, algebraic logic used part-whole theory and relied on a basically extensionalist conception of a collection, while in mathematical logic the theory of collections was based on Cantor’s Mengenlehre. In addition, there was, in his view, an important diﬀerence between the traditions concerning quantiﬁcation; the interpretation of the universal and existential cases as inﬁnite conjunctions and disjunctions with the algebraic analogies of inﬁnite products and sums was typical of the algebraic tradition. Grattan-Guinness also notes that the questions addressed in mathematical logic were more speciﬁc than those addressed in algebraic logic. Frege’s and Peirce’s logical views are discussed in several chapters of the present volume. Many contributors also touch on the more general question concerning the borderline between traditional and modern logic, the divisions between the traditions of modern logic, and the shift from the modern logic of the late nineteenth century and the early twentieth century till twentiethcentury logic. The periods of Western logic that are studied in the present

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The Development of Modern Logic

collection of articles extend from the thirteenth century to the end of the twentieth century. Unlike the rest of the contributions, the chapter on Indian logic covers several schools whose history reaches far back in the history but which are also living traditions in contemporary Indian logic.

3. Logic and the Philosophy of Logic Besides the term “logic,” the terms “philosophical logic” and “philosophy of logic” have various uses. Philosophy of logic can be understood as a subﬁeld of philosophy that studies the philosophical problems raised by logic, including the problem concerning the nature and the scope of logic. Those problems also include metaphysical, or ontological, and epistemological questions of logic, problems related to the speciﬁc features of logical formal systems (e.g., related to the basic vocabulary of logic) and logical validity, questions concerning the nature of propositions, judgments, and sentences, as well as theories of truth and truth-functions, and the questions concerning modal concepts and the alternatives of classical logic, which some call by the name “deviant logics.” The term “philosophical logic” is often used as a synonym of “philosophy of logic”; occasionally it means the same as “intensional logic,” or it is used as an opposite to “mathematical logic.” By metalogic, one normally means the study of the formal properties of logical systems, such as consistency and completeness, and thus distinguishes it from the philosophy of logic, which studies their philosophical aspects. The present volume deals with the history of modern logic and pays attention both to the core area of logic and to the philosophy of logic. Such terms as “classical logic,” “modal logic,” “alternative logics,” and “inductive logic” are also used and explained in the chapters of the volume. The variety of logics raises the problem of demarcation that is essential to the philosophy of logic: which formal systems belong to the objects of logical research, and which ought one to exclude from the ﬁeld of logic? For example, the program of logicism, which was supported by Frege, among others, was a position taken in the discussion concerning the demarcation of logic. Logic and philosophy have complicated relations. Nowadays logical tools are often used as the methods of philosophy. Logical discoveries have also been motivated by philosophical views, and philosophers have changed their opinions because of logical discoveries. Logic can be said to have a philosophical basis, and likewise there are philosophical doctrines that rely on developments of logic. The present collection of articles studies some of those relations. To some extent, it also pays attention to the relations between logic and mathematics and logic and linguistics. Logic and rationality are often tied together, but the concept of rationality has many uses in everyday language and in philosophical discussion. We talk about logical or argumentative rationality and refer to one’s ability to reason or to give arguments, and we also think that one who is rational is able to

Introduction

9

evaluate various views critically and independently of authorities; in this latter meaning, logic is considered to play a signiﬁcant role. Moreover, rationality is both theoretical and practical, the latter form of rationality being related to a person’s actions, and philosophers also tend to regard one’s ability to control one’s volitional and emotional impulses as a sign of rationality. There is no one concept or “essence” of reason that can be detected in philosophical or in everyday discussion. However, what we can ﬁnd in most uses of the concept is the general idea of control (control of thought, actions, passions, etc.), which is also central in logical rationality. Even if rationality as control or as rule-following seems to be crucially important, rationality as a faculty of judgment is also in everyday use in the practice of logicians as in all science. In the tradition of logic, it has been important both to be able to follow rules or repeat patterns and to be able to evaluate the commands and prohibitions. It is important both to be able to think inside a given system and to be able to evaluate the very system from the outside. The history of modern logic is a history of these two huge projects. Philosophers and logicians have used the volume titled The Development of Logic by William and Martha Kneale (1962) for decades. The ambitious idea behind the present work was to write a book on the development of modern logic that would bring the history of modern logic till the end of the twentieth century and would also pay attention to the philosophy of logic and philosophical logic in modern times. The idea was not to bring about a handbook but a volume that would be as close as possible to a one-author volume, that is, a balanced whole without serious gaps or overlaps. It was taken for granted in the very beginning that that goal cannot be reached in all respects. Each author has chosen his or her style, some wish to give detailed references, others are happier with drawing the main lines of development with fewer details; some express their ideas in many words, while others prefer a concise manner of writing. However, what has been reached is a story that covers a number of themes in the development of modern logic. The history begins with late medieval logic and continues with logic and philosophy of logic from humanism to Kant, that is, with two chapters whose scope is chronologically determined. Chapters 4–7 cover the nineteenth century and early twentieth century in certain respects, namely, they focus on the emergence of symbolic logic in two ways, ﬁrst, by paying attention to the relations between logic and mathematics, second, by emphasizing the connections between logic and philosophy. That discussion is completed by a chapter that focuses on the themes of judgment and inference from 1837 to 1936. The volume contains an extensive chapter of the development of mathematical logic 1900–1935, which is continued by a discussion on main trends in mathematical logic after the 1930s. The subﬁelds of logic that are called modal logic and philosophical logic are discussed in two separate chapters, one dealing with the history of modal logic from Kant until the late twentieth century and the other discussing logic and semantics in the twentieth century. Separate chapters are reserved for the philosophy of alternative logics, for the

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The Development of Modern Logic

philosophical aspects of inductive logic, for the relations between logic and linguistics in the twentieth century, and for the relations between logic and artiﬁcial intelligence. Eastern logic is not covered, but the main schools of Indian logic are presented in the last chapter of the volume. While the former part of the volume is chronologically divided, the chapters of the latter part follow a thematic division.

Note I have used extracts from my article “Peirce and the Logic of Logical Discovery,” originally published in Edward C. Moore (ed.), Charles Peirce and the Philosophy of Science (University of Alabama Press, Tuscaloosa, 1993), 105–118, with the kind permission of University of Alabama Press. The chapter also contains passages from my review article “Perspectives on Peirce’s Logic,” published in Semiotica 133 (2001), 157–167, which appear here with the kind permission of Mouton de Gruyter.

References Frege, Gottlob. [1879] 1964. Begriﬀsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. In Frege (1964), 1–88. Frege, Gottlob. 1964. Begriﬀsschrift und andere Aufsätze, ed. Ignacio Angelelli. Hildesheim: Georg Olms. Frege, Gottlob. 1893. Grundgesetze der Arithmetik, begriﬀsschriftlich abgeleitet, I. Band. Jena: Verlag von H. Pohle. Goldfarb, Warren D. 1979. Logic in the Twenties: the Nature of the Quantiﬁer. Journal of Symbolic Logic 44: 351–368. Grattan-Guinness, Ivor. 1997. Peirce between Logic and Mathematics. In Studies in the Logic of Charles Sanders Peirce, ed. Nathan Houser, Don D. Roberts, and James Van Evra, 23–42. Bloomington: Indiana University Press. Hintikka, Jaakko. 1979. Frege’s Hidden Semantics. Revue Internationale de Philosophie 33: 716–722. Hintikka, Jaakko. 1981. Semantics: A Revolt against Frege. In Contemporary Philosophy, vol. 1, ed. Guttorm Fløistad, 57–82. The Hague: Martinus Nijhoﬀ. Kneale, William and Martha Kneale. [1962] 1984. The Development of Logic. Oxford: Clarendon Press. Peirce, Charles Sanders. 1931–1935. Collected Papers of Charles Sanders Peirce, vols. 1–6, ed. Charles Hartshorne and Paul Weiss. Cambridge, Mass.: Harvard University Press. van Heijenoort, Jean. 1967. Logic as Calculus and Logic as Language. Synthese 17: 324–330. Whitehead, Alfred North, and Bertrand Russell. [1910, 1912, 1913] 1925–1927. Principia Mathematica I–III. London: Cambridge University Press.

2

Late Medieval Logic Tuomo Aho and Mikko Yrjönsuuri

1. The Intellectual Role and Context of Logic Our aim is to deal with medieval logic from the time when it ﬁrst had full resources for systematic creative contributions onward. Even before that stage there had been logical research and important logicians. The most original of them, Abelard, achieved highly signiﬁcant results despite having only a very fragmentary knowledge of ancient logic. However, we shall concentrate on the era when the ancient heritage was available and medieval logic was able to add something substantial to it, even to surpass it in some respects. A characterization such as this cannot be adequately expressed with years or by conventional period denominations; we hope though that the grounds for drawing boundaries will become clearer during the course of our story.

1.1. Studies It was characteristic of later medieval logic that it was pursued as an academic discipline, as a major component in an organized whole of studies. Indeed, after the Middle Ages, logic has never been allotted so large a share in the activities of the universities. Moreover, logic was connected to certain classical texts that were seen as natural foundations of this science. Thus, it is reasonable ﬁrst to say something about the system of studies in general and about the nature of these works in particular. Ever since Rome, school teaching had always centered on the trivium of grammar, rhetoric, and dialectic. When schools developed and the most prominent clusters of schools began to turn into universities, these disciplines 11

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The Development of Modern Logic

found their place in the faculty of arts (artes). Dialectic, the art of arguing and reasoning, was largely concerned with logical issues, and was often taken to be the most important art of the trivium. Thus, the outcome was that every student had to take extensive courses in logic. Perhaps the dialectical background can throw some light upon the linguistic and semantic tone of medieval logical thought. The faculty of arts was always much bigger than the higher faculties (theology, law, medicine). If there was a theological faculty in a university, it was associated with advanced studies and required a preliminary education in arts. But philosophical and logical research was pursued by theologians even after proceeding to the higher faculty; in fact, the most competent scholars often preferred the privileged higher faculty. Thus the history of logic must take into account the production of both faculties. Many commentaries on Peter Lombard’s theological Sentences contain important passages on logic, and topics related to logic are often dealt with in the so-called quodlibetal disputations, to mention just two examples. We cannot pay much attention to the history of universities, though we can say that the process of university education started in Italy in the twelfth century, Bologna being the oldest university. Paris, however, was undoubtedly the most important university for philosophy, and it received its oﬃcial statutes in 1215. Paris was a permanent international center for current philosophical and theological discussion. Another place where logical research was often especially popular was Oxford. These were the two capitals of medieval logic, although the center of gravity shifted to Italy in the less innovative period toward the end of the fourteenth century. During the fourteenth century, universities spread to the east and to the north. There were 15 universities in 1300, 30 in 1400, and about 60 in 1500, naturally of very diﬀerent size and quality, though one component of studies was standard everywhere, and that was logic.

1.2. The Growth of Logic Medieval philosophers normally made use of an array of authoritative classical texts, which were taken to be trustworthy, though not infallible. The curriculum was organized around these texts, and very often the problems discussed were put forward as questions of interpretation and explication of the texts. Hence the general breakthrough of Aristotelianism in the thirteenth century represented a great change, establishing Aristotle as the main source of academic studies. But in logic Aristotle had even before that been regarded as the greatest of authors, and anti-Aristotelian reactions did not seriously extend to logic. Rather than being rejected in the Middle Ages, Aristotle’s own work in logic was built upon and developed ever further toward the end of the period. The famous standard translation for most of Aristotle’s texts was that by William Moerbeke. With the logical works the case was diﬀerent: Though

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Moerbeke translated some of them in the 1260s, the authority of the old Roman translation by Boethius (c. 480–524) remained unquestioned. The Organon that late medieval logicians used was the Latin text of Boethius. (For Posterior Analytics, no translation by Boethius is known; its standard rendering was made by James of Venice before 1150.) These translations are actually quite accurate, although written in a very formal and literal idiom. Three concise basic works belonged to the kernel of logic throughout the Middle Ages. These were Aristotle’s own short Categories and De interpretatione, and Porphyry’s introduction, Isagoge. In addition to these, the so-called old logic (logica vetus) used Boethius’s logical works and a few minor ancient texts (by, e.g., Apuleius and Augustine). The shape of logic changed considerably when Aristotle’s complete works of logic became known in the middle of the twelfth century. That opened the way for the new logic, logica nova, and in a relatively short time the corpus of logica vetus was practically replaced by new works. Even Boethius’s treatises on syllogisms fell into disuse. Except for Aristotle and Porphyry, the only work that retained its place was Liber sex principiorum, a treatise explaining the categories that Aristotle himself does not dwell upon. The period of logica nova used as its authoritative corpus all the six works in Aristotle’s Organon: Categories, De interpretatione, Prior Analytics, Posterior Analytics, Topics, and Sophistici elenchi. At ﬁrst, dialecticians were especially fascinated by fallacies and sophisms (Soph. el.), but gradually the investigation turned more toward the formal theory of syllogism (Pr. Anal.). During the thirteenth century, they encountered problems that could not be answered by straightforward Aristotelian principles, and were thus drawn to new ﬁelds of logic. After the introduction of such new subjects, logic came to be called logica moderna, in contrast to logica nova, now called logica antiqua. This way of speaking, however, did not imply any break with the earlier Aristotelian tradition, only an expansion of investigation. The ﬁrst complete handbooks of logica moderna date from the second quarter of the thirteenth century. The earliest known overview is Introductiones in logicam by William of Sherwood from the 1230s, but the greatest success of all was the Tractatus, also called Summulae logicales, by Peter of Spain (probably from the 1240s). This comprehensive work maintained its status as a famous standard textbook throughout the later Middle Ages and the Renaissance, even in the time of printed books. It also served as the source for numerous shorter courses. Similar ambitious textbooks were written by Roger Bacon (Summulae dialectices, 1250s) and Lambert of Auxerre (Logica, 1250s). In a way, these works can be seen as a synthesis of the founding period of logica moderna: On the one hand, they were the ﬁrst systematic presentations of whole logic, on the other hand, they completed the new so-called terminist logic. Simultaneously a more profound philosophical discussion was started by the inﬂuential Robert Kilwardby, who wrote one of the ﬁrst commentaries on Prior Analytics (1240s).

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The Development of Modern Logic

As to the logical content in the overall presentations of logic, signiﬁcant advance comes only much later, in the generation of Walter Burley (c. 1275– 1344) and William Ockham (c. 1285–1347). Ockham’s Summa logicae is from the early 1320s, Burley’s De puritate artis logicae from the late 1320s. These works manifest a turn in logical literature toward new problems and to a more theoretical way of thinking. The greatest representative of the next period is John Buridan (c. 1300–1361?); a comprehensive picture of his teaching in Paris is given in his Summulae de Dialectica. In the latter half of the fourteenth century, logic was already highly technical. In particular, a series of Englishmen distinguished themselves, among them William Heytesbury (d. 1372?), Ralph Strode (d. 1387?), and Richard Lavenham (d. 1399). A kind of summary of this stage is the enormous Logica magna (c. 1400) by the Italian Paul of Venice (c. 1369–1429).

1.3. Non-Latin Traditions Our account will be only about the Latin West. The signiﬁcance of Arabic philosophy must be emphasized, and yet we shall not discuss the Arabic logic per se, as it had its creative phase long before the time of Western late medieval philosophy. Aristotle’s Organon was translated into Arabic in the ninth century in Baghdad, and a commentary tradition started soon after that. Logic was honored as a kind of grammar of reasoning, and for example, al-Farabi (c. 870–950) underscored its importance as the “forecourt of all philosophy.” Avicenna (Ibn S¯ın¯a, 980–1037), on the other hand, was already a brilliant, independent exception: During his time, logical research was already declining, and commentaries were replaced by handbooks. His work had a profound inﬂuence on Western theories of meaning. In the twelfth century, the Spanish Arabic school revived commentaries, and the last commentator, Averroes (Ibn Rushd, 1126–1198), was also the greatest. The works of Averroes, “the Commentator,” were soon translated and became highly appreciated in Europe. In logic he was not as dominant as in metaphysics or in natural philosophy, but undoubtedly his works belong to the background that was always present. Averroes’s thought survived mainly in the West. In the Islamic world, logic was integrated into studies of theology and law, and even handbooks were gradually replaced by more or less elementary textbooks. During the period we describe, from the thirteenth century on, Arabic logic no longer produced anything but new versions and editions of established textbooks. On the other hand, a rich tradition of Jewish philosophy was alive in Europe through the late Middle Ages. Logic was not its favorite ﬁeld, but some Jewish authors paid considerable attention to logical questions. However, these studies had little interaction with Latin logic, and thus had to rely solely on Aristotle as commented by Averroes and al-Farabi. Still, there were some innovations, the most interesting ﬁgure being Gersonides (1288–1344). Writing in a rigorous manner, he made a number of criticisms of traditional doctrines; among other

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things, he rejected the old Averroistic construction of modal syllogistics in Prior Analytics (thus paralleling contemporaneous Latin developments). Even today, very little is known about Byzantine logic. Apparently an uninterrupted interest in logic, “the instrument of philosophy,” existed among Byzantine scholars. It produced mainly Aristotelian commentaries, often in the neo-Platonist spirit. Its independent progress was severely hindered by a conservative, philological approach to Greek sources, and occasionally also by religious scruples against the pagan heritage.

1.4. Texts Aristotle was the essential basis of later medieval logicians, but other classical ideas also played their part. First, Greek Aristotelian commentators had discussed various problems in Aristotle’s logic and its correct systematization, and their work became partly known (either directly or through Arabic sources). Second, the Stoics had argued that Aristotelian predicate logic was insuﬃcient and required some background from the propositional logic that they studied as the real logic. No complete Stoic works were preserved, but these Stoic themes were transmitted, for example, by Augustine’s Dialectica and by Boethius. We shall meet similar problems in the medieval theory of demonstration in topics and consequences. Because of the Stoic inﬂuence, medieval logicians were always in a diﬀerent position from the ancient Peripateticians in that they were aware of the necessity of essentially nonsyllogistic inference. Furthermore, logic was obviously inﬂuenced by classical grammar, which provided it with categories like nouns, verbs, and other parts of speech, as well as central syntactical notions, the main authority being Priscian’s grammar of Latin. Finally, some logical material had found its way into the work of famous ancient authors, among them Cicero, and the Christian fathers. From the middle of the thirteenth century, there was a rapid increase both in logical studies and in Aristotelian studies in general. Soon the obligatory logical curriculum included the whole of the Organon. Aristotle’s text is so concise and diﬃcult that it was always accompanied by commentaries and explanatory texts. It was required that students mastered this material thoroughly, and practical logical exercises became very popular as a supplement to lectures. A major and growing part of studies was dedicated to logic. If we understand logic in the widest sense, it appears that more than half of the program of an arts faculty could be about logic. At least we may note that logic had an undisputed place in medieval learning, and that it was not a specialist subject since almost all leading philosophers wrote about logic. Most logical works were closely connected to university teaching. The usual teaching method in medieval universities was that a text was lectured on and explained in detail. The intention was to build a consistent interpretation of the text, to eliminate ambiguities and to resolve the problems and conﬂicts the text gave rise to, and a typical medieval method of study included disputations where some theses were argued for and against. The character of university

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The Development of Modern Logic

teaching goes some way to explaining the literary types that became widespread. In addition to simple lecture drafts, there were all kinds of commentaries, ranging from elementary glosses to large systematic books. There were treatises (tractatus), that is, manuals or more advanced surveys of some ﬁeld, which gradually became more independent of the underlying texts. The liking for argumentation and disputation produced quaestiones, analytic works where some speciﬁc question is resolved or a thesis defended. (Later, systematic studies were organized in the form of a series of questions even though they were often referred to as commentaries.) We must also remember that logical subjects are often encountered as digressions in other works, for example, in the extensive sentence commentaries of theologians. There has been quite a decisive improvement in the accessibility of medieval logic over the last three decades, when numerous texts have been published. However, a large amount of material still remains unpublished and even completely unstudied. In fact, it is quite possible that our whole view of the outlines of medieval logic will undergo a change; indeed, such changes have occurred before, and systematic historical research of this logic is still a very young enterprise.

1.5. Interdisciplinary Relations Obviously, logic had a well-established place in the system of disciplines in the Middle Ages. But what kind of interaction did logic have with the other sciences? Unfortunately, it is not easy to say anything deﬁnite about this. First of all, formal philosophy of science was studied by logicians in connection with the Posterior Analytics, which discusses the correct form and nature of deductive theories. In this way, the methodology and philosophy of science were a part of medieval logic. Also, the occasional attempts to create calculative scientiﬁc speculations used heavy logic, but in general there was little concrete connection of logic to particular natural sciences, which took care of their own subjects. On the other hand, metaphysics—universally considered a real science—was always relevant for logic. Thus, semantic theory, so prominent in medieval logic, is immediately bound to metaphysical questions. Just as early supposition theory employs a metaphysical basis, so in the latemedieval nominalist trend it is impossible to separate logical from ontological thought. The role of theological matters is less transparent. Obviously theology needed systematic thought and conceptual analysis, and was hence favorable to logic. The conceptual examples and diﬃculties that logicians examined were very often drawn from theology. Generally, the signiﬁcance of theology for logic must have been positive. Their union was made problematic, however, when many philosophers began to think that some mysteries of faith, such as the Trinity, were not only inscrutable but literally beyond logic—even if their exact formulation could be a task for logic. Thus Ockham and Buridan thought that certain theological notions had to be explicitly declared unsuitable for use as

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substitution instances in ordinary logical principles, and a few authors were even more radical on this point. On the other hand, there was—of course—some religious hostility which regarded logical reasoning as an unhealthy method in theological matters.

2. Language as the Subject Matter of Logic Thinkers in the Middle Ages were anxious to discuss the correct system and classiﬁcation of sciences. Since their philosophy of science was realist, they believed that the classiﬁcation should be based on the order of nature. Logic, however, clearly has special features that make its place in this scheme problematic. Is it a science that has as its subject matter some part or aspect of reality? Or is it merely the art of using linguistic idioms? Or is its function something else altogether?

2.1. A Science “of Words” or “of Reason” The ﬁrst known medieval textbook of logic, Garlandus Compotista’s Dialectica from the late eleventh century, already sets the discussion of this topic on a track that was to have crucial inﬂuence on the kinds of innovations that were to be achieved in medieval logic. Throughout the Middle Ages, logical theories had a very intimate relation to actual language use. According to Garlandus, logic is concerned with actual utterances (voces). After Garlandus, Abelard, for example, continues on the same track, but reﬁnes the position: As he sees it, statements are not built from mere spoken sounds but from words that have a signiﬁcation (sermones). Thus, they also constitute the subject matter of logic. Logic is “a science of words” (scientia sermonicalis). It seems that well into the thirteenth century the idea that logic studies actual language use remained basically unchallenged. Teachers and students of logic considered that their studies helped in the acquisition of argumentative skills for actual scientiﬁc disputations. Given the status of Latin as the language of all medieval learning, it was natural to make the appropriate logical distinctions from the viewpoint of spoken Latin. This gave an important status to essentially linguistic structures even in the later developments of medieval logic. In approaching many particular features of medieval logic, it is crucial to remember this pragmatic way of looking at the subject matter of logic. In the Middle Ages, the art of logic was not taken to be concerned with abstract structures in the way modern logic and modern mathematics are, but with actual linguistic practices of reasoning. It was generally accepted that logic is, at least in some sense, a practical science giving advice on how to understand and make assertive statements and how to argue and reason in an inferential manner—though opinions varied whether this practical characterization of logic was accurate in any deeper sense.

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In the Arab world logic was thought of in a diﬀerent manner, and thus toward the thirteenth century under Arabic inﬂuences the Latin world became aware of a diﬀerent way of looking at the character of logic as a ﬁeld of research. According to al-Farabi, the logos (in Arabic, al-nutq) discussed in logic occurs on two levels, one inscribed in the mind, and the other existing externally in spoken sounds. Thus, we may even separate diﬀerent senses of the Greek word logos in accordance with the level of discourse at issue. Avicenna was also inﬂuenced by al-Farabi’s discussion, and gave even further impetus to the idea that logic is concerned with intellectual structures rather than with what we do in spoken discourse. Thus, logic should be called “a science of reason” (scientia rationis), as the Latin world translated the idea. In the thirteenth-century Latin tradition, both the idea of logic as “a science of words” and as “a science of reason” had a foothold. In his major classiﬁcation of all the university disciplines, De ortu scientiarum, Robert Kilwardby (c. 1215–1279) gave a deﬁnition of the nature of logic that combined the two views. It is worth taking a closer look at his deﬁnition, because it also clariﬁes the medieval way of locating branches of logic in terms of Aristotle’s logical works in the Organon. According to Kilwardby, logic is “a science of words” (scientia sermonicalis) in the sense that “it includes grammar, rhetoric and logic properly so-called.” But as Kilwardly immediately points out, “in the other sense, it is a science of reason,” and in this sense it is “distinguished from grammar and rhetoric.” It may seem that here Kilwardby would be demarcating two diﬀerent disciplines both ambiguously called “logic.” But this is not really his intention, as he hastens to explain: Logic properly so-called must in his opinion be listed as one of the “sciences of words”; it is the science of words that attends to their rational content. As he sees it, logic does not study arguments as mere words nor as mere rational structures, but as rational structures presented in linguistic discourse. The grammatical and rhetorical features of these arguments, for example, do not pertain to the art of logic. Logic studies the rational structures expressed and understood in linguistic discourse—neither rational structures as such, nor linguistic structures as such. The core of logic can in Kilwardby’s view be found in Aristotle’s Prior Analytics. This is because at its core, logic is concerned with reasoning, and this is the main topic of Prior Analytics and its system of syllogistic reasoning. It is of some interest to note that Kilwardby is very Aristotelian in claiming that all forms of valid reasoning can be reduced to the categorical syllogism discussed in Prior Analytics. This was not the received view at the time, and Kilwardby’s position did not win unconditional approval. Abelard had discussed the theory of conditional inference and clearly would not have accepted such a principle. Indeed, conditional inferences were throughout the Middle Ages a standard part of logical curriculum. Soon after Kilwardby, toward the end of the thirteenth century, the theory of consequences (consequentiae) grew into a self-conscious general theory of inference that had no speciﬁc reference to the syllogistic system; syllogism was increasingly presented as a special case of inference.

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Kilwardby pushes aside one aspect of Aristotle’s discussion in the Prior Analytics. According to Kilwardby, only dialectical and demonstrative syllogisms are relevant to logic, while the rhetorical syllogisms discussed by Aristotle fall out of the scope of logic because they take “the form that is suited to the singular, sensible things considered by the orators.” Logic as a science is concerned with universal rational structures as captured in discursive reasoning. In Kilwardby’s presentation of the structure of logic, the system developed in the Prior Analytics is put to further use in the Posterior Analytics and the Topics. As Kilwardby sees it, the division into diﬀerent works is based on the matter to which the syllogistic structures are applied. The Posterior Analytics discusses the way in which the syllogistic form is applied to “speciﬁc matter” and yields scientiﬁc demonstrations. For its part, Topics is concerned with “common matter” and shows how we can construct good inferences relying on generic considerations. Aristotle’s Sophistici Elenchi, for its part, plays in Kilwardby’s view the role of considering what can go wrong in constructing an inference. As Kilwardby shows, the role of De interpretatione and Categoriae can also be considered in terms of the syllogism. De interpretatione considers the propositional structures that are essential for constructing syllogisms. A syllogism must be construed so that it has a middle term, and for this purpose it is necessary to see how assertive statements usable as premises can be built to consist of two terms conjoined aﬃrmatively or disjoined negatively. Categoriae goes into the structure even more deeply, considering the terms and their signiﬁcation in reality. From the mid-thirteenth century onward, Avicenna’s conception of logic as a science of reason gained increasing currency in philosophical discussions on the subject matter of logic. To some extent this happened at the expense of the earlier view of logic as a science of words. As we have already seen, Kilwardby restricts the meaning in which logic is a science of words so that it no longer carries much weight. Albert the Great’s general position concerning the nature of logic is similar, but in the beginning of his commentary on Aristotle’s Categories he takes the explicit position that logic is strictly speaking not a “science of words” at all. Rather, logic is concerned with argumentation, and argumentation should be referred to reason rather than to words. Albert’s student Thomas Aquinas (1224?–1274) followed him in this matter. In his more elaborate system, the subject matter of logic consists of three conceptual operations of the mind, namely, formation of concepts, of judgments, and of inferences. This systematization can be traced back to Plotinus and the neo-Platonic commentators of Aristotle’s logic in a more explicit way than Kilwardby’s system. The ﬁrst two operations are discussed, respectively, by Aristotle’s Categories and De interpretatione, and the third by the other four works included in the Organon. As Aquinas saw it, making a judgment—and, in fact, anything that logic is concerned with—requires an intellectual act of understanding. Thus, making a judgment is not primarily to be understood as a speech act but as a mental act. According to Aquinas, externally

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expressed linguistic structures should be seen as results and representations of intellectual acts, and only in this intermediate way does logic come to be concerned with linguistic structures. In the ﬁrst place, logic is concerned with the intellectual operations by which the universal features of material reality are understood. The detailed structure of Aquinas’s presentation of the subject matter of logic has the crucial feature that it relies heavily on the Aristotelian idea that all inferences can be presented as syllogisms. As Aquinas saw it, all of logic can be understood in terms of syllogistic structures. Since he thought that logic deals with the three basic operations of the intellect, any inference would have to be based on them. However, there are understandably quite stringent limitations on the extent to which logic can be derived from these basic operations. For example, with the claim that all assertions are made by the composition of a predicate with a subject, Aquinas was almost forced to reject conditionals as assertions. However, hypothetical propositions had a long tradition deriving from Stoic logic and had been dealt with already by Boethius, and thus Aquinas was compelled to comment on them. As he put it in the ﬁrst section of his commentary on De interpretatione, hypothetical propositions “do not contain absolute truth, the understanding of which is needed in demonstration . . . but they signify something to be true on condition.” According to Aquinas’s logic, conditionals could not be used as premises in scientiﬁc demonstrations. Neither Albert nor Aquinas worked much with the actual details of logical systems, and their discussion has more of the character of the philosophy of logic. However, the distinctive ﬂavor of medieval logic showed itself in its close connections to actual language use, and it incorporated analysis of a much wider variety of linguistic structures than the simple predications included in the syllogistic presented in the Prior Analytics. Moreover, Abelard’s work had already made medieval logicians acutely aware of a concept of inferential validity that was essentially unconnected to the syllogistic structure. While Kilwardby, Albert the Great, and Aquinas defended the strong Aristotelian program of reducing all inferential validity and thereby all logic to an analysis of the syllogistic system, actual work in logic was taking another course. In the subsequent development, Aquinas’s three operations of the mind were often referred to, but usually understood in a loose and suggestive manner. It became standard to treat logic with the organizational principle that Categories studies concepts and De interpretatione propositions, while Prior Analytics and the three subsequent Aristotelian works concentrate on inferences. It was, furthermore, commonly accepted that there are many traditional logical genres inherited from the twelfth century that do not ﬁt into this basic scheme. For example, there was an abundance of literature on the so-called syncategorematic terms, analyzing the logical properties of words such as “except” (praeter), “begins” (incipit), “whole” (totum) and “twice” (bis). Such problems had little connection to the development of syllogistic systems. Furthermore, it remained

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a problem to explain how such logical genres could ﬁt into the description of logic as a science of reason, because many of them were quite clearly motivated by the analysis of linguistic structures.

2.2. Mental Language An interesting alternative way of characterizing logic as concerned with mental concepts rather than Latin words was being developed at the time Aquinas was working, and it gained momentum among logicians in the latter half of the thirteenth century. It was based on quite a diﬀerent understanding of the workings of the human mind from that of Aquinas’s Aristotelian outlook. Roger Bacon (c. 1214–c. 1293) rejected the idea that the human understanding works only with real universals existing intentionally in the mind. Rather, the mind should be understood in terms of a discourse consisting of singular acts of intellection whereby diﬀerent singular things are understood in diﬀerent ways. According to Bacon, logic is not concerned with an external discourse but with the internal discourse of the mind, with “mental expressions and terms” (dicciones et termini mentales). In other words, Bacon posits a mental language to serve the role of the subject matter of logic. As we shall see, this approach was to play a major role in later developments. First, however, we must take a closer look at the content of Bacon’s suggestion. One of the central classical texts that Bacon refers to was Boethius’s distinction between three levels of discourse (oratio): intellectual, spoken, and written. In making the distinction, Boethius was commenting on Aristotle’s De interpretatione 16a10, and Boethius’s way of reading the passage was well known in the late Middle Ages, but it remained a debated issue how one should understand the intellectual level of discourse and how one should relate logic as a discipline to these levels. It seems clear, though, that Bacon understood the intellectual discourse in a way that can with good reason be called linguistic. He even takes pains to show how word order functions in this discourse. Without going into details, it is suﬃcient here to point out that he looked at the structure of mental sentences in terms of Aristotelian predication: The subject comes ﬁrst, then the predicate, both with their “essential determinations.” They are then followed with the various “accidental parts” of the composition. Especially his way of dealing with these “accidental parts” shows how looking at thought as a linguistic phenomenon gives Bacon a clear advantage in comparison to Aquinas from the logician’s point of view. Through his theory of mental language, Bacon is able to attribute considerably more logically relevant linguistic structure to the intellectual level. One of the aims of this enterprise was—as is evident to any logician—to show how to solve ambiguities of scope arising in Latin through the relatively loose rules concerning word order. In this way, Bacon worked toward a theory of an ideal language to serve logical functions as early as the 1240s, if the current scholarly opinion of the date of his Summa de sophismatibus et distinctionibus is correct.

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The mental discourse that Bacon was after is abstracted from spoken languages like Latin and overcomes their arbitrariness. However, there is also the other side of the coin: He speciﬁcally wanted to ﬁnd the basis for certain logically relevant Latin structures from mental discourse. Although he thought that grammatical gender has no correlate in mental discourse, the subject-predicate structure and many syncategorematic expressions have. Indeed, Bacon seems to ﬁnd from the mental discourse even more than a logician would need. In many issues, it becomes apparent that he was working more as a linguist than as a logician. His aim was a universal grammar rather than a universal language suitable for logic. Commentators have, accordingly, connected Bacon to the movement of speculative grammar emerging in the latter half of the thirteenth century. The approach to linguistic analysis employed by this school is often called “modist.” The label reﬂects the speciﬁc use of a threefold series of concepts: “Modes of being” (modi essendi) in reality were paralleled in language by “modes of signiﬁcation” (modi signiﬁcandi) and in the mind by “modes of understanding” (modi intelligendi). The movement was more closely connected to language theory than logical theory, and accordingly we will only discuss it brieﬂy here. The main idea of modist theory was to approach Latin expressions as generated from a universal grammatical structure accurately reﬂecting the structure of reality. That is, they thought that grammar is (in the words of Bacon) “substantially one and the same in all languages, although varied in its accidents.” Other central ﬁgures of this movement include Boethius of Dacia, Martin of Dacia, and Radulphus Brito. At the beginning of the fourteenth century, the program lost ground, although much of the terminological innovations, including the term “mode of signifying,” survived until the Renaissance in the standard vocabulary of logicians. According to the modists, all words have two levels of meaning. Words have in addition to their own speciﬁc meanings certain more general meanings, or so-called modes of signifying. To be more exact, a phonological construction gains a special meaning when it is connected to a referent that it “is imposed” (imponitur) to mean (in the so-called ﬁrst imposition). Furthermore, the word is also “imposed” (in the second imposition) to mean its referent in a certain grammatical category with certain modes of signifying. For example, pain can be referred to by a variety of Latin words in diﬀerent grammatical categories: dolor refers to it as a noun, doleo as a verb, dolens as a participle, dolenter as an adverb, and heu as an interjection. In all these words the special signiﬁcation is the same, but the modes of signifying are diﬀerent. The modists found no theoretical use for the most central logical term of the terminists, “supposition” (suppositio; it will be described with more detail in the next section). In their view, the varieties of ways in which words are used in sentential contexts are based on modes of signifying contained in the words, and thus they were not willing to admit that the sentential context as such would have an eﬀect on how the term functions—which is one of the leading principles

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of the supposition theorists. Rather, their approach was generative in the sense that the sentences were to be generated from words that have their signiﬁcation independently. This approach made it unnatural to distinguish the sentential function of a term from its signiﬁcation. It may, however, be noted that the term “consigniﬁcation,” meaning the function of syncategorematic terms in the terminist approach, was used by modists to express the way in which phonological elements of actually used words mean modes of signiﬁcation: For example, the Latin ending -us “consigniﬁes” nominative case, singular number, and masculine gender. The thirteenth-century grammarians recognized the congeniality of syncategorematic terms and modes of signifying: Both are understood as the elements of discourse that show how the things talked about are talked about and what in fact is said about them. From the viewpoint of the history of logic, it is important to recognize that from the twentieth-century viewpoint, the modist conception of grammar can be characterized as making the subject a “formal science.” The criteria of congruence were taken to depend solely on the grammatical structure, or the consigniﬁcations of the elements of the sentence, regardless of the special signiﬁcations of the terms used in the sentence. Modists thought of the generation of language as putting semantically signiﬁcant elements into grammatical structures. It seems that at least the Parisian master Boethius of Dacia wanted to develop also logic into this direction and wanted to make a distinction somewhat like the twentieth-century distinction between logical form and semantic content. Nevertheless, it was only some decades later at the time of John Buridan that the substance of logic was thoroughly reconsidered from this viewpoint.

2.3. The Universality of Logic From the viewpoint of practicing logicians, the debate concerning the subject matter of logic at the end of the thirteenth century probably seemed like a search for a credible account of the universal basis of the invariable features of argumentation found in the logical analysis of actual use of language. That is, what is the universal basis on which the validity of an inference formulated in a particular language is grounded? It was accepted as relatively clear that logic is about actually or potentially formulated tokens of terms, propositions, and arguments that are linguistic in some sense of the word. It was clear that such discursive arguments existed in such external media as spoken or written Latin expressions. However, logic aimed at, and appeared to have found, some kind of universality, and such universality apparently could not be achieved if logic was tied to a particular spoken language. Instead, thirteenth-century discussions converged in ﬁnding the universality of logic in intellectual operations. But what are these intellectual operations? Can we speak of a mental language serving as the domain of logic? In particular, is a mental proposition linguistic in any relevant sense? And because it was assumed that an aﬃrmative predication is

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based on or performs a composition, one had to ask what exactly does this composition put together. At the turn of the fourteenth century, we ﬁnd diﬀerent logicians giving diﬀerent answers to these speciﬁc questions. At that time, the most common answer was the one inspired by Bacon. It was based on looking at the mental discourse from the viewpoint of “imagined spoken words,” and accepting it as the privileged medium of logical arguments. This kind of explanation is straightforward and relatively acceptable from the metaphysical viewpoint, but is, of course, less satisfactory in explaining the kind of universality achieved in logic. If mental language is nothing but imagined Latin words, there seems to be little reason for assuming it to have any more universal status than Latin has. Yet that appears to be what Bacon wished to propose. The realist Walter Burley seems to have approached the problem from the viewpoint of the universality of logic. Given his realist metaphysics, it is understandable that he contributed the concept of “real proposition” (propositio in re). He aimed at explaining mental propositions as consisting of real external things, which are conceived and propositionally combined in the mind. This model of the metaphysical basis of mental language of course works only if conceptual essences are understood in a realist way without separating them from the things themselves. Also, such “real propositions” are not very language-like. The nominalist William Ockham formulated the most innovative and by far the most inﬂuential theory of mental language. He ridiculed the position of Burley by asking how it could be that the subject of a proposition formulated in Oxford is in Paris while the predicate is in Rome. A suitable example of such a proposition would be “Paris is not Rome.” Ockham seems to have gone back to Bacon’s theory, but with the awareness of some of its shortcomings. With his nominalist metaphysical outlook, he strongly held the view that all the metaphysically real things involved in mental propositions are particular mental acts or states. But the substantial logical strength of his theory of mental language was really that it was formulated in a way that was suﬃciently neutral from the metaphysical point of view. Indeed, Ockham himself started with the idea that mental language consists of ﬁcta (that is, of intellectually imagined objects of thought that do not have any kind of existence outside the mind but are simply “made up” by the mind) but ended with the view that mental language is better understood as consisting of intellectual acts intentionally directed at real or possible things. At one stage of his career he was working on the theory of mental language without being able to make up his mind which of these two rather diﬀerent views would provide the appropriate metaphysical foundations. In the ﬁrst chapters of Summa Logicae, William Ockham addresses the Boethian idea of three levels of language. In opposition to Aquinas’s treatment of the same topic, Ockham claims that written language is subordinated to spoken language rather than signiﬁes it. Similarly, spoken language is subordinated to mental language rather than signiﬁes it. That is, according

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to Ockham, all three languages similarly signify things in the external world. They are, furthermore, all equally languages. Written language is inscribed on external material things, and spoken language exists as a continuum of sounds. Similarly, mental language consists of real qualities of the thinking mind. Furthermore, in Aquinas’s picture intellectual acts were the signiﬁcations of linguistic expressions and by their nature could not serve as a medium of communication. For Ockham, mental language could by its nature serve equally as a medium of communication if only there were beings who could perceive its expressions apart from the “speaker” him- or herself. In fact, Ockham thought that we have every reason to suppose that the angels described in the Christian doctrine communicate in the same language in which we think. The main diﬀerence between mental language and the two other kinds of language is the naturalness of mental language. Unlike spoken ordinary languages, which we nowadays call natural, Ockham’s mental language is natural in the sense of not being conventional. The expressions of mental language have their signiﬁcations naturally, without explicit or implicit consent or any other kind of conventionality involved. A mental word is capable of signifying only the things it really signiﬁes, and it signiﬁes exactly those things to all competent users of the word. (It may be noted that Ockham admits that in angelic communication some mental expression may be unfamiliar to the perceiver and thus unintelligible to him.) In principle, there are no ambiguous terms in the mental language. This is one of the central features that make Ockham’s mental language an ideal language, which is then suitable for the purposes of a discipline like logic. There are also two other senses in which Ockham aims at description of an ideal universal language. On the one hand, he tries to describe in general terms what must be required of any language that is used for thinking, and assumes that mental language has only such necessary features without any accidental “ornaments of speech.” Since these features are necessary requirements of thought, all thought must comply with them. Thus, Ockham constructs a theory of a language that is universal in the sense of being used by all intellects that think discursively. On the other hand, according to Ockham, mental language is directly related to the constitution of the world. It reﬂects accurately mind-independent similarities between real things. Thus, a fully developed mental language would be universal in its expressive power: There cannot be any feature of the world that could be conceived by an intellectual being but not expressed in mental language. Everything that can be thought can also be cast in terms of mental language. From this principle it also follows that all linguistic diﬀerences between expressions of spoken languages that result in diﬀerent truth values (which are not “ornaments of speech”) have their correspondents on the level of mental language. From the logical point of view, perhaps the most interesting ideal feature of Ockham’s mental language is its compositionality, which makes it a recursive system. Complex expressions get their meaning from their constituent parts

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in a systematic way. In this respect, mental language shows similarities to twentieth-century formal calculi, although it is much more complex. According to Ockham, the expressions of mental language consist of categorematic and syncategorematic parts with speciﬁc linguistic roles (we will return to this distinction more fully in the next section). A categorematic term (e.g., “animal”) signiﬁes real individuals and refers to them as the other elements of the propositional context determine. A syncategorematic term (e.g., “every”) does not signify any external things but rather, as Ockham puts it, “performs a function with regard to the relevant categorematic term.” Typically, syncategorematic terms aﬀect the way in which the signiﬁcations of the categorematic terms result in reference (or suppositio) in the sentential context. We may say that the categorematic terms of a sentence determine which things are talked about, whereas the syncategorematic terms determine how they are talked about and what is actually said about them. The number of basic categorematic terms of the ideal mental language accords to the variety of things that could exist in the world; they express the natural kinds of possible things. Ockham’s view of the number and selection of syncategorematic terms is more diﬃcult to determine. On the one hand, it is clear that he is thinking of a much wider variety of such logical constants than twentieth-century logic used. On the other hand, it is equally clear that most of his logical rules concern the eﬀects of syncategorematic terms on logical relations between sentences. Because the compositional characteristics of mental language depend on the distinction between categorematic and syncategorematic terms, Ockham’s mental language seems to conform to the twentieth-century ideal principle of logical formalism, namely, the idea that all sentences directly reveal their logical form. This seems to be one of the features of the mental language that Ockham is most interested in, and much of his logic is devoted to systems elaborating on the functions of syncategorematic expressions. However, Ockham’s theory has interesting details that reﬂect a conscious decision not to accept logical form (as we nowadays understand it) as the guiding universal principle in determining the logical validity of an inference. The theory of mental language was also discussed and developed after Ockham, but without major revisions. The most important innovator was John Buridan, who altered much of the terminology used in deﬁning language and gave a rather diﬀerent account of how the simple terms of language are learned, but these revisions resulted in few changes that would be relevant to our purposes here. After Buridan, some minor topics like the role of proper names and individual terms, and the nature of word order as explanatory of issues of scope were discussed. These can hardly be called revolutionary with regard to the nature and purposes of logic. At the peak of its success, medieval logic had thus found a deﬁnition of its subject matter that provided a relatively reasonable explanation both of its universality and of its dependency on discursive linguistic structures. For the logicians of the second quarter of the fourteenth century, logic was the art

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of constructing and using mental propositions. It studied the basic syntactic features of mental language, the ways and forms of assertions that can be produced in it, and the ways these assertions can be organized in inferential relations. Because mental language was understood as capable of expressing all possible universal structures of discursive thought, logic studied the universal art of reasoning.

3. Terms 3.1. The Notion of a Term Textbooks of “traditional logic” used to divide their material into three sections: the doctrines of terms, propositions, and inferences. This practice is based on ancient grounds, of course, but Aristotle nowhere says that all logic should be so divided, and medieval logic did not at ﬁrst do so. In thirteenth-century logic books, often the chapters are still relatively independent, or at least not organized according to such a general plan. But then, at the turn of the century, this idea soon became dominant. We ﬁnd it, for example, in both Burley and Ockham, in spite of their sharp disagreement. We are going to follow this familiar order, starting with terms. Everybody agreed that terms were the ultimate units of discourse. In a way this is obvious, but the emphasis on this fact in logical contexts also has a nontrivial sense which shows the Aristotelian character of medieval logic. For the logic that Aristotle had developed was term logic, unlike that of the Stoics. But Aristotle gave two diﬀerent explanations of terms. In Categories he speaks about noncomposite expressions (“such as ‘man’, ‘ox’, ‘runs’, or ‘wins’ ”). In Prior Analytics he says (24b16–18): “I call that a term into which a proposition is resolved, i.e., the predicate or that of which it is predicated, when it is asserted or denied that something is or is not the case.” These explanations lead to very diﬀerent uses of the word “term.” In the ﬁrst sense, a term is simply any word. Many medieval logicians mentioned even meaningless words, like “ba,” “bu,” but only to concentrate on ordinary words. In this sense, which is that of grammarians, it is only required that a term is a noncomposite signiﬁcant element of the language. Or it can be a composite expression signifying one thing. In the second sense, which is more exciting for the logicians, a term is something that can stand as a subject or a predicate of a proposition. This excludes wide classes of words from the status of terms. According to the strictest deﬁnition, a term is only that type of nominal expression that can ﬁgure as S or P in a categorical proposition “S is P.” This leads to a question concerning the structure of terms because S and P can be complex expressions. Buridan, for instance, took a strictly propositional view and argued that a simple proposition has exactly two terms. In this usage a term is identiﬁed with an extreme (extremum) of a proposition. But

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since the various words occurring within such terms can be extremes in other propositions, authors kept on saying that propositions can have complex terms that are composed of simple terms. The theory of terms is obviously connected to grammar, and Priscian’s classiﬁcations had a strong inﬂuence on earlier writers. But logically it was important to eliminate Latin contingencies and consider as general cases as possible. However, that is a problematic requirement regarding terms: What could be those language-independent terms? Diﬀerent ways to tackle this question systematically were oﬀered ﬁrst by so-called speculative grammar, and then by the mentalistic interpretation of language, which was ﬁnally victorious, but both approaches emphasized the universality of language. For late medieval logicians, the terms were in the ﬁrst place mental terms that occurred in mental propositions.

3.2. Categorematicity A distinction that is especially important for logic was made between categorematic and syncategorematic terms. This distinction was well known to all logicians, and they usually introduced it immediately after the deﬁnitions of terms. The source of these notions was in grammar, but logicians gave them a new function, following a hint from Boethius. Priscian had written about “syncategorematic, i.e., consigniﬁcant, parts of speech”: Most words are grammatically categorematic since they can occur as subjects or predicates, but for instance, conjunctions, prepositions, adverbs, and auxiliary verbs cannot. They are syncategorematic and signify only together with other words. Logicians proceeded from this picture to distinguish two ways of meaning and to describe the logical behavior of philosophically interesting syncategorematic words. Syncategorematic words were ﬁrst studied in special treatises. This genre of Syncategoremata was popular from the last quarter of the twelfth century to near the end of the thirteenth century. Well-known treatises of this kind were written by Peter of Spain, William of Sherwood, Nicholas of Paris, and even the famous metaphysician Henry of Ghent. Later, the subject was incorporated into general textbooks of logic. The distinction itself had its systematic place at the outset of the exposition of the theory of terms, since it was utilized in many questions; particular syncategoremata were then discussed in their due places. Even in the fourteenth century most authors apparently based their deﬁnitions of syncategoremata on diﬀerent ways of signifying. According to Ockham, “categorematic terms have a deﬁnite and determinate signiﬁcation. . . . Examples of syncategorematic terms are ‘every’, ‘no’, ‘some’, ‘all’, ‘except’, ‘so much’, and ‘insofar as’. None of these expressions has a deﬁnite and determinate signiﬁcation.” Buridan states: “Syncategorematic terms are not signiﬁcative per se, as it were, but only signiﬁcative with another.” Paul of Venice still defended this view against “a common deﬁnition” that a syncategorematic term cannot be the subject or the predicate or a part of either. Such a purely syntactical criterion had been supported by Albert of Saxony (1316–1390).

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Syncategorematic expressions were usually counted as terms. A theoretical reason for this was a slogan that was in use at least from Peter of Ailly onward: A term is a sign that in a proposition represents something or somehow. Syncategoremata, indeed, signify “somehow” (aliqualiter), for thirteenth-century treatises had already pointed out that syncategoremata serve to show how the categorematic terms ought to be understood. It is thus essential for their signifying that they are joined with other terms to elaborate their meanings. Present-day readers will easily associate syncategorematic terms with logical constants. This is partly correct but must not be taken too literally. For one thing, the class of syncategoremata of language is much wider than the small sets of logical constants nowadays. However, the medievals ignored most syncategoremata and studied only those which seemed to be philosophically interesting. These were just words with special logical peculiarities, and hence, for these terms, the comparison with logical constants may be justiﬁed. The lists of diﬀerent logicians varied greatly, but several dozens of words were thus discussed. Among them belonged sentential connectives; words like “only” and “except”; quantiﬁers; modal operators; words like “whole” and “inﬁnite”; some verbs like incipit and desinit (“begins” and “ends”); and the copula est, that is, the copulative use of the verb “to be,” esse. General textbooks listed them but did not usually go into details of particular syncategoremata. In the fourteenth century, such closer study often took place by means of sophismata: In this literature it was typical to analyze sentences that were problematic or ambiguous because of syncategorematic words (see section 6). Buridan expressly said that the matter of a proposition consists of purely categorematic terms while syncategoremata belong to its form. From this point of view, it is interesting to notice that the notion of syncategorematicity proved diﬃcult because it did not determine a precise class. Thus Buridan had trouble with attitude operators: Verbs like “to know” and “to promise” clearly have a formal function and yet they are independently meaningful. The two criteria, the semantical and the grammatical, did not always coincide, and Peter of Ailly suggested that they should be wholly separated. A term could therefore be syncategorematic either “by signiﬁcation,” or “by function,” or in both ways.

3.3. Predicables In a proposition something is said of something, as Aristotle taught. It is therefore logically important to have some idea of the various types of things that can be thus predicated, the predicables (praedicabilia). Medieval logicians based their classiﬁcation here on Porphyry and Boethius. Obviously, a predicable is something that can be said (predicated) of something else, but in a stricter sense, it is only a universal term that can be predicated of many things. This distinction was made already in thirteenth-century textbooks, and it is easy to see that predicables have a close connection to the most famous medieval metaphysical problem, the problem of universals.

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Explaining Aristotle’s Categories, Porphyry mentioned ﬁve types of universal terms: species, genus, diﬀerentia, proprium, and accidens. These were the “ﬁve universals” (quinque voces) that recur in medieval discourses. They reveal various relations of the predicate to the subject: What kind of information does the predicate give us about the subject? When it is said that S is P, the predicate P may express a species to which S belongs, or a genus to which every S belongs, or a characteristic essential feature of them (diﬀerentia), or a nonessential but necessary property of every S and only them (proprium), or their accidental feature (accidens). (The P of species is a somewhat obscure case here because it can be predicated of individuals, too, unlike the others.) Added to a genus, a speciﬁc diﬀerence (diﬀerentia speciﬁca) deﬁnes a species, which in turn can be a genus for lower subspecies. In this way, the famous “Porphyrian tree” is generated, ranging from uppermost genera down to individuals. The doctrine of predicables was a standard part in medieval logic texts, and it was a relatively unproblematic part: The diﬃculty, of course, is metaphysical and concerns the essential, necessary, and accidental qualities. Logicians, however, used the ﬁve universals as metalinguistic tools to classify predicates. A more ontological question is that of categories, or praedicamenta, as logicians preferred to call them. The ﬁrst category is substance; the other categories are ways in which something can belong to a substance. Aristotle studied quality, quantity, and relation in his Categories, and more brieﬂy he discussed even place, time, position, habit (having), passion, and action. With some variants, medieval praedicamenta treatises give the same list of 10 members. As Buridan says, “this treatise is found in many summulae, but in many it is not.” Indeed, it is not obvious why logicians need to discuss a question that seems purely metaphysical. But there was a motive for those who included this treatise in their summulae—an assumption of the parallelism between predication and being. Except substance, all categories both “are said of things” and “are in things.” Thus, a classiﬁcation of ways of being in a substance also produces a classiﬁcation of questions and answers that can be made concerning an entity, and this is a logically relevant achievement. Later, nominalists give up the assumed parallelism and analyze categories simply metalinguistically, as classes of terms. Ockham, for instance, has a long discussion in which he wishes to show how terms of other categories are secondary to substance and quality. We may note in passing that predicables and categories have a very diﬀerent role among the speculative grammarians of the late thirteenth and early fourteenth centuries. For them, terms are intelligible because they manifest the same characters and structures as the entities of the world; the “modes of signifying” belonging to grammatical features of lexical meaning and inﬂection are functions that reﬂect categorial features of objects. Such an approach leads to a special view of metalinguistic issues. Hence it is also natural that these authors, the modists, concentrated on rather abstract lexical contents and were not very interested in the semantic properties of concrete occurrences of terms in particular sentences.

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3.4. Signiﬁcance The main body of the theory of terms consisted of proprietates terminorum. The tireless analysis of these “properties of terms” displays the intense interest in philosophical semantics that was characteristic of later medieval philosophy. This is a ﬁeld that seems to be a medieval invention. In Aristotle and other ancient sources, there were only scattered remarks on semantic questions, and it can hardly be said that they attempted to establish any self-conscious theory of semantics. On the other hand, after scholastic philosophy these problems were often considered futile, and explicit philosophical semantics was largely rejected. But the medieval theory has had a striking revival in the latter half of the twentieth century, when philosophical semantics has again grown into a complex discipline, often struggling with questions that bear an obvious resemblance to medieval themes. Undoubtedly the two most important properties of terms are signiﬁcation and supposition. They have often been compared to present-day “meaning” and “reference,” but this comparison must not be taken literally. For one thing, the emphasis was on the words and signs: Unlike many accounts of meaning and reference, the medieval doctrine viewed signiﬁcation and supposition mainly as something that the words do or as something that is done by means of words. Let us start with signiﬁcation. Logicians were aware of the ambiguity of this word. Usually, instead of interpreting signiﬁcation as a signiﬁed entity of some sort, they started from “acts of signifying” and assumed that terms had a property of being signiﬁcant. (In this respect, terms diﬀered from other words which had no signiﬁcation by themselves.) A word signiﬁes, or has signiﬁcation, because of its “institution,” or according to another common account, because of its “use” in language. In short, signiﬁcation is the role of the term in language. The same idea acquires a new slant with the introduction of mental terms. It then becomes standard to claim that spoken words have their signiﬁcations because of linguistic conventions, whereas the mental terms are natural signs that have their signiﬁcations necessarily, without any stipulation. Signiﬁcation is generally connected to mental acts of understanding: A linguistic term signiﬁes that of which it makes a person (a speaker or a hearer) think, a mental term is itself an act of thinking of something, a representation. (To quote John Aurifaber: “signifying is an accident of the intellect, but a word is the thing by means of which the intellect signiﬁes.”) The thing thus signiﬁed has “intentional being.” Even before the mentalistic turn, it was usual to ﬁnd the essence of words in signiﬁcation. Thus Thomas Aquinas said that “signiﬁcation is like the form of a word”—the matter was the phonological shape, the form was its signifying capacity. Later, it was said that mental concepts have their signiﬁcations “formally,” and spoken and written words essentially function as instruments of this signiﬁcation. Signiﬁcation is the deﬁning property of all terms; thus it is natural that it can be deﬁned no further. Late medieval philosophers seem to agree that

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signiﬁcation is a basic notion that can only be explained by illustration. For them, it was obvious that a term signiﬁes something, but there was a great, partly metaphysical controversy about what this something is. Boethius had already said that words signify concepts, that is, corresponding mental entities. This gave an impulse for the view that words signify concepts immediately and objects indirectly. (Such a “semiotic triangle” had been discussed earlier by Greek Aristotelian commentators.) This opinion became prevalent among the Thomists. Aquinas himself had pointed out that a term signiﬁes a general nature that is abstracted from individual entities. The later Thomists emphasized that the concepts were signs, too: Thus the words do signify objects “principally” (most important), although they do it only “indirectly” (through the concepts). A contrary position was championed by Bacon, and it won general support at the end of the thirteenth century. It started from the obvious fact that terms are used in propositions, and the propositions are about objects and not concepts. Thus all terms must signify objects. However, nonexistent objects cause problems which compel logicians to make reservations concerning that general principle. What is signiﬁed is, for instance, the object “regardless of its being or not being” (Kilwardby), or the object “secundum quod the intellect perceives it by itself” (Duns Scotus). Moreover, spoken words and mental terms signify the same objects. According to Ockham, it is a basic fact that words are “subordinated” to the corresponding mental terms in such a way that they signify the same things. He apparently did not think that this use of language could be further explained. Buridan was not satisﬁed with this kind of answer and again interpreted the subordination as a type of signiﬁcation: Words also signify concepts, in some sense. Later discussion became rather complex when diﬀerent positions were combined and reﬁned. Admitting then that terms signify something extramental, it is still not clear what this signiﬁcatum is for general terms. The question is inevitably connected to the theory of universals. The realist answer is that the term signiﬁes something general; “man” signiﬁes a universal, a species, a property, or a common nature of “man in general.” The nominalist answer is that the term signiﬁes all relevant individuals; “man” signiﬁes each man. Both answers cause trouble, which shows the uneasy union of signiﬁcation and denotation. For it was assumed, after all, that a term signiﬁes what it is true of, and this characterization would better suit denotation. Syncategorematic words have no signiﬁcation in the strict sense. However, most logicians were not as rigorous as Ockham, who said that they do not signify at all. Even Buridan was willing to admit that they did not signify things but ways of thinking. And both realists and nominalists agreed that syncategorematic words could “consignify,” that is, participate in forming signiﬁcant wholes. There is even another sense of the word “consigniﬁcation.” In addition to its basic signiﬁcation, a word can have some consigniﬁcation that further determines its content. Especially thirteenth-century authors often use this

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approach to explain the role of features like case endings and—the most discussed example—tenses. The idea is that the actual occurrences of words get richer contents than bare lexical words.

3.5. Supposition Denotation was ﬁrst discussed by means of appellation, a notion borrowed from the “appellative nouns” of grammar. Appellation is the relation between a general term and the things actually belonging under it at the moment of utterance. Often this notion was applied only to the predicates of propositions, but at least from William of Sherwood onward it had unrestricted use. The “property of terms” that caused the most extensive study was supposition. The word derives from grammatical contexts. According to Priscian, a word has a supposition when it is placed as the subject of a proposition. This meaning was usual in the twelfth century. On the other hand, grammar had also formed the idea that a word supposits because it refers to an individual. Gradually this became the central aspect, and the supposition of terms was their way to denote individuals. As the supposition theory expanded, logicians had to seek for suppositions even for other terms than the subjects of propositions—for predicates and parts of complex terms. The question of supposition began to concern the denotation of terms quite generally, and at the same time appellation lost much of its importance, turning into a special case of supposition. Supposition theory was a challenging subject especially because the supposition of terms depends on their position in a proposition. Each word that is not equivocal always has the same signiﬁcation, but its supposition varies in diﬀerent propositions. As Ockham said, “supposition is a property of a term, but only when it is in a proposition.” This compelled the logicians to develop classiﬁcations for the several kinds of supposition. As many scholars have pointed out, precisely this propositional approach was characteristic of the theory of supposition. It must, however, be noted that Peter of Spain admitted even a “natural supposition” (suppositio naturalis) that belongs to a term immediately because of its own signiﬁcation, and this idea was preserved by many Parisian logicians. Thirteenth-century terminist textbooks already include a detailed and clearly developed doctrine of supposition. In Paris during the second half of the century, this tradition had to give way to the modistic inﬂuence, but it survived largely undisputed in Oxford. Subsequently, the ideology of mental terms made it again generally accepted in the beginning of the fourteenth century. After this it became part of the permanent apparatus of late medieval logic. “To supposit” is obviously a technical term; it means something like “to stand for,” and this indeed was an alternative expression. Early terminists like Sherwood thought that supposition belongs only to substantives that are posited as subjects (i.e., subposited under predicates), whereas the denotative

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function of verbs and adjectives is copulation. Soon, however, it became the rule to merge these cases and connect supposition to every categorematic term. The deﬁnition of this general supposition is not evident. Perhaps it is clearest simply to quote concise deﬁnitions from two authors: “When a term stands for something in a proposition in such a way that we use the term for the thing and the term (or its nominative case, if it is in an oblique case) is truly predicated of the thing (or a pronoun referring to the thing), the term supposits for that thing” (Ockham). “All and only those terms supposit which, when something is pointed out by the pronoun ‘this’ or several things by the pronoun ‘these’, can truly be aﬃrmed of that pronoun” (Buridan). We shall try to sketch an overview of the divisions of supposition. First of all, in some cases the supposition is “improper” because the word is used in a nonliteral or metaphoric way; let us concentrate on “proper supposition” only. The deﬁnition of its various types displays both semantic and syntactic factors. It seems that the suppositum of a word can be of three fundamentally diﬀerent semantic kinds, and the supposition is accordingly called either material, simple, or personal.

supposition

material

simple

personal

A term has material supposition (suppositio materialis) when it stands for itself. It must be kept in mind that people in the Middle Ages did not use quotation marks, and material supposition is an alternative way to cope with some problems of use and mention. Sherwood notes that material supposition can be of two types: The word supposits itself either as a sheer utterance or as something signiﬁcant. His examples are “man” in “Man is monosyllabic” and “Man is a noun.” The supposition is simple (simplex) when a word stands for a concept. The classical elementary example is “Man is a species.” To realists, the suppositum then should be equated with some extramental conceptual signiﬁcatum. “If ‘man is a species’ is true, the term ‘man’ supposits its signiﬁcatum. . . . The word ‘man’ does not primarily signify anything singular; thus it signiﬁes primarily something general, and this is a species” (Walter Burley). According to nominalists, the simple suppositum is a mental entity, such as an intention. In the most common case, the word supposits some things that it signiﬁes. For historical reasons, this was called by the surprising name of personal supposition (suppositio personalis). Because both simple and personal supposition are related to the meaning, unlike material supposition, they were often together called formal supposition. On the other hand, nominalists liked to reduce concepts to mental words, so in a sense Buridan and Peter of Ailly are

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more straightforward than Ockham when they do not admit simple supposition as an independent class, counting it as material. The main task is the classiﬁcation of personal supposition, and here syntactic matters interfere. Let us start by providing the next diagram, representing the early state of the classiﬁcation, and then proceed to explanations of its titles.

personal supposition

idscrete

common

edterminate

merelyconfused

confused

confusedandidstrib utiv e

This scheme was in fact given by William of Sherwood, except that he makes the diﬀerence between common and discrete supposition in another context. Discrete supposition (suppositio discreta) belongs to discrete terms: that is, to proper names and demonstrative expressions, like “this man” or “this.” Then the suppositum is the unique object that is signiﬁed. All other terms have suppositio personalis communis. This common supposition is further divided into determinate and confused kinds. The supposition is determinate (determinata) when it allows instantiation, as we might say. But medieval logicians had no such notion. Early authors thought that a determinately suppositing term stands for one determinate object. Ockham improved on this, saying instead that determinate supposition supports descent to singulars, that is, to sentences that are got by substituting singular terms in place of a general term. Thus, in “A man is running” the term “man” supposits determinately because we can legitimately infer that “This man is running or that man is running or. . . ,” and each member of this disjunction in its turn allows ascent back to the original sentence. The supposition is merely confused (confusa tantum) if the proposition does not allow instantiation but is instead implied by its particular instances. As Ockham puts it, “in the proposition ‘Every man is an animal’, the word ‘animal’ has merely confused supposition; for one cannot descend to the particulars under ‘animal’ by way of a disjunctive proposition. The following is not a good inference: every man is an animal, therefore, every man is this animal or every man is that animal or every man is. . .” But it is also worth noticing that “Every man is this animal or that animal or that. . .” does indeed follow. Finally, the word has confused and distributive supposition (confusa distributiva) if it allows descent to all singulars but does not support any ascent. Here the reference concerns “distributively” each and every one of the individuals. For example, in “Every man is an animal” the term “man” has confused

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distributive supposition. It is correct to infer “This man is an animal and this man is an animal and. . .”; or as we write nowadays, “The man a is an animal and the man b is an animal and. . .” On the other hand, none of these singular propositions implies the original sentence. For confused supposition—and especially for the descent to singulars—it is important to decide what the adequate class of individuals ought to be. There was some debate on this point about the correct formulation until it was agreed that the terms had to be duly ampliated, in other words, extended from the basic case of all individuals presently belonging under the concept, such as all actual men, to include all past and future men as well, and in later logic even all possible instances (all possible men). So terms could be examined either with their actual supposition or with an extended supposition. It is not obvious what the motive behind supposition theory really was. Early authors possibly just wanted to capture various kinds of referring. But when Ockham and his followers started to build a more complex theory, with rules of descent and ascent, they probably did pursue something else. Thus, the supposition theory has been compared to the modern framework of quantiﬁcation theory, and clearly it has something to do with the problems of multiple quantiﬁcation and scope—problems that had no explicit place in Aristotle’s logic. Also, it can be seen as an attempt to warrant certain inference types, like those of descent and ascent. The interpretation here is still a matter of controversy.

4. Proposition The core of the medieval theory of judgment centers around the standard deﬁnition of proposition (propositio), deriving from the late ancient period through Boethius. The deﬁnition runs as follows: A proposition is an expression that signiﬁes something true or false. Propositio est oratio verum falsumve signiﬁcans. This deﬁnition accords with the classical theory of deﬁnition. It consists of the generic part (expression) and the distinguishing characteristic (signifying something true or false). For our purposes, however, it seems more useful to divide it into three parts and look at the concepts of truth and falsity separately from the problem of what it is that the proposition exactly signiﬁes.

4.1. Propositions Are Expressions As we have already seen, medieval authors understood logic as a discipline whose subject matter is linguistic discourse. It is well in line with this general approach that they also thought of the propositions studied in logic as sentences actually uttered in some language, typically either spoken or written. As we saw in section 2, a central issue in the determination of the subject matter

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of logic was whether (and in what sense) we could distinguish a special class of mental propositions. Thus, thoughts can be propositional only in so far as they have linguistic structure. A proposition, as the medievals thought of it, is something that is put forward as a sentence, and thus it has actual existence in time and typically also in space. As we will soon see in more detail, it was not the case that medieval logicians would have failed to make a distinction between the actual utterance and that which it expresses. Rather, they simply thought of propositional truth as an issue that comes up in connection with claims actually put forward, not as a property of abstract entities. From the viewpoint of twentieth-century logicians, this feature of medieval conceptual practice has some implications which are worth pointing out, although they are ultimately superﬁcial. According to medieval parlance, a proposition has to exist (i.e., has to be actually put forward in some language) to have a truth value, and it has its truth value in respect to some speciﬁc instant and context. Thus, a proposition like “there are no negative propositions” cannot be true, since it falsiﬁes itself, though it is clear that the case could be as it claims. Also, the same proposition can have diﬀerent truth values in diﬀerent situations. The truth value of “Socrates is seated” varies when Socrates either stands up or sits down. Furthermore, the truth of “this is a donkey” varies depending on what the demonstrative pronoun refers to. Indeed, all logical properties that a proposition has presuppose that it exists; thus medieval logicians often pointed out that their study applies to propositions, not eternally, but on all occasions in which they are put forward.

4.2. Propositions Carry Truth Values Not all signiﬁcative expressions are propositions. Boethius’s textbook distinguishes between “perfect” and “imperfect” expressions with the idea that an imperfect expression does not make complete sense but the hearer expects something more. More important, Boethius continues by listing questions, imperatives, requests, and addresses in addition to indicative sentences that make an assertion and count as propositions. This listing of the kinds of expressions is based on grammatical categories, and similar strategies were also followed in subsequent discussions. It may be of some interest to note that Buridan, for example, takes it to be worth an argument to reject Peter of Spain’s claim that sentences in the subjunctive mood (like “if you were to come to me, I would give you a horse”) do not count as propositions. It seems that medieval logicians disagreed on whether a proposition that is just mentioned without being asserted carries a truth value. The distinction between apprehensive and judicative uses of a propositional complex was rather standard. Ockham, for example, argues that it concerns propositions so that even an apprehended proposition has a truth value, although no stance is taken to it in the apprehension. Judgments, as he sees it, take stances on truth values, but propositions have them by themselves. Buridan, for his part, seems to rely on similar considerations to show that the sentential complex at issue

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is not a proposition. He seems to have thought that sentences are able to carry truth values through being asserted. We will return to this issue in connection with molecular propositions. For the most part, medieval logicians accepted the laws of noncontradiction and the excluded middle. Thus, every proposition has one and only one truth value. But neither of the two principles remained unchallenged. Aristotle’s famous sea battle in De interpretatione chapter 9 was widely discussed and within that debate it was also suggested that contingent propositions about the future perhaps do not yet have a truth value. This did not become the standard view. Similarly, in the widespread discussions concerning limitdecision problems and particularly the instant of change, it was suggested that perhaps contradictories are both true at the instant of change. Instead of accepting this, the standard line was to provide elaborate analyses of limit decision relying on mathematical considerations concerning inﬁnitesimal magnitudes. As is well known, in the more philosophical discussions concerning the nature of truth medieval logicians often put forward the principle of correspondence: Truth is adaequatio rei et intellectus. This deﬁnition was not, however, much used in the speciﬁc context of logic. There the term “truth” was mostly used with the more limited meaning of propositional truth, and it proved diﬃcult to exemplify from the real world anything that corresponded to a propositional complex. Thus, truth could hardly be explained as a relation between a real thing and a proposition. In the Aristotelian approach, things are referred to by using simple terms, and no simple expression—a mere term—can have a truth value. Truth rather arises from “composition” or “division” of terms in a predication, and depends on how this composition or division accords with how things really are. In his Syncategoreumata, Peter of Spain gives an elaborate suggestion that there is some kind of real composition, typically explicable with reference to the way in which everything in the world is composed of matter and form. According to Peter’s suggestion, the truth of a sentence depends on whether this “real composition” is expressed adequately. The standard Aristotelian dictum, “it is because the actual thing exists or does not that the statement is called true or false” (Cat. 12; 14b21–22), was not always understood in this manner. A typical way of explicating the claim that a proposition is true was to say that it “signiﬁes as it is” (signiﬁcat sicut est) or something to the same eﬀect. By such formulas logicians tried to avoid committing themselves to positing any real entity with which the true proposition would have direct correspondence. Instead, the expression often worked in a way analogous to what has lately been called “disquotational”: allowing transformation of the claim “p is true” into the simple claim “p.” Ockham’s Summa logicae (I, 43) contains an interesting discussion of in what sense truth is predicated of a proposition. In his opinion, it is not a real quality of the proposition. This can be proved by the fact that a proposition may change from truth to falsity by fully external change. For example, when something ceases to move, the truth value of the proposition “this thing

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is moving” changes without the proposition itself changing. According to Ockham’s explanation, “true” is a connotative term signifying that things are as the proposition signiﬁes. However, this remark leaves open the issue of what it is for things to be as the proposition signiﬁes.

4.3. Are There Any Propositional Signiﬁcates? Stoic logic, and in its wake important early medieval authors like Boethius and Peter Abelard, made a distinction between a declarative sentence and its dictum, or that which “is said.” Thus, the dictum expresses, or it simply is, the content of the proposition without being itself a proposition. For example, the proposition “Socrates is seated” (Socrates sedet) says or puts forward the dictum “that Socrates is seated,” which in Latin is typically expressed as an accusative plus inﬁnitive construction (Socratem sedere). Over the centuries, many logicians discussed the status of the dictum. Also, the related distinction between a proposition (as an expression) and its total signiﬁcate (in distinction from the separate signiﬁcates of its constituents) became a topic of an interesting dispute toward the second quarter of the fourteenth century. In his early work, Commentary on the Sentences, Ockham puts forward a theory according to which belief always concerns a proposition formulated in mental language. That is, when a person assents to something, he has to formulate a mental proposition expressing that which he assents to. He then reﬂexively apprehends the proposition as a whole and assents to it. It seems that Ockham’s motivation for this theory was the view that there is no way to grasp propositional content apart from formulating a proposition in mental language. Thus, if objects of beliefs are true or false, they must be formulated in mental language. Several contemporaries of Ockham did not straightforwardly accept the idea that the object of belief must always be an actually formulated proposition. Even Ockham himself shows some hesitation toward this theory in his Quodlibetal questions, which he composed later. It seemed to many authors that when one believes, for example, that God exists or that a man is running, the object of belief is somehow out in the world and not merely a proposition in the mind. The idea is that people do not always believe in sentences, but at least sometimes it should rather be said that they believe things to exist in a certain way. This consideration made medieval logicians search for something like propositional content outside the mind and a number of diﬀerent theories of how it could be found emerged. In Walter Chatton’s theory, the object of the assent has to be some extramental thing. If you believe that a man is running, the object of your belief is the man at issue. Thus, the signiﬁcate of the proposition “a man is running” is the man. Chatton recognized that his theory has the problematic consequence that the simple term “a man” and the propositional complex “a man is running” signify the same thing. As Chatton saw it, the diﬀerence in these two expressions is not in what they signify but in how they signify it.

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The terminology he used in this connection refers back to modist grammatical theories. It seems that both Ockham and a younger contemporary, Adam Wodeham, reacted against Chatton’s theory. In his Quodlibetal disputations, Ockham makes a further distinction concerning propositional assent, in eﬀect allowing it to be the case that you give assent without reﬂexively considering a mental proposition. In such a case, you simply form the proposition and give your assent in an unreﬂective way as connected to rather than directed at the mental proposition. As Ockham curiously points out, this kind of assent is not at issue in scientiﬁc knowledge, only in beliefs of ordinary life. According to Ockham’s obscure remarks, nothing is the object of this kind of assent. Wodeham seems to have continued from this basis in his theorizing. He wanted, though, to allow that even the nonreﬂective kind of assent is about something, and the signiﬁcate of the proposition appeared to be a suitable candidate for an object. However, its metaphysical status seemed quite unclear to the medieval mind. According to Wodeham, the signiﬁcates of propositions need to be categorically diﬀerent from the signiﬁcates of the terms. As he put it, propositions do, of course, signify the things signiﬁed by their terms, but no thing or combination of things is the adequate total signiﬁcate of the propositional complex. The adequate signiﬁcates of propositions are such that they can only be signiﬁed by propositions; even further, they do not belong to any of the Aristotelian categories nor can they be called things. Wodeham’s theory became known as a theory endorsing “complex signiﬁables” (complexa signiﬁcabilia). Such entities were rejected by most subsequent logicians, including major ﬁgures like John Buridan, but accepted by some, most famously by Gregory of Rimini—in subsequent discussion, the theory became known as his theory. In the third quarter of the fourteenth century, discussion of what propositions signify was abundant. Is it something like a mode of being? Or just a mental act of composition? Do propositions in fact signify anything more than just the things denoted by the terms, or perhaps even just the thing denoted by the subject? The fourteenth-century discussion concerning complex signiﬁables seems to have made it clear to late medieval logicians that their logic was based on a metaphysical view of the world as consisting of things and not of states of aﬀairs. The constituents of the world could be referred to by terms, but to make claims about the world, a diﬀerent kind of mental act was needed. Paradigmatically, one had to construct a complex expression asserting a composition of multiple entities.

4.4. Predication In Aristotelian logic, the ground for all judgments is laid by the predicative structure, where two terms are either joined or disjoined as the subject and the predicate. After Boethius, it remained customary in the Middle Ages to treat aﬃrmative predication and negative predication as two diﬀerent kinds

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of statement, and also to take negation simply as “destroying the force of the aﬃrmation.” Thus, it is not necessary here to treat negative predication distinctly from the basic aﬃrmative case. The aﬃrmative predication consists, as already Boethius recognized, not only of the two terms but also of the copula. Thus, when Aristotle remarks that a predication can be constructed either with a verb (e.g., “a man runs,” homo currit) or with a participle (e.g., “a man is running,” homo currens est), this was normally interpreted as meaning that the latter form is to be taken as primary. In the latter, the copula “is” was said to be added as a third part (tertium adiacens). In Latin, the copula was of course the standard verb “to be” (esse), which was also used in the simple existential claim “a man exists” (homo est). This use of est as secundum adiacens had to be explained since it appeared to lack either the copula or the predicate. As Boethius saw it, the verb serves here a double role. This solution was accepted in the Middle Ages, and thus there was no need to see it as an altogether diﬀerent kind of statement. Buridan even argued against ordinary linguistic practice that logically one should prefer the formulation “a man is a being” (homo est ens). Given that the copula joins the two terms into a predicative proposition and gives the sentence its assertive character, it still remains unclear exactly how it joins the terms together. It seems that this was one of the most fundamental points of disagreement among medieval logicians. For modern scholars it has proved rather diﬃcult to ﬁnd a satisfactory description of how the simple predication was understood in the Middle Ages. One crucial nontrivial issue seemed clear, though. Throughout the Middle Ages, it was commonly assumed that in the absence of speciﬁc contrary reasons, the verb “to be” even as the copula retains its signiﬁcation of being. Thus, all aﬃrmative predications carry some kind of existential force, while negative predications do not. In an aﬃrmation, something is aﬃrmed to exist; a negation contains no such aﬃrmation of existence. But beyond this simple issue, interpretations of the nature of predication seem to diverge widely. Most of the twentieth century discussions of the exact content of the diﬀerent medieval theories of predication have been based on the Fregean distinction between the diﬀerent senses of “to be.” Scholars have distinguished between inherence theories and identity theories of predication, despite the evident threat of anachronism in such a strategy. For want of a better strategy, we also have to rely on that distinction here. But instead of trying to classify authors into these two classes, let us simply look at the motivations behind these two rather diﬀerent ways of accounting for what happens in a predication. The idea of the inherence theory is that the subject and the predicate have crucially diﬀerent functions in the predication. While the function of the subject is to signify or pick out that which is spoken of, the function of the predicate is to express what is being claimed of that thing. The idea is, then, that the Aristotelian form signiﬁed by the predicate inheres in the thing signiﬁed by the subject. Peter of Spain seems to defend this kind of theory

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of predication when he tries to show that the copula signiﬁes that relation of inherence obtaining between matter and form, or between a subject and its accident. Aquinas seems to follow this account. Scholars have disagreed about Abelard’s theory, and it indeed seems that his rich discussion of the topic provided grounds for several kinds of diﬀerent subsequent theories. On the one hand, he seems to lay the basis for the inherence theory. On the other, he defends the idea that to look at the exact truth conditions of a predication like “a man is white” (homo albus est), it should be analyzed into a fuller form “that which is a man is that which is white” (idem quod est homo est id quod album est; Logica ingredientibus 60.13). With such a formulation he seems to have in mind the idea that for the aﬃrmative predication to be true, the subject and the predicate must refer to the same things. This is commonly called the identity theory of predication. Abelard’s “that which is” (quod est) formulation remained part of the actual practice of logical writing for several centuries. It can be found commonly from logical texts throughout the Middle Ages, although it was not always oﬀered as an explanation of the truth conditions of predication in general. The formulation has the special feature that it appears to give the subject and the predicate of a predication a similar reading. Both are to be understood as referring to some thing, and then the assertion put forward in the proposition would be the identity of these two things. This seems to amount to the identity theory of predication in Fregean terms. In the fourteenth century, both Ockham and Buridan seem to have quite straightforwardly defended the idea that the Aristotelian syllogistic is based on identity predications. As they put it, the simple predication “A is B” is true if and only if A and B supposit for the same thing. For the most part, truth conditions of diﬀerent kinds of propositions can be derived from this principle. Somewhat interestingly, Ockham nevertheless recognizes the need of basic propositions expressing relations of inherence. For Ockham, the predicate “white” is a so-called connotative term, and therefore a somewhat special case. According to his analysis, the predication “Socrates is white” (Sortes est albus) should be analyzed into “Socrates exists and whiteness is in Socrates” (Sortes est et Sorti inest albedo). In his metaphysical picture, Ockham allows both substances and qualities to be real things, and if one is allowed to use only so-called absolute terms that supposit in a sentence only those things which they signify, the relation of inherence (inesse) is not expressible with an identity predication. Qualities inhere in substances, but they are not identical with substances. The whiteness at issue in the claim “Socrates is white” is not Socrates, it is a quality inhering in Socrates. Socrates is not whiteness even if he is white. In his Summa logicae, Ockham has some special chapters on propositions involving terms in oblique cases (in cases other than the nominative). The just-mentioned proposition “whiteness is in Socrates” is a paradigm case of

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such a proposition (in Latin, the subject has to be in the dative case Sorti; in English, the eﬀect of the case is represented with the preposition “in”). Furthermore, all propositions involving the terms that Ockham calls “connotative” require in their logical analysis that oblique cases are used. The main claim of the short chapters of Summa logicae addressing propositions containing such terms is that their truth conditions cannot be given by the simple rule of thumb that the subject and the predicate must supposit for the same thing in an aﬃrmative sentence. Consequently, the rules for syllogisms formulated with such propositions are also abnormal. In eﬀect, Ockham excludes propositions with oblique terms from the ordinary syllogistic system, thus leaving a surprising gap in his logical system. In his logic, Buridan proceeds diﬀerently. For the purposes of the syllogistic system, he requires that all propositions should be analyzed into a form where truth conditions can be given through variations of the rule that in aﬃrmative sentences the subject and the predicate supposit for the same thing. This allows him to apply the standard syllogistic system to all propositions. The solution is at the price of greater semantic complexity. Buridan has to allow so-called connotative terms (including, e.g., many quality terms like “white”) as logically simple terms despite their semantic complexity. Both Ockham and Buridan apparently thought that identity predication is the logically privileged kind of predication. Nevertheless, they also both accepted the Aristotelian substance-accident ontology to such an extent that they had to ﬁnd ways of expressing the special relation of inherence. While Ockham allowed exceptions to the syllogistic through irreducible propositions expressing inherence, Buridan opted for a syllogistic system with obviously complex terms expressing inherential structures.

4.5. Negations As the medieval logicians saw it, the simple predication “A is B” contains altogether four diﬀerent places where a negation can be posited: 1. It is not the case that A is B. 2. A is not B. 3. Not-A is B. 4. A is not-B. It is of course clear that 1 is closest to the negation used in twentieth-century logic. In it, the negation is taken to deny the whole proposition. According to Boethius’s commonly accepted formulation, the force (vis) of the predication is in the copula, and hence denying the copula denies the whole proposition. Thus, the negation in 2 has the same eﬀect as in 1. (As Buridan notes, for quantiﬁers and other modiﬁers, the location of the negation may still make a diﬀerence.)

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2 is the standard negation of medieval logic. It is the direct contradictory of the corresponding aﬃrmative predication. In particular, it is noteworthy that this negation does not carry any existential presuppositions. Thus, “a chimera is not an animal” is true simply because no chimeras exist. 3 and 4 are aﬃrmative statements containing an inﬁnite term, as terms of the type “not-A” were called. In these cases, the negation is connected directly to a term and not to a proposition. An inﬁnite term was taken to refer to those things to which the term itself does not refer. Thus, not-man refers to anything that is not a man. Because these negations do not make the proposition negative, 3 and 4 carry existential content: Some B must exist for 3 to be true, and some A for 4 to be true. Although the syntactic idea of attaching a negation to a term was universally accepted in the Middle Ages, logicians seem to have disagreed about whether the term-negation should be taken to be essentially the same negation as the propositional one but in a diﬀerent use. In his Syncategoreumata, Peter of Spain seems to reject this idea. He presents the two negations as genuinely diﬀerent in themselves. His discussion is connected to a theory where even simple names and verbs signify in a composite sense. Thus, the idea is that “man,” for example, means a composition of a substance with a quality, a substance having the quality of being human. Thus, the inﬁnite term “not-man” signiﬁes a substance that has not entered into a composition with the quality of being human. Ockham, for his part, preferred to reduce negating a term to ordinary propositional negation, claiming that the meaning of “not-man” can be explained as “something which is not a man.” Buridan allows inﬁnite terms a signiﬁcant role in his syllogistic system, and thus seems to go back to thinking that the negation involved in them is fundamentally distinct from that which he calls “negating negation”—that is, the propositional negation that has power over the copula. Given that negations can be put in many places even in a simple predication, medieval logicians gained skill in handling combinations of diﬀerent negations. The idea that two negations cancel each other (provided that they are of the same type and scope) was also well known.

4.6. Quantiﬁers Aristotelian predications typically have so-called quantity. Medieval logic commonly distinguished between universal (“every A is B”), particular (“some A is B”) and indeﬁnite propositions (“A is B”). As Boethius already pointed out, the indeﬁnite predication that lacks any quantiﬁer is equivalent to the particular one. Some logicians did specify certain uses that violate this rule of thumb, but such exceptions are rare. In addition to quantiﬁed and indeﬁnite predications, singular predications were also discussed (e.g., “Socrates is running”). They had a subject term that was a proper name or some suitable demonstrative pronoun.

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Basic quantiﬁed predications were given vowel symbols as mnemonic labels from the ﬁrst two vowels of the Latin verbs “aﬃrm” (aﬃrmo) and “deny” (nego). Thus, the universal aﬃrmative was shortened as AaB, where A is the subject and B the predicate. Similarly, the particular aﬃrmative was AiB, the universal negative AeB and the particular negative AoB. These four predications were further organized into the so-called square of opposition to show their interrelations.

AaB

AeB

AiB

AoB

The upper two, the universal aﬃrmative and the universal negative, were called contraries; they cannot be true simultaneously, but could both turn out to be false. Similarly, the particular aﬃrmative and particular negative were subcontraries; they cannot be false simultaneously, but both could turn out true. The relation between the universal and the particular was called subalternation on both sides; the particular follows from the universal but not vice versa. The propositions in the opposite corners were called contradictories, since one of them had to be true and the other false. In the Middle Ages, a substantial amount of ink was used discussing whether a universal aﬃrmative could be true when only one thing of the relevant kind exists. The paradigm example was “every phoenix exists,” and many logicians rejected it with the requirement that there must be at least three individuals to justify the use of “every.” Toward the end of the thirteenth century this discussion seems to disappear, apparently in favor of the view that one referent is enough; the existential presupposition was never dropped, however. Another issue of detail that was also widely discussed later was the case of universal predications of natural sciences, which capture some invariable that does not appear to be dependent on the actual existence of the individuals at issue. A suitable example is “every eclipse of the moon is caused by the shadow of the Earth.” According to a strict interpretation of the existential presupposition, such predications prove false most of the time—which seems somewhat inconvenient. Two fundamentally diﬀerent suggestions for a better reading of the predication were put forward. Ockham seems to favor the idea that what really is at issue here is the conditional proposition “if the moon is eclipsed, the eclipse is caused by the shadow of the Earth.” This solution draws on the traditionally recognized idea that the conditional is implied by the universal aﬃrmation. However, Buridan opted for another solution. As he reads the universal aﬃrmation at issue, its verb should be read in a nontemporal sense. In such a reading, past and future eclipses also provide instances satisfying the diluted existential presupposition.

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4.7. Complex Terms Medieval logicians allowed that not all terms of a predication are simple. A predication can, of course, always be divided into the subject and the predicate together with the copula and the appropriate quantitative, qualitative, and modal modiﬁers. But the two terms may possibly be further analyzable (e.g., “a just man is talking,” where the subject “just man” consists of two parts), and indeed this was a topic that attracted much attention during the Middle Ages. Toward the middle of the fourteenth century, discussion on this topic resulted in a detailed theory on the interaction between diﬀerent kinds of combinations of categorematic and syncategorematic elements that can be found in a predication. To tackle with issues of scope an elaborate system of word order rules was introduced for the technical Latin used by logicians. It seems that thirteenth-century logicians did not take it to be a serious problem that complex predications do not behave in ways that suit the needs of syllogistics. Following Aristotle’s remark (Analytica priora I, 36; 48b41–49a5), syllogisms with oblique terms in the various cases were usually discussed separately, and thus it seems that the thirteenth-century logicians probably thought that more complicated predications do not necessarily ﬁt into the ordinary syllogism. As we already noted, Ockham makes this slight inconvenience clear in his Summa logicae. It seems that Ockham fully understood that the traditional syllogistic logic does not always work if actually used linguistic structures are given full logical analyses. Also, he explicitly allows that there is no general way of giving the truth conditions of the various kinds of complex predications; in particular, he points out that even as simple a construction as the genitive case makes the standard truth conditions of identity predications inapplicable. “The donkey is Socrates’s” is an aﬃrmative predication. However, its truth requires, but it is not suﬃcient for it, that the subject and predicate supposit for diﬀerent things (“donkey” for a domestic animal owned by a person, and “Socrates” for the owner of the animal). More generally, Ockham thought that mere identity predications are not suﬃcient to explain the expressive power of the actually used language. A richer variety of propositions had to be accounted for, but in fact they found no place in syllogistic logic. Thus, syllogistic logic was not a complete system covering all valid inferences. After Ockham, Buridan took another approach. As he saw it, all categorical propositions can be reformulated as straightforward Aristotelian predications ﬁtting the needs of the ordinary syllogism and having the rule of identity or nonidentity of supposition as the criterion of truth. For this purpose, he had to modify the traditional systems of combining diﬀerent categorematic and syncategorematic elements so that they appear as geared toward building up terms whose suppositions can be decided. Perhaps most important, he saw that he could not assume that standard Aristotelian predications would be found as the end results of logicolinguistic analysis. Rather, he understood the building blocks of the syllogistic system—identity predications—to be more

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or less artiﬁcial constructions built from complex terms. For example, for the purposes of syllogistic logic the sentence “the donkey is Socrates’s” must be read as “the donkey is Socrates’s thing,” although the predicate “Socrates’s thing” clearly is not a simple term of the ideal mental language. Buridan did not assume that all mental or spoken propositions would be identity predications. Rather, he assumed that for the purposes of syllogistic logic, any proposition could be transformed into an equivalent identity predication. By such means, syllogistic logic could serve as a complete system containing all inferences. Buridan’s strategy involved, therefore, a massive expansion of the syllogistic system toward incorporating increasingly complex terms. Whereas logicians up to Ockham had accepted that a wide variety of propositions are nonstandard from the viewpoint of syllogistic logic, Buridan builds rules on how the content of these nonstandard propositions can be expressed by standard structures involving very complex terms. Buridan provides elaborate rules concerning complex terms. The idea is to show how nouns and verbs interact with diﬀerent syncategorematic expressions and produce terms that ﬁt into standard Aristotelian predications. In Buridan’s view, all propositions can be transformed so that the truth conditions can be expressed through the criteria of an identity predication. In aﬃrmative sentences, the terms must supposit for the same thing, while in negative sentences, they must not supposit for the same thing. To see the full strength of Buridan’s new system, let us consider a somewhat more complicated example. Buridan analyzes “Each man’s donkey runs” (cuiuslibet hominis asinus currit) in a new way. Traditionally, this Aristotelian sentence was understood as a universal aﬃrmation consisting of the subject “man” in the genitive case, and a complex predicate. This analysis makes the subject supposit for men, and the predicate for running donkeys so that the assertion cannot be read as an identity predication. Thus, standard syllogistics are not applicable to a proposition like this. Most logicians up to and including Ockham seem to have been satisﬁed with the implied limitations of the syllogistic system. Buridan, however, analyzes the proposition as an indeﬁnite aﬃrmation. It has a complex subject “each man’s donkey,” which includes two simple categorematic terms (“man,” “donkey”), a marker for the genitive case (the genitive ending “’s”), and a quantiﬁer (the universal sign “each”). The quantiﬁer does not make the proposition universal, because it has only a part of the subject in its scope and must therefore be understood as internal to the subject term. As a whole, the subject supposits for sets of donkeys such that each man owns at least one of the donkeys in the set. The predicate of the proposition is a simple term, “running.” It supposits for sets of running things. Construed in this way, the predication can be evaluated with the standard criteria of truth, and standard syllogistics can be applied to it. It seems clear that Buridan took seriously the programmatic idea that the Aristotelian syllogistic system should provide a universal logical tool which did not allow major exceptions to behave in nonstandard ways. But instead of analyzing complex propositions into combinations of predications with simple

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terms, Buridan provides elaborate rules concerning the ways in which complex terms are built.

4.8. Hypothetical Propositions In the Middle Ages, not only conditionals but also conjunctions and disjunctions were called hypothetical (hypothetica) propositions. Otherwise the treatment of conditionals and disjunctions causes no surprises to a modern reader familiar with basic propositional logic. Walter Burley, for example, gives the following account. Conjunctions are propositions consisting of two further propositions that are joined with the conjunction “and” or something equivalent. Their truth conditions require that the propositions thus joined are true. Negating a conjunction makes reference to another type of hypothetical, namely disjunction, because denial of a conjunction requires only that one or the other of the conjuncts is denied. Disjunction, for its part, is deﬁned in the inclusive manner: Its truth conditions require that one of the parts is or both of them are true. Denial of a disjunction produces a conjunction, and as Burley notes, denial of a disjunction of contradictories (e.g., “Socrates runs or Socrates does not run”) produces a conjunction which includes contradictories. Certain interesting issues are raised in more detailed discussions of conjunctive and disjunctive propositions. One such is the nature and exact content of conjunctive and disjunctive terms used in propositions like “every man runs or walks.” Are they reducible to conjunctive and disjunctive propositions and why not exactly? How ought they be accounted for in inferential connections? Another, more philosophical issue was the question of whether the parts of conjunctions and disjunctions are strictly speaking propositions. As Buridan notes, the “force of the proposition” (vis propositionis) in a disjunction is in the connective, and thus not in either of the disjuncts. Hence, it is only the whole and not the parts that carry truth value in the composition. When someone utters a disjunctive proposition consisting of contradictories, he does not, according to Buridan, say anything false, although one of the parts would be false if uttered as a proposition. Thus, hypothetical propositions do not, strictly speaking, consist of categorical propositions but of linguistic structures exactly like categorical propositions. It seems that medieval logicians treated conjunctions and disjunctions in a straightforwardly truth-functional manner. It seems equally clear that their treatment of conditionals diﬀers from the twentieth-century theory of material implication. Indeed, in the Middle Ages theory of conditionals was mainly developed in connection with a general theory of inference, under the label “consequences.” Conditionals were taken to express claims concerning relationships of inferential type. Medieval logicians also distinguished further types of hypothetical propositions. In Buridan’s discussion (1.7), altogether six kinds of hypothetical propositions are accounted for, including conditional, conjunctive, disjunctive,

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causal (“because the sun shines above the Earth, it is daytime”), temporal (“Socrates runs when Plato disputes”), and local ones (“Socrates runs to where Plato disputes”). Buridan even vaguely suggests that perhaps other Aristotelian categories may also give rise to hypothetical propositions in a way similar to temporal and local hypotheticals. It is clear that this approach to hypothetical propositions relies on other ways of combining propositions than just the truth-functional ones. The connective may express something more than just a truth function.

4.9. Modal Operators Logical issues connected to possibility and necessity, which in the twentieth century have been studied as alethic modal logic, were a central research topic in late medieval logic. These modal terms were usually discussed together with other modiﬁers operating in similar syntactic roles. For example, the twentiethcentury ﬁelds of study known as deontic logic (dealing with permissibility and obligation) and epistemic logic (dealing with concepts of knowledge and belief) have their counterparts in the Middle Ages, where these issues were discussed together with possibility and necessity. Most medieval logicians discussed altogether four modal operators crucial to modern alethic modal logic: possibility, impossibility, contingency, and necessity. These were deﬁned in relation to each other so that the necessary was usually taken to be possible but not contingent, whereas the impossible was taken to be neither possible nor contingent. Like the square of opposition of simple predications, modal predications were often organized into a square of modal opposition following Aristotle’s presentation in De interpretatione (ch. 13). It is particularly noteworthy that following Aristotle’s model, the square of modal opposition typically contained just the modal operators, not complete sentences. In a somewhat schematized way, the basic square can be illustrated as follows:

N ¬M¬

N¬ ¬M

¬N¬ M

¬N M¬

In this square, the relations of contrariety, subcontrariety, subalternation, and contradiction were said to behave as they would in the basic square of simple predications. Following Aristotle, medieval logicians made a distinction between two ways of understanding a modal predication to make sense of examples like the

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possibility that someone sitting walks. Understood de dicto, there is no such possibility. The sentence “someone sitting walks” is impossible. But understood de re, there is such a possibility, since the person who is sitting may be able to walk. Thus, if the modal predicate “can walk” is understood de re, or as concerning the person who is actually sitting now, the sentence might be true. All subsequent major logicians discussed this distinction in some form or other. In the sections concerning modal propositions in Ockham’s Summa logicae, it is clear that the de dicto reading is given logical priority. Using another traditional terminology, Ockham prefers to call it the composite sense (sensu composito) and does not oppose it to a de re reading but to the roughly similar divided sense (sensu diviso). Ockham apparently thinks that modality is a property of propositions rather than terms, and aims at reducing readings sensu diviso to sensu composito through analyzing modal propositions in sensu diviso into propositions sensu composito. There are three main models by which modern scholars have been able to account for the way in which medieval logicians understood what it means to say that something is possible: the statistical model, the potency model, and the consistency model. During the medieval period, the modal concepts used by particular logicians typically ought not to be explained through reference to a single model. Rather, these three diﬀerent strands of thought have inﬂuenced to varying degrees the modal thinking of diﬀerent medieval logicians. The basic intuition explained by the statistical model is that all and only those things seem to be possible which sometimes occur. If something never happens, it means that it can’t happen. The potency model, for its part, explains the intuition that whether something is possible depends on whether it can be done. For something to be possible it is required that some agent has the potency to realize it, though it is not required that the thing is actually realized. However, because normally there are no generic potencies that remain eternally unrealized (why should we say that humans can laugh if no one ever did?), this model becomes clearly distinct from the statistical model only when God’s omnipotence is understood to reach wider than just actual reality. God could have created things or even kinds of things which he never did, and these things remain therefore eternally unrealized possibilities. It seems that throughout the Middle Ages, God’s omnipotence was thought to be limited only by the law of noncontradiction. Contradictions are not real things, and therefore God’s power is not limited, although we can say that he cannot realize contradictions. This consideration seems to have been one of the reasons why logicians in the thirteenth century increasingly used the criterion of consistency to judge claims about possibility. But it seems that the development of syntactic logical techniques also made it natural to demarcate a class of propositions that are impossible in the traditional sense but nevertheless seem to involve no contradiction (e.g., that man is irrational, or that man is not an animal). Especially the traditional technique of laying down a false or even impossible thesis for an obligational disputation (see the

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following) seems to have encouraged consideration of consistent propositions or sets of propositions that are in some sense impossible. Some authors, like Boethius of Dacia, even use the special expression compossibilitas to refer to this kind of concept of consistency as distinct from possibility. As is well known, from Duns Scotus onward, several logicians made this kind of concept of consistency crucial for possibility in general. The medieval discussion can be characterized as aiming at ﬁnding a way to account for these rather distinct intuitions of what it means to say that something is possible. A certain shift in emphasis is visible. Whereas earlier authors pay more attention to the statistical idea at the expense of consistency, later authors tend to neglect or argue against intuitions captured in the statistical model while emphasizing consistency as the criterion of possibility.

5. Classical Forms of Inference 5.1. Syllogisms We must next turn to the “theory of inference.” Ignoring probable inferences for now, we can say that this part of logic tries to describe how some propositions necessarily follow from others, from their premises. The propositions of a certain sequence have such properties that the last one must follow necessarily from its predecessors. An important type of inference is the syllogism—the inference on which Aristotle concentrated in his Prior Analytics. The syllogism was the best-known and paradigmatic type of inference throughout the Middle Ages. In the thirteenth century, when logicians studied demonstrative inference, they were almost exclusively concerned with syllogistics; but afterward, when a more general inference theory developed, the policies of various authors diﬀered widely. Thus Ockham still devotes the main part of his inference theory to a detailed analysis of syllogistics, and so does Buridan, whereas Burley regards it as a well-known special case of the more interesting subject of inference in general. The syllogism is probably the most famous item of “traditional” logic, but actually it has a not very dominant place in the works of medieval logicians. (For instance, in the Logica magna of Paul of Venice it is the subject of only one of 38 treatises.) However, it is systematically and historically so important that we must discuss it in relatively more detail. All authors start by presenting or elaborating the highly condensed deﬁnitions in the beginning of the Prior Analytics. The often-quoted characterization in An. Pr. 24b19–20 says: “A syllogism is a discourse (oratio) in which, certain things having been supposed, something diﬀerent results of necessity because these things are so.” In a broad sense, any formally valid inference could be called a syllogism. But in the stricter sense, a syllogism has precisely two “things supposed,” two premises. There has been quite a lot of discussion on

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whether Aristotelian syllogisms are better understood as conditionals (“if p and q then r”) or as deductive inferences (“p, q; therefore r”). The latter interpretation is perhaps more popular nowadays, and apparently it is also the correct way to see medieval syllogistics, at least in its classical stage. (Obviously the two things have a systematic correspondence, the relation that we nowadays call the deduction theorem, and many medieval authors were fully aware of it.) This means that syllogisms are like natural deductions of present-day logic. Though medieval syllogistics followed Aristotle closely, there were some formal diﬀerences. Thus Aristotle—for special reasons—had formulated his syllogistic propositions as “Y belongs to X,” “mortal belongs to man.” This manner was never adopted in Latin; the medieval logicians wrote just “X is Y ,” “man is mortal.” Aristotle himself had brought the theory of nonmodal syllogistics to such perfection that there was little room left for initiative or disagreement. However, medieval texts produced a more systematic form for the theory, obviously aiming at didactic clarity. A syllogism consists of two premises and one conclusion; the ﬁrst premise is called major, and the second premise minor. Each proposition has two terms, a subject and a predicate, connected by a copula. But the two premises have a term in common, the so-called medium, and the terms of the conclusion are identical with the two other terms of the premises. Syllogisms can then be classiﬁed according to their conﬁguration Subj–Pred into four diﬀerent ﬁgures as follows:

major minor

I

II

III

IV

M –B A–M

B–M A–M

M –B M –A

B–M M –A

Further, each syllogistic proposition belongs to its type a, e, i, or o because of its quality and quantity: They are aﬃrmative or negative, universal or particular. If we proceed by deﬁning that the conclusion must always have the structure A–B, then it is obvious that each ﬁgure includes 43 = 64 alternative combinations, and the total number is 256. But this is not exactly the classical method, so let us have an overview of the syllogism as it was usually presented. Medieval logicians have a full and standard apparatus for syllogistics as early as the ﬁrst terminist phase. They list the same valid syllogisms, usually in the same order, and also call them by the same names. The textbooks of William of Sherwood and Peter of Spain supply these names, which stem from some unknown earlier source and even the famous mnemonic verse composed on them. The names have three syllables, one for each sentence, containing the logical vowels a, e, i, and o. (In the following list of syllogisms, we mention these names that have recurred in all later logic.) The valid syllogisms were known as moods.

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The ﬁrst ﬁgure includes the four famous syllogisms from which Aristotle starts: every M is B, every A is M , therefore every A is B

(Barbara)

no M is B, every A is M , therefore no A is B

(Celarent)

every M is B, some A is M , therefore some A is B

(Darii)

no M is B, some A is M , therefore some A is not B

(Ferio)

The second ﬁgure has four moods: no B is M , every A is M , therefore no A is B

(Cesare)

every B is M , no A is M , therefore no A is B

(Camestres)

no B is M , some A is M , therefore some A is not B

(Festino)

every B is M , some A is not M , therefore some A is not B

(Baroco)

Furthermore, the third ﬁgure contains six moods: every M is B, every M is A, therefore some A is B

(Darapti)

no M is B, every M is A, therefore some A is not B

(Felapton)

some M is B, every M is A, therefore some A is B

(Disamis)

every M is B, some M is A, therefore some A is B

(Datisi)

some M is not B, every M is A, therefore some A is not B

(Bocardo)

no M is B, some M is A, therefore some A is not B

(Ferison)

After the Renaissance, logicians continue by giving the ﬁve moods of the fourth ﬁgure: Bramantip, Camenes, Dimaris, Fesapo, and Fresison. That, however, is not the orthodox Aristotelian way. Aristotle knew inferences like these but did not include a fourth ﬁgure in his theory. Instead, he wanted to place these syllogisms into the ﬁrst ﬁgure. Following his remarks, Theophrastus developed a clear account of the matter, and it was well known in the Middle Ages through Boethius. In Theophrastus’s account, the major term need not be the predicate in the conclusion, which can also have the inverted order B–A. This gives us the ﬁve so-called indirect moods of the ﬁrst ﬁgure: every M is B, every A is M , therefore some B is A no M is B, every A is M , therefore no B is A every M is B, some A is M , therefore some B is A

(Baralipton) (Celantes) (Dabitis)

every M is B, no A is M , therefore some B is not A

(Fapesmo)

some M is B, no A is M , therefore some B is not A

(Frisesomorum)

This method can replace the fourth ﬁgure, though it does introduce a certain unsatisfactory asymmetry.

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The problem of the missing ﬁgure has caused much scholarly debate that we cannot enter into here. Medieval logicians were quite aware of the problem since they had seen at least Averroes’s comments on the fourth ﬁgure. Arguments were often given to refute “objections” questioning the suﬃciency of three ﬁgures. Apparently medieval authors were unanimous in thinking that the fourth ﬁgure could be eliminated with the indirect moods of the ﬁrst ﬁgure. They either said that there were only three ﬁgures, or more precisely, like Albert of Saxony, that the fourth is superﬂuous. It is noteworthy that they did not regard the order of premises as essential. Thus there are 19 valid syllogistic moods. A small addition was obtained by allowing the ﬁve “subaltern” moods, which yield a particular conclusion though a universal one would be valid too. For example, Barbari instead of Barbara leads to “some A is B.” This step would not be accepted in modern logic where universal implications have no existential import, and it indicates clearly that medieval syllogistics assumed that every term really had existential reference. Aristotle had only implicit allusions to singular propositions in syllogisms, and it was a good achievement that medieval logicians constructed a full and systematic theory of singular syllogisms. Ockham was the most active worker here. He emphasizes that the singularity of terms makes no diﬀerence for the validity of inference. This amounts to a considerable reinterpretation of the whole notion of a propositional term. Moreover, he gives explicit cases of singular syllogisms in each ﬁgure, for example, the third ﬁgure “expository syllogisms” like “x is B, x is A, therefore some A is B.” (For nominalists like him, the question had special epistemological relevance because of the basic status of truths about individuals.) Some later Ockhamists even drew a dichotomy across the whole syllogistics between expository syllogisms and those with general mediums.

5.2. Theory of Syllogistics Syllogistics, undoubtedly, is just a small portion of logical inferences, but systematically it is extremely important. The unique thing in classical syllogistics is that it was a formal theory. Its results are not separate truths achieved by trial and error; instead, they are derived in a deductive manner. This had largely been achieved already in the Prior Analytics and continued by ancient commentators. Medieval logicians were very interested in this project. The most important tool here is conversion. It is a completely general method that pertains to all propositions of the S–P form, but it ﬁnds good use in syllogistic theory. Brieﬂy, in a conversion the subject and the predicate change places, and conversion rules tell when such a transposition is legitimate. The following set of (nonmodal) conversion rules was universally accepted. First, in simple conversion AeB converts with BeA, and AiB converts with BiA. In other words,

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some A is B if and only if some B is A, and no A is B if and only if no B is A. Second, in conversion per accidens AaB implies BiA, and AeB implies BoA (this negative one is the only rule that was not in Aristotle): if every A is B then some B is A, and if no A is B, then some B is not A. These are only per accidens, because they change the quantity and do not hold in the opposite direction. Third, ever since Boethius even contraposition was taken as a type of conversion. It preserves the quality and quantity but “changes the ﬁnite terms into inﬁnite ones.” For example, “if every A is B then every nonB is non-A.” Fourteenth-century logicians noticed that contraposition need not be valid when any of the terms is empty—an existential assumption is required. With conversions, some syllogistic moods can be derived from others. The idea is that if certain syllogisms are selected as basic, others can be derived from them by a clever use of ﬁxed methods. Aristotelians called this process “reduction,” present-day logicians would call it proof. Conversion was the most important method of reduction. The other method was reductio ad impossibile: A mood is valid because the negation of the conclusion leads to the negation of a premise. With these methods, all syllogistic moods could be reduced to the direct moods of the ﬁrst ﬁgure—in fact even further, to Barbara and Celarent. This was basic stuﬀ in all textbooks, and the consonants in the names of moods refer to the methods of reduction. (S: convert simply; P: convert per accidens; M: transpose the premises; C: reduce ad impossibile.) These privileged syllogisms are cases of dici de omni et nullo, in which the conclusions can be seen as immediate corollaries of simple aﬃrmation or negation. As Buridan explains, “dici de omni applies when nothing is taken under the subject of which the predicate is not predicated, as in ‘Every man runs’. Dici de nullo applies when nothing is taken under the subject of which the predicate is not denied.” So direct ﬁrst ﬁgure syllogisms are immediately self-evident, and medieval logicians, like Aristotle, called them “perfect.” Others are imperfect in the sense that their validity needs to be shown. The growth of syllogistic theory naturally leads to the philosophical question of its foundations. Such a problem can arise from two perspectives: One may wonder about the status of syllogistics in the totality of logic, or one may ask if particular syllogistic inferences depend on some other principles. a. The question about the general status of syllogistics became current when the theory of consequence developed in the beginning of the fourteenth century (see section 6). Aristotle had started from syllogisms and proceeded to a brief discussion of other inferences; now logicians took the opposite direction. In the thirteenth century, some logicians’ attitude seems to be that all strict demonstrative logic is syllogistical, but the more people were concerned with logical research, the clearer it became that other inferences are valid, too; and

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this was then explicated by means of the concept of consequence. However, the relation between syllogistics and consequences is not very clear. Syllogistics is a part of consequence theory, in the sense that one particular type of consequences are “syllogistic consequences.” (This is especially clearly said by Buridan, whereas Ockham prefers to keep the titles unconnected.) And syllogisms hold because they are good or solid consequences, in our words, logically valid ones. But does syllogistics depend somehow on other parts of the theory? It seems that medieval logicians did not think so. They were aware of the importance of propositional logic—after all, the Stoic heritage had survived—but they did not work in the present-day fashion and build ﬁrst a propositional calculus, then a predicate logic on it. Burley is an interesting case here: He really starts from the simple consequences of propositional logic. But he had no followers in this respect, and contrary to what has been suggested, even he does not apparently aim at any stratiﬁcation of logics here. b. More concretely, one might ask if the validity of a particular syllogistic mood is based on some principles, or if a syllogism involves the use of other logical laws. This problem does not appear in terminist manuals, but it is discussed in the 1240s by Robert Kilwardby. He insists that the necessity of dici de omni et nullo is of such a self-evident nature that it cannot be regarded as a genuine inference step. Many logicians agreed with him. Kilwardby even asks if syllogisms presuppose separate inferences of conversion, and argues that it is not so. Suppose that no B is A; just add “every A is A” as the second premise, and you get the converted sentence, “no A is B,” by Cesare. Similarly in other cases, we see that conversion reduces to syllogism. This idea was not generally accepted, but conversion was occasionally considered so immediate a transformation that it could not be called an inference at all. Soon, however, an alternative view was articulated. About 1270, Peter of Auvergne refers to loci, the governed steps of argumentation theory, and says that “every syllogism holds because of a locus from a more extensive whole to its part.” Simon of Faversham, Radulphus Brito, and others then developed this thought that a syllogism must involve a “principle of consequence.” The conclusion is somehow included in the premises. But the remarks are brief and obscure. In any case, they anticipate the fourteenth-century view of logically necessary consequence relation that is not peculiar to syllogisms.

5.3. Modal Syllogisms Aristotle devotes a large part of his Prior Analytics to modal syllogisms. But unlike nonmodal syllogistics, this area remains in a very unsatisfactory state. The modalities he there studies are necessity, impossibility, and contingency. He wishes to produce a complete set of syllogisms in which some propositions have such modalities; further, he tries to systematize these syllogisms like the nonmodal ones. Here he needs conversion, reductio, and a third method, ekthesis, based on deﬁning new predicates. Medieval logicians replaced ekthesis with a more elegant method of expository syllogisms.

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The main problem is that Aristotle’s theory looks incoherent. His set of accepted syllogisms might be the outcome if all modal propositions were read de re only, as concerning the modal properties of individuals. But then the conversion rules do not hold: Obviously “every A is something necessarily B” does not convert to “some B is something necessarily A.” Moreover, his choice of valid syllogisms contains some oddities. Ancient commentators struggled with these puzzles, and medieval Aristotelians could not avoid them. Peter of Spain’s Summulae does not really discuss modal logic, but Kilwardby, Lambert, and Albert the Great try to save Aristotle’s doctrine. They resort to a very strong interpretation of necessity, proposed by Averroes, which concerns only necessities which hold per se because of essences. Even this technique demands some arbitrary decisions, and in any case it amounts to a severe restriction of modal syllogistics. A similar approach seems to have continued through the thirteenth century. The ﬁrst known work that introduces new methods is the commentary by Richard of Campsall, written about 1308. Campsall’s own theory is conservative, since he wants to maintain the Aristotelian syllogisms and conversions by means of a strict and somewhat confused de re reading. But the novelty is that he makes a systematic distinction between divided and composite readings. It is connected to the idea, initiated by Duns Scotus, of simultaneous alternative states of aﬀairs. This new semantics of modal notions made possible a new and diﬀerent approach to modal logic. From this point of view, modal logic was seen to be much wider than the part that Aristotle had developed, and the relations of modes could be systematized in a new way. The basic notions were now necessity and possibility, which could be understood as realization in all and some alternatives respectively. The ﬁrst exact presentation of the resulting syllogistics was the very thorough account in Ockham’s Summa logicae. In Paris, the orthodox Aristotelian model survived much longer, but Buridan’s Tractatus de consequentiis (1335) provides a modern theory, which is almost as full as Ockham’s. A third and more concise classical text is in Pseudo-Scotus’s commentary on Prior Analytics (c. 1340). The new modal logic gave plenty of room for the notion of contingency, and it caused some disagreement, but for simplicity we bypass this and concentrate on the syllogistics of possibility and necessity. The composite and divided readings of them were strictly distinguished. The composite readings are easier, and accordingly they were less discussed. They were indeed de dicto in the sense that strictly speaking they only make a singular nonmodal claim about a dictum; for example, “it is necessary that some A is B” is interpreted as “the dictum ‘some A is B’ is necessary.” The syllogistic for such propositions follows from the general consequence theory. Ockham and Buridan agree that in every mood, if both premises are preﬁxed with necessity N, the conclusion is necessary too. On the other hand, a syllogism MMM, with all the three propositions modalized as possible, does not hold because the premises need not be compatible. Ockham also remarks on NMM and MNM.

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Much more problematic were divided premises, that is, propositions with genuinely modalized copulas. The main device for dealing with them was ampliation (see section 3.5), which extends the subject term to refer to supposita that occur in alternative nonactual states of aﬀairs; thus “every A is possibly B” will be read “everything which is or possibly is A is possibly B.” But ampliation may be blocked by adding quod est A, “what (actually) is” A. Now it is striking that ampliation was understood in two diﬀerent ways. Ockham assumed that ampliation is good for possibilities (and contingencies)—but he did not accept it for necessities. In other words, only actual things could be said to have necessary properties. The reasons for this are not clear; perhaps he thought that necessities always involve some existence postulate. Buridan, in his turn, said clearly that all modalities amplify the subject in the same way, and this became the common view, that is, if the subject of a modality is not explicitly restricted to what is, it is freely ampliﬁed. (We must therefore be cautious if we wish to use present-day possible world apparatus here.) Buridan drew an octagonal diagram of the propositions “Every/Some A is necessarily/possibly B/not B” and analyzed all the 56 logical relations between them. This made the map of modalities much clearer. Combinations of syllogistic moods, modalities, and restrictions produce a huge number of cases, and logicians could not mention every case explicitly, although they did pursue a full theory of them. They also comment on cases where some propositions are nonmodal. We can only sketch some outlines now. In the direct ﬁrst ﬁgure, everybody accepted MMM syllogisms as valid. Buridan and Pseudo-Scotus accept NNN, MNM, and NMN. The seemingly surprising NMN here shows the eﬀect of ampliation. (For instance, every M is necessarily B, some A is possibly M , therefore some A is necessarily B.) Ockham accepts NNN only when restricted to actuals; for Buridan’s school this is another valid syllogism, like several other moods resulting from a restriction of subjects of N or M. Buridan also accepts, for example, _NM with an assertoric major. In the second ﬁgure, Buridan mentions NNN, NMN, and MNM (and Pseudo-Scotus mistakenly adds MMM). These again have restricted versions (in the style of: if every actual B is necessarily M and every actual A is possibly not M , then no A is B). But Ockham allows no valid syllogisms here. In the third ﬁgure, all accept MMM. Buridan and Pseudo-Scotus accept NNN, NMN, and MNM, while Ockham accepts only restricted versions of these. Some of them, not precisely the same ones, are in Buridan. Ockham’s theory looks somewhat unﬁnished: His view of ampliation causes trouble, and he derives a great number of results by discussing individual examples one by one. Buridan, on the other hand, uses a very elegant deductive method with, for example, cleverly formulated conversion rules. His theory is the summit of medieval modal logic. His pupils Albert of Saxony and Marsilius of Inghen continued to give comprehensive accounts of modal syllogistics, with some usually unsuccessful innovations, but after them modal syllogisms seem to have fallen out of fashion.

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5.4. Topics and Methodology An important part of medieval logic was topics. The dialectics of the old trivium mostly belonged to it. The ultimate source was Aristotle’s Topics, but a second and simpler authority that replaced it for a long time was Boethius’s De diﬀerentiis topicis. The main subject in this inquiry was loci, locus being Latin for Aristotle’s topos (literally “place,” here something like “consideration”). Aristotle does not deﬁne his topos, whereas Boethius gives two meanings for locus. It can be a “maxim,” a self-evident sentence that needs no further proof, but it can also be a logically relevant feature that distinguishes two sides. Confusingly, the distinction can be between sentences, like aﬃrmative and negative, antecedent and consequent, or between concepts, like genus and species, part and whole. For example, the distinction between genus and species supports the maxim: What belongs to the genus belongs to the species. Boethius’s double notion of loci long guided medieval topics. On the other hand, Aristotle emphasized an aspect which was not so prominent in Boethius: Topics concerns dialectical argumentation, the ﬁnding, testing, and examining of plausible theses. Hence it is not restricted to methods of demonstrative scientiﬁc proof of necessary results. Treatises as early as the eleventh century discuss topics, and this interest culminates with the Aristotelian revival of the thirteenth century. Thus, Peter of Spain gives a detailed list of various loci which follows Boethius closely. An important idea in such lists is that loci are supposed to guarantee the validity of an inference or argument that was not immediately valid because of its form. For instance, Peter’s inference “The housebuilder is good, therefore the house is good” is surely not formally valid—and not even quantiﬁed—but it is “conﬁrmed” by the locus of cause and eﬀect: “That whose eﬃcient cause is good, is itself also good.” We see that the result is still not conclusively proved, but the addition connects the argument to syllogistics. This need of support is characteristic of “enthymematic” arguments, demonstrative or not. Nowadays we are accustomed to think that they are valid because of some suppressed deductive premises, but medieval authors did not always see the matter so. Often they thought that the support came from a rule and not from an implicit premise. The terminists were inclined to think that all valid arguments are reducible to syllogisms; topics gives metalogical directions for ﬁnding suitable middle terms for the reduction. After the early terminists, topics was still constantly discussed. After all, the Topics was a big book in the Organon and belonged to the obligatory courses, at least in part. But the heyday of topics was over when logica moderna was developed. It was no longer a really inspiring ﬁeld, although it undoubtedly had some importance: Topics apparently inﬂuenced the growth of consequence theory (see section 6.1), and the doctrine of loci was also relevant in discussions concerning the foundation of syllogisms.

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When the consequentia theory developed, both syllogisms and nonsyllogistic inferences could be seen as cases of the same general patterns. As a result, topics lost an important function. The arguments that were formerly studied in topics were, in the fourteenth century, normally included in consequences. Also, it is signiﬁcant that topics was no longer connected to enthymemes but to dialectical arguments, that is, its special character was seen as epistemic. Usually, the leading logicians no longer treated topics as a separate subject at all—Ockham, for instance, studied topical arguments only as a relatively uninteresting special case. On the other hand, Buridan still painstakingly devoted a whole treatise to topics. Later the interest in topics diminished even more; Paul of Venice did not speak of it. However, commentaries on Aristotle’s Topics were written throughout the fourteenth and ﬁfteenth centuries, but no new ideas were presented. The Aristotelian theory of science was highly abstract; while it had little contact to concrete problems, it did have a close connection to logic. The basic source for medieval discussion was Posterior Analytics, though direct commentaries on this diﬃcult work were not very common. In the Aristotelian picture, developed for example by Aquinas, an ideal science consists of a system of demonstrative syllogisms. Their premises must be true, necessary, and certain. Premises can be derived by other syllogisms, but ultimately they rest on evident necessities. As Kilwardby says, “the demonstrator considers his middle term as necessary and essential, and as not possibly otherwise than it is; and so he acquires knowledge, which is certain cognition that cannot change.” Science is thus a system of syllogisms about causes and essences; it can use logical principles, but logic itself is obviously not a science. Much of this grandiose view later had to be given up, when ﬁrst Scotus problematized the notion of necessity and then Ockham problematized the notion of evidence.

6. New Approaches to Inferences During the thirteenth century, four new domains of logical research broadly falling into the scope of propositional logic emerged: consequences, obligations, insolubles, and sophisms. In overall treatments of logic like Ockham’s Summa logicae and Buridan’s Summulae dialectica, these new branches of logic were discussed in the place traditionally occupied by treatments of dialectical topics in the sense in which they referred to what Aristotle discusses in his Topics. This is not to say that the traditional theory of dialectical topics, for which Cicero and Boethius had provided the classical texts, had disappeared altogether. Nor can we say that these new areas of logic had replaced the tradition of dialectical topics. Rather, the purposes aimed at by research in these new areas were seen to be approximately similar to those traditionally aimed at by the theory of dialectical topics, and consequently the new ﬁelds were taken to complement traditional discussions. In modern terms, we can say that the point of gravity was moving from the theory of argumentation toward formal logic.

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Let us start with consequences, considering four diﬀerent issues pertaining to this crucial area of logic. Late medieval discussion of consequences aimed at giving clear and speciﬁc determinations of (1) what is a consequence, (2) the deﬁnition of the validity of a consequence, (3) how they should be classiﬁed, and (4) rules concerning valid consequences.

6.1. What Is a Consequence? In general, late medieval treatments of consequences understood them as inferences. That is, they were not called “true” (vera) or “false” (falsa), but rather were said to be “good” (bona), or simply “to be valid” (valeo) or “to hold” (teneo), or in the opposite case “to fail” (fallo). Despite an acknowledged close connection to conditional propositions, consequences were usually discussed separately as belonging to a diﬀerent place in the overall structure of logic. Ockham, for example, discusses conditionals within his theory of propositions, and turns to consequences as a theory of nonsyllogistic inference in the beginning of III, 3 in Summa logicae: “After treating syllogism in general and demonstrative syllogism, we now have to turn to the arguments and consequences that do not apply the syllogistic form.” The genre of logical writings on theory of consequences seems to have arisen in the thirteenth century from recognition of the fact that a general theory of inferential validity can be formulated in addition to, or as an extension of, the traditional syllogistic system. As such, medieval logicians had been aware of the idea at least since Abelard’s work, and Boethius had already composed a special treatise on what he called “Hypothetical syllogisms,” that is, on propositional logic. Nevertheless, Walter Burley’s De puritate logicae seems to have been the ﬁrst overall presentation of logic to discuss the theory of inference systematically starting from general issues of consequences and moving toward more particular issues after that, allowing syllogistic only the minor position of a special case. That most medieval logicians saw consequences as inferences and not as propositions is reﬂected in the fact that they aimed at formulating general rules (regulae) of valid inferences; traditional dialectical topics were also seen to belong to this set in addition to a number of more formal ones. The outstanding exception in this picture is John Buridan and his Tractatus de consequentiis. He explicitly deﬁnes consequences as hypothetical propositions consisting of two parts, the antecedent and the consequent, joined by a connective like “therefore” (ergo). Thus, Buridan’s consequences amount to conditional propositions with speciﬁc content. He treats consequences as pieces of discourse that assert the validity of an inference from the antecedent to the consequent: “One follows from the other” (una sequatur ad aliam). Accordingly, Buridan does not discuss or lay down metalinguistic rules (regulae) of consequences in this treatise, but instead asserts “conclusions” (conclusiones) concerning what can be truly said about the kinds of sentences following from each other.

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It seems clear that all prominent medieval logicians saw the distinction between the acceptability of performing an inferential step and the assertion that a valid inferential relation obtains. Whereas most logicians thought that consequences should be understood as inferences, Buridan made the opposite decision. For him, a consequence was a proposition, a conditional claim concerning an inferential relation between the antecedent and the consequent. He seemed to have had no followers in this opinion, but because of his prominent position in late medieval logic, his surprising stand has caused a number of misunderstandings concerning the issue both for medieval authors and for modern commentators.

6.2. Criteria of Validity The simplest way to formulate the deﬁnition of inferential validity was to ground it on the idea that it is impossible for the antecedent to be true and the consequent false. Indeed, it seems that all late medieval deﬁnitions of validity can be seen as variously qualiﬁed or modiﬁed versions of this principle. In the ﬁrst known treatise directly dedicated to consequences, Burley’s De consequentiis, we ﬁnd the deﬁnition that a consequence is valid if “the opposite of the consequent is repugnant to the antecedent.” The problem with this deﬁnition is that it seems unclear in which sense we are to take the word “repugnant,” since it is often used in a way that already contains reference to inferential connections. Indeed, Burley elsewhere opts for alternative deﬁnitions closer to the modal criterion. In Buridan, we ﬁnd the following list of three alternative descriptions concerning when some proposition “is an antecedent to another” or, in other words, a consequence is valid: (a) “that is antecedent to something else which cannot be true while the other is not true” “illa alia non existente vera”; (b) “that proposition is antecedent to another proposition which cannot be true while the other is not true when they are formed simultaneously” “illa alia non existente vera simul formatis”; (c) “that proposition is antecedent to another which relates to the other so that it is impossible that howsoever it signiﬁes, so is the case, unless howsoever the other signiﬁes, so is the case, when they are formed simultaneously” “sic habet ad illam quod impossibile est qualiterqumque ipsa signiﬁcat sic esse quin qualiterqumque illa alia signiﬁcat sic sit ipsis simul propositis.” Buridan ﬁnds each of these three descriptions problematic, but accepts the last, if it is understood in a suitably loose manner. The problem with the ﬁrst deﬁnition is related to the standard medieval requirement that a proposition must be actually formulated to be true. This makes it clear that almost no consequence would be valid according to the ﬁrst criterion, since the consequent need not be

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formulated when the antecedent is. The second aims at correcting this problem through the simple addition “when they are formed simultaneously,” but falls prey to it as well. A consequence like “no proposition is negative, therefore no donkey runs” should be invalid, but turns out to be valid on criterion (b) as well as on (a), because the antecedent is never true when it is actually put forward. Thus, it cannot be true without the consequent being true even if they were simultaneously formed. With criterion (c), Buridan takes another strategy. He recognizes that the consequential relation should not be seen to obtain with the sentences themselves, not even between potentially formulated ones, but rather between their contents. However, Buridan did not believe that such sentential contents would exist (see the section about propositional signiﬁcates, complexa signiﬁcabilia), and therefore the formulation of the criterion (c) makes problematic ontological commitments. Apparently he could not ﬁnd a formulation that would avoid them, and thus we are left without a satisfactory description of inferential validity. It seems, nevertheless, that Buridan’s strategy of transporting criteria of validity from the actual sentences to their signiﬁcations or contents became a generally accepted one. In some interesting sense, which still puzzles modern scholars, Buridan’s further discussions on the topic take a “mentalistic” turn in the conception of logical validity. He considered that logical validity depended on the mind in a more crucial sense than many of his predecessors. Some formulations by his followers made this mentalistic turn even more obvious in ways that we shall see in the next section.

6.3. Classiﬁcations of Valid Consequences The most traditional medieval distinction among kinds of valid consequences was the distinction between those valid “as of now” (ut nunc) and those valid “simply” (simpliciter). Validity ut nunc was taken to mean something like validity given the way things now are: From “every animal is running,” it follows ut nunc that Socrates is running, at the time in which Socrates exists as an animal. After his death, the consequence ceases to be valid. Simple validity, on the other hand, meant validity in all circumstances. In this sense, from “every animal is running,” it follows that “every human is running.” It is noteworthy that validity ut nunc also contains some kind of necessity, and thus it cannot be compared to twentieth-century material implication. Late medieval logicians put their main interest in two other, philosophically more interesting distinctions. Somewhat confusingly the concepts “form” and “matter” were used in both distinctions, so that when we come to Paul of Venice, a consequence may be, for example, “formally formal” or “materially formal,” since he combines the two distinctions into one systematic presentation. In both distinctions the issue was to separate a class of consequences that were valid in a privileged manner: not only valid, but “formally valid.” In one sense, formal validity meant a substitutional kind of validity, where a consequence is formally valid, if it “is valid for all terms” (tenet in omnibus

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terminis), and only materially valid if its validity is based on the special content of some of the terms used in the inference. In this sense, the paradigm examples of formally valid inferences were syllogisms in the Aristotelian ﬁgures, but also examples like modus ponens could be put forward. In the other sense, an inference was called formally valid only if the consequent was “formally included” (includit formaliter) in the antecedent or in the “understanding” (intellectus) of the antecedent; this kind of formal validity was often called “natural” or “essential” validity. It seems that the roots of both distinctions can be traced back to the early Middle Ages. At least Kilwardby gives ground for both distinctions. Nevertheless, the two distinctions seem to have had a somewhat diﬀerent history. Furthermore, the concept of material validity remained in most treatments rather obscure. It seems, however, that especially as related to the latter deﬁnition of formal validity based on inclusion, material validity was often understood as having to do with certain properties of the propositions used. The paradigm cases of materially valid inferences followed the rules “from the impossible anything follows” and “the necessary follows from anything.” Let us ﬁrst look at the latter kind of formal validity, the one based on the idea that the antecedent must “formally include” the consequent. The concept of “formal inclusion” seems to have been developed by late thirteenthcentury theologians, such as Henry of Ghent, Godfrey of Fontaines, and Duns Scotus. In many texts the topic comes up in a discussion concerning the role of the third person in the divine Trinity, employing the special technique of obligations (see following). These discussions resulted in elaborated theories of what it means to say that a concept is included in another concept, or that an assertion conceptually includes and thus entails another claim. The primary examples studied by medieval logicians included inferences like “a human exists, therefore, an animal exists,” and the explanation of their “formal” validity was based on the necessary conceptual or essential relation between the species “human” and the genus “animal.” The concept “human” was said to “formally include” the concept “animal,” and thus the inferences based on this relation were said to be “formally valid.” In twentieth-century terms, we would rather describe them as analytically valid inferences. William Ockham was aware of this discussion and aimed at bringing the results into the systematic context of logical theory. In the classiﬁcation of Summa logicae, a consequence is formally valid if it is valid by general rules of a speciﬁc kind. They must concern the syntactic features of the propositions (forma propositionis) involved in the consequence. Also, the rules must be self-evident (per se nota). This part of the deﬁnition is in eﬀect identical with, or at least comes very close to, the substitutional type of deﬁnition of formal validity. But on the same page Ockham also admits as formally valid consequences that are valid by something he calls an “intrinsic middle.” His example is “Socrates does not run, therefore a man does not run,” which is valid by the “intrinsic middle,” “Socrates is a man.” It seems that Ockham wanted to present this type of formal validity to allow also inferences based

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on something like conceptual inclusion within this group, the inclusion being expressible as an intrinsic middle. Some 10 years later, Ockham’s student Adam Wodeham explicitly distinguished between two diﬀerent ways of understanding the concept of formal validity. One of them uses only the substitutional criterion, while the other accepts as formally valid all consequences based on truths known in themselves (per se nota). Insofar as Wodeham’s per se nota refers to all analytic truths and not only conceptual inclusion, the deﬁnition is wider than that derived from the traditional slogan “formally includes,” but it is clearly on the same track. Material validity is deﬁned by Ockham with reference to something he calls “general conditions of the propositions,” and he gives the ex impossibile quodlibet rule as an example. In Ockham’s case this is strange, since he clearly knew that from a contradiction it is possible to derive anything with rules which he allows to be formal. Do we, thus, have inferences that are both material and formal? In his deﬁnition of formal validity, Buridan presents only the substitutional principle, without mentioning the idea of conceptual inclusion. His examples of inferences which are valid but not formally so, however, show that he was aware of the criterion but did not want to use it. He straightforwardly claims that those inferences, which are valid so that all substitutions of the categorematic terms with other terms are also valid, are formally valid. Among formally valid inferences, Buridan explicitly counts inferences from contradictions, though of course not from weaker impossibilities like from “a man is not an animal.” These he classiﬁes as material. It seems that in the latter half of the fourteenth century, Buridan had few followers in his classiﬁcation principles. Only Albert of Saxony seems to have accepted the substitution principle as the sole criterion of formal validity. The majority of logicians seem to have wanted to develop an idea which is closer to what was later in the twentieth century called analytic validity. The criterion of formal validity was, therefore, formal or conceptual inclusion of the conclusion in the premises. As an interesting special case, Paul of Venice presents a system that uses both concepts of formal validity, thus producing a very elaborate system.

6.4. Rules of Consequences Usually medieval discussions of the theory of consequences also included a selection of rules warranting valid inferences. Instead of anything close to a complete listing of such rules, we must here satisfy ourselves with a look at the types of such rules presented in the medieval discussions. We have already encountered two such rules: “from the impossible anything follows” and “the necessary follows from anything.” These rules were practically never completely rejected in the later Middle Ages. However, their applicability in speciﬁc contexts was often limited, and as they were typically classiﬁed as

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materially valid, they were understood as belonging to a somehow inferior kind. Medieval authors knew the main rules of so-called classical propositional logic. For example, detachment is put by Burley as concisely as possible with a term variable: “If A is, B is; but A is; therefore B is.” Transitivity rule is presented by Burley as the consequence “from start to ﬁnish” (a primo ad ultimum). He also discusses other examples of basic propositional logic of the kind, but when we turn to the later fourteenth century, the selection of rules of this type leaves nothing to be hoped for. One type of rules of consequences that seems to have interested medieval authors quite widely is based on epistemic operators. These were often discussed by direct comparison with modal rules; if something is necessary, it is the case, and if something is known, it is the case. More interesting (and more disputable) examples of relevant inference schemes are more complicated. The rule “if the antecedent is known, the consequent is also known” was often held to be valid only on the further condition that the consequence itself is known. The ﬁrst rule of consequences in Ockham’s Summa logicae is that “there is a legitimate consequence from the superior distributed term to the inferior distributed term. For example, ‘Every animal is running; hence every man is running’.” All medieval logicians accepted this example as valid, though they often formulated the rule diﬀerently, and the explanation of the kind of validity varied. Buridan, who relied on the substitution principle, thought that the consequence is valid in a standard syllogistic mood with the help of a suppressed premise. But almost all other logicians thought that something like the rule given by Ockham suﬃces for showing the validity. The reference to the relation between a superior and an inferior term given in the rule was understood in terms of the criterion on conceptual containment. It seems that in twentieth-century terms, the rules of this type could be characterized as regulating analytic validity. These types of rules already bring us close to Aristotle’s program in the dialectical topics presented in the Topics. This work was indeed much used in compilations of the listing of the rules for consequences. Also, in many works the lists contain rules that have more of the character of the theory of argumentation than of formal logic. The rules for consequences are indeed one of the places where the diﬀerences between modern and medieval conceptions of logic are most clearly visible.

6.5. Obligations The genre of late medieval logical literature that has perhaps been the most surprising for modern commentators carries the title obligations (obligationes). The duties or obligations at issue in the treatises carrying this title were of a rather special kind. The basic idea was based on the Socratic question/answer game as described and regulated by Aristotle in the Topics. In the speciﬁc medieval variant of the game the opponent put forward propositions that had

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to be granted, denied, doubted, or distinguished by the respondent. In giving his answers, the respondent was expected to pay recognition to the truth, but especially to some special obligation given to him in the beginning of the exchange by the opponent. This duty was understood to override the general duty of following the truth, but not the general logical duty of respecting arguments and avoiding contradictions. Here we cannot go into details concerning the diﬀerent variants of the system, although a number of interesting logical issues arose through the study of the particular kinds of possible duties. The main type of an obligational disputation, as medieval authors knew it, was based on a positum, a sentence put forward by the opponent in the beginning as something that the respondent has to grant. This sentence was typically false, and often even impossible in some way not directly implying a contradiction (conceptually impossible, naturally impossible, etc.). Then the opponent put forward further propositions, and in answering them the respondent had to pay attention to inferential relations between the positum and these later proposita. Altogether four main alternative sets of exact rules of how the inferential connections ought to be recognized were developed in the Middle Ages. According to one late thirteenth-century system, described by the Parisian logician Boethius of Dacia in his commentary on Aristotle’s Topics, the respondent must grant everything that the opponent puts forward after the positum, with the sole exception of propositions that are inconsistent (incompossibile) with the positum or the set of posita, if there are several. Boethius divides propositions into “relevant” and “irrelevant” ones with the criterion of an inferential connection to the positum. Those inconsistent with the positum are called repugnant (repugnans), and those following from it are called sequent (sequens). The repugnant ones must be denied and the sequent ones must be granted. Others are irrelevant, and Boethius claims that the respondent must grant them, since this implies nothing for the positum. In his discussion, Boethius relied on an already traditional terminology, but not all of the earlier authors would have agreed with his rules. The early fourteenth-century discussion took place mainly in England, and there a diﬀerent set of rules came to be accepted as the traditional system. According to these rules, the respondent should of course grant the positum and anything following from it. Similarly, he should deny repugnant propositions. But he should grant true irrelevant propositions and deny false ones. After having granted or denied such propositions, he should take them into account in the reasoning. He should grant anything that follows from the positum together with propositions that have been granted earlier or whose negations have been denied. Thus, the respondent must keep the whole set of his answers consistent, but otherwise follow the truth. Duns Scotus claimed that in an obligational disputation based on a false positum one need not deny the present instant, but one can understand the counterfactual possibility at issue in respect to the present instant. (Unlike many of his predecessors, Scotus denied the principle “what is, is necessary,

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when it is.”) After Scotus, it became customary to think of the set of answers after the disputation as a description of some consistently describable situation. This brought obligational disputations close to counterfactual reasoning and thought experiments. Richard Kilvington suggested in his sophismata an interesting revision of the rules apparently based on the idea that the disputation ought to describe the situation that would obtain if the false positum were true. He claimed that this principle ought to be taken as the rule guiding answers, giving the respondent a duty to grant what would be true and deny what would be false if the positum was true. Kilvington’s suggestion did not gain many followers. Most authors kept to the traditional rules, probably because Kilvington’s rules seemed too vague. Formally valid inferential connections were taken to provide a better foundation for obligational disputations. But another revision was also suggested, and for some time it gained more followers. Roger Swineshed suggested that all answers ought to be decidable solely on the basis of the positum without recognition of any subsequent exchange. Swineshed’s suggestion was that the respondent ought to grant the positum and anything following from it, and deny anything repugnant with it. Other propositions were to be taken as irrelevant, and they were not to be respected in the reasoning. This had the implication that irrelevant propositions would have to be kept separate from the mainline of the disputation, as a kind of second column in the bookkeeping. As Swineshed explicitly recognized, contradictions between the two columns could arise so that, for example, a conjunction may be denied when one of its conjuncts is granted as the positum and the other is granted as true and irrelevant. The main logical topic studied in obligational disputations was logical coherence. The disputations were in essence structures allowing propositions to be collected together into a set, with evaluation of the coherence of the set as the crucial issue at each step. The diﬀerent rules formulated the alternative exact structures for such a procedure.

6.6. Insolubles Early treatises on obligations are often connected with treatises carrying the title “insolubles” (insolubilia). In these treatises something is laid down in a way similar to how the obligational positum is laid down, but the crux of the discussion is that the given propositions appear to describe a possible situation and yet they entail a contradiction. The case is thus paradoxical. As a common example from the obligational treatises themselves, we may mention the rule that the respondent ought not accept “the positum is false” as his positum. The case is, of course, closely analogous to what is nowadays known as the Liar Paradox. It is not clear that obligational disputations were the original context of the genre of logic that came to be called insolubilia, since the ﬁrst treatments of such paradoxes in their own right seem to be equally early and have other

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sources, too. But the way medieval logicians formulated their versions of the Liar Paradox comes with an obligational terminology and context. If we turn to the mature treatises of the early fourteenth century, the paradigmatic insoluble is the proposition “Socrates is saying what is false,” and the assumed situation is that Socrates utters this and only this proposition. Then it is shown that if the sentence is true it is false (because if it is true, what it signiﬁes is the case), and if it is false it is true (because it signiﬁes that it is false, and that was assumed to be the case). Because these results cannot stand together—every proposition is true or false but not both—a contradiction seems to follow from what is clearly possible, for the only assumption seems to be that Socrates makes a simple understandable claim. Medieval logicians discussed a wide variety of carefully formulated analogous paradoxes, and it seems that some of them were formed to counter speciﬁc purported answers to the paradox. For example, if the paradox is claimed to result from direct self-reference, we may be asked to consider other examples. For example, medieval logicians considered cases where two or more people make assertions about the truth or falsity of each other’s claims and thus produce a paradoxical circle. A paradox reminiscent of the Liar Paradox can be produced without any proposition referring to itself—the paradox is not dependent on direct self-reference. It is also interesting to note that some practical analogs of the paradox were considered. Assume, for instance, like Buridan, that Plato is guarding a bridge when Socrates wants to cross it. Then Plato says, “If you utter something false I will throw you into the river, and if you utter something true I will let you go.” Socrates replies, “You will throw me into the river.” Now, what should Plato do? Cervantes makes Sancho Panza face a similar problem when he is the fake governor of an island, and indeed, Cervantes probably got the paradox from some medieval treatment of logic. The variety and the history of the diﬀerent solutions of the insolubles is too wide and complicated to be even summarized here. Some main alternative solutions presented in the medieval discussion must suﬃce for now. In the early discussions, the so-called nulliﬁers (cassantes) claimed that the one who utters a paradoxical sentence “says nothing.” If Socrates says only the sentence “Socrates says what is false,” he has not really uttered a proposition at all, and thus no truth value is needed. The problem, of course, is to explain precisely why the utterance fails to be a proposition. Some authors gave the reason that a part, like “false,” cannot refer to its whole; but this thesis is too generalized. In his Sophistical Refutations, Aristotle mentions the case where somebody says something that is simultaneously both true and false. This remark occurs in connection to the fallacy of confusing truth in a certain respect and absolute truth (secundum quid and simpliciter). Thus applications of this fallacy were often tried in solving insolubles, but understandably the results were not very convincing. One related suggestion was that insolubles were to be treated not as cases of genuine self-reference but instead as cases where a certain shift of reference (transcasus) takes place. When Socrates says that he is lying, he

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simply cannot mean that very utterance itself, and therefore we must look for some other utterance in the immediate vicinity. In the assumed case, this approach makes the insoluble false simply because there is nothing else that Socrates says. Fourteenth-century logicians found all these suggestions too simple-minded. In the early 1320s, Thomas Bradwardine used symbolic letters for propositions and assumed that every proposition a signiﬁes, in addition to its ordinary signiﬁcation, even “a is true.” (Strictly speaking, this was formulated as a general doctrine only later.) Substituting a = “a is false,” we get “a is false and a is true,” a contradiction that shows that a is false. A similar strategy is further reﬁned by William Heytesbury (1335). He puts the issue within the framework of obligation theory, discussing cases where insolubles are pressed on the respondent. All insolubles turn out to be false, but he admits that there is no general solution; what is needed is a careful study of what exactly is extraordinary in the signiﬁcation of each relevant sentence. Some authors, like Swineshed round 1330, argued that an insoluble proposition “falsiﬁes itself.” This requires a new opinion about truth: For the truth of a proposition, it does not suﬃce that it signiﬁes what is the case, but it also must not falsify itself. This fundamental novelty may have been one reason why the theory was not generally accepted—and, moreover, its applications soon lead to obscurities. Later, Gregory of Rimini and Peter of Ailly tried to utilize the doctrine of mental language in this context. The complex theory that Peter developed (in the 1370s) argues that spoken insolubles correspond to two conﬂicting mental propositions, whereas a mental proposition cannot ever be insoluble. This idea became well known but did not gain general acceptance. To sum up, we may say that the common view was that certain propositions were called insoluble not because of logical puzzles that could not be solved but because providing a solution “is diﬃcult,” as many authors remark. It was generally agreed that insolubles were false. Only a few authors took seriously the possibility that the paradox might be a genuine one, one that did not allow any satisfactory solution. But even they did not think of insolubles as a threat to the system of logic as a whole. Insolubles were not considered to undermine the foundations of logic but simply to be one interesting branch of logical studies. One might surmise that this can derive from the idea of looking at logic as an art dealing with the rational structures embedded in the mental basis of ordinary language, rather than as a calculating system based on special foundations.

6.7. Sophismata Buridan’s Summulae de Dialectica concludes with an almost 200-page section containing sophisms (sophismata), which are examples construed in a rather distinct way so that they make the need of logical distinction clearly visible. Buridan’s work is no exception; diﬀerent kinds of collections of such sophisms

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are commonly found in medieval logic manuscripts. It seems that they were used in medieval logic teaching as exercises to show how general logical systems could be applied in practical contexts. But often they also contain interesting material that is not discussed in systematic treatises. Separate collections of sophisms circulated throughout the Middle Ages. Perhaps the most famous of the early examples of such aids of teaching was known as the magister abstractionum. Little is known of the person, and he may not have been a single person. It is possible that we simply have a collection of examples which circulated among teachers of logic, who would each add their own examples and drop out others. Later, many authors of logical textbooks compiled their own collections of sophisms. This is what we ﬁnd for example in the case of Buridan. In early fourteenth-century Oxford, such a textual genre gained new signiﬁcance by assuming a relatively speciﬁc independent role not only in the university curriculum (where undergraduate students in their ﬁrst years of university were called “sophists” [sophistae]) but also in logical study. The collections of sophisms composed by Heytesbury, Kilvington and some other members of the so-called group of Oxford calculators were an important locus of logicolinguistic and mathematical study providing important results that were later used by pioneers of early modern science. A sophism in this sense of the word consists of (1) the sophisma sentence; (2) a casus, or a description of an assumed situation against which the sophisma sentence is evaluated; (3) a proof and a disproof of the sophisma sentence based on the casus; and (4) a resolution of the sophism telling how the sophisma sentence ought to be evaluated and how the arguments to the contrary should be countered. In the discussion of sophisma 47 in his collection, Kilvington assumes that the procedure in solving a sophisma must abide with the rules of obligational disputations. That is, the casus is to be understood as having been posited in the obligational sense, and thus anything following it would have to be granted and anything repugnant to it would have to be denied. From this viewpoint, the proof and disproof can be articulated as obligational disputations. Although an explicit commitment to using obligational rules such as Kilvington’s is rare in the collections of sophisms in general, obligational terminology is omnipresent. In many sophisms, the problematic issue was to show how the sophisma sentence was to be exactly understood. For this reason, sophismata became an especially suitable place for determining exact rules of scope and the interpretation of words serving important logical roles. Indeed, this is the context where late medieval logicians developed the exactitude in regulating logical Latin that was ridiculed by such Renaissance humanists as Juan Luis Vives. Heytesbury’s Rules for Solving Sophisms (1335) is a good representative of the genre and can thus be used as an example here. It consists of six chapters. The ﬁrst is on a topic we have already mentioned, so-called insolubles. The second discusses problems of epistemic logic with sophisms based on the words

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“to know” and “to doubt.” The third tackles problems connected to the use of pronouns and their reference. In the remaining three parts, Heytesbury turns to problems that may be better characterized as natural philosophy rather than logical analysis of language. The fourth part considers a traditional topic, the verbs “to begin” and “to cease,” and thereby issues connected to limit decision problems and temporal instants. The ﬁfth part, on maxima and minima, continues on the same tract from a diﬀerent viewpoint. The sixth and ﬁnal part is dedicated to “three categories,” referring to the Aristotelian categories of place, quantity, and quality. Especially this last part and its discussions of speed and acceleration proved very fruitful in the early development of modern science despite the fact that all the cases studied in it are purely imagined and lack any sense of experiment. For example, instead of real bodies in motion, medieval logicians considered imagined bodies in motion. In fact, this chapter and others of its kind show how the medieval secundum imaginationem method, relying only on logicolinguistic analysis, was able to provide results that have often been misguidedly attributed to experimental scientists working centuries later. One of the speciﬁc techniques used in solving sophisms deserves treatment of its own in a history of logic. In early thirteenth-century texts, a sentence like “Socrates begins to be pale” was analyzed as something like “Socrates was not pale and Socrates will be pale.” The analysis was accompanied with a discussion on which of the two conjuncts in the particular kind of change at issue should be given in the present tense, and how one should formulate the continuity requirement that Socrates, say, will be pale immediately after the present instant, even before any given determinate future instant. Such an analysis became a standard technique used in a large variety of cases and was called “exposition” (expositio). Without going into the particulars of the speciﬁc verb “to begin,” it is worth pointing out here that the idea in such an analysis is to break down the sentence containing the problematic “exponible term” into a conjunction or a disjunction that is equivalent in its truth conditions. For fourteenth-century logicians, it was a commonly accepted doctrine that there is a large number of terms that admit, or in the contexts of a sophism, demand such an analysis. Furthermore, this kind of analysis was taken to be necessary for practically all philosophically central terms if there was a need to treat them in a logically exact manner.

7. The End of the Middle Ages 7.1. Later University Logic Undoubtedly, the main plot of medieval Aristotelian logic lies in the development that began from the early terminists and led to the stage of Burley and Ockham, and then had its academic culmination in the systematic work of Buridan. But the discipline of logic survived after that, and some new special features appeared and new developments took place in the late Middle Ages.

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It is probably true to say that logicians were no longer very original during this time. But here it is necessary to emphasize that logic was a widespread and multiform discipline; the volume of material is very great, and much of it is still unexamined. A gradual change happened in philosophy in general during the fourteenth century, a change whose background is hard to explain. It has been pointed out that the whole cultural climate was no longer the same: The fourteenth century included great political upheavals; the Church had diﬃculties that led to the great schism; various protest movements appeared, and so on. All this contributed to the loss of the previous unity. It is customary to start the “autumn of the Middle Ages” from 1350, but this demarcation is largely symbolic; the only concrete thing that can support it is the Black Death, which killed many philosophers in 1349. After 1350, philosophy was still practiced in the old style, and logic has hardly ever been as prominent a part in philosophy as in the latter half of the fourteenth century. However, the overall authority of philosophy and logic started to diminish. Let us try to sketch an overview of the historical development. Ockham, a political dissident, had never made an uncontested breakthrough—in fact, he was considered an extremist even among nominalists. In logic, however, his thought had a wide inﬂuence. Buridan, then, had more indisputable prestige, and as regards logic, his inﬂuence became dominant in Parisian philosophy during the 1340s. In this ﬁeld he had two extremely competent pupils, Albert of Saxony (d. 1390) and Marsilius of Inghen (d. 1396). After the generation of Buridan’s students, the position of Paris weakened, although it was still the most famous university. England underwent a quite distinctive process. In the beginning of the fourteenth century the best logicians were English, and even after them there were original ﬁgures in Oxford, like Bradwardine, Heytesbury, and Billingham. Then, after 1350, logic turned to great technical sophistication but little essentially original appeared in the works of logicians such as Hopton, Lavenham, Strode, Feribrigge, and Huntman. Soon after 1400, a complete collapse took place in England, and only some elementary texts were produced during the ﬁfteenth century. But English logic was, however, very inﬂuential in the late Middle Ages on the Continent. English works of the fourteenth century were studied and commented on in Italy. Particularly Ralph Strode’s logic achieved great fame. Paul of Venice had studied at Oxford, and he transmitted the comprehensive English tradition to the Italian logicians of the ﬁfteenth century: Paul of Pergula, Gaetano of Thiene, and others. Moreover, the ﬁfteenth century is the era of the triumph of the university, which also involved a geographical expansion of philosophical studies. Hence we meet a number of new active centers of logic emerging in Central Europe, in universities like Prague, Cracow, and Erfurt. A typical feature in ﬁfteenth-century philosophy is a conscious turn toward old masters. Thus, philosophy formed into competing schools with their own

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clear-cut doctrines; this process was promoted by the commitment of religious orders to their oﬃcial authorities and by the allotment of chairs in philosophy. These Thomist, neo-Albertist, Scotist, and nominalist currents were not very innovative in logic, though some of their leaders were ﬁrst-rate logicians (like the Scotist Tartaretus). The form of logical works changed gradually. Instead of voluminous commentaries, two other types of work became popular: shorter discussions of individual subjects, and more general summulae expositions. A far-reaching step was the innovation of printing, which led to the promotion of textbooks in particular. (The ﬁrst printed logical book was the Logica parva by Paul of Venice, in 1472.) On the whole, we can say that logic was no longer very creative; there were few original results, and perhaps they were not even actively pursued. We can feel some signs of the later sentiment that the science of logic had already been completed. In statements like this, we must remember, though, that there has been particularly little historical research on ﬁfteenth-century logic. Attention was often concentrated on earlier results; thus there was much interest in all kinds of special cases and counterexamples, which we cannot discuss here. Generalizing crudely, we might say that the exponibilia, the sophismata, and the insolubilia became especially popular themes, whereas the fundamental questions of terms, propositions, and inferences were less debated. Modal logic seems to disappear, though it has a surprising revival at the end of the ﬁfteenth century (with Erfurtians like Trutvetter). At the same time there is also a revival of philosophical logic in Paris (e.g., Scotsmen around John Mair). The strictly formal part of older logic, such as syllogistics, was still taught everywhere, and occasionally even cultivated in so far as there was an opportunity to develop it. A famous example is the innovation of the so-called pons asinorum. Fifteenth-century authors formulated clearly this virtually mechanical method for ﬁnding a suitable minor premise by means of which a given conclusion can be syllogistically inferred from a given major. In less formal matters, we encounter an interesting line by examining the widely read Speculum puerorum (1350s) by Richard Billingham. He discusses the probatio, literally “proof” but also meaning “trial,” of propositions and concludes that it is only possible by a further probatio of its terms. “Immediate” terms are simple, but others can be submitted to some of the three forms of such a treatment. First, “exponible” terms can be replaced by several occurrences of simpler terms in a conjunction of simpler propositions which is equivalent to the original one. Thus “only a man” is exponible in the proposition “only a man runs,” and its exposition leads to the equivalent “a man runs and nothing but a man runs.” “Resoluble” terms are replaceable by simple terms, leading (not to equivalents but) to truth grounds; thus “a man” is resoluble to “this” since “a man runs” has a truth ground “this runs and this is a man.” And for “oﬃcial” terms, it can be shown that they hold an oﬃce together with dicta, like the modal and attitudinal operators do. By means of the probatio of terms,

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the proposition ought to acquire a logically elementary form, and problems arising from diﬃcult constructions can then be handled. Finally Billingham gives some grammatical rules for advancing without error in the probatio. Similar ideas can be found in a number of later English and Italian authors who discuss such basically non-Aristotelian themes. The attention turned to logical grammar. Logic courses often started from logica vetus, continued with material from the Aristotelian Prior and Posterior Analytics, and then concentrated on the new themes. This feature can be seen as a mark of a shift from logic in the strict sense toward conceptual analysis of logically diﬃcult items: problematic concepts, ambiguous linguistic constructions, and so on. Accordingly, much attention was awarded to questions of grammatical deep structure and its accurate expression by means of variants in lexical forms and word order. The serious nature of these problems can now be appreciated again, in the light of present-day grammatical theory, but it is of course true that ﬁfteenth-century authors did not have a suﬃcient technical apparatus for mastering their Latin sentences. It is also easy to understand that these undertakings seemed useless and annoying to many critics.

7.2. Reactions It is common to speak about “medieval logic,” and one easily thinks of it as a monolithic totality. Perhaps we have managed to say that the truth is much more complex. But all the authors we have discussed so far had a solid Aristotelian background. There were, however, even other tendencies, which started to grow during the ﬁfteenth century. We might prepare the way for the novelties by mentioning earlier dissidents. The Aristotelian methodology in science was rather restrictive, and for a long time repeated attempts had been made to ﬁnd a place for something more innovative. Bacon is perhaps the most famous among these authors: He showed great curiosity in matters of empirical science and made initiatives in the philosophy of science. But his logic seems to follow well-known Aristotelian lines. A much more perplexing case is Raimundus Lullus (Ramón Llull, c. 1235–1316). Having no academic training, he did not care about logica moderna; instead, he sought to create an original way of argumentation that would undeniably prove Christian dogmas to inﬁdels. This so-called Ars magna, to which he gave several formulations, uses various basically neo-Platonic sources. As basic concepts, he chooses some central divine attributes and cross-tabulates them with certain logicometaphysical aspects. This ought to produce, in the way of multiplication tables, a scheme of interesting manifestations. Lullus also suggested that concepts should be written on concentric circles and arguments performed by rotation of the circles. In fact, Lullus never achieved any logical results, and his program rests heavily on theological premises. But he introduced the idea of purely combinatorial procedure (with symbolic letters), and this was something that fascinated many later authors. “Lullists” reappeared during the ﬁfteenth century, and even Leibniz was interested in Lullus.

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Late medieval university logic acquired a respected opponent when Italian humanists began to propagate their new ideals. In the middle of the fourteenth century, Petrarch had violently attacked scholasticism and particularly logic, making it clear that professional logic was a corrupt and useless discipline that could not beneﬁt a literary civilization. His leading followers, such as Bruni and Bracciolini, were more detailed in their criticisms. According to them, what is sensible in logic is delivered through the studies of language and dialectics, whereas university logic is mostly incomprehensible sophistry. They also pointed out, correctly, that medieval logic consisted of additions made by barbarians to the classical heritage. The early humanists mainly expressed nothing but their discontent, but a more substantial alternative logic was developed by the famous philologist Lorenzo Valla (1407–1457) in his Dialecticae disputationes. He argued that a lot of the scholastic problems were actually illusory and resulted from obscure and abstract misinterpretation of questions that were essentially linguistic. Valla admitted that a small kernel of elementary logic was needed, as ancient Romans had already admitted, but for him formal validity was not as interesting as the informal convincing power of arguments. Thus he focused on the dialectical theory of reasoning and discussion, emphasizing matters of grammar and style. His work anticipates the revival of topics in a new form. A similar nonscholastic development was continued by many other authors. Gradually the humanist inﬂuence extended outside Italy to the whole of Europe, and there grew a conscious eﬀort to form a simple logic free of tradition. In this process, the new logic also found a place in the academic environment and much logical literature turned to dialectical issues, new ancient sources became known, and logic deﬁnitely entered the era of printed books. All this amounts to a basic transformation, and the next part can well start with it.

Selected Further Readings Ashworth, E. J. 1974. Language and Logic in the Post-Medieval Period (Synthese Historical Library 13). Dordrecht: Reidel. Ashworth, E. J. 1978. The Tradition of Medieval Logic and Speculative Grammar from Anselm to the End of the Seventeenth Century: A Bibliography from 1836 Onwards (Subsidia Mediaevalia 9). Toronto: Pontiﬁcal Institute of Mediaeval Studies. Bäck, Allan. 1996. On Reduplication: Logical Theories of Qualiﬁcation (Studien und Texte zur Geistesgeschichte des Mittelalters 49). Leiden: Brill. Biard, Joël. 1989. Logique et théorie du signe au XIV siècle. Paris: Vrin. Boh, Ivan. 1993. Epistemic Logic in the Later Middle Ages. London: Routledge. Braakhuis, Henk A. G., C. H. Kneepkens, and L. M. de Rijk, eds. 1981. English Logic and Semantics from the End of the Twelfth Century to the Time of Ockham and Burleigh (Acts of the Fourth European Symposium of Medieval Logic and Semantics). Leiden: Ingenium Publishers. Ebbesen, Steen, ed. 1995. Sprachtheorien in Spätantike und Mittelalter (Geschichte der Sprachtheorie, Teil 3). Tübingen: Narr.

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Green-Pedersen, Niels J. 1984. The Tradition of the Topics in the Middle Ages. München: Philosophia Verlag. Jacobi, Klaus, ed. 1993. Argumentationstheorie: Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns (Studien und Texte zur Geistesgeschichte des Mittelalters 38). Leiden: Brill. Knuuttila, Simo. 1993. Modalities in Medieval Philosophy. London: Routledge. Kretzmann, Norman, Anthony Kenny, and Jan Pinborg, eds. 1982. Cambridge History of Later Medieval Philosophy. Cambridge: Cambridge University Press. Kretzmann, Norman, and Eleonore Stump, eds. 1988. Cambridge Translations of Medieval Philosophical Texts, vol. 1: Logic and the Philosophy of Language. Cambridge: Cambridge University Press. Kretzmann, Norman, ed. 1988. Meaning and Inference in Medieval Philosophy. Studies in Memory of Jan Pinborg. Dordrecht: Kluwer. Lagerlund, Henrik. 2000. Modal Syllogistics in the Middle Ages (Studien und Texte zur Geistesgeschichte des Mittelalters 70). Leiden: Brill. Nuchelmans, Gabriel. 1973. Theories of the Proposition. Amsterdam: North Holland. Nuchelmans, Gabriel. 1980. Late Scholastic and Humanist Theories of the Proposition. Amsterdam: North Holland. Panaccio, Claude. 1999. Le discours intérieur de Platon à Guillaume d’Ockham. Paris: Seuil. Perler, Dominik. 1992. Der propositionale Wahrheitsbegriﬀ im 14. Jahrhundert (Quellen und Studien zur Philosophie 33). Berlin: Walter de Gruyter. Pinborg, Jan. 1972. Logik und Semantik im Mittelalter. Stuttgart: FrommannHolzboog. Pironet, Fabienne. 1997. The Tradition of Medieval Logic and Speculative Grammar: A Bibliography (1977–1994). Turnhout: Brepols. Read, Stephen, ed. 1993. Sophisms in Medieval Logic and Grammar (Nijhoﬀ International Philosophy Series 48). Dordrecht: Kluwer. de Rijk, Lambertus Marie. 1982. Through Language to Reality. Studies in Medieval Semantics and Metaphysics. Northampton: Variorum Reprints. Rosier-Catach, Irène. 1983. La grammaire spéculative des modistes. Lille. Spade, Paul Vincent. 1988. Lies, Language and Logic in the Late Middle Ages. London: Variorum Reprints. Spade, Paul Vincent. 1996. Thoughts, Words and Things: An Introduction to Late Mediaeval Logic and Semantic Theory, available at http://pvspade.com/Logic/ noframes/index.shtml. Stump, Eleonore. 1989. Dialectic and its Place in the Development of Medieval Logic. Ithaca N.Y.: Cornell University Press. Yrjönsuuri, Mikko, ed. 2001. Medieval Formal Logic: Obligations, Insolubles and Consequences (New Synthese Historical Library 49). Dordrecht: Kluwer.

3

Logic and Philosophy of Logic from Humanism to Kant Mirella Capozzi and Gino Roncaglia

1. Humanist Criticisms of Scholastic Logic The ﬁrst impression of a reader who “crosses the border” between medieval and Renaissance logic may be that of leaving an explored and organized ﬁeld for a relatively unexplored and much less ordered one. This impression is emphasized by the fact that while in the medieval period we can assume, despite relevant theoretical diﬀerences, some consensus about the nature and purpose of logic, such an assumption cannot be made with reference to the postmedieval and Renaissance period: The many “logics” coexisting and challenging each other were often characterized by deeply divergent assumptions, articulations, and purposes. As far as logic is concerned, we could almost be tempted to use this “explosion of entropy” as the very marker of the shift between the medieval and the Renaissance period. The development of humanism, with its criticism of the late medieval logical tradition, is not the only factor contributing to this situation, but surely is a relevant one. Excessive and artiﬁcial subtlety, lack of practical utility, barbarous use of Latin: These are the main charges that humanist dialecticians made against scholastic logic. Such charges do not simply point out formal deﬁciencies that could be eliminated within a common logical framework, but call for a change of the logical paradigm itself. The eﬀort to address such charges had a deep inﬂuence on the evolution of logic and resulted in a variety of solutions, many of which were based on contaminations between selected but traditional logical theories, on the one hand, and mainly rhetorical or Though we decided on the general structure of this chapter together, sections 1–4 and 8 are by Gino Roncaglia, while sections 5–7 and 9–11 are by Mirella Capozzi.

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pedagogical doctrines on the other. But the charges themselves were initially made outside the ﬁeld of logic: One of the very ﬁrst invectives against scholastic logic came from Francesco Petrarca (1304–1374), hardly to be considered a logician (Petrarca 1933–42, I, 7). The central point at issue is the role of language. The late medieval scholastic tradition used language as a logical tool for argumentation, and favored the development of what J. Murdoch (1974) aptly called “analytical languages”: highly specialized collections of terms and rules which—once applied to speciﬁc and deﬁnite sets of problems—should help guarantee the formal precision of reasoning. In this tradition, the use of a simpliﬁed and partly artiﬁcial Latin could help the construction of sophisticated formal arguments. The humanists, on the contrary, privileged the mastery of classical Latin. For them, language—together with a few simple and “natural” arguments taken from ancient rhetoric—was a tool for an eﬀective and well-organized social and pedagogical communication. Besides the diﬀerent theoretical standpoints, there is a social and cultural gap between two diﬀerent intellectual ﬁgures. Scholastic-oriented teachers are usually university professors who tend to consider logic, philosophy, and theology as specialized ﬁelds. For them, knowledge is reached through a self-absorbing (and largely self-suﬃcient) intellectual activity, whose formal correctness is regulated by logic. Many humanist dialecticians, on the contrary, do not belong to and do not address themselves to the academic world: They consider logic a tool to be used whenever language is used with rhetorical or practical purposes, and regard a broad “classical” culture more important than a specialized and abstract one (see Jardine 1982, 1988). One should be careful, however, in assessing the reasons for the privilege humanists accorded to rhetoric. For the humanists, logic—or rather dialectic, to use the term that, already present in the Ciceronian tradition and in the Middle Ages (see Maierù 1993), was preferred by most humanist and Renaissance authors—has to do with the use of arguments. But to be practically eﬀective, such arguments have to be natural, aptly chosen, easily stated and grasped, expressed in good, classical Latin. And they don’t need to be demonstrative arguments: Probable arguments are also included within the scope of dialectic. One should also be careful in considering humanism as a monolithic movement aimed at banishing all reminiscence of medieval logic. Humanism is not chronologically subsequent to scholasticism, and many humanists knew late scholastic logical texts fairly well, such as those by Paul of Venice. Some even praised them (Vasoli 1968, 20–23; Perreiah 1982, 3–22; Mack 1993, 14–15). Nevertheless, formally correct and truth-preserving arguments were considered as only some of the tools available to a good dialectician. The latter’s aim is to master the art of using language (ars bene disserendi), the Ciceronian disserendi diligens ratio, and this requires not only demonstrative skills but also the ability to persuade, to construct probable arguments, to obtain consensus. The deﬁnition of dialectic provided by Rudolph Agricola (1444–1485)—one of the many Renaissance variations on Cicero’s own—is representative of this

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point of view. According to it, dialectic is the “ars probabiliter de qualibet re proposita disserendi” (art of speaking in a probable way about any proposed subject). The explication of “probabiliter” clariﬁes the broad scope of the term (see Mack 1993, 169–173, where “probabiliter” is translated as “convincingly”): “probable (probabile) in speaking is not only what is actually probable, that is, as Aristotle states, what is accepted by all, or by the most part, or by the learned. For us, probable is what can be said about the proposed subject in an apt and adequate way” (Agricola 1967, 192). This meaning of the concept is broad enough to include good old-fashioned demonstrative arguments in the ﬁeld of dialectic (Risse 1964–70, I, 17–18), but they are no longer the only kind of arguments a dialectician should take into account. A ﬁrst introduction to sources, principles, and precepts of humanist-oriented logic is provided by the works of the prominent humanist dialectician Lorenzo Valla (1407–1457), who, signiﬁcantly, received his cultural training mostly within the humanist circles of the papal curia. While some of the earlier humanists were content with a dismissal of scholastic logic—Petrarca’s and Bruni’s invectives against the barbari britanni being the most often quoted testimony of this attitude (Garin 1960, 181–195; Vasoli 1974, 142–154)—in his Repastinatio dialecticae et philosophiae (Valla 1982), Valla added to heavy criticism of traditional logical doctrines a complete and systematic reassessment of the nature and purpose of dialectic from a humanistic point of view. According to Valla, dialectic deals with demonstrative arguments, while rhetoric deals with every kind of argument—demonstrative as well as plausible ones. Therefore dialectic is to be considered as a part of rhetoric, and rhetoric has to provide the widest spectrum of argumentative tools to all branches of learning. Moreover, dialectic should be simple and disregard all the questions that, though discussed by logicians with technical logical tools, actually pertain to Latin grammar. During the Middle Ages the relation between logic and grammar had been closely investigated by the so-called modist logicians. They worked at a sophisticated speculative grammar, based on an ontologically grounded correspondence between ways of being, ways of thinking, and ways of signifying. Valla’s grammar, on the contrary, is based on the Latin of classical authors, and therefore on a historically determined consuetudo in the use of language. Valla thus carries out what has been described as a “deontologization” of language (Camporeale 1986; Waswo 1999). Valla devotes the ﬁrst of the three books of his Repastinatio to the foundations of dialectic and to a discussion of the Aristotelian doctrine of the categories. Here, too, Valla applies his general rule: simpliﬁcation through reference to concrete uses of Latin, rather than to an abstract metaphysical system. The 10 Aristotelian categories are thus reduced to 3—substance, quality, and action—and examples are given to show how the remaining categories can be reduced to quality and action. Similarly, the transcendental terms, which according to the medieval tradition “transcend” the division among the 10 categories and are reciprocally convertible, are reduced to the only term “res.” The reason why Valla prefers the term “res” to the traditional

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“ens” is that “ens” in classical Latin is not a noun but a participle that can be exposed as “that thing (res) which is.” Therefore the term “res” is the true fundamental one. This example shows how Valla explains problematic terms or sentences by oﬀering a reformulation considered more precise and easier to analyze. The practice of explanation through reformulation was familiar to medieval logicians under the name of expositio, but Valla uses expositio to reach linguistic, rather than logical clariﬁcation. Valla’s second book is devoted to proposition and addresses the question whether all propositions should be reduced to the basic tripartite form: subject– copula–predicate (“A est B”). This question was the object of a long debate, continued during the whole period we are dealing with (Roncaglia 1996), and had usually been investigated under the assumption that it was the logical structure of the proposition at issue. Valla, on the contrary, perceives the problem as related to the grammatical structure of the proposition, and accordingly oﬀers a negative answer, since in the use of Latin the construction “est + participle” (Plato est legens) is not equivalent to the use of an indicative form of the verb (Plato legit). The Spanish humanist Juan Luis Vives (1492– 1540) will share the same attitude (see Ashworth 1982, 70). To support his contention, Valla considers propositions like Luna illuminatur, which—in Latin—can be transformed into a tripartite form only through a shift in meaning. A further argument is drawn from the idea that the participle form of the verb may be seen as somehow derivative with respect to the indicative form. Therefore—if something is to be reduced at all—it should be a participle like legens, to be reduced to qui legit (Valla 1982, 180). Logicians should not superimpose their logical analysis to the “good” use of language, but should rather learn from it. Language should be studied, described, and taught, rather than “corrected” from an external point of view. Valla did not consider the study of modal propositions as pertaining to logic (hence his complete refusal of modal syllogistic). This refusal—common to most humanistic-inﬂuenced Renaissance philosophy—is once again defended on linguistic rather than purely logical grounds. Why should we attribute to terms like “possible” and “necessary” a diﬀerent status from that of grammatically similar terms like “easy,” “certain,” “usual,” “useful,” and so on? (Valla 1982, 238; see Mack 1993, 90; Roncaglia 1996, 191–192.) Valla’s third book, devoted to argumentation, preserves the basic features of Aristotelian syllogistic, but dismisses the third ﬁgure and, as already noted, modal syllogisms. Owing to his desire to acknowledge not only demonstrative but also persuasive arguments, Valla pays great attention to hypothetical and imperfect syllogisms and to such nonsyllogistic forms of argument as exemplum and enthymemes. The ﬁnal section of Valla’s work is devoted to sophistic argumentations. Medieval discussions of sophisms allowed logicians to construct interesting, complex, and borderline situations to test the applicability and the eﬀectiveness of their logical and conceptual tools. Valla is fascinated by the persuasive and literary strength of “classical” problematic arguments, such as the sorites (a

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speech proceeding through small and apparently unavoidable steps from what seems an obvious truth to a problematic conclusion) or the dilemma, in which all the alternatives in a given situation are considered, only to show that each of them is problematic. Valla does distinguish “good” and “bad” uses of these kinds of “arguments,” but his criterion is basically that of practical usefulness in persuasive rhetoric. Valla’s Repastinatio is also a typical example of the importance humanists assigned to the “invention” (inventio) of arguments, connected with topics. Renaissance dialecticians considered Aristotle’s Topica as a systematic treatment of practical reasoning, and complemented it with Cicero’s Topica and with the treatment of topics included in Quintilian’s Institutio oratoria, which— rediscovered in 1416—had become one of the most popular textbooks on rhetoric by the end of the century, while Boethius’s De diﬀerentiis topicis, widely used in the Middle Ages (Green-Pedersen 1984), had only few Renaissance editions (Mack 1993, 135). Both Cicero’s and Quintilian’s treatment of topics helped shift the focus from “formal” disputations to rhetorical and persuasive ones. The most complete and inﬂuential Renaissance study of topics is contained in Agricola’s De inventione dialectica (Agricola 1967, 1992). Agricola grounds his conception of topics on his realist conception of universals (Braakhuis 1988). In his opinion, things are connected by relations of agreement and disagreement, and topics are orderly collections of common marks, which help us organize and label relations, and ﬁnd out what can or cannot be said about a given thing in an appropriate way. While being systematically arranged, topics, according to Agricola, are not a closed system: The very possibility of viewing things from diﬀerent angles and perspectives, of relating them in new ways, not only enables us to draw or invent arguments but also allows us to ﬁnd new common marks. We have already considered Agricola’s deﬁnition of dialectic. In his opinion, topics are the method of dialectical invention, while the discourse (oratio) is its context. There are, however, two diﬀerent kinds of dialectical discourse: exposition (expositio) and argumentation (argumentatio). The former explains and clariﬁes, and is used when the audience doesn’t need to be convinced, but only enabled to understand what it is said. The latter aims at “winning” assent, that is, at persuading. Although argumentation is connected with disputation, necessary arguments are not the only way to win a disputation: Plausible and even emotionally moving arguments should be considered as well. Agricola’s concept of argumentation is thus connected with rhetoric, a connection strengthened by the fact that both use natural language. This explains why Agricola has no use for the kind of highly formalized, analytical language used by medieval and late medieval logic. However negative Valla’s and Agricola’s attitude toward the logical tradition, it was never as negative as that of Petrus Ramus (Pierre de La Ramée, 1515–1572). According to his biographer Freigius, Ramus’s doctoral dissertation (1536) defended the thesis: “everything that Aristotle said is misleading

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(commentitium).” This does not imply—as many assumed—that Ramus considers all Aristotelian theories to be false: In his opinion, Aristotle is guilty of having artiﬁcially complicated and corrupted the simple and “natural” logic which Aristotle’s predecessors—notably Plato—had devised before him (Risse 1964–70, I, 123–124). Scholastic logic is obviously seen by Ramus as a further step in the wrong direction. Various versions of Ramus’s logic (including the 1555 Dialectique, in French: Ramus 1996; for a survey of the diﬀerent editions of his works and of the stages marking the complex development of Ramus’s dialectic, see Bruyère 1984) were published between 1543 and 1573. After his conversion to Protestantism in 1561, his library was burned, and he had to ﬂee from Paris. Ramus died on August 26, 1572, killed on the third day of the St. Bartholomew’s massacre. His being one of the Huguenot martyrs undoubtedly boosted the fortune of his already popular works in Calvinist circles. Ramus’s concept of dialectic is based on three main principles: Dialectic should be natural (its foundations being the “eternal characters” which constitute, by God’s decree, the very essence of our reasoning), it should be simple (it deals with the correct way of reasoning, but disregards metaphysical, semantic, and grammatical problems as well as unnecessary subtleties), and it should be systematically organized, mainly by means of dichotomic divisions. Therefore, Ramus’s books extensively used diagrams, usually in the form of binary trees: A feature that may be connected—as argued by Ong (1958)—with the new graphical possibilities oﬀered by printed books, and that will inﬂuence a huge number of sixteenth- and seventeenth-century logic textbooks, not only within the strict Ramist tradition. The ﬁrst and foremost division adopted by Ramus is Cicero’s division between invention (inventio) and judgment (iudicium or dispositio). They are the ﬁrst two sections of logic. A third section, devoted to the practical and pedagogical exercise of dialectic (exercitatio), is present in the ﬁrst editions of Ramus’s logical works but disappears after 1555. The inventio deals with the ways arguments are to be found. Because arguments are to be found and classiﬁed by means of topics, according to Ramus, the treatment of topics should precede, rather than follow (like in Aristotle), that of judgment. Ramus’s table of topics, organized by means of subsequent dichotomic divisions, is strongly inﬂuenced by Agricola and by Johannes Sturm (1507–1589), who taught dialectic and rhetoric in Paris between 1529 and 1537 and greatly contributed to the popularity of Agricola in France. Ramus’s treatment of judgment is also unconventional. While in traditional logic this section presupposes an extensive treatment of proposition, Ramus deals with this subject in a sketchy way and adds an independent (albeit short) section on the nature and structure of proposition only in the 1555 and successive editions of his work. In the last edition Ramus follows Cicero in using the term axioma to refer to a categorical proposition (having used earlier the term enuntiatio or enuntiatum), while he always gives the more speciﬁc meaning of major premise of a syllogism to the term propositio.

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Syllogism and its various forms (including induction, example, and enthymeme) constitute the core of the “ﬁrst judgment”: the ﬁrst of the three sections in which Ramus divides his treatment of judgment in the earlier editions of his dialectic. Ramus’s explicit eﬀort is that of simplifying Aristotelian syllogistic, but during the years between the 1543 edition of the Dialecticae institutiones and his death, his syllogistic underwent so many changes that it is impossible to give a faithful account of it in a few pages. Typical of Ramus’s syllogistic is his use of the terms propositio, assumptio, and complexio to refer to the major premise, minor premise, and conclusion of a syllogism, and his tendency to favor a classiﬁcation of syllogisms according to the quantity of the premises, considering as primary moods those with two universal premises. In the earlier editions of his dialectic, Ramus held that all moods with particular premises should be reduced to universal moods. He admitted some of them later on, but banned the reductio ad impossibile used to reduce second and third ﬁgure moods to the ﬁrst ﬁgure. But Ramus’s better known innovation in the ﬁeld of syllogistic is the so-called Ramist moods: syllogisms in which both premises are singular, accepted on the ground that individuals could be seen as (lowest) species. The discussion about Ramist moods will keep logicians busy for most of the subsequent century. The second section of Ramus’s treatment of judgment (called “second judgment” in the earlier editions of his work) deals with the ways to connect and order arguments by means of general principles. Ramus attributes great importance to this “theory of method,” which he further develops in the later editions of his logical works, and which in his opinion shapes the whole system of science (also oﬀering the conceptual foundation for an extensive use of dichotomies). According to Ramus, the dialectical method (methodus doctrinae) goes from what is most general to what is most particular. This is done by means of divisions that, in turn, are drawn on the base of deﬁnitions expressing the essence of the concepts involved. Division and deﬁnition are thus the two main tools of method. The opposite route, going from particular instances to more general concepts (methodus prudentiae), might be used when either the lack of a more general conceptual framework or reasons of practical convenience force us to dwell on single or partial pieces of information. However, it cannot guarantee certainty; and is therefore mainly used in rhetorical discourse aiming at persuasion, rather than in demonstrative reasoning. In Ramus’s opinion, however, the distinction between methodus doctrinae and methodus prudentiae does not imply that we have two methods: We have only one method—based on an ideal “knowledge space” organized by means of deﬁnitions and divisions— that, in given and concrete situations, also allows for tentative and partial bottom-up routes. Thus conceived, the dialectical method is governed by three laws, which constitute the Ramist counterpart of the Aristotelian-Scholastic de omni, per se and universaliter primum principles. Ramus calls them the laws of truth, justice and wisdom: in the ﬁeld of science every statement (i) should be valid in all its instances; (ii) should express a necessary (essential) connection of

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the concepts involved; (iii) should be based on subject and predicate that are proper and proportionate (allowing for simple conversion). Ramus’s logic was very inﬂuential in the second half of the sixteenth and in the ﬁrst half of the seventeenth century (Feingold, Freedman, and Rother 2001). However, “pure” Ramist scholars—mostly active in the Calvinist areas of Germany, in Switzerland, in Holland, and in England—were to face an almost immediate opposition not only in Catholic but also in Lutheran universities, and saw their inﬂuence decrease after the beginning of the seventeenth century. Much more inﬂuential (and more interesting) were the many “eclectic” logicians who either tried to reconcile Ramus’s and Melanchthon’s logical views (PhilippoRamists) or introduced some Ramist themes within more traditional (and even Aristotelian) contexts.

2. The Evolution of the Scholastic Tradition and the Inﬂuence of Renaissance Aristotelianism Despite humanist criticisms, the tradition of scholastic logic not only survived during the sixteenth and seventeenth centuries but evolved in ways that are much more interesting and articulated than most modern scholars suspected until a few decades ago. Our knowledge of this evolution is still somehow fragmentary, but the scholarly work completed in recent years allows some deﬁnite conclusions. We can now say that in this evolution of the late scholastic logical tradition, six factors were particularly relevant: (i) the work of a group of Spaniards who studied in Paris at the end of the ﬁfteenth and at the beginning of the sixteenth century and later taught in Spanish universities, inﬂuencing the development of logic in the Iberian peninsula; (ii) a renewed attention toward metaphysics, present in the Iberian second scholasticism and most notably in the works of Francisco Suárez (1548–1617), whose Disputationes Metaphysicae (Suárez 1965) inﬂuenced many authors all across Europe; (iii) the crucial role of the newly formed (1540) Society of Jesus, whose curriculum of studies (Ratio Studiorum) was to shape institutional teaching in all of Catholic Europe; (iv) the complex relations with humanism, and the inﬂuence of logicians like Agricola, whose doctrines, while taking as their starting point a humanist conception of logic, were nevertheless susceptible of somehow being absorbed or integrated within a more traditional framework; (v) the “new Aristotelianism” of authors like Jacopo Zabarella (1533–1589) and Bartholomaeus Keckermann (1572?–1609); and (vi) the renewed interest in scholastic logic, discernible in reformed Europe (and most notably in Germany) as a consequence of the doctrinal and theological conﬂicts with the catholic ﬁeld and within the reformed ﬁeld itself. In the following pages, we provide some details on this complex development. At the end of the ﬁfteenth century and in the ﬁrst decades of the sixteenth, the Paris college of Montaigu became a center of logical research in which the late medieval logical (especially nominalist) tradition survived and to

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some extent ﬂourished. A group of Spanish and Scottish logicians, lead by the Spaniard Jeronimo Pardo (d. 1505) and by the Scottish John Mair (1467/9– 1550), debated themes such as the nature of supposition and signiﬁcation, the distinction between categorematic and syncategorematic terms, the role of beings of reason (entia rationis), the nature of proposition (further developing the late medieval discussions on mental propositions), modality, and the theory of consequences. Somehow connected to this Paris group, or active there at the beginning of the sixteenth century, were the Spaniards Antonio Núñez Coronel (d. 1521), Fernando de Encinas (d. 1523), Luis Núñez Coronel (d. 1531), Juan de Celaya (1490–1558), Gaspar Lax (1487–1560), Juan Dolz (ﬂ. 1510), the Frenchman Thomas Bricot (d. 1516), the Belgian Pierre Crockaert (Pierre of Brussels, d. 1514), and the Scot George Lokert (d. 1547). Particularly interesting is their discussion about the nature of complexe signiﬁcabile (propositional complex), a subject already debated by medieval logicians. The medieval defenders of this theory, associated with the name of Gregory of Rimini (c. 1300–1358), held that the object of science is not the proposition itself but what is signiﬁed by it (and determines its truth or falsity); such total and adequate meaning of the proposition is neither a physical nor a purely mental being and is not reducible to the meaning of its parts. It is rather similar to a state of aﬀairs, which can be signiﬁed only by means of a complex (the proposition) and is therefore called complexe signiﬁcabile. The discussion on the nature (and usefulness) of the complexe signiﬁcabile was connected to the discussion on the role of the copula, since the copula was usually considered as the “formal” component of the proposition, “keeping together” subject and predicate. The copula was thus considered as a syncategorematic term: a term that does not possess an autonomous meaning but helps determine the meaning of the proposition as a whole. The defenders of a “strict” complexe signiﬁcabile theory did not need a separate discussion of the mental copula, because in their opinion the complexe signiﬁcabile is a unity and cannot be analyzed in terms of its parts. But many authors—among them John Buridan (c. 1295–1356)—assigned a much more relevant role to the copula, seen as the (syncategorematic) mental act that, in connecting subject and predicate, establishes the proposition. It is this very theory that was discussed by many of the above-mentioned late ﬁfteenth- and early sixteenth-century Paris-based logicians (see Ashworth 1978, 1982; Muñoz Delgado 1970; Nuchelmans 1980; Pérez-Ilzarbe 1999). Pardo’s position in this discussion was the most original. In his opinion, the copula is not purely syncategorematic: It is subordinate to a conceptual schema that represents something (i.e., the subject) as related in a certain way to something else (i.e., the predicate) or to itself (Nuchelmans 1980, 49). In this way the copula, while retaining its formal function, also signiﬁes something (aliquid), that is, the subject, as considered in a given way (aliqualiter), namely as modiﬁed by the relation with its predicate. The idea of the copula signifying aliquid aliqualiter, and not simply aliqualiter, and the special relevance attributed to the subject in determining the meaning of

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the copula and of the proposition as a whole, were discussed, and generally criticized, by Pardo’s successors. They especially investigated the role of impossible propositions, as well as propositions with a negative, privative, or impossible subject, and the problem of whether a quasi-syncategorematic nature could be attributed to the proposition as a whole. The Iberian Peninsula was one of the strongholds of Catholicism. Moreover, as we have seen, it inherited many features (as well as textbooks and Paristrained professors) from Parisian late scholasticism. This made the inﬂuence of the humanist movement—albeit discernible—less radical than elsewhere. Therefore, the Iberian Peninsula was the ideal context in which Catholic logicians—dwelling on the scholastic (chieﬂy Thomist) logical and philosophical tradition—could pursue the work of doctrinal and pedagogical systematization that was required by the struggle against the reformed ﬁeld. The Carmelite universities of Salamanca (Salamanticenses) and Alcalà (Complutenses) and the Jesuit university of Coimbra (Conimbricenses) each produced a complete philosophical course, including speciﬁc volumes devoted to logic. Of these the most inﬂuential was probably the Coimbra Logic, compiled by Sebastian Couto (1567–1639) but partially dependent on Pedro da Fonseca (1528–1599), who had been teacher at that university. Fonseca, the “Portuguese Aristotle,” published the Institutionum Dialecticarum Libri VIII (Fonseca 1964) in 1564, a logical treatise built on the model of Peter of Spain and widely read throughout Europe. Fonseca’s logic interprets the traditional emphasis on terms by giving a theoretical priority to the conceptual moment over the judicative one (truth and falsity are in concepts rather than in judgment) and among concepts, to singulars over abstracts and universals. To reconcile God’s foreknowledge and human free will, and to handle the problem of future contingents—a theme of special interest for all Iberian philosophers—Fonseca developed, independently from Luis de Molina (1535–1600), a theory of the scientia media, or, as he says, of “conditioned futures,” by which God foreknows all the consequences of any possible free decision. Placing Fonseca’s theories within a wider and more systematic treatment, the Coimbra logic oﬀers a translation and a detailed commentary of Aristotle’s Organon, which, in the form of questions, includes a discussion of most of the topics debated by sixteenth- and seventeenth-century logicians. The Conimbricenses reject the idea that beings of reason are the object of logic (in the scholastic tradition logical concepts such as “genus” and “species” were considered to be entia rationis, and the Thomist tradition considered them as the formal object of logic): Dwelling on the idea of logic as ars disserendi, they prefer to characterize it as a “practical science” dealing with the construction of correct arguments. Argumentation is, therefore, the ﬁrst and main object of logical enquiry. Particularly interesting is the long section devoted to the nature of signs at the opening of the commentary on Aristotle’s De Interpretatione (see Doyle 2001). The concept of sign is here taken in a broad meaning, as to include not only spoken, written, and mental “words,” but also iconic

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languages and arithmetical signs. It is to be remarked that the inﬂuence of Coimbra logic was not limited to Europe: Jesuit missionaries used it in Latin America and even in China. If the teaching of logic in Coimbra is connected to Fonseca, another important ﬁgure of Iberian logic and philosophy, Domingo de Soto (1494/5–1560), is connected to Alcalà and Salamanca, where he taught. Soto made important contributions to a plurality of ﬁelds, so much so that it was said qui scit Sotum, scit totum (who knows Soto, knows everything). Despite his endorsement of Thomism—testiﬁed by his defense of the theory that the object of logic are beings of reason—Soto was open to Scotist, nominalist, and even humanist inﬂuences, and his commentary on Aristotle’s logic (Soto 1543) criticizes the “abstract sophistries” of the late scholastic logical tradition. This, however, did not prevent him from discussing and adopting many late scholastic logical theories, including large sections of medieval theories of terms. His Summulae (Soto 1980) are a commentary on one of the key works of medieval logic, Peter of Spain’s Tractatus (best known as Summulae Logicales; see previous chapter), and include an ample discussion of signiﬁcation, supposition, and consequences (see d’Ors 1981; Ashworth 1990; Di Liso 2000). Soto adopts an apparently Ciceronian deﬁnition of dialectic, considered as the art of discussing probabiliter. As remarked by Risse (1964–70, I, 330), however, this should not be considered a rhetorical attempt to establish apparent plausibility, but rather as an attempt to establish rational assertibility. Among the interesting points of the Summulae are the treatment of induction in terms of ascensus (the passage from a conjunction of singular propositions—or from a proposition with a copulative term as subject or predicate—to a universal proposition, or to a proposition with a general term as subject or predicate) and a complex square of modalities, which takes into account the quantity of the subject. Soto’s discussion of second intentions oﬀers what has been interpreted as a sophisticated theory of higher-level predicates (Hickman 1980). One of Soto’s students in Salamanca was Franciscus Toletus (1533–1596), who later taught both in Zaragoza and Rome, in the Jesuit Collegium Romanum, and was the ﬁrst Jesuit to be appointed cardinal. Toletus wrote both an Introduction and a Commentary on Aristotle’s logic (Toletus 1985). Like Soto, Toletus adopts some humanist theories—he takes the deﬁnition of logic as ratio disserendi from Boethius and divides it into invention and judgment— but his logic is actually a synthesis of Aristotelianism and Thomism, deeply inﬂuenced by the late medieval logical tradition. He considers beings of reason as formal objects of logic—thus partly endorsing the Thomist position—but maintains that logic’s material object is constituted by our concepts of things and, ultimately, by things themselves, for logical beings of reason are only second intentions, based on ﬁrst-order concepts—thus partly endorsing the position of Arab commentators of Aristotle (Ashworth 1985b, xli). Of special interest is his extensive use of physical and geometrical examples within the discussion of categories, and his long discussion of contingent futures within the commentary on De Interpretatione.

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The most important Jesuit philosopher working in Spain at the end of the sixteenth century was Francisco Suárez (1548–1617). His Disputationes Metaphysicae, ﬁrst published in 1597 (Suárez 1965), constituted a reference text and a model for further works both in the Catholic and in the Reformed ﬁelds. According to Suárez, metaphysics oﬀers a general and uniﬁed theory of real being (ens reale) and of its divisions, whereas logic deals with the way of knowing and explaining such divisions. Though the Disputationes Metaphysicae is not a logic textbook, it discusses many issues relevant to the philosophy of logic. Suárez pays great attention to relations, subdivided into real relations (only conceptually and not really distinct from the things on which they are grounded, but nevertheless to be considered as a category of beings) and conceptual relations, which are only a product of the mind and as such do not have any ontological status. Suárez’s detailed discussion of both kinds of relations helps to explain the special interest that many scholastic-oriented logicians devoted to this topic in the seventeenth century. The last of the Disputationes—disputation LIV—is devoted to a subtle discussion about beings of reason (entia rationis) and relations of reason. According to Suárez, beings of reason are not “real” (actual or possible) beings and do not share a common concept with real beings; their only reality is that of being object of the understanding (they only have objective existence in the intellect). Therefore, they are not to be included within the proper and direct object of metaphysics. They can nevertheless be dealt with within the context of metaphysical research, given their nature of “shadows of being” (Suárez 1996, 57) and given their usefulness in many disciplines, especially logic and natural philosophy. Suarez’s opinion on entia rationis is thus diﬀerent both from that of those—like the Scotist Francis of Mayronnes (1280?–1327?)—who simply denied their existence, and from that of those—like many Thomists, including Cardinal Cajetanus (Tommaso de Vio, 1469–1534)—who thought that there is a concept common to them and to real beings. Suárez included impossible objects in the range of entia rationis: His discussion is thus especially relevant to the history of the logical and ontological status of impossible entities (Doyle 1987–88, 1995). The discussion on the nature of entia rationis was a lively one in sixteenth-century Spain and was bound to continue in Catholic Europe during most of the seventeenth century. An interesting example is that of the Polish Jesuit Martinus Smiglecius (1564–1618). In his opinion, the opposition between ens reale and ens rationis is not grounded on the fact that the ens rationis is not a form of being, but on the fact that it is by deﬁnition a being which is not, and cannot possibly be, an ens reale. A being of reason is thus, according to Smiglecius, one whose essence implies the impossibility of its real existence. The fact that entia rationis cannot have real existence is, according to Smiglecius, a logical and not just a physical impossibility. They, however, can have conceptual (and hence intentional) existence. In the later Middle Ages, English logicians had been famous for their subtleties: The logical, physical, and epistemic sophisms discussed by the socalled calculatores, working at Merton College in Oxford, deeply inﬂuenced late

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fourteenth- and early ﬁfteenth-century logic both in Paris and in Italy, and were exactly the kind of logical subtleties rejected by humanist logicians. During the ﬁfteenth and in the ﬁrst decades of the sixteenth century, however, the English logical tradition declined (see Giard 1985). This did not prevent a slow penetration of humanist ideas, testiﬁed by the 1535 statutes or the university of Cambridge, recommending the reading of Agricola and Melanchthon as substitutes for late medieval scholastic texts, and by the Dialectica published in 1545 by the Catholic John Seton (c. 1498–1567). The latter oﬀers a drastically simpliﬁed treatment of traditional topics such as signiﬁcation, supposition, categories, syllogism, but liberally uses nonformal arguments and literary examples, divides dialectic into invention and judgment, adopts Agricola’s deﬁnition of dialectic as well as his classiﬁcation of topics, and quotes, beside Cicero and Quintilian, modern humanists like Erasmus and Vives. In the last decades of the sixteenth century, the debate on Ramism was to shake both English and continental universities. In England, Ramus found in William Temple (1555–1627) a learned defender and commentator, who, despite the strong opposition of his fellow Cambridge teacher and former master Everard Digby (1550–1592), managed to make of Cambridge, albeit for a short time, a stronghold of Ramism. The penetration of Ramism in Oxford was less substantial, and by the beginning of the new century the anti-Ramist positions were predominant in both universities. The defeat of Ramism was accompanied by the propagation of Aristotelianism—tempered by humanist-oriented attention toward classical literary examples rather than purely logical ones and toward rhetorical practices such as the declamatio— and by the circulation of the leading logic books published in the continent (among them Zabarella and Keckermann). The Logicae Artis Compendium by the Oxford professor Robert Sanderson (1587–1663; Sanderson 1985) is a good example of this new situation. Sanderson abandons the division of logic into invention and judgment, favoring a threefold division according to the three acts of the mind: The ﬁrst, dealing with simple concepts, is associated with the treatment of simple terms; the second, dealing with composition and division, is associated with propositions; and the third, dealing with discourse, is associated with argumentation and method. Though this threefold division is present in the medieval and late medieval tradition and is discussed by the Conimbricenses, Zabarella, and Keckermann, Sanderson and other Oxford logicians seem to have been among the ﬁrst to use it as the main division for logic textbooks (Ashworth 1985b, xli). In his logic, Sanderson includes medieval topics such as the theory of supposition and consequences, but their presentation is straightforward and not very elaborated. His discussion of method is more articulate and gives a foremost role to pedagogical concerns. We have already mentioned the Padua professor Jacopo Zabarella, who, advocating a renewed, “pure,” and philologically accurate Aristotelianism, absorbs both some humanist instances—visible in the pedagogical organization of his works and in the inclusion of Aristotle’s Rhetoric and Poetic within a broad treatment of logic, on the ground of their dealing with probable

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arguments, such as rhetorical syllogisms and examples—and some features of the so-called Paduan Aristotelianism: a distinctive attention to the Arab interpretations of Aristotle (notably Averroes) and to Galen’s concept of science. Zabarella wrote commentaries on Aristotle’s logical works as well as autonomous logic tracts: Among the latter are the De Natura Logicae (in Zabarella 1966), the De Methodis, and the De Regressu (both in Zabarella 1985). According to Zabarella, logic deals with second intentions, that is, with the (meta)concepts produced by our intellect in reﬂecting on the ﬁrst notions, those derived from and referring to real things. Because second intentions are products (and ﬁgments) of our intellect, logic is not a science but an instrument, or, to be more precise, an instrumental intellectual discipline, aimed at devising conceptual tools for correct reasoning and for discriminating truth and falsity. Because of its instrumental nature, logic is somehow similar to grammar: Just as grammar provides the tools needed to write and speak in an appropriate way, logic provides the tools needed to reason in an appropriate way. According to Zabarella, order (ordo) and method (methodus) are among the main tools oﬀered by logic: The ﬁrst organizes the subject matter of a discipline and the knowledge we have acquired; the second gives the rules and procedures to be followed to acquire new knowledge, going from what we know to what we do not know (on the Renaissance concept of method, see Ong 1958; Gilbert 1960). In dealing with contemplative sciences, the ordo goes from the universal to the particular and to the singular (“compositive order”), while in dealing with practical and productive arts it goes from the desired eﬀects to the principles that produce them (“resolutive order”). Although order concerns a discipline as a whole, method always has to do with the handling of speciﬁc problems, of speciﬁc “paths” going from what is known to what is unknown. Those paths are basically syllogistic demonstrations: The method is thus somehow a special case of syllogism. And since a syllogism can only go from cause to eﬀect (compositive method or demonstratio propter quid) or from eﬀect to cause (resolutive method or demonstratio quia), the same will hold for method. The resolutive method is used in the “hunt” for deﬁnitions, and is most needed in natural sciences; the compositive method is used in mathematics, where we start from already known, general principles and try to demonstrate all their consequences. Both methods presuppose necessary connections and are therefore only valid within contemplative sciences: Practical and productive arts, dealing with contingent truths, will have to content themselves with rhetorical and dialectical arguments, which, being only probable, are not subject to a rigorous application of method. However, even in contemplative sciences (especially in natural sciences), our knowledge of eﬀects and of their causes is often far from clear, and we need a process of reﬁnement, which Zabarella calls regressus and which involves the use of both compositive and resolutive methods: (1) We ﬁrst use the resolutive method to go from a confused knowledge of the eﬀect to a confused knowledge of its cause; (2) we then examine and clarify the knowledge of the cause (examen)—an activity that Zabarella connects to a speciﬁc ability of the human mind, interpreted by

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some modern scholars in terms of the construction of a model; and (3) we ﬁnally use the compositive method to go from a clear knowledge of the cause to a clear knowledge of the eﬀect. This last stage is the highest sort of demonstration (demonstratio potissima), a notion already present in the Thomist tradition.

3. Logic in Reformed Europe: From Humanism to “Protestant Scholasticism” It is unfortunate that most historical accounts of logic devote relatively little attention to Philipp Melanchthon (1497–1560), “Germany’s teacher” (praeceptor Germaniae), prominent reformer and close collaborator of Luther. Actually, in the overall context of European logic in the mid-sixteenth century, the role played by Melanchthon is one of the highest signiﬁcance. From 1520, when at the age of 23 he published his Compendiaria dialectices ratio (Melanchthon 1520), to 1560—the year of his death—there are records of more than 60 different editions of his logic works. The last version of Melanchthon’s dialectics, the Erotemata dialectics (Melanchthon 1846), was to be the standard reference for protestant logic until the beginning of the seventeenth century. What makes Melanchthon’s logic interesting and explains its inﬂuence is above all the very evolution of his works. In the Compendiaria Dialectices Ratio, a young, strongly antischolastic Melanchthon oﬀers a simpliﬁed and rhetorically oriented treatment of dialectic, purged of many “superﬂuous” scholastic subtleties. Like many humanist dialecticians, here Melanchthon rejects the third syllogistic ﬁgure (which he considers “remote from common sense”) and the treatment of modality (the scholastic theories on modality are considered “tricky rather than true”). A few years later, however, Melanchthon’s opinions on both matters (as well as on many others; see Roncaglia 1998) radically changed. In the De Dialectica libri IV (Melanchthon 1528) the third ﬁgure is accepted and discussed at length, and Melanchthon bitterly criticizes Valla for rejecting it, while in the Erotemata (Melanchthon 1846) the discussion of modal propositions is considered to be “true and perspicuous, useful in the judgment of many diﬃcult questions.” The evolution of Melanchthon’s logic is thus marked by a progressive rejection of humanistic-rhetorical models and by a return to the Aristotelian and scholastic tradition. Two further aspects of the evolution of Melanchthon’s dialectic deserve attention: the gradual shift from a bipartite toward a tripartite conception of the structure of the proposition, and the growing interest in fallacies. In 1520, Melanchthon endorses the theory that every proposition has two main components: subject and predicate. In 1528, the question is seen from a grammatical perspective, and noun and verb are considered as being the two main components of the proposition. The verb, however, is further subdivided: It may be a proper verb or a construct made up of the substantive verb (the copula “est”) and a noun. The copula acquires a fully autonomous role in the Erotemata, where every proposition is seen as having not two but three main

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parts: the subject, the predicate, and an (explicit or implicit) copula, seen as the formal sign of the connection between subject and predicate. Such a theory will play an important role in subsequent logic, because the copula will be considered to be not only the logical “glue” of the proposition but the actual bearer of its modal and quality modiﬁcations (see Nuchelmans 1980, 1983; Roncaglia 1996, 2003). The discussion of fallacies also testiﬁes Melanchthon’s increasing use of scholastic doctrines. Absent in 1520, a short section on fallacies appears in 1528, accompanied, however, by the observation that anyone who has fully understood the precepts supplied for the construction of valid arguments doesn’t need special rules to avoid paralogisms. But in the following editions of his dialectics, Melanchthon systematically adds new divisions and new examples. He distinguishes between fault of matter and fault of consequence, corresponding to the traditional division of fallacies in dictione and extra dictionem, and presents the principal fallacies of both sorts. In the Erotemata, fallacies undergo a still closer scrutiny within a systematic framework clearly derived from scholasticism. Many elements indicate that there was one main and primary reason for this return to scholasticism: the perception that the Reformed ﬁeld—engaged in the sharp debate with Catholic theologians, and in the equally sharp debate among diﬀerent Reformed confessions—desperately needed eﬀective logical tools. Rhetoric could be useful in winning popular support, but was much less eﬀective in winning subtle theological debates. In the complex theological and political struggle that was under way in Europe, universities were to become crucially relevant players. Logic was to become a weapon in the theological struggle, and Melanchthon was probably the ﬁrst to perceive that clearly. Melanchton’s works on dialectic, together with the Dialectica by Johannes Caesarius (1460–1550), another interesting and inﬂuential mixture of humanist and Aristotelian elements, had thus the ultimate eﬀect of paving the way that was to be followed by Protestant logicians: endorsement of some humanist doctrine (ﬁrst and foremost the pivotal role of topics and inventio), and great attention to the pedagogical organization of their work, but within a context that retained many tracts of traditional logic; and that—given the relevance of logic for the theological debate—was to devote a renewed attention even to some of the once deprecated scholastic subtleties. It is therefore hardly surprising that the attempt, made by the so-called Philippo-Ramist logicians, to conjugate Ramus’s drive for simplicity and for systematic, method-oriented classiﬁcation, with Melanchthon’s humanistinﬂuenced but somehow more conservative treatment of logic, was not destined to have a long success. Given the renewed role of logic in the interconfessional theological debate, the two paths were bound to diverge and the Ramist component was to succumb: At the end of the sixteenth century, the antiRamist pamphlet was to become a well-established literary genus in the logic production of Protestant Germany. A ﬁerce battle against Ramism was led by Cornelius Martini (1568–1621), among the founders of the so-called Protestant

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Scholastic. Martini endorses Zabarella’s deﬁnition of logic as mental habitus dealing with “second notions,” that is, concepts used to represent and classify, rather than immediately derived from perception. Martini divides logic into formal and material, with formal logic seen as dealing with the pure form of (syllogistic) consequences. Zabarella’s inﬂuence is also apparent in the work of Bartholomaeus Keckermann (ca. 1571–1608). In his Systema logicae (1600, in Keckermann 1614), the eﬀort to organize logic as a discipline (largely on the basis of topics) is clear from the very deﬁnition of logic, which can be considered as a human ability—and is then to be regarded as a mental habitus—but can also be considered as the corpus of doctrines resulting from the use of this ability (ars externa): that is, as a system. In this perspective, knowledge of the historical constitution of this doctrinal corpus becomes important: Therefore, it is not by chance that the short section on the history of logic, present in many sixteenthand seventeenth-century treatises, acquires in Keckermann status, accuracy, and completeness. Keckermann’s interest in the history of logic is also connected with the eclectic tendency of many early seventeenth-century logicians: Given that in a Zabarella-oriented perspective logic is a human activity (and is also the systematically arranged, historical product of this activity), it is natural to try to collect the “logical tools” developed by diﬀerent logicians in diﬀerent times and contexts. This eclectic tendency is usually implicit, and is not necessarily connected with the endorsement of Zabarella’s positions (some aspects of which were actually criticized by many systematic-oriented or eclectic logicians), but it is clearly present in the encyclopedism of authors such as Johann Heinrich Alsted (1588–1638) or Franco Burgersdijk (1590-1636), whose Institutionum logicarum libri duo (1626) was the standard logic handbook in the Netherlands, and, like Keckermann, included a large section on the history of logic (see Bos and Krop 1993). A remarkable feature of this eclecticism is the tendency to reabsorb, within a context usually marked by Renaissance Aristotelianism, even some of Ramus’s doctrines, notably the emphasis on the practical utility of logic and on the need of a well-arranged, easily graspable, and pedagogically oriented method. In the ﬁrst half of the seventeenth century, in reformed Europe, despite the terrible destruction of the Thirty Years War, the university system was expanding, and acquiring a political relevance that was bound to transform any doctrinal diﬀerence in the occasion of sharp conﬂicts (see Wollgast 1988b). This complex situation enhanced logical research and produced some new and interesting theories. In discussing the structure of the proposition, the Berlin-based Johannes Raue (1610–1679) proposed a new theory of the nature and role of the copula. In his opinion, the standard proposition of the form “S is P” should be analyzed as “that what is S is that what is P” (id quod est S est id quod est P), that is, as having three copulas. The role of the main copula (the middle one, which Raue calls “real copula”) is then diﬀerentiated from that of the auxiliary ones: It can be used only in the present tense, while time and modal modiﬁcations are seen as operating on the auxiliary copulas.

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The pronoun “id” stands for the “third common entity” (tertium commune) in which subject and predicate are joined, and it has been observed that “Raue delights a Fregean reader when he emphasizes that ‘S’, the subject, . . . in his analysis is predicated of the tertium commune just as the predicate ‘P’ ” (Angelelli 1990, 188). This “newest theory,” of which Raue is very proud, was criticized by Johannes Scharf (1595–1660): a polemical exchange that was well known to Leibniz. Leibniz had the highest opinion of another famous logician of the time, Joachim Jungius (1587–1657). Jungius’s Logica Hamburgensis, one of the most clear and complete logical works of the seventeenth century, deals at length with such relevant and “advanced” topics as the theory of relations and the use of nonsyllogistic consequences (Jungius 1957, 1977). Jungius is not the only one to deal with such theories, which were considered useful in theological disputations, but his treatment of them is always clear and insightful. This is especially true of his investigation of the inversio relationis (from “David is the father of Solomon” to “Solomon is the son of David”) and of the consequence a rectis ad obliqua (from “the circle is a ﬁgure” to “he who draws a circle draws a ﬁgure”). Jungius’s discussion of the latter—which he considers a simple consequence (the consequent is inferred from the antecedent without the need of a middle term)—was the subject of a detailed analysis in the correspondence between Leibniz and Jungius’s editor Johannes Vagetius (1633–1691), who tried to oﬀer a formal representation of its structure (see Mugnai 1992, 58–62 and 152–153).

4. Descartes and His Inﬂuence When I was younger I had studied, among the parts of philosophy, a little logic, and, among those of mathematics, a little geometrical analysis and algebra. . . . But, in examining them, I took note that, as for logic, its syllogisms and the greater part of its other teachings serve rather to explain to others the things that one knows, or even, like the art of Lull, to speak without judgment about those of which one is ignorant, than to learn them. . . . This was the reason why I thought that it was necessary to seek some other method, which, comprising the advantages of these three, were free from their defects. (Descartes 1994, 33–35) This passage, from René Descartes (1596–1650) Discours de la méthode (1637), oﬀers a good synthesis of Descartes’s attitude toward traditional logic. Descartes’s criticism of syllogism does not concern its validity but its power as a tool for scientiﬁc research, and is clearly expressed in his Regulae ad directionem ingenii: “dialecticians are unable to devise by their rules any syllogism which has a true conclusion, unless they already have the whole syllogism, i.e. unless they have already ascertained in advance the very truth which is deduced in that syllogism” (Descartes 1964–1976, X, 406). The core of the argument is a classic one, advanced in diﬀerent forms at least since

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Sextus Empiricus (see Gaukroger 1989, 6–25): Syllogism is a circular form of reasoning, since it only holds if both its premises are already known to be true, but if both premises are already known to be true, the conclusion is already known to be true, too. Therefore, to discover something new, we cannot depend on syllogism. Pierre Gassendi (1592–1655) advanced a similar criticism. He observed that the evidence needed to accept one of the premises of a syllogism is provided or presupposed by its conclusion. Thus, in the Barbara syllogism “All m are p, all s are m, therefore all s are p,” the truth of “all m are p” can only be established by generalization of the fact that all instances of m—including s—are p: The truth of the conclusion is presupposed by, rather than inferred from, the truth of the premises. Descartes discusses a further argument against syllogism: The validity of a syllogism does not guarantee the truth of its conclusion, which depends on the truth of the premises. The syllogism alone—while giving us the false impression of dominating the concepts we are dealing with—cannot establish it. This argument too is fairly traditional; in the period we are dealing with, we ﬁnd a similar one in Francis Bacon (1561–1626): We reject proofs by syllogism, because it operates in confusion and lets nature slip out of our hands. For although no one could doubt that things which agree in a middle term, agree also with each other (which has a kind of mathematical certainty), nevertheless there is a kind of underlying fraud here, in that a syllogism consists of propositions, and propositions consist of words, and words are counters and signs of notions. And therefore if the very notions of the mind (which are like the soul of words, and the basis of every such structure and fabric) are badly or carelessly abstracted from things, and are vague and not deﬁned with suﬃciently clear outlines, and thus deﬁcient in many ways, everything falls to pieces. (Bacon 2000, 16) While in Bacon this argument is used to advocate the need of “true induction” (progressive generalization accompanied by the use of his “tables of comparative instances”), in Descartes it is used to advocate the role of intuition. According to Descartes, the process of knowledge acquisition depends on (1) intellectual intuition, that is, the intellectual faculty that allows a clear, distinct, immediate, and indubitable grasp of simple truths; and (2) deduction, that is, the grasp of a connection or relation between a series of truths. According to Descartes, deduction is therefore not to be seen as an inferential process governed by logical rules, but rather as the exercise of an intellectual faculty that is ultimately based on intuition. The process of mastering a long or complex deduction is a sort of intellectual exercise, consisting in the recursive application of intuition over each of its steps. The aim of this process is certainty: The idea of degrees of certitude or probability is totally alien to Descartes’s intuitionbased conception. And since for Descartes intellectual intuition is a natural

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faculty, there are no abstract rules or inference patterns governing intuition or deduction: We can only give precepts—like the well-known four regulae given in the Discours—helping us in the better use of this faculty. One aspect of Descartes’s concept should be stressed: The combined use of intuition and deduction allows us to attain knowledge, but does not suﬃce by itself to guarantee that the knowledge we attain is true. If a proposition p is intuitively clear and evident for us, we are entitled to claim that it is true. But while this claim is justiﬁed, its correctness is not grounded on the fact that p is perceived by us as clear and evident, because a deceptive God could give us a clear and distinct intuition of something that is not true (i.e., something that does not correspond to reality; from this point of view, Descartes is now generally considered as holding a correspondence theory of truth; see Gaukroger 1989, 66). Therefore, the well-known cogito argument is needed to ensure God’s external guarantee of our knowledge. According to Descartes, we only need (and we can only achieve) this external guarantee: God’s knowledge is not a model for our knowledge, and there is no set of eternal truths binding God’s knowledge and ours in the same way, since eternal truths themselves result from the joint action (or rather from the uniﬁed action) of God’s will and understanding. Descartes’s resort to intellectual intuition as ultimate foundation of certainty is somehow at odds with his work in the ﬁeld of algebra and geometry and with his discussion on the relevance of analysis. In Descartes’s opinion, analysis is associated with the discovery of new truths (while synthesis has to do with presenting them in such a way as to compel assent), and its function is apparent in mathematics and in analytical geometry, when we use variables (general magnitudes) instead of particular values. Descartes, however, doesn’t seem to perceive the possible connection of this method with deductive reasoning: on the contrary, he seems to associate deduction with the less imaginative, painstaking word of synthetically computing individual magnitudes. The inﬂuence of scholasticism on Descartes’s philosophy is greater than one might suspect at ﬁrst sight (see already Gilson 1913). For instance, hints at a “facultative” concept of logic (see section 6) were present in authors (among them Toletus, Fonseca, and the Conimbricenses) he knew. But Descartes’s concept of deduction diﬀered very much from traditional logic. This did not prevent some Cartesian-Scholastic logicians to reconcile them. Johann Clauberg (1622–1665), in his aptly named Logica vetus et nova (Clauberg 1658), defended Descartes’s methodical rules against the charge of being too general or useless, attributing them the same kind of rigor and strength of Aristotle’s logical rules. Johann Christoph Sturm (1635–1703) made a similar attempt. More articulate was the position of Arnold Geulincx (1624–1669), who published a Logica Fundamentis suis restituta (1662), and a logic more geometrico demonstrata, the Methodus inveniendi argumenta (1663; both in Geulincx 1891–1893). Like most Dutch logicians, Geulincx was deeply inﬂuenced by the eclectic Aristotelianism of Burgersdijk (see section 3). He thus merged late scholastic, Aristotelian, and Cartesian themes in a logic that, with some hyperbole, he labelled “geometric.” Its treatment includes the so-called De Morgan

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rules (well known in medieval scholastic logic, but less frequently dealt with by Renaissance logicians). He also devised a “logical cube” whose faces represented all the axioms and argument forms of his logical system. Descartes’s inﬂuence is also evident in the Port Royal Logic, which we will discuss in the next section. Before dealing with it, however, there are two authors—somehow diﬃcult to classify by means of traditional historiographic labels—which are worth mentioning: the French Jesuit Honoré Fabri (1607– 1688) and the Italian Jesuit Gerolamo Saccheri (1667–1733). Neither of them was an “academic” logician, and they both had wide-ranging interests. Fabri corresponded with most of the major philosophers and scientists of the time (including Descartes, Gassendi, and Leibniz); was interested in philosophy, mathematics, astronomy (he discovered the Andromeda nebula), physics, and biology; and wrote on calculus and probabilism (his book on this subject was condemned by the Church). His Philosophia (1646) is inﬂuenced by Descartes, but the section on logic is pretty original: He developed a combinatorial calculus which allowed him to classify 576 syllogistic moods in all the four ﬁgures; he also used a three-valued logic (based on truth, falsity, and partial falsity) which he applied to the premises and conclusions of syllogisms, and used disjunctions to express hypothetical judgments. Wide-ranging were also the interests of Saccheri, who, besides working on logic, also wrote on mathematics and geometry. In trying to prove the parallel lines postulate, in the Euclides ab Omni Naevo Vindicatus (1733) he hints—against his will—to non-Euclidean geometries. Both in logic and in geometry he makes use of the consequentia mirabilis (well known to the mathematicians of the time): If p can be deduced from non-p, then p is true. In his Logica demonstrativa (published anonymously in 1697)—a treatise on logic organized “more geometrico”—he applies the consequentia mirabilis to syllogistic. One of his proofs refers to the rejection of AEE syllogism in the ﬁrst ﬁgure. Saccheri shows that this very rejection (stated in E-form: “no AEE syllogism in ﬁrst ﬁgure is valid”) can be the conclusion of a ﬁrst ﬁgure AEE syllogism with true premises. If such a syllogism is not valid, then it constitutes a counterexample to the universal validity of AEE syllogisms (which are thus to be rejected). If it is valid, the truth of its premises implies the truth of its conclusion. Such an elegant demonstration has been correctly seen as the mark of an argumentation strategy based on the skillful use of confutations and dilemmas (see Nuchelmans 1991, 133–137).

5. The Port-Royal Logic A mixture of ancient and new doctrines characterizes the Logique ou l’Art de penser published anonymously in 1662 but written by Antoine Arnauld (1612– 1694) and Pierre Nicole (1625–1695). The authors belonged to the Jansenist movement of Port-Royal, hence the current denomination of their work as the Port-Royal Logic (Arnauld and Nicole [1683]). According to widespread

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opinion, the authors endorse Descartes’s philosophy. This is in many respects true, especially as regards the origin of ideas and the account of the scientiﬁc method. Indeed, in the 1664 edition of the book, the authors declare that the section on the analytic and synthetic method is based on the manuscript of Descartes’s Regulae ad directionem ingenii. However, the Port-Royal Logic is not a straightforward Cartesian logic because it relies on many sources. Apart from the inﬂuence of Augustine and Pascal (1623–1662), the authors, though condemning scholastic subtleties, acknowledge the utility of some scholastic precepts and are not always adverse to Aristotle. True, they reject the Aristotelian categories and topics, but describe these doctrines and make them known to their readers. The Port-Royal Logic is also diﬀerent from a humanistic ars disserendi, and even more from an ars bene disserendi, as Ramus would have it, for it is intended to be an ars cogitandi, an art for thinking. The authors maintain that, since “common sense is not so common a quality as people think” (First Discourse 17, trans. 6), people ought to educate themselves to be just, fair, and judicious in their speech and practical conduct. Such an education should be oﬀered by logic, but traditional logic pays too much attention to inference, whereas it should concentrate on judgment because it is in judging that we are liable to make errors compromising our rational and practical conduct. So, because judgment is a comparison of ideas, a detailed study of ideas must precede it. In the Port-Royal Logic, “idea” is an undeﬁned term: “The word “idea” is one of those that are so clear that they cannot be explained by others, because none is more clear and simple” (I, i, 40, trans. 25). “Idea,” therefore, is a primitive term that can only be described negatively. Accordingly, the authors maintain that ideas are neither visual images nor mere names, and are not derived from the senses, because, although the senses may give occasion to forming ideas, it is only our spirit that produces them. Once ideas are produced, logic investigates their possible relations and the operations one can perform on them. Such relations and operations are founded on a basic property of universal or common ideas (as diﬀerent from singular ideas): the property to have a comprehension and an extension. The comprehension of an idea consists of “the attributes that it contains in itself, and that cannot be removed without destroying the idea. For example, the comprehension of the idea of a triangle contains extension, shape, three lines, three angles, and the equality of these three angles to two right angles, etc.” (I, vi, 59, trans. 39). The fact that the comprehension of “triangle” contains not only three lines but also the property proved by the theorem that the sum of its angles is equal to two right angles, gives way to speculations as to what extent humans dominate the comprehensions of their own ideas (Pariente 1985, 248ﬀ). The extension of an idea “are the subjects to which the idea applies. These are also called the inferiors of a general term, which is superior with respect to them” (I, vi, 59, trans. 40). The notion of extension is ambiguous. The subjects to which an idea applies can be intended either as the class of individuals of whom that idea can be predicated, or as the ideas in whose

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comprehension that idea is contained. In the latter sense, extension is clearly deﬁned in terms of comprehension; therefore, comprehension is considered a primitive notion in comparison with extension. The ambiguity of the notion of extension, already noticed by some interpreters (Kneale and Kneale 1962, 318–319), is somehow intended, for it serves, as we will see, to deﬁne diﬀerent properties of the operations that can be performed on ideas, as well as to solve classical problems of quantiﬁcation in the doctrine of judgment and reasoning. If we subtract an attribute from the comprehension of an idea, by deﬁnition we obtain a diﬀerent idea, in particular a more abstract one: Given the idea “man,” whose comprehension certainly contains “animal, rational,” by subtracting “rational” we destroy the idea “man” and obtain a diﬀerent and more abstract idea, which is the residual idea with respect to the original comprehension of “man.” The operation just described is abstraction that, if reiterated, produces an ascending hierarchy of increasingly abstract ideas. The operation of abstraction is the means by which the Port-Royal Logic introduces an inverse relation between comprehension and extension, often called the “Port-Royal Law” in subsequent literature. For the authors maintain that in abstractions “it is clear that the lower degree includes the higher degree along with some particular determination, just as the I includes that which thinks, the equilateral triangle includes the triangle, and the triangle the straight-lined ﬁgure. But since the higher degree is less determinate, it can represent more things” (I, v, 57, trans. 38). This means that the smaller the comprehension, the larger the extension, and vice versa. To move from a higher to a lower idea in the hierarchy we have to restrict the higher one. Restriction can be of two kinds. The ﬁrst kind of restriction is obtained by adding a diﬀerent and determined idea to a given one: If to the idea A (animal) we add the idea C (rational), so as to have a new idea composed by the joint comprehensions of A and C, and if we call B (man) the idea thus composed, then B is a restriction of A and C and is subordinated to them in the hierarchy of ideas. This restriction cannot be obtained by adding to an idea some idea it already contains in itself, for the alleged restriction would be a mere explication of the given idea: If B contains A, then by adding A to B we obtain B, that is BA = B. The second kind of restriction consists in adding to a given idea “an indistinct and indeterminate idea of part, as when I say ‘some triangle.’ In that case, the common term is said to become particular because it now extends only to a part of the subjects to which it formerly extended, without, however, the part to which it is narrowed being determined” (I, vi, 59, trans. 40). The possibility of two kinds of restriction shows that comprehension and extension do not enjoy the same properties. While the ﬁrst restriction modiﬁes the comprehension of the restricted idea so that we get a diﬀerent idea having a richer comprehension and a smaller extension, the second restriction concerns only the extension of the restricted idea with no modiﬁcation of its comprehension: It remains the same idea. But this depends on the fact that the notion of extension of the Port-Royal Logic is,

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as already noticed, ambiguous. In this second case, the extension is obviously constituted by the individuals to whom the idea applies. The Port-Royal Logic, also due to Augustine’s inﬂuence, is very attentive to the linguistic expression of ideas. The authors maintain that if reﬂections on our thought never concerned anyone but ourselves, it would be enough to examine them in themselves, unclothed in words or other signs. But we can make our thoughts known to others only by associating them to external signs, and since this habit is so strong that even in solitary thought things are presented to the mind by means of the words we use in speaking to others, logic must examine how ideas are joined to words and words to ideas. (untitled preface 38, trans. 23–24) This means that thought is prior to language and that a single thought can underlie diﬀerent linguistic forms. This view of the relation between thought and language is one of the guidelines behind the project of a universal grammar contained in the Grammaire générale et raisonnée, published in 1660 by Arnauld in collaboration with Claude Lancelot (1616–1695) (Arnauld and Lancelot [1676]). This view, which has received great attention since Noam Chomsky’s (1966) much-debated claim that it preﬁgures transformational generative grammar, is also relevant to logic (see Dominicy 1984). Given that logic studies the properties of ideas, their mutual relations, and the operations that can be performed on them, and given that ideas are designated by words which can be equivocal, the authors establish the convention that, at least in logic, they will treat only general or universal ideas (as diﬀerent from singular ideas) and univocal words (I, vi, 58, trans. 39). Such are the words associated to ideas by way of a nominal deﬁnition, meant as the imposition of a name to an idea by way of a free, public, and binding baptismal ceremony, of the kind used in mathematics and whose model is found in Pascal ([1658 or 1659], 242ﬀ). Nominal deﬁnitions make it possible to use words (particularly substantives) of ordinary language as if they were the signs of a formal language in which everything is explicit: The best way to avoid the confusion in words encountered in ordinary language is to create a new language and new words that are connected only to the ideas we want them to represent. But in order to do that it is not necessary to create new sounds, because we can avail ourselves of those already in use, viewing them as if they had no meaning. Then we can give them the meaning we want them to have, designating the idea we want them to express by other simple words that are not at all equivocal. (I, xii, 86, trans. 60) We can not only make abstractions and compositions of ideas but also compare them and, “ﬁnding that some belong together and others do not, we

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unite or separate them. This is called aﬃrming or denying, and in general judging” (II, iii, 113, trans. 82). Since judging is a comparison of ideas, and since the activity of making syllogistic inferences can be considered as a comparison of two ideas through a third one, syllogistic inferences lose much of their importance, while judging is established as the most important of our logical activities (Nuchelmans 1983, 70–87). This does not mean that the Port-Royal Logic neglects syllogisms. On the contrary, it contains an articulate doctrine of syllogism based on the fundamental principle that, given two propositions as premises, “one of the two propositions must contain the conclusion, and the other must show that it contains it” (III, xi, 214, trans. 165). It also contains a nontrivial treatment of syllogistic moods that was to be implemented by the young Leibniz in his De Arte Combinatoria (see section 8). Such a treatment is based on “the law of combinations,” applied to “four terms (such as A.E.I.O.)” giving 64 possible moods, and on a set of rules that make it possible to select the well-formed ones, so that, given rules for the valid moods in each ﬁgure, one can dispense with the doctrine of the reduction of other ﬁgures to the ﬁrst: Each mood of any ﬁgure is proved valid by itself (III, iv, 88–89, trans. 143–144). Indeed, though the ﬁrst edition of the book contained a chapter on the reduction to the ﬁrst ﬁgure, that chapter is left out of all subsequent editions. Nevertheless, though the authors seem competent in pointing out a frequent confusion between the fourth ﬁgure and the ﬁrst ﬁgure with transposed premises (III, viii, 202, trans. 155), they are very traditional in other respects. For instance, they reduce the consequentiae asyllogisticae, called complex and composed syllogisms, to the classical categorical moods, thus provoking a reproach from Vagetius in the preface to the second edition of Jungius’s Logica Hamburgensis (see section 3). The Port-Royal Logic attributes great importance to method, corresponding to the operation of the spirit called “ordering.” Method is divided into two major sections: The ﬁrst, devoted to demonstration and science, follows Descartes’s methodical rules, thus giving an outline of the methods of analysis and synthesis; the second, devoted to opinion and belief, contains interesting observations about epistemic modalities and probability, as well as the outlines of Pascal’s wager on the existence of God. By introducing the question of probability, the Port-Royal Logic breaks away from one of the major tenets of Descartes’s philosophy, and opens new perspectives for a probability not limited to games of chance but extended also to events valuable on the basis of frequencies (Hacking 1975b). The Port-Royal Logic was highly successful, as can be gathered from its numerous editions (Auroux 1993, 87). Its inﬂuence, also thanks to Latin, English, and Spanish translations (Risse 1964–70, II 79), is apparent in most of the subsequent European logical literature. Obviously some scholars still preferred the Aristotelian model. For instance, John Wallis (1616–1703), one of the best mathematicians of the time, though the Port-Royal Logic agreed with the argument he had already produced in 1638 that in syllogisms singular propositions must be considered as universal, rejects the Ramist syllogistic

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moods (Wallis 1687) on Aristotle’s authority alone. A direct reaction to the Port-Royal Logic is presented by Henry Aldrich (1648–1710), whose Artis Logicae Compendium (Aldrich [1691]), published anonymously, reprinted many times and still widely used in the Victorian era, preserves scholastic doctrines and an account of the syllogism which “is the best available” (Ashworth 1974, 237). In some concluding remarks, Aldrich criticizes the fundamental principle of syllogism of the Port-Royal Logic, which he considers as a disguised version of the dictum de omni et nullo. Yet the very existence of such a criticism implicitly proves the fame the Port-Royal Logic had achieved (Howell 1971, 54–56). As for Port-Royal’s seminal theory of probability, the young Leibniz already acknowledged its merits in 1667 (Leibniz 1923–, VI, i, 281n), while Jakob Bernoulli wrote his Ars conjectandi, published posthumously (Bernoulli [1713]), as a development of that theory and as a complement to the Ars cogitandi, the Latin version of the Art de penser (Hacking 1975b, 78; Daston 1988, 49).

6. The Emergence of a Logic of Cognitive Faculties Toward the end of the seventeenth century, many logicians developed an interest in the analysis of cognitive faculties. Descartes moved in that direction when he focused his attention on the operations of intuition and deduction, but also the Port-Royal Logic considered the reﬂection on the nature of the mind as the means for a better use of reason and for avoiding errors. The study of cognitive faculties was not simply meant to provide an expositive framework for logical doctrines. As a matter of fact, dealing with the nature and object of logic and with the justiﬁcation of the traditional partitions of logical treaties through a reference to mental operations had been a well-established practice since Aristotle’s Organon: The operations of simple apprehension, judgment, and reasoning had been mentioned as mental counterparts to the logical doctrines of concepts, judgments, and inferences (for a similar approach see Sanderson, section 2). The novelty of what has been called the “facultative logic” of the late seventeenth and eighteenth centuries (Buickerood 1985) is that the cognitive operations involved in the formation and use of ideas become a central concern of logicians. The most important author working on a logic of cognitive faculties is John Locke (1632–1704). The Essay concerning Human Understanding (Locke 1690) is often quoted as a primary example of indiﬀerence, if not contempt, for logic. This is not true if it is intended to describe Locke’s attitude to logic in general, rather than his attitude toward the doctrine of syllogism. Locke, who had been provided at Oxford with a sound scholastic logical education (Ashworth 1980), asks logicians to give up their claim that they teach humans, who are born rational, how to reason by means of syllogisms (Essay, IV, xvii, 4). Logical research should rather investigate the way we form, designate, combine and, in general, use ideas. This concept of logic attributes fundamental importance to language. If logic is the study of the faculties that produce and work with ideas, then it

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becomes impossible to ignore that we can work with ideas only in so much as we connect them to linguistic signs. This very connection of ideas to linguistic signs, although very useful, is a main source of our errors; therefore logic must provide a detailed treatment of the use and of the abuse of words. Indeed Locke compares logic to semiotic: “σημειωτικ , or the Doctrine of signs, the most usual whereof being Words, . . . is aptly enough termed also λογικ , Logick” (Essay, IV, xxii, 4). Locke does not intend to privilege the study of words over that of ideas, for he considers both ideas and words as signs, with the diﬀerence that ideas are signs of things and words are signs of ideas (see Ashworth 1984). Nevertheless it cannot be doubted that the success of Locke’s philosophy contributed to the extraordinary importance language was to have for a large section of later philosophy and logic (see Hacking 1975a). One could say that a logic centered on the study of faculties that produce ideas includes a good deal of epistemological, psychological, and linguistic research (see Hatﬁeld 1997). This is true, but one should consider that this is a consequence of a double eﬀort. On the one hand, Locke wanted to continue the old battle against the ontological basis of the Aristotelian logical tradition by eliminating all talk about natural essences. On the other hand, he wanted to win the new battle against Cartesian innatism (accepted by the Port-Royal Logic) by investigating the empirical origin of our thought. A study of such questions and of the correct use of our ideas and their signs could provide a guide to man’s intellectual conduct in the exercise of judgment, especially in an age characterized by a strong skeptical movement. But providing men with a guide in making judgments was seen as the purpose of logic: In this respect, Locke had many predecessors, notably the authors of the Port-Royal Logic. It must also be considered that it was still left to logic, once the observation of cognitive operations was completed and a careful reﬂection on them was performed, to establish norms for the correct use of those very operations. Defenders of the Aristotelian tradition, such as John Sergeant (1622–1707), criticized Locke’s approach to logic (see Howell 1971, 61–71). But many more were Locke’s admirers, and the impact of his views was impressive. Particularly successful was his refusal of innate ideas (Essay, I, ii), and his emphasis on probable knowledge, though he was far from considering probability from a quantitative point of view (see Hacking 1975b, 86–87). What interests us is that from a very early stage, Locke’s doctrines were included in logic textbooks, frequently in association with direct or indirect references to the Port-Royal Logic. This seems strange if one considers that Locke and the Port-Royalists were so far apart on the question of the origin of ideas. But this important diﬀerence was overlooked by taking into account what Locke and the PortRoyal Logic had in common: the purpose of logic as the guide for correct judgment, the idea that logicians should reﬂect on human understanding, the importance of the linguistic expression of our thoughts. Moreover, the Port-Royal Logic, whose intended readers included the students of the petites écoles of Port-Royal, who had to be prepared for the curricula of European

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universities, was a good source of information about traditional logical topics (such as syllogisms, fallacies, method) that the Essay lacked and that the Port-Royalists had treated without deference toward Aristotle. Perhaps the ﬁrst of such logic handbooks was the Logica sive ars ratiocinandi (Le Clerc 1692) written by the Swiss Jean Le Clerc (1657–1736), who adopted Locke’s doctrine of judgment, his classiﬁcation of ideas, and his philosophy of language, and proposed a mixture of Locke’s and Port-Royal Logic’s theses as concerns the question of probability. A similar approach to logic is to be found in another Swiss scholar, Jean-Pierre Crousaz (1663–1750), who published a series of books on logic, the latest of which was Logicae systema (Crousaz 1724). As it can be expected, a number of British authors published Locke-oriented logic handbooks, most of which had several editions in Britain and elsewhere. Isaac Watts (1674–1748), in his Logick (Watts 1725) and in a popular supplement to it (Watts 1741), followed Crousaz and Le Clerc in oﬀering a Lockean analysis of human nature (especially perception) with a preference for judgment and proposition, rather than syllogism. William Duncan (1717–1760), in his Elements of Logick (Duncan 1748), stated that the object of logic is the study of the faculties of the human understanding and of cognitive procedures, and divided logic into four parts according to the model of the Port-Royal Logic (see Yolton 1986). Francis Hutcheson (1694–1746) too, in his posthumous Logicae compendium (Hutcheson 1756), considered logic as the study of cognitive faculties, but also oﬀered a kind of axiomatic presentation of syllogistic. The Essay was promptly translated into Latin and French. By 1770, professors of Prussian universities were oﬃcially asked to follow Locke in their lectures on metaphysics (von Harnack [1900], I, i, 373). But much before that date Georg Friedrich Meier (1718–1777), the author of the text adopted by Kant for his logic courses (see section 11), already used the Essay in his lectures. The reception of the Essay as a book of logical content was made easier by the inclusion of Locke’s doctrines in logic handbooks such as those we have mentioned. But it was left to Locke’s posthumous Of the Conduct of the Understanding (Locke [1706] 1993), originally intended as an additional chapter to the fourth edition of the Essay, to enter directly into the ﬁeld of logical literature. For in this small book, Locke explicitly declares his views to be an improvement over the standard logic of his time (Buickerood 1985, 183). Of the Conduct of the Understanding was widely read not only in Locke’s own country (Howell 1971, 275ﬀ.) but also in Germany. In the second decade of the eighteenth century, the Thomasian philosopher Johann Jakob Syrbius (1674–1738) used it as a guide for his lectures, and Wolﬀ referred to it in his German Logic (Wolﬀ [1713], Preface). Later on Georg David Kypke translated it into German (Locke [1755]), but at the time of this translation Locke had already been favorably discussed in the incipient German literature on the history of logic (Budde 1731). In France, Locke’s views were well received by authors who also supported Port-Royal’s doctrines. This is the case of Jean Baptiste d’Argens (1704–1771)

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who, in the section devoted to logic of a philosophical treatise addressed to ladies, repeated the Lockean argument that all ideas originate from the senses, at the same time referring to the Port-Royal Logic (d’Argens 1737, Log. §§3, 1) in very positive terms. Among French scholars, Étienne Bonnot de Condillac (1714–1780), the inﬂuential promoter of a radically empiricist philosophy usually referred to as “sensationism,” deserves a special mention. In works of logical content written in the later part of his life—La Logique (Condillac [1780]) and the posthumous La langue des calculs (Condillac [1798])—he developed a concept of logic that owes much to Locke, although he proudly maintains that it is similar to no one else’s. Condillac describes logic as consisting of an analysis of experience by which we study both the origin of ideas and the origin of our own faculties (Condillac [1780], Preface). For, diﬀerently from Locke, who admitted sensation and reﬂection as sources of our ideas, Condillac admits sensation only and maintains that not only ideas, but also all our faculties originate from the use of the senses. This is possible because our senses are complemented by our fundamental linguistic capacity: We would not have complex ideas nor would we be capable of operating with them if we had no language. Consequently, Condillac maintains that any science is a well-made language and such is also the art of reasoning, a conviction that made him reverse the priority order of grammar and logic (Auroux 1993, 93). He adds that, to build a well-made language, we need a method because language, despite its role in the acquisition of our cognitive faculties, has also been used to produce a jargon for false philosophies. The method Condillac recommends for the construction of a well-made language in any science is analysis, in particular analysis as it is used in mathematics, that he considers the paradigm of a well-made science whose language is algebra.

7. Logic and Mathematics in the Late Seventeenth Century At the end of the seventeenth century, another image of the discipline began to emerge. It was borne out of a comparison of logic with mathematics, a comparison intended to prove the superiority of mathematics over logic. Some authors ascribed the superiority of mathematics to its axiomaticdeductive method. This conviction had enjoyed considerable success (see De Angelis 1964), and was enhanced by an interpretation of Descartes’s rules of method as recommendations to begin with a few simple and already known notions (axioms) and then proceed to unknown notions (theorems). Those who endorsed this interpretation seemed to draw the conclusion that the old logic centered on syllogism should be replaced by a new logic intended as a method to ﬁnd and order truths according to the model of the mos geometricus. Some authors attributed the superiority of mathematics to the problemsolving and inventive techniques of algebra. In this perspective, the search for equations relating unknown to given elements, exempliﬁed in Descartes’s

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Géométrie (1637), was interpreted as the true Cartesian logic and was absorbed into the tradition that viewed mathematics as a universal science of invention. In the seventeenth century, algebra was still a new technique, independent of logic, that many considered a rediscovery, but also an improvement of ancient mathematical doctrines. A reference to Descartes was somehow inevitable in this ﬁeld, too, since Descartes appreciated algebra as an intermediate step toward his more abstract mathesis universalis. (On the origins of algebraic thought in the seventeenth century, see Mahoney 1980.) Naturally enough, some authors suggested that algebra could be a useful model for logic. This is the case of Ehrenfried Walter von Tschirnhaus (1651– 1708). He left syllogisms and other traditional parts of logical treatises out of his logic, but, as the title of his major work declares, he held that logic must provide a Medicina mentis, a remedy against the illness represented by our errors, and an aid for the healthy art of invention (Tschirnhaus [1695]). In particular, he claimed that his method of invention would have, in all ﬁelds of knowledge, the same function of algebra in the mathematical sciences. What he actually did, however, was to give an exposition of the methods of analysis and synthesis and a comment on Descartes’s rules of method, albeit with a new attention to empirical sciences for which he envisaged a mixed method of a priori and a posteriori elements (Wollgast 1988a). The assessment of the positive role that algebra could have for logic outlived the idea that logic should imitate, or even be substituted by, the axiomaticdeductive method. The latter was either reduced to a mere synthetic (top-down) order of exposition, or was declared inadmissible outside mathematics, either because of intrinsic diﬀerences between mathematics and other sciences (and philosophy), or because it was held responsible for the degeneration of Cartesianism into Spinozism. But also the algebraic model underwent profound changes. For the algebraic model, followed by many logicians from the late seventeenth century up to almost the end of the eighteenth century, is not the same as the algebraic model used in problem solving. Algebra is no longer considered as a methodical paradigm to be followed analogically by logic to restore the latter’s function as an intellectual guide but as a tool for logic. Many logicians now try to apply algebraic techniques directly to logical objects, that is, to ideas and propositions. In other words, they try to build a logical calculus based on a symbolic representation of logical objects and on rules for manipulating signs, on the assumption that an adequate symbolism has been used. From this point of view, doctrines of ideas such as those of the Port-Royal Logic and of the emergent logic of cognitive faculties, usually considered extraneous to the development of mathematically oriented logic, instead acted as stimuli and provided a ﬁeld of application for the ﬁrst tentative logical calculi. On the one hand, as it has been pointed out (Auroux 1993, 94), a calculus of ideas needs a theory of ideas. On the other hand, scholars who had welcomed a logic of cognitive faculties professed the highest esteem for algebra. We have mentioned Condillac’s positive reference to algebra as the language of mathematics, but decades earlier Nicolas Malebranche (1638–1715)

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had claimed that “algebra is the true logic” (Malebranche [1674] 1962, VI, i, v). Also Locke, who belittled syllogistic and the axiomatic method, did not hide his admiration for algebra (Essay, IV, xii, 15). Locke, however, did not even think of applying the powerful tool of algebra to ideas. Nor did Thomas Hobbes (1588–1679), although in his De corpore, published in 1655, whose ﬁrst part is signiﬁcantly entitled Computatio sive Logica, he maintained that reasoning is computation, where computing means adding several things or subtracting one thing from another in order to know the rest (Hobbes [1839] 1961, I, i, §2). But other scholars were ready to attempt the actual construction of logical calculi. We examine some of such attempts, but ﬁrst consider a declared failure to establish a parallelism between logical and algebraic reasoning, that is, the Parallelismus ratiocinii logici et algebraici (Bernoulli [1685] 1969) of the above-mentioned Jakob Bernoulli (1655–1705). This is an academic dissertation in which Bernoulli was the Praeses (and therefore the real author) and his younger brother Johann (1667–1748) was the Respondens, a circumstance that has often brought to the attribution of the work to the “Bernoulli brothers.” The parallelism concerns the objects, the signs, and the operations of both logic and mathematics. The objects of logic are ideas of things, while the objects of mathematics are ideas of quantity. Likewise, the signs of ideas of things are words (“man,” “horse”), while the signs of quantity are letters of the alphabet: a, b, c for known quantities, and x, y, z for unknown quantities. Bernoulli does not use literal symbols for ideas of things because, on a par with the Port-Royal Logic, he assumes that every idea of thing is (at least in logic) univocally designated by a word, so that every idea of thing has its own sign. Bernoulli then introduces the operations we perform on both kinds of ideas: (1) to put together, (2) to take away, (3) to compare. 1. Ideas of quantity are put together by the sign “+”, as in “a + b.” Ideas of things are put together by the connective “and”, as in “virtue and erudition.” 2. From an idea of quantity one can take away a smaller quantity, thus obtaining the diﬀerence: Given a and b, where a is greater than b, the taking away of b from a is denoted by “a − b.” Similarly, from a complex idea of thing, one can take away one of the less complex ideas it contains, thus obtaining the diﬀerence: From the complex idea “man” one can take away “animal” and the diﬀerence is “rational.” 3. Given two ideas of quantity, if the mind perceives an equality between them, it unites them by the sign “=”, as in “a = b.” If the mind perceives an inequality between them, it uses the signs “>” and “ b,” “a < b.” Given two ideas of things, the mind can ﬁnd (1) agreement or identity between them, and in this case it will aﬃrm one of the other; (2) disagreement or diversity between them, and in this case it will deny one of the other. Aﬃrmation and negation take place thanks to an enunciation (enunciatio), and are expressed by “it is” (est) and “it is not” (non est), as in “man is animal,” “man is not brute.”

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While there is a parallelism between algebra and logic with respect to the operations of putting together and taking away, the parallelism breaks down in the case of comparison. Bernoulli ﬁrst considers the case of agreement. Two ideas of quantity agree when a common measure, applied to them the same number of times, exhausts both, that is, when they are equal. Two ideas of things agree, so that it is possible to aﬃrm a predicate of a subject, provided that a third idea, common to both, exhausts at least the predicate: It is possible to aﬃrm “theft is sin,” provided that some common idea is found in “theft” (no matter if “theft” contains some other ideas besides) and exhausts “sin.” It is clear that Bernoulli intends the comparison of ideas of things as the comparison of their comprehensions. This is conﬁrmed by the fact that “theft is sin” is an indeﬁnite proposition, that is, a proposition whose subject is not quantiﬁed. Now, Bernoulli explains, in an indeﬁnite proposition the predicate is found in the nature of the subject, which means that it cannot be taken away from the idea of the subject in which it is contained without destroying it, according to the Port-Royal Logic deﬁnition of comprehension. We have seen that for Bernoulli the agreement or identity of subject and predicate subsists even if it is incomplete, that is, if the third idea common to both exhausts the predicate without exhausting the subject. Here a problem arises: While the equality of quantities is mutual (if a = b, then b = a), an aﬃrmative indeﬁnite proposition expressing an incomplete agreement is not convertible: “Man is rational” is true, but “Rational is man” is false. Moreover, what happens if the predicate is not found in the nature of the subject but is constituted by some accidental attribute? The answer is that it would be impossible to establish even a partial agreement and, strictly speaking, it would be impossible to aﬃrm that predicate of the subject. Bernoulli decides to overcome these problems by quantifying over the subjects, that is, by taking extensions into account. In this way it becomes possible to form true aﬃrmative propositions such as “All men are sinners” and “Some men are learned,” that is, propositions that in the indeﬁnite form (“Man is sinner” and “Man is learned”) are false. The proposition “All men are sinners” is particularly interesting because it is a true universal proposition although “sinner” is not found in the nature of “man” (Jesus is [also] man, but is not sinner). Therefore, Bernoulli states that the subject of universal and particular propositions are “the species or the individuals that are contained under that [subject]” (Bernoulli [1685] 1969, §11, trans. 176). Is it practical to consider which are the essential attributes of the subject and which are the accidental ones and, in the ﬁrst case, be allowed to make indeﬁnite judgments while, in the second case, resort to quantiﬁed ones? And how to overcome the problem of the impossibility of converting true indeﬁnite propositions, which are exactly those that most resemble algebraic equations? Bernoulli suggests that one should always quantify all aﬃrmative propositions, including true indeﬁnite ones. Consequently, one will be allowed to say “Some men are learned,” which can be converted simpliciter into “Some learned beings are men,” as well as “All men are sinners” and “All men are rational,”

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which can be both converted per accidens into “Some sinners are men” and “Some rational beings are men.” Diﬀerences between algebra and logic also appear when a comparison of ideas shows their disagreement. In algebra, the disagreement of two ideas of quantity means that between them there is an inequality, a relation designated by the sign “ a. In logic, the disagreement of two ideas of things is expressed by a negative indeﬁnite proposition. But the subject and the predicate of negative indeﬁnite propositions can be converted only if the disagreement depends on the opposition of the ideas considered, as in the case of “man is not beast.” This means, as Bernoulli explained in later essays (see Capozzi 1994), that by converting “Animal is not a man” one obtains “Man is not animal,” which is false, because man and animal are not opposite ideas. Also in this case, logic has to resort to the quantiﬁed propositions of old syllogistic, but this means that there are no real logical equations between the ideas themselves. Not so in algebra, as it can be proved by the fact that every inequality is perfectly convertible. The conclusion is that no complete parallelism exists between algebra and logic. As a substitute, Bernoulli recommends the direct use of algebra in science by arguing that in science everything can be quantiﬁed and all that can be quantiﬁed can undergo algebraic treatment. His pioneering mathematical treatment of probability goes in that direction. Bernoulli’s case is instructive. It shows that this is not a lethargic period of logic, as some historians have maintained (Blanché 1996, 169–178), but it also makes one wonder what made Bernoulli fail where other logicians—at the same time or a few decades later—made progresses. In our opinion, the main reason for Bernoulli’s failure was the doctrine of ideas he choose. We have already pointed out that Bernoulli depends on Port-Royal’s view that every idea of thing can be univocally designated by a word (at least in logic), and that every idea of thing is endowed with an indestructible comprehension, conveyed by the word. In the case of aﬃrmation, this makes Bernoulli consider only the relation of containment of the predicate in the subject as basic. Consequently, he is unable to deal with possible predicates that do not disagree with the content of the subject but are not contained in its comprehension. To build a calculus it is not enough to have a rudimentary algebra and a doctrine of ideas. One has to choose a suitable doctrine of ideas.

8. Leibniz The German logician and philosopher Gottfried Wilhelm Leibniz (1646–1716) has a foremost role in the history of formal logic. However, it is almost impossible (and probably misleading) to represent his contributions to logic as a single and coherent set of theories nicely inserted within a linear path of development. There are at least three reasons that rule out such a reassuring view. In the ﬁrst place, Leibniz contributed ideas—often through scattered and

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incomplete fragments rather than through structured and polished writings—to a plurality of logically relevant subjects: from the arithmetization of syllogistic to the theory of relations, from modal logic (and semantic of modal logic) to logical grammar—and the list could be easily extended. Moreover, relevant contributions are often found in works, fragments, or letters not explicitly devoted to the ﬁeld of logic. In the second place, most of Leibniz’s writings testify to a work in progress in the deepest meaning of the expression: Diﬀerent and sometimes incompatible strategies are explored in fragments dating back to the same years or even to the same months, corrections and additions may substantially modify the import of a passage, promising and detailed analysis remain uncompleted or are mingled with sketchy hints. Nevertheless, there is a method in Leibniz’s passionate and uninterrupted research: a deeper unity that is given by a set of recurring problems and by the wider theoretical framework in which they are dealt with. Last but not least, it should always be kept in mind that Leibniz’s works known by his contemporaries and immediate successors constitute a very limited subset of his actual production. It is only during the twentieth century that Leibniz’s role in the history of logic came to be fully appreciated, and this appreciation is connected to at least two diﬀerent moments: the publication by Louis Couturat, in 1903, of the Opuscules et fragments inédits de Leibniz (Leibniz 1966; see also Couturat 1901), many of which were devoted to logic, and the progress made during the second half of the century in the publication of the complete and critical edition of Leibniz’s texts (Leibniz 1923–). Almost three centuries after his death, this edition (the so-called Akademie-Ausgabe) is, however, still to be completed. Leibniz’s interest in logic, and the amplitude of his logical background, is already evident in his youthful Dissertatio de Arte Combinatoria (Leibniz 1923–, VI, i, 163–230). This work is subdivided into 12 problemata (problems), mainly devoted to the theory of combinations and permutations, accompanied by a discussion of some of their usus (applications). Of greatest logical relevance are the combinatorial approach to syllogistic and the discussion of a symbolic language (characteristica) based on a numerical representation of concepts. In dealing with syllogistic, Leibniz takes the work done by Hospinianus (Johannes Wirth, 1515–1575) as his starting point. Like him, Leibniz considers four diﬀerent quantities—singular (S) and indeﬁnite (I) propositions are added to universal (U) and particular (P) ones—and the two traditional qualities given by aﬃrmative (A) and negative (N) propositions. Given that a syllogism consists of three propositions (the two premises and the conclusion), we have 43 possible combinations of the four diﬀerent quantities and 23 possible combinations of the two diﬀerent qualities. The number of possible diﬀerent simple moods of the syllogism, valid and invalid, is therefore according to Leibniz (43 × 23 ) = 512: the same result obtained by Hospinianus. If we take into account the four diﬀerent syllogistic ﬁgures (Leibniz includes and explicitly defends the fourth ﬁgure, which was rejected by Hospinianus), we get a total of (512 × 4) = 2048 “moods in ﬁgure.” It is still through a combinatorial

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method, based on the exclusion of the syllogisms conﬂicting with four classic rules (nothing follows from pure particulars, no conclusion can be of stronger quantity than the weaker premise, nothing follows from pure negatives, and the conclusion follows the quality of the weaker premise) and of eight further moods conﬂicting with the rules given for the four syllogistic ﬁgures, that Leibniz gets the number of 88 valid syllogistic moods. Whereas Hospinianus assimilated singular propositions to particular ones, Leibniz considers them similar to universal propositions, while indeﬁnite propositions are connected to particular ones. In this way the number of valid moods in ﬁgure can be reduced to 24 (6 in each of the four ﬁgures): the 19 “classical” ones, plus 5 new ones that are actually the result of applying subalternation to the conclusions of the 5 “classic” moods with a universal conclusion. The interest of Leibniz’s treatment of syllogistic in the De Arte Combinatoria is not to be found in radical innovations concerning the number of moods of valid syllogisms, but rather in the fact that they are obtained through the systematic use of a combinatorial calculus, used as a sort of deductive device. The syllogistic “deduction” of the rules of conversion is also part of this attempt, based—as in Ramus and in a number of sixteenth- and seventeenthcentury German logicians, including Leibniz’s former teacher Jakob Thomasius (1622–1684)—on the use of identical propositions. In a later fragment, the De formis syllogismorum mathematice deﬁniendis (Leibniz 1966, 410–416), identical propositions are used to obtain a syllogistic demonstration not only of conversion but also of subalternation. Since in this demonstration a ﬁrst ﬁgure syllogism is used, Leibniz can “deduce” all valid moods of the second and third ﬁgure using only the ﬁrst four moods of the ﬁrst ﬁgure, together with subalternation and the rule according to which if the conclusion of a valid syllogism is false and one of its premises is true, the second premise should be false, and its contradictory proposition should therefore be true (methodus regressus). The valid moods of the fourth ﬁgure can be deduced using conversion (the syllogistic proof of which only required moods taken from the ﬁrst three ﬁgures). In this way, Leibniz will complete his construction of syllogistic as a sort of “self-suﬃcient” deductive system. The second result of the De Arte Combinatoria worth mentioning is the construction of a symbolic language in which numbers are used to represent simple or primitive concepts, and their combinations (subdivided in classes according to the number of primitive concepts involved) are used to represent complex or derivate concepts. Fractions are used to simplify the representation of complex concepts, with the numerator indicating the position of the corresponding term within its class and the denominator indicating the number of the class, that is, the number of primitive concepts involved. In Leibniz’s opinion, such a language would oﬀer a solution to the main problems of the logica inventiva (logic of invention): ﬁnding all the possible predicates for a given subject, all the possible subjects for a given predicate, and all the possible middle terms existing between a given subject and a given predicate. This would also allow a mathematical veriﬁcation of the truth of propositions and of the correctness of

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syllogistic reasoning. As we shall see, this project was to ﬁnd a more developed and logically satisfactory form a few years later in 1679. The idea of constructing a symbolic language in which numbers represent concepts is not new. Such an idea was already present in a number of attempts to construct a “universal language,” attempts that were often inﬂuenced by the combinatorial works of Ramon Llull. Leibniz himself makes reference to the works by Johann Joachim Becher (1635–1682?; Character pro notitia linguarum universali, 1661) and Athanasius Kircher (1602–1680; Polygraphia nova et universalis ex combinatoria arte detecta, 1663); similar attempts were made by Cave Beck (1623–1706?; The Universal Character, 1657) and others, and are described by Kaspar Schott (1608–1666; Technica Curiosa VII—Mirabilia graphica, sive nova aut rariora scribendi artiﬁcial, 1664). In the same period, the Spanish Jesuit Sebastian Izquierdo (1601–1681; Pharus Scientiarum, 1659), aiming at a sort of “mathematization” of the ars lulliana, substituted numerical combinations for the alphabetical ones used by Llull, and something similar to a “numerical alphabet”—highly praised by Leibniz—was developed by George Dalgarno (c. 1626–1687) in his Ars Signorum (1961). In Leibniz, however, the construction of a numerical characteristica is not only a handy representational device; it is strictly connected with the idea of the inherence of the predicate in the subject in every true aﬃrmative proposition (predicate-in-subject or predicate-in-notion principle). This idea was already present in the scholastic tradition: In his Commentary on Peter of Spain’s Tractatus, Simon of Faversham (c. 1240–1306) writes that propositions “are called complex because they are founded on the inherence of the predicate in the subject, or else because they are caused by a second operation of the intellect, namely the composition and division of simples” (Simon of Faversham 1969). During the Middle Ages, however, the inherence theory of the proposition was confronted with the idea according to which “in every true aﬃrmative proposition the predicate and the subject signify in some way the same thing in reality, and diﬀerent things in the idea” (Thomas Aquinas 1888–1889, I, xiii, 12). The predicate-in-notion principle was to become a cornerstone of Leibniz’s logical work, and Leibniz was to apply it not only to analytical but also to contingent propositions: “always, in every true aﬃrmative proposition, necessary or contingent, universal or singular, the concept of the predicate is included in some way or other in that of the subject” (Leibniz 1973, 63). In the De Arte Combinatoria, however, Leibniz only deals with propositions made of general terms, and—as it has been already mentioned—the principle is mainly used for the discovery of subjects and predicates within the context of the logica inventiva. Its explicit use as a method for checking the truth of a proposition given its subject and its predicate is to be found only in the 1679 essays (see Roncaglia 1988), where Leibniz chooses to represent simple or primitive terms by means of prime numbers. The advantages of this notation were already stressed in a fragment, dated February 1678, known as Lingua generalis: “The best way to simplify the notation is to represent things using multiplied numbers, in such a way that

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the constituting parts of a character are all its possible divisors. . . . Simple elements may be prime or indivisible numbers” (Leibniz 1923–, VI, iv, 66). Just as compound (reducible) terms can be traced back—by means of deﬁnitions—to the simple, irreducible terms constituting them, the “characteristic numbers” of compound terms will be obtainable from the multiplication of the characteristic numbers of the simple terms constituting them, so that the characteristic number of a compound term can always be univocally broken down into those of the simple (relative) terms composing it. Even at the time of the De Arte Combinatoria, Leibniz was conscious of the diﬃculty in ﬁnding terms that are really simple, and he had to be satisﬁed with simple terms of a relative and provisional nature. This diﬃculty was gradually to become, for Leibniz, an actual theoretical impossibility dependent on the limits of human understanding. While we can cope with abstract systems that are the result of human stipulation (and in which simple terms are established by us), only God can handle the much more complex calculus representing the inﬁnite complexity of the actual world (and, as we shall see, of the inﬁnite number of possible worlds among which the actual one has been chosen). To give an example of his notation, Leibniz uses the deﬁnition of “homo” as “animal rationale,” to which the following characteristic numbers are assigned: animal = 2 rationale = 3 homo = (animal rationale = 2 × 3) = 6 To verify the truth of a proposition, one has just to check whether the prime factors of the characteristic number of the predicate are or not all included among those of the characteristic number of the subject. The proposition “Homo est animal” is thus true, since the characteristic number of “animal” (i.e., 2) is a prime factor of the characteristic number of “homo” (i.e., 6). A network of relations is thus established between the ﬁeld of logic and its numerical “model,” allowing an actual logical “interpretation” to be assigned to the numbers and arithmetical operations employed: Number

Term

Prime number

Simple term

Prime factorization of number

Analysis of term

Number expressed in factorial notation

“Real” deﬁnition of term by means of its component simple terms

Multiplication (calculation of the least common multiple)

Conceptual composition

Exact divisibility of a by b (where a and b are the characteristic numbers of the terms A and B)

Veriﬁcation of the truth of the proposition “A is B”

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This calculus, however, presents some diﬃculties if it is used as a device by which to verify all forms of syllogistic reasoning, as Leibniz intended to do. While it permits adequate representation of UA and PN propositions, it is clearly unsuitable—despite Leibniz’s repeated attempts (including the use of fractions and square roots) to get around the problem—to represent UN and PA propositions. The diﬃculties Leibniz encounters here are connected to the representation of the incompatibility between terms: Though it is always possible to ﬁnd the least common multiple of the characteristic numbers of two terms, it should not always be possible to construct a term that includes any two given predicates. Some predicates are simply incompatible. It is then hardly surprising that without a way to aptly “restrict” the combinations of the terms’s characteristic numbers (i.e., to restrict conceptual composition), Leibniz ﬁnds it “too easy” to verify PA propositions and “too diﬃcult” to verify UN propositions. To solve this problem, Leibniz modiﬁes his notation, making it more complex but much more powerful. Instead of using only one characteristic number for each term, he uses a pair of numbers—one positive and one negative—with no common prime factors. The use of “compound” (i.e., with a positive and a negative component) characteristic numbers allows for the following correspondences: “Compound” characteristic number

Term

Positive component of characteristic number

“Aﬃrmative” component of term

Negative component of characteristic number

“Negative” component of term

Prime factorization of “compound” characteristic number

Analysis of term

“Compound” characteristic number expressed in factorial notation, where no common prime factor is present in its positive and negative components

“Real” deﬁnition of term, demonstrating its possibility by the absence of contradictions within its deﬁnition

Presence of a common prime factor in the positive and negative components of a characteristic number

Logical impossibility of the corresponding term

Now, assuming that a(+) and a(−) represent the positive and negative components of the compound characteristic number assigned to the term A, that b(+) and b(−) have the same function with respect to the term B, and that A and B are possible terms (i.e., that no same primitive factor is present in either a(+) and a(−) or in b(+) and b(−)), the following rules for the veriﬁcation of propositions can be stated:

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Presence in a(+) and in b(−) or in a(−) and in b(+) of at least one common prime factor

Incompatibility between A and B: veriﬁcation of the proposition “no A is B” (UN)

Absence in a(+) and in b(−) or in a(−) and in b(+) of common prime factors

Compatibility of A and B: veriﬁcation of the proposition “some A is B” (PA)

Exact divisibility of a(+) by b(+) and of a(−) by b(−)

Veriﬁcation of the proposition “every A is B” (UA)

Non exact divisibility of a(+) by b(+) and of a(−) by b(−)

Veriﬁcation of the proposition “some A is not B” (PN)

This more elaborate attempt is not without ﬂaws, the most relevant being the problem of representing conceptual negation. Leibniz’s proposal to obtain the characteristic number of a negative term like “non-A” by simply changing the sign of the two components of the characteristic number of the corresponding positive term “A” is ill founded and leads to inconsistencies in the calculus. The question raised here—the nature of conceptual negation—has always raised problems for Leibniz (see Lenzen 1986), and is also connected to the more “philosophical” problem of establishing the nature of incompossibility, a problem clearly stated in a famous passage of the fragment known as De veritatibus primis: “This however is still unknown to men: from where incompossibility originates, or what can make diﬀerent essences conﬂict with each other, given the fact that all the purely positive terms seem to be compatible the one with the other” (Leibniz 1875–1890, VII, 195). Nevertheless, Leibniz’s logical essays of April 1679 represent one of the most interesting and complete attempts of arithmetization of the syllogistic, and oﬀers a well-developed—albeit not fully satisfactory—account of traditional logic by means of an intensional calculus. In a sense, they also represent a turning point in Leibniz’s logical works. The unsolved diﬃculties in ﬁnding a numerical model for his still mostly combinatorial calculus, and the problems associated with the representation of negation, probably led him to a twofold shift in his strategies. On the one hand, despite the interest that notational systems will have for him during all his life, Leibniz became increasingly aware that the research of an apt notation should be accompanied by a closer investigation of the logical laws and principles that should constitute the structure of the calculus. On the other hand, semantic acquires a deeper role: Leibniz perceives that the rules governing conceptual composition cannot be reduced to a sort of “arithmetic of concepts,” and are much more complex. Negation, modality (with special emphasis on compossibility and incompossibility among concepts), relations (with special emphasis on identity), complete concepts of individual substances—much of Leibniz’s logical and philosophical work in the following years will deal with these subjects.

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The ﬁrst tendency—a closer investigation of the logical laws and principles to be used in the calculus—is already clear in the Specimen calculi universalis and in its Addenda (both probably dating around 1680). Here, Leibniz abandons the exposition “by examples,” favoring the much more powerful algebraic notation, which uses letters to represent concepts. The perspective is still intensional, and the inclusion of the predicate in the subject remains the cornerstone of the system. According to the Specimen, the general form of a proposition is “a is b,” and a per se true proposition is of one the three following forms: “ab is a,” “ab is b,” or “a is a.” A per se valid conclusion is of the form “if a is b and b is c, then a is c” (principle of syllogism), and according to Leibniz all true propositions can be derived from (or rewritten as) per se true propositions. The Specimen also contains the ﬁrst clear formulation of one of Leibniz’s key principles, that of substitution salva veritate: “Those are ‘the same’ if one can be substituted for the other without loss of truth” (Leibniz 1973, 34; for a discussion of this principle, see Ishiguro 1991, 17–43). Among the principles used are those according to which in conceptual composition the order of terms and the repetition of a term are irrelevant (“ab = ba,” and “aa = a”). The Addenda oﬀer a short discussion of negation, which was not considered in the Specimen itself, and add some further proofs, including the theorem “if a is b, and d is c, then ad will be bc.” Leibniz calls it “praeclarum theorema” and proves it in this way: “a is b, therefore ad is bd (by what precedes); d is c, therefore bd is bc (again by what precedes), ad is bd, and bd is bc, therefore ad is bc” (Leibniz 1973, 41). In the following years Leibniz will often use the signs “+” (or “⊕”) and “−” to indicate logical composition and logical subtraction, stressing that the rules governing operations with concepts are diﬀerent from those of arithmetical addition and subtraction: Whereas in arithmetic “a + a = 2a,” in the case of conceptual composition “a+a = a.” Moreover, Leibniz will carefully distinguish between conceptual subtraction and logical negation: While in an abstract conceptual calculus it is always possible to “subtract” from the concept of man that of rationality, seen as one of its intensional components, the result of denying it (“men non-rational”) is a simple impossibility, given the fact that rationality is an essential part of the concept of man (Non inelegans specimen demonstrandi in abstractis, Leibniz 1923–, VI, iv, n. 178). These principles are among the ones governing the so-called plus-minus calculus that Leibniz developed in a number of fragments dating around 1687. The basic assumption of the plus-minus calculus is that “A + B = L” is to be interpreted as “A (or B) is included in L,” where the relation of inclusion is—as usual—the intensional inclusion between concepts. “L − A = N ” is to be interpreted as conceptual subtraction: N is the intensional content of the concept L that is not included in the concept A. If “A + B = L,” the terms A (or B) and L are said to be subalterns; two terms, neither of which is included in the other, are said to be disparate, and two terms that have a common component are said to be communicating. It should be stressed that, here as elsewhere, Leibniz uses “term” to designate not a linguistic

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entity but a concept: His calculus is thus directly an “algebra of concepts.” The calculus uses “nihil” to represent a term with empty intensional content, and the rules “A + nihil = A” and “A − A = nihil” are introduced. Leibniz also uses “nihil” as a way to obtain “privative” concepts: if “E = L − M ,” “L = nihil,” and M is a nonempty concept, E will be a privative concept. This assumption has been criticized due to the fact that it introduces inconsistencies in the calculus (Lenzen 2004), but can be seen as a further indication of the relevance that Leibniz attributed to the representation of negative or privative concepts within his logic, and of the diﬃculties connected with the diﬀerence between conceptual subtraction, arithmetical subtraction, and logical negation. The plus-minus calculus has recently been the subject of much interpretive work, also due to the publication of the long-awaited vol. VI, iv of the Akademie Ausgabe, which oﬀers the ﬁrst critical and complete edition of the relevant texts (Leibniz 1923–, VI, iv, vols. 1–3). Among the problems debated (see Lenzen 2000, 2003, 2004; Schupp 2000) are the possibility of a set-theoretical representation of the calculus, and the relation between its intensional and extensional interpretations. As we have seen, Leibniz’s approach is—in most of his logical writings—clearly intensional. However, Leibniz himself was well aware of the diﬀerence between intensional and extensional approaches, and considers the one as the reversal of the other: the method based on concepts is the contrary of that based on individuals. So, if all men are part of all animals, or if all men are included in all animals, it is true that the notion of animal is included in the notion of man. And if there are animals that are not men, we need to add something to the idea of animal to get the idea of man. Since when the number of conditions grows, the number of individuals decreases. (Leibniz 1966, 235) This thesis may be (and has been) criticized, since the number of actually existing individuals falling under a given concept is usually contingent: From a Quinean point of view, “it might just happen that all cyclists are mathematicians, so that the extension of the concept being a cyclist is a subset of the extension of the concept being a mathematician. But few philosophers would conclude that the concept being a mathematician is in any sense included in the concept being a cyclist” (Swoyer 1995, 103). Nevertheless, as Lenzen (2003) correctly observes, this criticism cannot be applied (or at least not in such a naive form) to Leibniz’s logic: The (extensional) domain of Leibniz’s logic is consistently characterized by Leibniz himself as one of possible rather than of actual individuals. The (possible) contingent coincidence of the sets of actually existing cyclists and mathematicians would by no means imply, from a Leibnizian point of view, that the two concepts have the same extension and therefore should have the same intensional content. In a later fragment, known as Diﬃcultates quaedam logicae (Leibniz 1875–1890, VII, 211–217), Leibniz will even use the idea that the domain of his logic is one of possible rather than of actual individuals to justify the conversion per accidens of UA propositions

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(from “All A are B” to “Some B is A”), thus avoiding the problem of the existential import of PA propositions. Despite the fact that they probably precede most of the texts on the plusminus calculus, the 1686 Generales Inquisitiones (Leibniz 1982) are generally considered Leibniz’s most developed and satisfactory attempt of logical calculus; Leibniz himself considered them a “remarkable progress” over his earlier works. The main feature of the Generales Inquisitiones is the attempt to oﬀer a uniﬁed framework for a calculus of terms and a calculus of propositions. As far as terms (or concepts) are concerned, Leibniz distinguishes between integral terms (terms that can be the subject or the predicate of a proposition: the categorematic terms of scholasticism) and partial terms (terms like “same” or “similar,” which are to be used only in conjunction with one or more integral terms, and specify or modify an integral term or a relation among integral terms: the syncategorematic terms of scholasticism). The introduction of partial terms and the discussion of oblique cases clearly testify to the new interest Leibniz devoted to relations. Being discussed at the term level (and therefore at the level of concepts), relations and oblique cases are clearly not considered by Leibniz as mere linguistic accidents. The problem of the possible “reduction” of partial or relational terms and of relational propositions to nonrelational ones is therefore not one of simple “surface-structure” reformulation of a spoken or written sentence, but rather one of logical analysis of the proposition and of its constituent terms. Leibniz was to devote much eﬀort and a large number of texts and fragments to this analysis, clearly inﬂuenced by the late scholastic discussion on relations and on the connection between the relation itself and its fundamenta, that is, the concepts or the things among which the relation is established. Starting with Russell (1900), who attributed to Leibniz a straightforward and uniform reductionistic approach with respect to relations, criticizing it, Leibniz’s treatment of relations has been the subject of much interpretive work. It is now clear that Leibniz oﬀered diﬀerent accounts for diﬀerent kinds of relations and that, while he consistently denied relations an extramental reality independent from that of the related concepts, he thought that at least some relations (among those involving diﬀerent individuals) are not reducible in a straightforward way to simple and nonrelational monadic predicates. However, this does not imply, according to Leibniz, the need of propositions which are not in subject-predicate form, but rather the need (1) to consider within the properties pertaining to a given subject also those expressing relational accidents, and (2) to recognize the logical role of reduplicative terms, which can be used in connecting propositions referring to the diﬀerent fundamenta of a same relation. Thus, the proposition “Paris loves Helen” can be reduced, according to Leibniz, to “Paris is a lover, and eo ipso (for this very reason) Helen is a loved one,” rather than to the simple and independent propositions “Paris is a lover” and “Helen is a loved one” (Mugnai 1992). Reduplicative terms like quatenus, eo ipso, and so on, which were already studied by scholastic and late-scholastic logicians, in this way acquire a special role within Leibniz’s logic.

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In the Generales Inquisitiones, integral terms are further subdivided into simple, complex, and derivative. The discussion of simple terms shows a clear shift when compared to the earlier combinatorial attempts: While general abstract terms like “ens,” derived from the scholastic tradition and from the discussion on transcendental terms, are still present, Leibniz adds to the class of simple terms also terms connected to individuals and perceptions, like “Ego” (“I”) or the names of colors; a passage that somehow anticipates the discussion about simple and innate ideas that will be at the core of the much later Nouveaux Essais (Leibniz 1923–, VI, vi). However, as in the earlier attempts, the choice of simple terms remains provisional and is strongly inﬂuenced by the limits—both necessary and contingent—of our knowledge. A special case is that of the privative term non-ens: Like nihil in the plus-minus calculus, non-ens corresponds here to a term with empty intensional content, and plays an important role in the axiomatization of the calculus. Complex terms are obtained by composition from simple terms, while derivative terms are obtained from partial terms “completed” by integral terms, that is, from integral terms modiﬁed or connected by means of syncategorematic and relational terms or by the use of oblique cases. In the Generales Inquisitiones, for the ﬁrst time, Leibniz includes a discussion of complex terms referring to individuals (Leibniz 1982, 58–62) in his logical calculus. According to Leibniz, they are based on complete concepts, that is, concepts that include all which can be said of that individual. Their complexity, however, is such that only God can carry out their complete analysis: Men can only rely on experience to assert the possibility of a given complete concept (i.e., the absence of contradictions within its intension) and the inclusion of a given contingent predicate within a given complete concept. Complete concepts (corresponding to individual substances) are another theoretical cornerstone of Leibniz’s philosophy, and it is no coincidence that in the very same year in which he was working at the Generales Inquisitiones Leibniz also wrote the Discours de métaphysique (Leibniz 1923–, VI, iv B, 1529–1588), the text that oﬀers for the ﬁrst time and in a structured way the philosophical framework in which the theory of complete concepts is to be placed. Much of the interpretive work done on Leibniz’s philosophy and philosophy of logic in the last three decades deals in one way or another with the discussion of complete concepts1 : from the possibility of distinguishing within them a “core set” of essential properties, which could also allow for transworld identiﬁcation of individuals across possible worlds (each complete concept, if considered in its integrity, is bound to a given possible world, and possible worlds themselves are seen by many interpreters as maximal sets of mutually compossible complete concepts), to the presence within complete concepts of relational predicates; from the discussion of Leibniz’s conception of contingency and of individual freedom to that of preestablished harmony. Shifting from terms to propositions, Leibniz states in the Generales Inquisitiones that “ ‘A is B’ is the same as ‘A is coincident with some B’ or A = BY ” (Leibniz 1973, 56), where B is part of the intensional content of A:

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a formulation close to the ones we have already discussed, which, however, can be of interest if we consider the role attributed here to “Y ,” seen as a sort of existential quantiﬁer applied to the predicate. Leibniz oﬀers a wider discussion of predicate quantiﬁcation in an undated fragment known as Mathesis Rationis (Leibniz 1966, 193–206; see Lenzen 1990), and in some of his (many) attempts of graphical representation of the basic notion of conceptual inclusion and of the four forms of categorical propositions, mainly based on the use of lines or circles (the fragment known as De formae logicae comprobatione per linearum ductus probably being the most notable among them; Leibniz 1966, 292–321). In the Generales Inquisitiones, the connection between the treatment of propositions and that of terms is, if possible, even stronger than in the preceding essays, since Leibniz observes that the four traditional forms of categorical propositions can be rewritten in the following way (Leibniz 1982, 112; his thesis is clearly indebted to the late-scholastic treatment of the passage from proposition “tertii adjecti” to propositions “secundi adjecti”): (PA)

Some A are B

AB est res

(PN) (UA) (UN)

Some A are not B All A are B No A is B

A(non-B) est res A(non-B) non est res AB non est res

Here—if the proposition is not one about contingent existence—“est res” is to be interpreted as “is possible” (Leibniz 1982, 110), and possibility is in turn to be interpreted as absence of contradiction within the intension of the composed term. A similar conception is to be found in the Primaria Calculi Logici Fundamenta, dating to August 1690 (Leibniz 1903, 232–273). This treatment of propositions leads to a term-oriented treatment both of syllogistic inferences and of hypothetical propositions. According to Leibniz, just like in a categorical proposition the subject includes the predicate, in a hypothetical proposition the antecedent includes the consequent. Therefore, an implication of the form “If p, then q” is to be interpreted as “If (A is B) then (C is D),” which in turn can be rewritten as “(A is B) is (C is D),” or “(A includes B) includes (C includes D).” This idea, already present in a fragment known as Notationes Generales probably written between 1683 and 1685 (Leibniz 1923–, VI, iv, 550–557), will return in many later texts and leads Leibniz to hold that the forms and modes of hypothetical syllogisms are the same as those of categorical syllogisms. Leibniz never devoted a detailed analysis to the logic of propositions, but in many of his works and fragments refers to propositional rules derived from the medieval tradition of consequences and from the late-scholastic discussion on topical rules; clearly, in his opinion, an “algebra of propositions” can only be grounded on the algebra of concepts. As already noted, most of the logical texts and many of the most remarkable achievements of Leibniz’s logic were not known to his contemporaries and

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his immediate successors. Nevertheless, Leibniz’s logic cannot be considered simply an isolated product of a genial mind: He had a deep knowledge of the late-scholastic logical tradition, from which he derives not only many topics he deals with but often also the approaches adopted in dealing with them. He also had a wide net of relationships—both through letters and by personal acquaintance—with many of the most prominent ﬁgures of the European learned world (among them Arnauld, Tschirnhaus, Jakob Bernoulli, and, as we will see, Wolﬀ, to name just some scholars mentioned in this chapter). His logical and philosophical theses are also the result of those interactions, and probably some of them circulated even without the support of publication. Despite the great interest of Leibniz’s logic from a contemporary point of view, Leibniz was a seventeenth-century logician, not a twentieth-century logician in disguise.

9. Logic in Germany in the First Half of the Eighteenth Century In the period we are considering, German logic deserves special attention. Since logic was a subject included in most academic curricula, it became a privileged ﬁeld of study and a great number of logical texts were published (see Risse 1965). Many German logicians enter the debate on Cartesianism, are fully aware of Bacon’s exhortation to work at a logic of empirical sciences, pay attention to the notion of probability, examine the relationship between logic and mathematics, and seem open to the suggestions of facultative logic. If one had to name a single author who takes a stand on all these questions, one should mention Leibniz. But, as already said, in this period Leibniz’s logic enters marginally in the oﬃcial picture of German logic, not only because most of his strictly logical production was unknown at that time but also because he did not belong to the academic world. What was known of Leibniz’s philosophy and logic inﬂuenced a number of German logicians of that time, but the logical scene of the ﬁrst two generations of the German Enlightenment was dominated by Christian Thomasius and Christian Wolﬀ. Christian Thomasius (1655–1728) was the son of Jakob Thomasius, Leibniz’s teacher. However, he does not share Leibniz’s view of logic, in as much as he agrees with the humanists in criticizing schoolmen for having instructed generations of students in the making of useless subtleties. At the same time he advocates the study of logic. This is less paradoxical than it sounds. While the Port-Royal Logic recommended a logical instruction because common sense is not so common as people believe, Thomasius oﬀers a moral and religious justiﬁcation. He maintains that because of the original sin, mankind has darkened its natural light (lumen naturale) and has to achieve a healthy reason through a process of puriﬁcation, so as to avoid the errors it usually makes. This cathartic process is entrusted by Thomasius to logic because logic can teach how to counteract errors in judgment and their main source, namely,

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prejudice (in particular, the prejudice of authority and the prejudice of self love). In this way Thomasius agrees with Tschirnhaus and the Port-Royal Logic that logic is a medicine and its primary aim is the correctness of judgments (Thomasius [1691b], dedication). From the Port-Royal Logic—translated into Latin (Arnauld and Nicole 1704) by one of his followers—Thomasius also borrows arguments for rejecting Aristotelian categories (Thomasius 1702, VII §25). Such a concept of logic requires that technicalities should be abolished: Only an “easy” logic can dispel prejudices and teach how to proﬁt from the few precepts needed for the rational conduct of common human beings—and not only learned scholars—in the search of truth and in the practical exercise of prudence. Thomasius’s precepts consist of two basic rules of method: (1) to proceed from what is easy and known to what is more diﬃcult and unknown, (2) to connect remote conclusions to principles only through near (propinquae) conclusions. In these two rules one can ﬁnd an echo of Descartes. But Thomasius is not a Cartesian because he opposes the doctrine of innate ideas, convinced that in the intellect there is nothing that has not been in the senses (a thesis he argues for independently of Locke). He also rejects the tradition of the mos geometricus because he holds it responsible for the degenerate Spinozistic version of Cartesianism so that, from this point of view, he diﬀers also from Tschirnhaus. Thomasius is rather an eclectic (Beck [1969], 247–256) who encourages the study of other philosophers’s ideas, thus promoting studies in the history of philosophy and also in the history of logic: From 1697 to the ﬁrst decades of the eighteenth century it is possible to register a number of essays on the latter subject (Risse 1964–70 II, 507). Thomasius’s eclecticism can be easily appreciated if one considers that, on the one hand, he adds precepts derived from the tradition of humanistic dialectic to his apparently Cartesian rules of method and, on the other hand, he insists that logic should concentrate on the problem of certainty in empirical knowledge. Like many others in this period, Thomasius believes that, although in empirical matters complete certainty is not attainable, it is still possible to avoid skepticism by working on the notion of probability. However, Thomasius’s interest for probability is not to be overestimated, since he inclines to a notion of probability still strongly connected with Aristotelian dialectic and with the doctrine of topical syllogism. Thomasius’s ideas were well received by the incipient age of the Enlightenment that looked favorably to a logic meant for ordinary people (and this favored the proliferation of textbooks) and, from a more theoretical point of view, approved of his antiskeptic battle regarding empirical knowledge. Nevertheless, Thomasius not only advocated a rigid separation between mathematics and philosophy, but also opposed any formalism in logic and deplored the enormous growth of syllogistic, convinced that the ﬁrst ﬁgure is suﬃcient, though incapable of guaranteeing the truth of the conclusions (Thomasius 1702, IX, §12; [1691a], XII, §§19–21). Many of his followers agreed on these matters, with a notable exception. In Halle, where a Thomasian circle was

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ﬂourishing, Andreas Rüdiger tried to reconcile Thomasius’s views on logic and philosophy with his own research on the nature and scope of logic. Rüdiger (1673–1731) agrees with Thomasius that, due to the original sin, mankind no longer participates in God’s archetypal logic and is prey of prejudices, thus making it necessary to conquer a recta ratio through the study of logic (Rüdiger 1722, I, i, 1). Rüdiger also agrees with Thomasius that logic should deal with probability as a response to skepticism, and to this eﬀect he produces an articulated doctrine of probability that obtained remarkable diﬀusion through the ﬁrst edition of the Philosophisches Lexicon (Walch [1726]) published by Johann Georg Walch (1693–1775). But what seems to interest Rüdiger most is that logic should be recognized as a legitimate means for ﬁnding truths and should be proved capable of attaining this purpose with a procedure as diﬀerent as possible from the procedure of mathematics. Rüdiger believes that Spinozism rests on two pillars: the doctrine of innate ideas and the illegitimate application of the mathematical method outside mathematics. Therefore, he rejects both. He is perfectly aware that mathematics is inventive, but he makes this depend on the fact that mathematical proofs can resort to sensibility. Rüdiger does not ascribe the sensibility of mathematical proofs to the use of “visual” aids, such as geometric ﬁgures, but to the demonstrative procedures based on numeration. In his opinion: (1) mathematics is the science of quantity, (2) all quantities are measurable, (3) we can measure only in so far as we can numerate, (4) “all numeration is of individuals, in so much as their terms are perceived by the senses.” His conclusion is: “Therefore all numeration is sensible: but the entire way of mathematical reasoning is numeration, then this entire way [of reasoning] is sensible Q.E.D.” (Rüdiger 1722, II, iv, 1a). The possibility to avail themselves of this kind of sensible reasoning (ratiocinatio sensualis) enables mathematicians to refer to sensible data that would escape their attention if they could rely on intellect alone, whereas sensible data oﬀer them the basis for the discovery of unknown (mathematical) truths (Rüdiger 1722, II, iv, 3c; see Cassirer 1922, 525–527). The heuristic capacity of ratiocinatio sensualis rests on Rüdiger’s theory of truth. According to Rüdiger, the truth of our judgments (which he calls logical truth) consists in the agreement of our knowledge with our sensation (“convenientia cognitionis nostrae cum sensione”; Rüdiger 1722, I, i, 12). But we can trust the agreement of our knowledge with sensation only because there is a metaphysical truth that consists in the agreement of sensation with its objects (“convenientia ipsius sensionis cum illo accidente, quod sentitur”). This means that the metaphysical truth, which makes us trust our logical truth, presupposes that our senses are not fallible (Rüdiger 1722, I, i, 11). Because our senses are not fallible (under God’s guarantee), the certainty and inventive power of mathematics can rest on sensible reasoning. Outside mathematics, however, and in particular in the ﬁeld of philosophy, we work only with ideas and cannot resort to the ratiocinatio sensualis. It is nevertheless possible to use a logical way of reasoning meant for ideas

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(ratiocinatio idealis) and as inventive as the ratiocinatio sensualis: the syllogism. By claiming that syllogism is inventive, Rüdiger openly challenges a long series of scholars, including Thomasius, who had criticized syllogism for being sterile. He argues that the inventive function of syllogism has not been appreciated because syllogism has been used for ﬁnding the premises of a given conclusion. This means that syllogism has been used analytically, whereas syllogism can be inventive if it is used synthetically, as a means for searching an unknown conclusion beginning from a given premise (Rüdiger 1722, II, vi, 1). To show how a synthetic syllogistic is possible, Rüdiger assumes that every proposition (of the four kinds that can enter into syllogisms, A, E, I, O) expresses a precise relation between the subject and the predicate, a relation belonging to a set Rüdiger carefully classiﬁes: subordination, opposition, partial diversity (he considers identical ideas as the same idea; see Rüdiger 1722, I, xii, 2, 3). On this assumption he maintains that we make a synthetic syllogism beginning with a premise whose two terms stand in one of the admissible idea-relations. We then obtain a conclusion by connecting one of the terms of the premise with any unknown idea that stands in a deﬁnite relation (included in the set of classiﬁed idea-relations) with the other term. For instance, given a universal aﬃrmative proposition “All A are B” (where A is subordinated to B) as premise, we can connect the predicate B with any idea C of which we only know the relation it entertains with the subject A. Let such a relation be that of subordination: We can validly conclude that the unknown idea C, being subordinated to A is also subordinated to B, so that, since the relation of subordination can be expressed by a universal aﬃrmative proposition, we obtain the “new” conclusion “All C are B” (Rüdiger 1722, II, vi, 55 1– 4). This single example makes it clear that Rüdiger’s synthetic syllogistic is founded on the old technique of the pons asinorum (Thom 1981, 72–75), traditionally used for ﬁnding premises, a technique that Rüdiger could ﬁnd in works of the Peripatetic tradition he knew well, pace Thomasius who had ridiculed it (Thomasius [1691a], XII, §11). Rüdiger simply reverses the pons asinorum in the search for a conclusion, as it can be appreciated from the graphical representations he gives of his synthetic syllogisms (reunited in a single representation by Schepers 1959, 99). Rüdiger expressed his views on ratiocinatio idealis in a logical environment in which they must have been unpopular. It is not surprising, therefore, that his direct followers, who approved of his separation of mathematics from logic, were no longer interested in his reason for separating them, that is, his passionate defense of the inventive capacity of syllogism. Adolph Friedrich Hoﬀmann (1703–1741) and Christian August Crusius (1715–1775) choose to refer to Rüdiger as the philosopher who did not simply denounce, like Thomasius, the ill eﬀects of the application of the mos geometricus to philosophy, but had argued that mathematics and philosophy should never be mixed because they have diﬀerent objects and diﬀerent methods of reasoning (Tonelli 1959). It may seem paradoxical, therefore, that Rüdiger’s classiﬁcation of idea-relations, supporting his doctrine of syllogism (as well as his doctrine of conversion and

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other nonsyllogistic inferences) and connected to his thesis of the independence of logical from mathematical reasoning, was to stimulate later logicians to once again take up projects of an algebraic calculus of ideas. But such projects were not resumed until the Thomasian school, including its peculiar Rüdigerian variant, was no longer dominant, due to the emergence of Christian Wolﬀ and his school. Wolﬀ (1679–1754) has much in common with the ﬁrst generation of the German Enlightenment. Like the Thomasians, he believes that philosophy should improve human life and give due importance to empirical knowledge. He also shares some of their religious motivations and their appreciation for Locke (see section 6). What divides Wolﬀ from Thomasius on logical matters is, ﬁrst, a very diﬀerent evaluation of mathematics and of the tradition of the mos geometricus: Wolﬀ had received a mathematical education and was professor of mathematics in Halle. Second, Wolﬀ acknowledges the intellectual inﬂuence exerted on him by Leibniz, with whom he exchanged a correspondence that ended only with Leibniz’s death. In one of his letters Leibniz had recommended to Wolﬀ to pay due attention to syllogism: “I absolutely never dared to say that the syllogism is not a means for ﬁnding truths” (Leibniz [1860], 18). Consequently Wolﬀ, far from contrasting syllogistic and the mathematical method, makes a double revolution with respect to the Thomasian school. He (a) assumes mathematical reasoning as an example to be followed in any research ﬁeld, (b) claims that the allegedly empty and useless syllogism is the inner fabric of any reasoning, including the exemplary mathematical one. In diﬀerent places—but especially in his logical works, the so-called German Logic of 1713 and the so-called Latin Logic ﬁrst published in 1728—Wolﬀ considers geometrical demonstrations as chains of common syllogisms in the ﬁrst ﬁgure (Wolﬀ [1713], IV, §§20–25, [1740] §551), and concludes that, where understanding and reason are concerned, there is a single rational procedure, a single method, a single logic valid for mathematics and philosophy alike: “both philosophy and mathematics derive their method from logic” (Wolﬀ [1740], Preliminary Discourse §139 note). The importance of syllogism is justiﬁed by Wolﬀ by the argument that syllogism mirrors the natural way of reasoning. But he does not ground logic on empirical psychology alone (better: on the empirical psychology of a privileged set of men, the mathematicians, see Engfer 1982, 225). He declares that logic has a solid foundation also in ontology (Wolﬀ [1740], Preliminary Discourse §89). Both pillars of this foundation of logic conspire in favoring syllogism: (1) Ontology, as it was established in the scholastic tradition, justiﬁes the dictum de omni et nullo that presides over syllogism in the ﬁrst ﬁgure (Wolﬀ [1740], §380); (2) empirical psychology ensures that the model of natural inference is the simplest syllogistic inference, that is, syllogism in the ﬁrst ﬁgure. Wolﬀ’s foundation of syllogism—and indeed of logic—on ontology and empirical psychology grants an absolute privilege to syllogisms in the ﬁrst ﬁgure. To this eﬀect, Wolﬀ maintains that ﬁgures diﬀerent from the ﬁrst (he does not admit

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the fourth ﬁgure) are not simply reducible to it but are already syllogisms in the ﬁrst ﬁgure in disguise: They are cryptic ﬁrst ﬁgure syllogisms (Wolﬀ [1740], §§382–399; see Capozzi 1982, 109–121). He also maintains that noncategorical syllogisms, consequentiae immediatae and any other kind of inference are reducible to syllogisms in the ﬁrst ﬁgure. In this way, Wolﬀ contrasts antisyllogistic conceptions, but makes no concessions to Rüdiger: The syllogisms he refers to are absolutely standard and in no way synthetic. In a letter to Leibniz, he scorns Rüdiger’s synthetic syllogistic and refers to the idea of a mathematical ratiocinatio sensualis as to one of the “paradoxa” of Rüdigerian logic (Leibniz [1860], 117). Despite his meager syllogistic, Wolﬀ oﬀers a deeply “logicist” philosophy of logic, for not only does he bring logic and mathematics together, he also considers logic prior to mathematics. This makes one wonder why he did not try a mathematical calculus of ideas. This problem is clearly related to the possibility of a heuristic. Wolﬀ knows that mathematicians use heuristic devices not reducible to syllogisms. In his German Logic he denies that “the whole algebraic calculus . . . takes place only according to syllogisms in form” (Wolﬀ [1713], IV, §24). A similar statement is to be found in his Latin Logic: “logic [i.e., logic centered on syllogism] has a notable and famous use in the art of discovery, but nevertheless it does not exhaust it” (Wolﬀ [1740], §563). Elsewhere he explains that, to discover hidden truths, it is sometimes necessary to resort to heuristic artiﬁces such as, in the a priori invention, the artiﬁces of the ars characteristica (see Arndt 1965, 1971). For, he says, this art helps separate geometric and arithmetic truths from images, so as to obtain truths from the data by means of a calculus. He grants that such an art is the most perfect science, but he believes that we only have a few examples of that art in algebra, yet none outside it (Wolﬀ [1713] IV, §22). The latter statement is revealing: To Wolﬀ the establishment of a suitable alphabet of thoughts as a prerequisite of the ars characteristica combinatoria must have appeared too great an obstacle. We know that this was a problem for Leibniz, and we know how he dealt with it. But apparently Wolﬀ was convinced that a calculus in logicis is utopian and considered it a mere desideratum. As to Leibniz’s interest in a logic of probability to be used when deliberating about political, military, medical, and juridical matters (Leibniz [1860] 1971, 110), Wolﬀ seems to doubt that a mathematical ars conjectandi could be of practical use regarding such matters (Leibniz [1860] 1971, 109). Wolﬀ’s doubt was not so strong as to make him exclude that Leibniz’s wish could ever come true, but was strong enough to make him exclude that it could come true in the foreseeable future. Nevertheless, Wolﬀ includes probability in the practical part of his logic, paying attention to the features of probable propositions, in particular to the ratio of suﬃcient and insuﬃcient reasons that make it possible to consider probability a measurable degree of certainty. In this way, he deﬁnitely abandons not only Thomasius’s obsolete treatment of probability reminiscent of the old topic but also Rüdiger’s nonmathematical analysis of probable knowledge.

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Wolﬀ made a great impact on German philosophy, and some of his doctrines were well received in Europe at large, as can be appreciated from the evident Wolﬃan imprint of some entries in the French Encyclopédie (Carboncini 1991, 188ﬀ.), where Locke and Port-Royal Logic had already found much space (Risse 1964–70, II 528). However, at ﬁrst Wolﬀ met with harsh criticism, albeit not for his logical but for his metaphysical views. In 1723, the Halle circle succeeded in convincing King Friedrich Wilhelm to banish Wolﬀ from the city because his alleged determinism, inherited from Leibniz, was a peril to religion (Wundt [1945], 234–244; Beck [1969], 258). Wolﬀ went to Marburg, where he wrote a series of works in Latin, beginning with the Latin Logic. Wolﬀ’s Latin works increased the number of his followers, so much so that a second anti-Wolﬃan oﬀensive, launched in 1734, ended in a defeat when, in 1740, Wolﬀ was readmitted to Halle with great honors. The Wolﬃan era, as concerns logic, was very positive. Despite literature that considers him an exponent of the dark ages of logic, it is diﬃcult not to give him credit for proposing a positive image of the discipline resting not only on its function as a guide in making judgments and in avoiding errors but also on the power of its inferences. But above all, it is impossible to ignore that his revaluation of syllogism diﬀered from Rüdiger’s because he used it to counteract the idea of a gap between logic and mathematics. This explains why Wolﬀ’s success promoted a revival of logic that did not simply contribute to the production of new logical textbooks—given the importance he accorded to logic in academic curricula—but spurred logical investigations. It must be stressed that those who were encouraged by Wolﬀ’s philosophy to engage in logical research did not usually follow the details of his logic. Independent authors, as well as a number of Wolﬃans, referred to old logical literature, including sixteenth- and seventeenth-century Aristotelian and scholastic treatises, and even to the works of anti-Wolﬃans, especially those of Rüdiger, undoubtedly the most distinguished of them. In this sense one must agree with Risse that if one excludes the very ﬁrst generation of Wolﬀ’s followers, it is diﬃcult to draw precise boundaries between the Wolﬃan school and its opponents (Risse 1964–70, II, 615). An example of the new post-Wolﬃan logicians is Johann Peter Reusch (1691–1758). In his fortunate Systema logicum (Reusch [1734]), though in many respects faithful to Wolﬀ, Reusch admits the inﬂuence of Aristotle, Jungius, the Port-Royal Logic, Johann Christian Lange (1669–1756), and Rüdiger (Reusch [1734], preface). As to the question of syllogistic, Reusch informs his readers about traditional doctrines and about the combinatory of syllogistic moods with a reference to Leibniz’s De Arte Combinatoria (Reusch [1734], §530). Nevertheless, he also proposes a syllogistic that, he maintains, opens the gates of the syllogistic moods (modorum cancelli) (Reusch [1734], §543), being founded on a single rule to which all syllogisms of any ﬁgure must conform: “The entire business of reasoning is done by substitution of ideas in the place of the subject or of the predicate of the fundamental proposition, that some call equation of thoughts” (Reusch [1734], §510). In other words, Reusch conceives

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of syllogisms as consisting of a single premise (propositio fundamentalis) and a conclusion obtained by assuming a new idea and substituting it for either the subject or the predicate of the propositio fundamentalis by a substitution rule. Such a substitution rule—which he means as a version of the old dictum de omni et nullo—is governed by the principle of contradiction and presupposes a network of relations among ideas. The description of the admissible idea-relations is so important for Reusch’s syllogistic, and indeed for his whole logic, that two chapters of his Systema logicum—De convenientia et diversitate idearum and De subordinatione idearum—are devoted to it. This study is clearly inﬂuenced by Rüdiger (Capozzi 1990, lxvi), but Reusch did not refer to Rüdiger as the defender of a synthetic inventive syllogistic but as the author of a syllogistic based on a deﬁnite set of idea-relations. That this was the outstanding feature of Rüdiger’s logic was clear to the historian of logic von Eberstein who, though unfavorable to Rüdigerian philosophy, in 1794 stated that Rüdiger had been the ﬁrst to determine “syllogistic ﬁgures according to the relations of concepts and not according to the position of the middle term” (von Eberstein [1794–1799], I, 112–113). No wonder the independent Wolﬃan Reusch was attracted to this approach to syllogistic so as to prefer it to Wolﬀ’s idolatry for the ﬁrst ﬁgure and to Leibniz’s combinatory of moods in the De Arte Combinatoria. In Reusch, however, there is no hint of a separation between the ratiocinatio sensualis of mathematics and the nonmathematical ratiocinatio idealis advocated by Rüdiger as an argument in favor of the inventive power of his synthetic syllogisms. This is true not only of Reusch. After Wolﬀ, logic is acknowledged as the only argumentative structure used in every ﬁeld of knowledge, from mathematics to philosophy. This is why a few logicians felt entitled to take a further step: If, according to Wolﬀ, there is no longer a gap between logic and mathematics, nothing prevents one from disregarding Wolﬀ’s restriction of logic to an outdated syllogistic. These logicians felt entitled to apply mathematical tools to ideas according to the study of idea-relations made by the adversaries of Wolﬃan logic or by independent Wolﬃans.

10. Logical Calculi in the Eighteenth Century In Jena, where Reusch was professor of logic and metaphysics since 1738, his attitude toward logic was not an exception, as can be seen in the logical work of Joachim Georg Darjes (1714–1791). But the outstanding work written in this logical context is an essay that Reusch recommended in the 1741 edition of his Systema logicum to his more specialized readers. This work is the Specimen Logicae universaliter demonstratae (Segner [1740] 1990) written by the mathematician and scientist Johann Andreas Segner (1704–1777) with the explicit aim of treating syllogistic by way of a calculus (per calculum) based on the example of algebra.

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To this end, Segner builds an axiomatic system consisting of 16 deﬁnitions, 3 postulates, and 2 axioms. The deﬁnitions introduce ideas, their relations, their arrangement in a hierarchy of genera and species, and the operations for forming ideas. Segner deﬁnes idea as a mental representation of something. If the idea is simple, its contents are obscure ideas and the simple idea is confuse for us; if the idea is composite, its contents are clear ideas and the composite idea is distinct for us. Consequently, by deﬁnition, every idea contains some idea within itself. In this way, Segner can presuppose the relation of containment (viewed from an intensional perspective) as the basic relation between two ideas. But it must be clear that Segner does not identify the content of an idea with its comprehension in the sense of the Port-Royal Logic. He simply says that given two ideas A and B, A is contained (or involved) in B if, whenever B is posited, A is also posited. The notion of containment is used to deﬁne all the relations between two ideas that are relevant for the construction of a calculus. Segner designates such relations by special algebraic symbols “−”, “=”, “>”, “ B), if A does not contain B and B contains A. IV. A is inferior to B (A < B), if A contains B and B does not contain A. V. A is coordinated to B (A × B), if A does not contain B and B does not contain A. As can be seen from this list, Segner does not have a symbol for the relation of opposition, but expresses the opposition between A and B by saying that A contains −B and B contains −A. By the expression −A Segner refers to the idea that is inﬁnitely opposite to A, and deﬁnes A as inﬁnitely opposite to −A if A contains −−A, and −−A contains A. It must be stressed that Segner does not intend an idea designated by −A as a “negative” idea opposed to a “positive” one: If by A we indicate “nontriangle,” by −A we indicate “triangle.” Ideas that can be put in a hierarchy of subordination, can also be submitted to the operations of composition and abstraction, whose possibility Segner guarantees by two special postulates. A further postulate guarantees that if A is an idea, then −A is an idea. Segner ﬁnally states two axioms that express conditions satisﬁed by some of the operations: Axiom I: If A and B are opposite ideas, then there is no idea C such that C = AB. Axiom II: If A contains B, then AB = A. In this simple system (somewhat simpliﬁed here) Segner derives a number of propositions (either problems or theorems) strictly connected to his calculus. Among the most important are the following:

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1. Given an idea A, by abstracting some of its contents, ﬁnd an idea B such that B > A. 2. If, given two ideas A and B, there is an idea AB. diﬀerent from either A or B, then A × B. 3. Given a universal idea A and its coordinate idea B, AB < A and AB < B. 4. No inﬁnite idea can be inferior or identical to a ﬁnite idea. 5. The deﬁned relations among ideas are exhaustive and reciprocally exclusive: They are not reducible one to the other. A number of theorems establish how the relation (and therefore the sign) between two ideas changes if one of them is replaced by its opposite: 6. If A = B, then −A = −B. 7. If A < B, then −A > −B. 8. If A > B, then A × −B. 9. If A × B, then −A × B; if −A × B, then either −A × B or A > B. Further theorems give an exhaustive list of valid syllogisms: 10. If A = B and C = B, then C = A. 11. If A = B and C > B, then C > A. 12. If A = B and C < B, then C < A. 13. If A = B and C × B, then C × A. 14. If A > B and C < B, then C < A. 15. If A > B and C × B, then either C < A or C × A. 16. If A > B and C > B, then C is consentient with A (i.e., is not opposite). Segner also proves a theorem that singles out all invalid syllogisms. Then he pays attention to some nonsyllogistic inferences and claims that “they shine of their own light,” whether or not we can give them syllogistic form. In particular, he proves the following inferences by composition: 17. If A = B and C = D, then AC = BD. 18. If A = B and C < D, then AC < BD. 19. If A < B and C < D, then AC < BD. Segner then proves the following theorem concerning inferences by abstraction: 20. If A is consentient with B, if C has been abstracted from A, and if D has been abstracted from B (so that C > A and D > B), then C is consentient with D.

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Thanks to a number of further deﬁnitions and three further propositions, Segner applies his system to the verbal expressions of common logic (he also pays attention to singular ideas and strictly particular propositions whose subjects bear the preﬁx “only”): “All A are B” means either A = B or A < B. “No A is B” means only A < −B. “Some A is B” means either A = B or A < B or A > B or A × B. “Some A is not B” means either A < −B or A × −B. Segner’s work proves that not all attempts to construct a logical calculus in an intensional perspective were destined to failure. Segner succeeds where Bernoulli failed (see section 7) because he never assumes that every idea has a (Port-Royalist) comprehension made of necessary attributes that cannot be modiﬁed without destroying it. Therefore, given two ideas, he simply considers three possible cases: Provided that A and B are not opposite, either A contains B, and then A < B, or A is contained in B, and then A > B, or neither idea contains the other, and then A × B. Thus Segner, unlike Bernoulli (apparently unknown to him), is under no obligation to use nouns of ordinary language as signs of ideas so as to recall their unchangeable comprehension, but uses literal symbols. When he considers the expressions of idea-relations in verbal propositions, he is under no obligation to change his intensional perspective and to consider the extensions of ideas whenever a predicate is not contained in the comprehension of a subject. We can aﬃrm a predicate B of a subject A even if they are coordinate ideas, that is, ideas whose relation is, by deﬁnition, a relation at one time of consent and of noncontainment. Some interpreters have suggested that Segner is similar to Leibniz and even a “disciple of Leibniz” (Vailati 1899, 88). Actually, Segner and Leibniz diﬀer at least inasmuch as Segner’s relation of coordination—being a relation of noncontainment—does not respond to the Leibnizian predicate-in-notion criterion. But there certainly are striking similarities. In the Specimen calculi universalis and in its Addenda, Leibniz uses an algebraic notation and an intensional perspective (see section 8). Moreover, what for Leibniz is a per se true proposition of one of the forms “ab is a,” “ab is b,” or “a is a,” has a detailed treatment in Segner. According to Segner, “A is B” is an aﬃrmation that can rest on one of the following relations between A and B, all of which produce truths: AB = A, if A < B (according to Axiom I), AB < A or AB < B if A × B (no. 3 in the list of Segner’s propositions). Equally, in his list of valid syllogisms (10–16), Segner includes Leibniz’s per se valid conclusion (“if a is b and b is c, then a is c”). As to Leibniz’s principle by which the repetition of a term is irrelevant, Segner also maintains that “the idea of the subject composed with itself cannot produce a new idea”

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(Segner [1740], 149). Also Leibniz’s praeclarum theorema, “if A is B, and D is C, then AD will be BC” (see section 8) has an elaborate equivalent in Segner’s inferences by composition. Other similarities can be found if one compares Segner’s logic with the Generales Inquisitiones, notably with respect to the calculus with negative terms (on the problems posed by the use of negative or inﬁnite predicates instead of negative propositions, and on Segner’s skilful solution, see Capozzi 1990, clx–clxiv). Segner did not know these Leibnizian doctrines, but the undeniable similarities we have stressed are not due to a miracle. They have an explanation in the fact that Segner, like Leibniz, was able to unify a variety of existing doctrines in a single system, depending on his practice of mathematics and using only an intensional approach. But that means that Segner’s logical background, albeit unsupported by knowledge of Leibniz’s relevant texts, was rich enough to oﬀer him a ﬁrm ground on which to build his calculus. In this respect, the changes that took place in German logic in the ﬁrst three decades of the eighteenth century show that Segner is representative of the logic of his time and not an inexplicable exception. But as in the case of Leibniz, this does not detract from his merits: It only emphasizes them. Let us consider one of the most interesting features of his system: the ﬁve idea-relations. Segner was not the ﬁrst to consider such relations, but depended on Rüdiger and Reusch. He was well acquainted with Reusch’s logic (the Specimen is dedicated to him) and he also knew Rüdiger’s idea-relations. In an academic dissertation of 1734, discussed by one of his students under his guidance, he expressly quotes Rüdiger’s logical work on such matters (Capozzi 1990, xcix). This does not make him less original, for it is due to him that such idea-relations are proved exhaustive and exclusive and are used as part of an adequate calculus. In this respect, Segner can be compared to the later mathematician Joseph Diez Gergonne (1771–1859). In his Essai de dialectique rationelle (Gergonne 1816–17) Gergonne considers ﬁve idea-relations using the notion of containment as basic but giving it an extensional interpretation: “the more general notions are said to contain the less general ones, which inversely are said to be contained in the former; from this the notion of relative extension of two ideas originates” (Gergonne 1816–17, 192). This extensional interpretation of the notion of containment is used by Gergonne (1816–17, 200) to classify the relations between two ideas on a par with the circles of Leonhard Euler (1707–1783) (see later in this section) and to designate each of them by a symbol. Two ideas A and B, where A is the less general idea and B is the more general one, can (1) have nothing in common, so that they stand in the relation H; (2) intersect each other, so that they stand in the relation X; (3) coincide, so that they stand in the relation I; (4) be such that A is contained in B, so that they stand in the relation C; (5) be such that that B is contained in A, so that they stand in the relation C. Like Segner, Gergonne also gives a correspondence list between the standard A, E, I, O propositions and the relations of ideas expressed by them:

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Gergonne

Propositions

Segner

I C

All A are B

A=B A” between two letters means negation (see Venn [1894], 499). Ploucquet, who takes an extensional point of view, maintains that in an aﬃrmative proposition the predicate is taken particularly. He says that this kind of particularity holds “in a comprehensive

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sense [sensu comprehensivo]” (Ploucquet 1763b, 52). Thus “All men are rational” means “All men are some rational beings,” a proposition that in his symbolism can be written as “M r,” where the letters signify the initials of the subject and the predicate. The particularity “in a comprehensive sense” of the predicate does not exclude that there could be other individuals apart from those under consideration. As for negative propositions, they are usually meant as having a universal predicate, but here too, though this may seem absurd in common language, they can have a particular predicate, as is the case with “All men are not some animals,” meaning, for instance, that they are not irrational animals. In this logical framework, Ploucquet develops a doctrine of subalternation and conversion that contrary to accepted rules, allows the conversion of particular negative propositions, for “Some A is not B” correctly converts into “No B is some A.” As to the syllogistic calculus, all one has to do is to represent the premises by Ploucquet’s symbolism and check that the premises are not both negative and that they do not contain four terms. Under these conditions, one can draw the conclusion by deleting the middle term and relating the two remaining terms, taking care that they preserve the same quantity they had in the premises. In this calculus also a syllogism in the ﬁrst ﬁgure of the form “All M are P , no S are M , then no S are P ,” which is invalid according to traditional doctrines, becomes acceptable, for it can be symbolized as follows: M p, S > M , then S > p (see Menne 1969). Another author of logical calculi is Johann Heinrich Lambert (1728–1777). This famous and eclectic scientist contributed to the study of language, metaphysics and logic, in addition to important research in the ﬁelds of optics, geometry (conic sections, perspective, theory of parallel lines, writing on the latter subject one of the basic texts in the history of non-Euclidean geometries), astronomy (comets), physics, technical applications of his theoretical works, and cosmology. Lambert’s interest in logic dates back to the ﬁfties when he wrote the so-called Six essays of an art of the signs in logic [Sechs Versuche einer Zeichenkunst in der Vernunftlehre], published only after his death (Lambert [1782–1787], I). It is rather diﬃcult to explain why Lambert decided not to publish these essays at the time that he wrote them, especially as they contain the general outline of his calculus, as well as comments on the nature of deﬁnition and on the representation of relations (see Dürr 1945). According to some interpreters (Barone 1964, 88), this decision depended on the fact that as Lambert confessed, in these writings he was attracted to the idea of discovering what “was concealed in the Leibnizian characteristic and in the ars combinatoria.” While Segner did not enter into these matters and Ploucquet refused the very idea of a universal calculus, Lambert adopted Leibniz’s ideal and, given his ignorance of the latter’s relevant texts, wanted to pursue Leibniz’s end by his own means. Now, a calculus aiming at being both formal and real presupposes an alphabet of simple elements. Therefore,

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Lambert had to postpone the publication of his early technical results until his philosophical investigation could establish such an alphabet of simple and ﬁrst concepts which, not containing composition in themselves, could not contain contradictions. Meanwhile, Lambert wrote the Neues Organon (Lambert [1764]), an important and famous book in which, according to the idea of mathesis universalis, he devoted himself to searching for the basic concepts that could help insert already acquired knowledge into a rational system and promote new discoveries. The Neues Organon consists of four parts: Dianoiology, Alethiology, Semeiotics, and Phenomenology, to which we will refer beginning with Alethiology and ending with Dianoiology. Alethiology, the doctrine of truth and error, deals with elementary concepts. Lambert gives a list of the latter, including conscience, existence, unity, duration, succession, will, solidity, extension, movement, and force. In Lambert’s view, connecting simple concepts produces truths that are not subject to change, and therefore can be considered as “eternal truths.” Eternal truths provide a foundation for all a priori sciences, in particular arithmetic, geometry, and chronometry. Semeiotics studies the relation between sign and meaning, and therefore introduces both a theory of language and the project of a characteristic. Phenomenology is the doctrine of appearance. Here Lambert, in discussing certainty and its relations to truth and error, also considers the degrees of possible certainty, and the probability of cognitions of which we have no absolute certainty. Dianoiology investigates the laws of the understanding. This part of the Neues Organon contains diagrams representing concept relations in propositions and syllogisms. Lambert represents a concept—considered in extension, that is, with respect “to all the individuals in which it appears” (Lambert 1764, Dian. §174)—as a line that can either be closed or open. He then represents the relations that two concepts entertain in the four basic propositions of categorical syllogisms: All A are B ...B A

b... a

No A is B A

a B

Some A is B b

B b ...A...

Some A is not B B

b ...A...

In the diagram representing universal aﬃrmative propositions, what counts, in addition to the length of the lines, is that A is drawn under B. The diagram representing universal negative propositions is clear. As for particular aﬃrmative propositions, the diagram shows that we only know some individuals A that are B, or at least one individual A that is B. Therefore, it remains indeterminate if also all A are B, or even all B are A. In the case of particular negative propositions the diagram shows not only that A is indeterminate but also that A is neither under B nor completely beside it, as in the case of universal negative propositions. On this basis one cannot only immediately

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make all valid conversions, but also represent all syllogisms, with the advantage of dispensing with the reduction to the ﬁrst ﬁgure (see Wolters 1980, 129–166). When the Neues Organon was published diagrams were no novelty, though Lambert introduced his diagrams before the already mentioned circular diagrams by Leonhard Euler (Euler 1768–1772): All A are B

No A is B

Some A is B

Some A is not B

B A

A

B

A

B

A

B

In fact, representations of concepts, propositions, and syllogisms by means of circles, lines, and other ﬁgures had already been devised by Johann Christoph Sturm (1661), who also introduced circular diagrams representing new syllogisms having negative terms; by Johann Christian Lange (1712, 1714); by Ploucquet (1759); and by Leibniz himself (on the history of diagrams in logic see Gardner 1983; Bernhard 2001). From what we have seen of Lambert’s and Ploucquet’s logical work, we can understand why their contemporaries were intrigued by their diﬀerent approaches to the problem of a logical calculus and wanted to assess their comparative merits. In a public debate in which Lambert and Ploucquet took part directly—reported in Bök ([1766])—Ploucquet’s Methodus calculandi was compared with Segner’s logical work, while Lambert’s diagrams in the Neues Organon underwent severe criticism. On the occasion of this debate Lambert began a correspondence with Georg Jonathan Holland (1742–1784), a pupil of Ploucquet. In a letter to Holland, Lambert criticized Ploucquet’s use of the traditional rule that nothing follows from two negative premises to exclude nonconclusive syllogism. He also criticized Ploucquet for using letters standing for the initials of substantives in syllogisms, thus showing his lack of a true symbolism (Lambert [1782] 1968, 95–96). But in an article of 1765, Lambert, though claiming that his diagrams in the Neues Organon were an example of a characteristic, acknowledged that they were only a little thing with respect to his project of a general logical calculus (Bök [1766], 153). At last, Lambert’s logical calculus was published in his Disquisitio (Lambert [1765], dated 1765 but actually printed in 1767). Here he states the aim of his calculus and lists the requisites any calculus must satisfy. As to his aim, Lambert says that he wants to ﬁnd a method for treating qualities similar to the method used in algebra for treating quantities. Just as in algebra we employ the ideas of relation, equality, proportion, and so on, so in the logical calculus we have to employ the ideas of identity, identiﬁcation, and analogy. As to the requisites a calculus must satisfy, they are the following. (1) For every operation we introduce, there must be the inverse operation, in full analogy with algebra where, when two quantities are added, it is always possible to obtain either by subtracting it from the total. (2) Given the object,

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the relations and the operations of the logical calculus, an adequate symbolism must be found. The symbols must be a perfect replacement for the things they symbolize to be safely used in their place. This means that we need a characteristic that mirrors things, a real characteristic, in which simple signs stand for simple things and are capable, once composed, to stand for composed things, so that it is also possible to proceed inversely from a composite to its simple elements. (3) Lambert also requires that we have a clear knowledge of the simple elements and the basic relations of the calculus; we must therefore know the combinatorial part of the ars characteristica combinatoria. In his opinion, the simple elements are qualities, that is, the special aﬀections of things we can consider as their attributes. Qualities are simple elements because, according to established ontological doctrines, they can be considered as “absolute” attributes, whereas other attributes, notably quantity, must be thought only “relatively” (a similar conception occurs in Leibniz’s De Arte Combinatoria). Lambert’s calculus in the Disquisitio, as it was the case with his unpublished essays of the ﬁfties, is intensional, that is, “does not concern individuals but properties” (letter to Holland 21.4.1765, Lambert [1782] 1968, 37). After trying the extensional perspective in the Neues Organon, his return to his early preference for the intension of concepts is undoubtedly due to a conscious choice. For Lambert wants to ﬁnd what is “simplest” and “ﬁrst” in concepts, but to obtain what is simpler, it is necessary to consider what is more complex, and in the case of concepts, the more complex concepts are those containing the simpler ones. Therefore, it is necessary to consider concepts as properties, as concepts containing other concepts, thus disregarding the class of individuals to which they extend. When dealing with judgments and syllogisms, Lambert’s ﬁrst aim is to establish the identity of the subject and predicate of judgments. Therefore, given the judgment “All A are B,” where A and B are not already obviously identical, Lambert establishes their identity by considering the subject as containing the predicate plus other properties. Hence the following symbolism (Lambert [1765], 461–462): All A are B

No A is B

Some A is B

Some A is not B

A = nB

A:p = B:q

mA = nB

mA:p = B:q

In the universal aﬃrmative, A = nB, n stands for the qualities which can be found in the subject A but not in the predicate B. In the universal negative, A:p = B:q, the sign “:” stands for a logical division and expresses which qualities, p and q, must be subtracted from the subject and the predicate, because neither belongs both to the subject and the predicate, so as to obtain A = B. Similarly, in the particular aﬃrmative, mA = nB, and in the particular negative, mA:p = B:q. On this basis, Lambert obtains a general formula expressing any kind of judgment: A/p = nB/q (here Lambert substitutes the

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sign of fraction for the sign “:”). From this formula one can easily derive a formula expressing any kind of syllogism: mA/p = nB/q μA/π = νC/ρ . mνC/pρ = μnB/πq To give an example of how this general formula applies to particular syllogisms, a syllogism Barbara, whose premises are B = mA and C = B, has the conclusion C = mA, whereas a syllogism Celarent, whose premises are B/q = A/p and C = νB, has the conclusion C/q = νA/p (Lambert [1782–1787], I, 102–103, 107). Despite the fact that the Disquisitio’s treatment of syllogism is very diﬀerent from that of the Neues Organon, it was disappointing for Lambert’s most competent readers. Holland sent Lambert a letter in which (beside mentioning his own tentative calculus) he observed that, however good Lambert’s calculus was, it did not achieve the declared aim to ﬁnd symbols mirroring reality. What are A, B, m, n, symbols of? Above all, which are the primitives that they are supposed to be symbols of? A deﬁnite answer, Holland concluded, could perhaps be expected from a new work Lambert had announced (Lambert [1782] 1968, 259–266). The work Holland referred to, titled Architectonic (Lambert [1771]), was published a few years later. In this treatise, which promised to give a theory of what is simple and ﬁrst in philosophical and mathematical knowledge, the author collects the results of his philosophical research going back to the mid-forties. But for all its importance as the summa of Lambert’s thought, the Architectonic provides no formal treatment, nor gives a new and complete list of simple elements that could be used as basic elements of a real characteristic, because it contains the same elements already listed in the Neues Organon. The conclusion to be drawn is that Lambert’s project shared Leibniz’s ambitions and in this respect went far beyond Segner’s and Ploucquet’s calculi, but perhaps was too ambitious and, though providing interesting details in the application of algebra to logic, can be said to be unachieved. In a sense, Lambert admitted as much in a letter (14.3.1771) to Johann Heinrich Tönnies: “should the universal characteristic belong to the same class as the philosopher’s stone or the squaring of the circle, it can at least, just as these, induce other discoveries” (Lambert [1782] 1968, 411). Ploucquet’s refusal of Leibniz’s project and Lambert’s somber admission to Tönnies may sound too pessimistic if one considers how much they and other eighteenth-century logicians—not to mention Leibniz—had progressed since Bernoulli’s failed parallelism. But especially Lambert’s assessment of universal characteristic as something similar to the squaring of the circle makes it clear that these authors believed that the construction of a satisfactory logical calculus was hindered by a possibly insurmountable obstacle: the overpowering amount of philosophical analysis to be done in the ﬁelds of metaphysics,

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semiotics, and natural language to reach a suitable alphabet of thoughts. As a matter of fact, unknown to these eighteenth-century logicians, there was an obstacle, not only on the side of philosophical analysis but also on the side of mathematics. Nineteenth-century logicians will ﬁnd out that one had to reﬂect also on the nature of mathematics and algebra, especially on their apparently exclusive link with quantity, before an algebra of logic could come to life.

11. Kant Interest for logical calculi seems to vanish at the end of the eighteenth century. We have mentioned some of the reasons behind this phenomenon, but according to a still widely received opinion this was due to the inﬂuence exerted on logic by Immanuel Kant (1724–1804). This opinion is usually justiﬁed by saying that Kant introduced confusion in logic through his notion of transcendental logic. As a matter of fact, Kant had a deﬁnite concept of logic, related to his transcendental philosophy but not to be confused with it. To evaluate Kant’s concept of logic, one must take into account his 40 years-long activity as a logic teacher, using as a textbook Georg Friedrich Meier’s Auszug aus der Vernunftlehre (Meier 1752b), a short version of the latter’s Vernunftlehre (Meier 1752a) (on Meier’s logic see Pozzo 2000). We have several texts related to this teaching activity, which constitute the socalled Kantian logic-corpus. Apart from the programs of the courses, such texts are (1) Kant’s handwritten annotations on Meier’s Auszug (the so-called logical Reﬂexionen, in Kant 1900, XVI), (2) lecture notes taken by students (Kant 1900–, XXIV; Kant 1998a, 1998b), and (3) I. Kant’s Logik, a book published in 1800 by Gottlob Benjamin Jäsche by collecting a selection of Kant’s annotations on Meier’s Auszug with Kant’s consent (Kant 1900–, IX, 1–150). These texts must be used with care and must always be compared with Kant’s published production. Nonetheless, they are essential to assess his views on logic, allowing a deeper insight into the importance of logic for Kant’s philosophy, and testifying to his knowledge of the discipline he was due to teach. As it is impossible to give details here of Kant’s treatment of logical doctrines, we will only discuss his general concept of logic.2 A comparative study of the Kantian logic-corpus shows that Kant’s concept of logic is the result of a sustained eﬀort of reﬂection lasting several years. He began as an almost orthodox Wolﬃan, founding logic on empirical psychology and ontology (Logik Blomberg, Kant 1900–, XXIV, 28). In his mature conception, however, he took the opposite view and denied that logic could be founded on either empirical psychology or ontology. To this eﬀect Kant argues that a logic founded on empirical psychology could describe human logical behavior but not prescribe laws to it. In his opinion, logical rules do not mirror what we actually do when we think, but are the standard to which our thoughts must conform if they are to have a logical form: Logic considers “not how we do think, but how we ought to think”

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(I. Kant’s Logik, Kant 1900–, IX, 14). As to the formerly accepted foundation of logic on ontology, Kant simply suppresses it, to the dismay of many of his contemporaries and later idealist philosophers. In particular, opposing Kant, Hegel proposed a new logic identical to metaphysics which, like old metaphysics, would admit that “thought (with its immanent determinations) and the true nature of things are one and the same content” (Barone 1964, 202). Thus it is rather surprising that William and Martha Kneale claim that it was Kant “with his transcendentalism who began the production of the curious mixture of metaphysics and epistemology which was presented as logic by Hegel and other Idealists of the nineteenth century” (Kneale and Kneale 1962, 355). The independence of logic from ontology and empirical psychology raises the problem of the origin and justiﬁcation of logic. Kant gives an indirect answer to the problem of the origin of logic by way of a comparison of logic with grammar. Logic and grammar—he maintains—are similar in as much as we learn to think and to speak without previous knowledge of grammatical and logical rules, and only at a later stage we become conscious of having implicitly used them. Nonetheless logic and grammar diﬀer because, as soon as we become aware of grammatical rules, we easily see that they are empirical, contingent, and subject to variations. On the contrary, once we become conscious of the logical structure of our thought, we cannot fail to appreciate that without that structure we could not have been thinking at all. Therefore logic precedes and regulates any rational thinking and is necessary in the sense that we cannot consider it contingent and variable. Kant concludes that logic “is abstracted [abstrahirt] from empirical use, but is not derived [derivirt]” from it (Reﬂexion 1612, Kant 1900–, XVI, 36) so that it can be considered a scientia scientiﬁca, whereas grammar is only a scientia empirica (Logik Busolt, Kant 1900–, XXIV, 609). This is important because the logic considered by Kant is not a natural logic that could be investigated by psychology, but is an “artiﬁcial logic.” This being the origin of logic, its justiﬁcation can be reduced to the fact that, according to Kant, logical principles such as the law of contradiction are accepted without proof: “All rules that are logically provable in general are in need of a ground [Grund] from which they are derived. Many propositions (e.g. that of contradiction) cannot be proved at all, neither a priori nor empirically” (Logik Dohna-Wundlacken, Kant 1900–, XXIV, 694). In other words, logical rules, “once known, are clear by themselves” (Reﬂexion 1602, Kant 1900–, XVI, 32). This means that logic not only is necessary, scientiﬁc, and a priori, but also is capable of justifying itself. These features make logic one of the means Kant uses in carrying through his philosophical project of explaining the possibility of experience according to his Copernican revolution. An important part of this project consists in showing that it is possible to ﬁnd all the general forms of thought—categories—without having to fall back on metaphysics or experience. Now logic (rather, one of its most important parts, i.e., the functions of judgment), which Kant has made no longer dependent on empirical

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psychology and ontology, qualiﬁes as the perfect clue to the categories. But since categories have to be completely enumerated to be employed in a complete list of principles of the understanding, which deﬁne the ﬁeld of possible experience, logic has to satisfy a further requisite: It has to be complete. Hence Kant’s well-known statement that logic “seems to all appearance to be ﬁnished and complete” (Kant 1997, B viii). Kant has been criticized for this statement, and in our opinion he lacks conclusive arguments to support it. But one must consider that a proof of the completeness of logic would have been easy if Kant had preserved the foundation of logic on empirical psychology and ontology, both ultimately guaranteed by God. It is also fair to point out that Kant envisages the possibility, for a closed system, of growing “from within,” on a par with living organisms that grow with no addition of new parts (Kant 1997, A 832/B 860). Applying this to logic, one could say that some growth in logic is possible, although within the boundaries of a systematic structure. The scientiﬁc, necessary, and self-justifying nature of logic guarantees that it has great autonomy and the maximum spectrum of application. Such prerogatives are counterbalanced by precise limitations: “Nobody can dare to judge of objects and to assert anything about them merely with logic without having drawn on antecedently well-founded information about them from outside logic” (Kant 1997, A 60/B 85). Consequently, logic is the supreme canon of truth with respect to the formal correctness of thought, but must be indiﬀerent to its contents. In this way Kant makes his concept of logic more deﬁnite. Logic, having no speciﬁc subject matter, is general. Having nothing to do with human psychology, it is pure. Concerning only the form of thought, it is merely formal. The ﬁrst consequence of this conception is that logic has to be analytic, although not in the sense that it deals with analytic judgments only. For logic is not concerned with the analytic/synthetic distinction which is left to transcendental logic: “The explanation of the possibility of synthetic judgments is a problem with which general logic has nothing to do, indeed whose name it need not even know” (Kant 1997, A 154/B 193). Logic is analytic in two senses. (1) “General logic analyzes the entire formal business of the understanding and reason into its elements, and presents these as principles of all logical assessment of our cognition” (Kant 1997, A 60/B 84). (2) Logic is analytic inasmuch as it has nothing to do with dialectic, both intended as the rhetorical art of disputation and as the part of logic dealing with probability. The most evident and better known reason for Kant’s separation of logic from dialectic is the connection between dialectic and rhetoric. A rhetorical dialectic is an art for deceiving adversaries in a dispute and for gaining consent not only disregarding truth but also purporting to produce the semblance [Schein] or illusion of truth. Kant condemns this kind of “logic” as unworthy of a philosopher (see I. Kant’s Logik, Kant 1900–, IX, 17). As to the association of dialectic with probability, it goes back to the distinction made by Aristotelian logical treatises between analytic, that is, the part of logic dealing with truth and certainty, and dialectic, that is, the part of logic dealing with

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what is probable, according to Boethius’s translation of the Greek éndoxos with the Latin probabile. This distinction was adopted by many eighteenthcentury logicians, notably by Meier who divides logic into analytica or “logic of completely certain erudite cognition,” and dialectica or logica probabilium, deﬁned as “logic of probable erudite cognition” (Meier 1752b, §6, in Kant 1900–, XVI, 72). Like many philosophers (including Leibniz), till the early seventies Kant hoped that a general logic of the uncertain could be found. Such a logic, although diﬀerent from the Aristotelian and humanist doctrines of probability and attentive to the late seventeenth-century results in the ﬁeld of probability calculus, was intended to be capable of also dealing with qualitative matters concerning justice, politics, and so on. Later on, Kant completely changed his view. He considered probability as a measurable degree of certainty—in this he agreed with Wolﬀ—which “can be expressed like a fraction, where the denominator is the number of all possible cases, the numerator is the number of actual cases” (Logik Pölitz, Kant 1900–, XXIV, 507). This view restricts probability (Wahrscheinlichkeit, probabilitas) to matters that can be subjected to a numerical calculus (games of chance and statistically based events such as mortality indexes), and excludes the possibility of an instrumental art for weighing, rather than numbering, heterogeneous reasons pro and contra a given qualitative question. Against this alleged art Kant objects that it concerns the notion of “verisimilitude” (Scheinbarkeit, verisimilitudo) rather than probability. In his view, if such an art, under the name of dialectic, belonged to logic, the latter would no longer be a canon of truth but would become an instrument for producing an illusion of truth by assigning an alleged probability—in fact a mere verisimilitude—even to questions that are beyond possible experience, such as the existence of the soul. Hence Kant’s claim that only probability restricted to matters that can be subjected to a numerical calculus is worthy of this name and, because it is contiguous to truth and certainty, belongs to the analytic part of logic and need not be dealt with in a special part of logic called dialectic (Kant 1900–, A 293/B 349). Kant’s position is drastic: Logic and dialectic must part and go separate ways. The second consequence of Kant’s view that logic is a mere formal canon of truth is that the content of logic must be limited to the doctrine of elements: concepts, judgments, and inferences. Therefore, logic must not trespass into the domains of anthropology and psychology, nor give advice for the use of logic in the ﬁelds of the natural sciences or of practical life. This means that Kant breaks away from one of the main trends of European logic, which had tried to give new life to the discipline by stressing its usefulness either as a guide for judging, or as a kind of methodology for empirical research, or as a medicine against errors, or as an epistemological exercise. In particular, Kant breaks away from Locke’s view of logic, despite the fact that he had formerly praised it, for he maintains that the study of the origin of concepts “does not belong to logic, but rather to metaphysics” (Logik Dohna-Wundlacken, Kant 1900–, XXIV, 701).

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The fact that Kant separates logic from epistemology does not mean that the texts of the Kantian logic-corpus do not contain epistemological parts. On the contrary, these texts make very interesting reading on matters such as opinion, belief, knowledge, hypotheses, probability, and so on. But these matters are no longer intended as belonging to pure logic because, to deal with them, one must take into account the content of knowledge and the human cognitive constitution, including sensibility, or at least the form of sensibility, as well as practical aspects of human action, such as the interest we have for accepting something as true. If Kant had written a logic handbook himself, he probably would have treated such matters at length, in addition to other interesting questions, such as the doctrine of logical essence, in a doctrine of method. The third consequence of Kant’s view of logic is that it is only a canon for checking the correctness of our thoughts but is incapable of invention. Kant’s sharp distinction between logic and mathematics contributes to this view. He agrees with Wolﬀ that there is a single logic to be complied with by mathematicians and nonmathematicians alike, but logic is insuﬃcient to explain why mathematics is ampliative. According to Kant, mathematics is the science that constructs a priori its concepts, that is, exhibits a priori the intuitions corresponding to them. Thus, mathematics relies also on the form of sensible intuition, so that it has content and can be inventive with respect to it. This applies not only to arithmetic and geometry but also to algebra, which is inventive because it refers (albeit mediately) to a priori intuitions. Therefore, Kant rejects the view of those who “believe that logic is a heuristic (art of discovery) that is an organ of new knowledge, with which one makes new discoveries, thus e.g. algebra is heuristic; but logic cannot be a heuristic, since it abstracts from any content of knowledge” (Logik Hechsel, Kant 1998b, 279 = ms. 9). These statements are not borne out of ignorance. Kant knew the outlines of Leibniz’s ars characteristica combinatoria, on whose utopian nature he commented in an essay of 1755 (Nova dilucidatio, Kant, 1900–, I, 390) in terms that seem to anticipate analogous statements by Ploucquet and Lambert. Moreover, his logic-corpus, as well as his works and correspondence, provide evidence that (1) he was well acquainted with the combinatorial calculus of syllogistic moods; (2) he used Euler’s (whom he quotes) circular diagrams to designate concepts, judgments, and syllogisms; (3) he knew the linear diagrams of Lambert, with whom he corresponded; (4) he probably had some knowledge of Segner’s and Ploucquet’s works; and (5) he actively promoted the diﬀusion of Lambert’s posthumous works containing the latter’s algebraic calculus. But all this did not shake his conviction that an algebraic symbolism of idea-relations and the use of letters instead of words are not by themselves a means to invention. If we consider the development of logic from humanism onward, we see that one of the basic motivations of logical research in the whole period was the demand to make logic inventive. The humanist theories of inventio, Bacon’s studies on induction, Descartes’s theory of problem solving, the ars inveniendi and a large part of Leibniz’s ars characteristica combinatoria, and so on,

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can be viewed in this perspective. Kant objected that this kind of research, while claiming to be purely formal, was meant to deal with the content of knowledge. The condition (and cost) of his objection to an inventive logic was the separation of logic from mathematics, but in this way he achieved his aim of separating pure logic from metaphysics and psychology, as well as from any transcendent foundation. This aspect of Kant’s concept of logic reappears in the philosophy of logic of some later logicians. Thus, despite substantial diﬀerences, Frege’s concept of logic seems indebted to Kant’s in several respects, such as the idea that the only logic that really counts is scientiﬁc logic, rather than some natural logic; the contention that a scientiﬁc or artiﬁcial logic provides necessary and universal rules; the condemnation of any intrusion of psychology into logic by the argument that logic is normative on a par with moral laws; the idea that logic is used for justifying knowledge rather than for acquiring new knowledge. But even Venn, who claims that Kant had “a disastrous eﬀect on logical method” (Venn [1894], xxxv) begins his own system of logic by stating: “Psychological questions need not concern us here; and still less those which are Metaphysical” (Venn [1894], xxxix). Perhaps it would have been more diﬃcult for him to make such a statement if Kant had not already made that very same statement.

Notes 1. Reference to secondary literature devoted to Leibniz’s notion of complete concept could span over many pages. We will limit ourselves to the seminal papers by Mondadori (1973) and Fitch (1979), to the discussion included in Mates (1986), and—for two recent accounts based on diﬀerent interpretations—to Zalta (2000) and Lenzen (2003). Mondadori’s and Fitch’s papers are included, together with other relevant contributions, in Woolhouse (1993). 2. The body of literature on Kant is enormous, and also literature on Kantian logic is very extensive, ranging from the relation between general and transcendental logic to the doctrines of concepts, judgments, and inferences, not to mention topics such as the relations between logic and language and mathematics. We will mention only Shamoon (1981), Capozzi (1987), Pozzo (1989), Brandt (1991), Reich (1992), Wolﬀ (1995), Capozzi (2002), and Capozzi (forthcoming) (the latter containing an extensive bibliography). A precious research tool is provided by an impressive Kantian lexicon, still in progress, many of whose volumes are devoted to Kant’s logic-corpus (Hinske 1986–).

References: Primary Sources Agricola, Rodolphus. [1479] 1967. De inventione dialectica. Reprint of the ed. Köln 1539, with Alardus’s commentary. Frankfurt: Minerva. Agricola, Rodolphus. [1479] 1992. De inventione libri tres. Drei Bücher über die Inventio dialectica, ed. L. Mundt. Tübingen: Niemeyer. Aldrich, Henry. [1691] 1849. Artis Logicae Compendium. Oxford. Edited by H. L. Mansel as Artis Logicae Rudimenta. Oxford: Hammans.

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Aquinas, Thomas. 1888–1889. Opera omnia iussu impensaque Leonis XIII P. M. edita, t. 4–5: Pars prima Summae theologiae. Rome: Typographia Polyglotta S. C. de Propaganda Fide. Argens, Jean Baptiste d’. 1737. La philosophie du bon sens, ou réﬂexions philosophiques sur l’incertitude des connoissances humaines, à l’usage des cavaliers et du beau-sexe. London. Arnauld, Antoine, and Claude Lancelot. [1676] 1966. Grammaire Générale et raisonnée, critical ed. of the 1676 edition by Herbert E. Brekle (1st ed. 1660). Stuttgart-Bad Cannstatt: Frommann-Holzboog. Arnauld, Antoine, and Pierre Nicole. [1683] 1981. La Logique ou l’art de penser, critical ed. of the 5th ed., Paris 1683, eds. P. Clair and F. Girbal (1st ed. 1662). Paris: Vrin. English trans. in Arnauld and Nicole 1996. Arnauld, Antoine, and Pierre Nicole. 1996. Logic or the Art of Thinking. Trans. and ed. J. Vance Buroker. Cambridge: Cambridge University Press. Arnauld, Antoine, and Pierre Nicole. 1704. Logica sive Ars cogitandi. Trans. J. Fr. Budde. Halle. Bacon, Francis. [1620] 2000. The New Organon. Trans. L. Jardine and M. Silverthorne. Cambridge: Cambridge University Press. Becher, Johann Joachim. 1661. Character pro notitia linguarum universali. Frankfurt: Ammon and Serlin. Beck, Cave. 1657. The Universal Character, by which all the Nations in the World may Understand One Another’s Conceptions. . . . London: Maxey and Weekley. Bernoulli, Jakob. [1685] 1969. Parallelismus ratiocinii logici et algebraici. In Die Werke von Jakob Bernoulli, vol. I, ed. Naturforschende Gesellschaft in Basel. Basel: Birkhäuser. English trans. in Boswell (1990), 211–224. Bernoulli, Jakob. [1713] 1975. Ars conjectandi. In Die Werke von Jakob Bernoulli, vol. III, ed. Naturforschende Gesellschaft in Basel. Basel: Birkhäuser. Bök, August Friedrich, ed. [1766] 1970. Sammlung der Schriften, welche den logischen Calcul Herrn Prof. Ploucquets betreﬀen, mit neuen Zusätzen. Reprint, ed. A. Menne. Stuttgart-Bad Cannstatt: Frommann-Holzboog. Budde, Johann Franz. 1731. Compendium historiae philosophiae observationibus illustratum, ed. Johann Georg Walch. Halle. Burgersdijk, Franco. [1626] 1645. Institutionum logicarum libri duo. Leiden: Oﬃcina Commelini. Caesarius, Johannes. [1520] 1545. Dialectica. Mainz: Schoeﬀer. Clauberg, Johann. [1654] 1658. Logica vetus et nova. Amsterdam: Oﬃcina Elzeviriana. Condillac, Étienne Bonnot de. [1780] 1948. La Logique ou les premiers développements de l’art de penser. In Oeuvres philosophiques de Condillac, ed. G. Le Roy, vol. 2, 368–416. Paris: Presses Universitaires de France. Condillac, Étienne Bonnot de. [1798] 1948. La langue des calculs. In Oeuvres philosophiques de Condillac, ed. G. Le Roy, vol. 2, 418–529. Paris: Presses Universitaires de France. Crousaz, Jean Pierre de. 1724. Logicae systema. Genève. Dalgarno, George. [1661] 1961. Ars Signorum. London: Hayes. Reprint Menston: Scholar Press. Descartes, René. 1964–1976. Ouvres de Descartes, ed. C. Adam and P. Tannery. Paris: Vrin.

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Reich, Klaus. 1992. The Completeness of Kant’s Table of Judgments, trans. J. Kneller and M. Losonsky of Die Vollständigkeit der kantischen Urteilstafel, Berlin (2nd ed.) 1948 Hamburg 1986 (3rd ed.). Stanford, Calif.: Stanford University Press. Risse, Wilhelm. 1964–70. Die Logik der Neuzeit, 2 vols. Stuttgart-Bad Cannstatt: Frommann-Holzboog. Risse, Wilhelm. 1965. Bibliographia Logica: Verzeichnis der Druckschriften zur Logik mit Angabe ihrer Fundorte, vol. I: 1472–1800. Hildesheim: Olms. Roncaglia, Gino. 1988. Modality in Leibniz’ Essays on Logical Calculus of April 1679. Studia Leibnitiana 20/1, 43–62. Roncaglia, Gino. 1996. Palaestra rationis. Discussioni su natura della copula e modalità nella ﬁlosoﬁa ‘scolastica’ tedesca del XVII secolo. Firenze: Olschki. Roncaglia, Gino. 1998. Sull’evoluzione della logica di Melantone. Medioevo 24, 235– 265. Roncaglia, Gino. 2003. Modal Logic in Germany at the Beginning of the Seventeenth Century: Christoph Scheibler’s Opus Logicum. In R. L. Friedman and L. O. Nielsen 2003, 253–308. Russell, Bertrand. 1900. A Critical Exposition of the Philosophy of Leibniz. Cambridge: Cambridge University Press. Schepers, Heinrich. 1959. Andreas Rüdigers Methodologie und ihre Voraussetzungen, (Kant-Studien, Ergänzungsheft 78). Köln: Kölner Universitäts-Verlag. Schmitt, Charles B., Quentin Skinner, and Eckhard Kessler, eds. 1988. The Cambridge History of Renaissance Philosophy. Cambridge: Cambridge University Press. Schupp, Franz. 2000. Introduction, trans. and commentary to G. W. Leibniz. Die Grundlagen des logischen Kalküls. Hamburg: Meiner. Shamoon, Alan. 1981. Kant’s Logic, Ph.D. dissertation, Columbia University (1979). Ann Arbor: University Microﬁlms International. Swoyer, Chris. 1995. Leibniz on Intension and Extension. Nous 29/1, 96–114. Thom, Paul. 1981. The Syllogism. München: Philosophia Verlag. Tonelli, Giorgio. 1959. Der Streit über die mathematische Methode in der Philosophie in der ersten Hälfte des 18. Jahrhunderts und die Entstehung von Kants Schrift über die “Deutlichkeit.” Archiv für Begriﬀsgeschichte 9, 37–66. Vailati, Giovanni. 1899. La logique mathématique et sa nouvelle phase de développement dans les écrits de M. J. Peano. Revue de Métaphysique et de Morale 7, 86–102. Vasoli, Cesare. 1968. La dialettica e la retorica dell’umanesimo. Milano: Feltrinelli. Vasoli, Cesare. 1974, Intorno al Petrarca e ai logici “moderni.” Miscellanea Medievalia IX, Antiqui und Moderni, Berlin: De Gruyter, 142–154. Venn, John. [1894] 1971. Symbolic Logic. Reprint of the 2nd ed. New York: Chelsea Publ. Co. Waswo, Richard. 1999. Theories of Language. In The Cambridge History of Literary Criticism III—The Renaissance, ed. G. P. Norton, 25–35. Cambridge: Cambridge University Press. Wolﬀ, Michael. 1995. Die Vollständigkeit der kantischen Urteilstafel. Mit einem Essay über Freges Begriﬀsschrift. Frankfurt: Klostermann. Wollgast, Siegfried. 1988a. Ehrenfried Walther von Tschirnhaus und die deutsche Frühaufklärung. Berlin: Akademie-Verlag. Wollgast, Siegfried. 1988b (2nd ed.). Philosophie in Deutschland zwischen Reformation und Aufklärung 1550–1650. Berlin: Akademie-Verlag.

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Wolters, Gereon. 1980. Basis und Deduktion. Studien zur Entstehung und Bedeutung der Theorie der axiomatischen Methode bei J. H. Lambert (1728–1777). Berlin–New York: Walter de Gruyter. Woolhouse, Roger S., ed. 1993. Gottfried Wilhelm Leibniz: Critical Assessments, vols. I–IV. London–New York: Routledge. Wundt, Max. [1945] 1964. Die deutsche Schulphilosophie im Zeitalter der Aufklärung. Tübingen. Reprint Hildesheim: Olms. Yolton, John. 1986. Schoolmen, Logic and Philosophy. In The History of the University of Oxford, vol. V: The Eighteenth Century, eds. L. S. Sutherland and L. G. Mitchell, 565–591. Oxford: Clarendon Press. Zalta, Edward. 2000. A (Leibnizian) Theory of Concepts. Philosophiegeschichte und logische Analyse/Logical Analysis and History of Philosophy 3, 137–183.

4

The Mathematical Origins of Nineteenth-Century Algebra of Logic Volker Peckhaus

1. Introduction Most nineteenth-century scholars would have agreed to the opinion that philosophers are responsible for research on logic. On the other hand, the history of late nineteenth-century logic clearly indicates a very dynamic development instigated not by philosophers but by mathematicians. A central outcome of this development was the emergence of what has been called the “new logic,” “mathematical logic,” “symbolic logic,” or, from 1904 on, “logistics.”1 This new logic came from Great Britain, and was created by mathematicians in the second half of the nineteenth century, ﬁnally becoming a mathematical subdiscipline in the early twentieth century. Charles L. Dodgson, better known under his pen name Lewis Carroll (1832– 1898), published two well-known books on logic, The Game of Logic of 1887 and Symbolic Logic of 1896, of which a fourth edition appeared already in 1897. These books were written “to be of real service to the young, and to be taken up, in High Schools and in private families, as a valuable addition of their stock of healthful mental recreations” (Carroll 1896, xiv). They were meant “to popularize this fascinating subject,” as Carroll wrote in the preface of the fourth edition of Symbolic Logic (ibid.). But astonishingly enough, in both books there is no deﬁnition of the term “logic.” Given the broad scope of these books, the title “Symbolic Logic” of the second book should at least have been explained. The text is based (but elaborated and enlarged) on my paper “Nineteenth Century Logic between Philosophy and Mathematics” (Peckhaus 1999).

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Maybe the idea of symbolic logic was so widely spread at the end of the nineteenth century in Great Britain that Carroll regarded a deﬁnition as simply unnecessary. Some further observations support this thesis. They concern a remarkable interest by the general public in symbolic logic, after the death of the creator of the algebra of logic, George Boole, in 1864. Recalling some standard nineteenth-century deﬁnitions of logic as, for example, the art and science of reasoning (Whately) or the doctrine giving the normative rules of correct reasoning (Herbart), it should not be forgotten that mathematical or symbolic logic was not set up from nothing. It arose from the old philosophical collective discipline logic. It is therefore obvious to assume that there was some relationship between the philosophical and the mathematical side of the development of logic, but standard presentations of the history of logic ignore this putative relationship; they sometimes even deny that there has been any development of philosophical logic at all, and that philosophical logic could therefore justly be ignored. Take for instance William and Martha Kneale’s program in their eminent The Development of Logic. They wrote (1962, iii): “But our primary purpose has been to record the ﬁrst appearances of these ideas which seem to us most important in the logic of our own day,” and these are the ideas leading to mathematical logic. Another example is J. M. Bocheński’s assessment of “modern classical logic,” which he dated between the sixteenth and the nineteenth century. This period was for him noncreative. It can therefore justly be ignored in a problem history of logic (1956, 14). According to Bocheński, classical logic was only a decadent form of this science, a dead period in its development (ibid., 20). Authors advocating such opinions adhere to the predominant views of present-day logic, that is, actual systems of mathematical or symbolic logic. As a consequence, they are not able to give reasons for the ﬁnal divorce between philosophical and mathematical logic, because they ignore the seed from which mathematical logic has emerged. Following Bocheński’s view, Carl B. Boyer presented for instance the following periodization of the development of logic (Boyer 1968, 633): “The history of logic may be divided, with some slight degree of oversimpliﬁcation, into three stages: (1) Greek logic, (2) scholastic logic, and (3) mathematical logic.” Note Boyer’s “slight degree of oversimpliﬁcation” which enabled him to skip 400 years of logical development and ignore the fact that Kant’s transcendental logic, Hegel’s metaphysics, and Mill’s inductive logic were called “logic,” as well. This restriction of scope had a further consequence: The history of logic is written as if it had been the nineteenth-century mathematicians’ main motive for doing logic to create and develop a new scientiﬁc discipline as such, namely mathematical logic, dealing above all with problems arising in this discipline and solving these problems with the ﬁnal aim of attaining a coherent theory. But what, if logic was only a means to an end, a tool for solving nonlogical

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problems? If this is considered, such nonlogical problems have to be taken note of. One can assume that at least the initial motives of mathematicians working in logic were going beyond creating a new or further developing the traditional theory of logic. Under the presupposition that a mathematician is usually not really interested in devoting his professional work to the development of a philosophical subdiscipline, one can assume that theses motives have to be sought in the mathematician’s own subject, namely in foundational, that is, philosophical problems of mathematics. Today historians have recognized that the emergence of the new logic was no isolated process. Its creation and development ran parallel to and was closely intertwined with the creation and development of modern abstract mathematics which emancipated itself from the traditional deﬁnition as a science which deals with quantities and geometrical forms and is therefore responsible for imaginabilia, that is, intuitive objects. The imaginabilia are distinguished from intelligibilia, that is, logical objects which have their origin in reason alone. These historians recognized that the history of the development of modern logic can only be told within the history of the development of mathematics because the new logic is not conceivable without the new mathematics. In recent research on the history of logic, this intimate relation between logic and mathematics, especially its connection to foundational studies in mathematics, has been taken into consideration. One may mention the present author’s Logik, Mathesis universalis und allgemeine Wissenschaft (Peckhaus 1997) dealing with the philosophical and mathematical contexts of the development of nineteenth-century algebra of logic as at least partially unconscious realizations of the Leibnizian program of a universal mathematics, José Ferreirós’s history of set theory in which the deep relations between the history of abstract mathematics and that of modern logic (Ferreirós 1999) are unfolded, and the masterpiece of this new direction, The Search for Mathematical Roots, 1870–1940 (2000a) by Ivor Grattan-Guinness, who imbedded the whole bunch of diﬀerent directions in logic into the development of foundational interests within mathematics. William Ewald’s “Source Book in the Foundations of Mathematics” (Ewald 1996) considers logical inﬂuences at least in passing, whereas the Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, edited by Ivor Grattan-Guinness (1994), devotes an entire part to “Logic, Set Theories and the Foundation of Mathematics” (vol. 1, pt. 5). In the following, the complex conditions for the emergence of nineteenthcentury symbolic logic will be discussed. The main scope will be on the mathematical motives leading to the interest in logic; the philosophical context will be dealt with only in passing. The main object of study will be the algebra of logic in its British and German versions. Special emphasis will be laid on the systems of George Boole (1815–1864) and above all of his German follower Ernst Schröder (1841–1902).

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2. Boole’s Algebra of Logic 2.1. Philosophical Context The development of the new logic started in 1847, completely independent of earlier anticipations, for example, those by the German rationalistic universal genius Gottfried Wilhelm Leibniz (1646–1716) and his followers (see Peckhaus 1994a, 1997, ch. 5). In that year the British mathematician Boole published his pamphlet The Mathematical Analysis of Logic (1847).2 Boole mentioned (1847, 1) that it was the struggle for priority concerning the quantiﬁcation of the predicate between the Edinburgh philosopher William Hamilton (1788– 1856) and the London mathematician Augustus De Morgan (1806–1871) that encouraged this study. Hence, he referred to a startling philosophical discussion which indicated a vivid interest in formal logic in Great Britain. This interest was, however, a new interest, just 20 years old. One can even say that neglect of formal logic could be regarded as a characteristic feature of British philosophy up to 1826 when Richard Whately (1787–1863) published his Elements of Logic.3 In his preface Whately added an extensive report on the languishing research and education in formal logic in England. He complained (1826, xv) that only very few students of the University of Oxford became good logicians and that by far the greater part pass through the University without knowing any thing of all of it; I do not mean that they have not learned by rote a string of technical terms; but that they understand absolutely nothing whatever of the principles of the Science. Thomas Lindsay, the translator of Friedrich Ueberweg’s important System der Logik und Geschichte der logischen Lehren (1857, English translation), was very critical of the scientiﬁc qualities of Whately’s book, but he nevertheless emphasized its outstanding contribution for the renaissance of formal logic in Great Britain (Lindsay 1871, 557): Before the appearance of this work, the study of the science had fallen into universal neglect. It was scarcely taught in the universities, and there was hardly a text-book of any value whatever to be put into the hands of the students. One year after the publication of Whately’s book, George Bentham’s An Outline of a New System of Logic appeared (1827) which was intended as a commentary to Whately. Bentham’s book was critically discussed by William Hamilton in a review article published in the Edinburgh Review (Hamilton 1833). With the help of this review, Hamilton founded his reputation as the “ﬁrst logical name in Britain, it may be in the world.”4 Hamilton propagated a revival of the Aristotelian scholastic formal logic without, however, one-sidedly preferring the syllogism. His logical conception was focused on a revision of

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the standard forms by quantifying the predicates of judgments.5 He arrived at eight standard forms (Hamilton 1859–1866, vol. 4, 1866, 287): 1. A “All A is all B”

toto-total.

2. A “All A is some B”

toto-partial.

3. I “Some A is all B”

parti-total.

4. I “Some A is some B”

parti-partial.

5. E “Any A is not any B”

toto-total.

6. E “Any A is not some B”

toto-partial.

7. O “Some A is not any B”

parti-total.

8. O “Some A is not some B” parti-partial. Hamilton’s unconsidered transition from the collective “all” to the distributive “any” has already been criticized by William and Martha Kneale (1962, 353). Hamilton used a geometrical symbolism using wedges for illustrating the eﬀects of this modiﬁcation.6 The controversy about priority arose when De Morgan, in a lecture “On the Structure of the Syllogism” (De Morgan 1846) given to the Cambridge Philosophical Society on 9 November 1846, also proposed the quantiﬁcation of the predicates.7 Neither had any priority, of course. The application of diagrammatic methods in syllogistic reasoning proposed, for example, by the eighteenth-century mathematicians and philosophers Leonard Euler, Gottfried Ploucquet, and Johann Heinrich Lambert, presupposed a quantiﬁcation of the predicate.8 The German psychologistic logician Friedrich Eduard Beneke (1798–1854) suggested to quantify the predicate in his books on logic published in 1839 and 1842, the latter of which he sent to Hamilton (see Peckhaus 1997, 191–193). In the context of this presentation, it is irrelevant to give a ﬁnal solution of the priority question. It is, however, important that a dispute of this extent arose at all. It indicates that there was a new interest in research on formal logic. This interest represented only one side of the eﬀects released by Whately’s book. Another line of research stood in the direct tradition of Humean empiricism and the philosophy of inductive sciences: the inductive logic of John Stuart Mill (1806–1873), Alexander Bain (1818–1903), and others. Boole’s logic was in clear opposition to inductive logic. It was Boole’s follower William Stanley Jevons (1835–1882; see Jevons 1877–1878) who made this opposition explicit. As mentioned earlier, Boole referred to the controversy between Hamilton and De Morgan, but this inﬂuence should not be overemphasized. In his main work on the Laws of Thought (1854), Boole went back to the logic of Aristotle by quoting from the Greek original. This can be interpreted as indicating that the inﬂuence of the contemporary philosophical discussion was not as important as his own words might suggest. In writing a book on logic he was

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doing philosophy, and it was thus a matter of course that he related his results to the philosophical discussion of his time. This does not mean, of course, that his thoughts were mainly inﬂuenced by this discussion. In any case, Boole’s early algebra of logic kept a close connection to traditional logic, in the formal part of which the theory of syllogism represented its core.9 Traditional logic not only provided the topics to be dealt with by the “Calculus of Deductive Reasoning,”10 it also served as a yardstick for evaluating the power and the reliability of the new logic. Even in the unpublished manuscripts of a sequel of the Laws of Thought titled “The Philosophy of Logic,” he discussed Aristotelian logic at length (see Boole 1997, 133–136), criticizing, however, that it is more a mnemonic art than a science of reasoning.11

2.2. The Mathematical Context in Great Britain Of greater importance than the philosophical discussion on logic in Great Britain were mathematical inﬂuences. Most of the new logicians can be related to the so-called Cambridge Network (Cannon 1978, 29–71), that is, a movement that aimed at reforming British science and mathematics which started at Cambridge. One of the roots of this movement was the foundation of the Analytical Society in 1812 (see Enros 1983) by Charles Babbage (1791–1871), George Peacock (1791–1858), and John Herschel (1792–1871). Joan L. Richards called this act a “convenient starting date for the nineteenth-century chapter of British mathematical development” (Richards 1988, 13). One of the ﬁrst achievements of the Analytical Society was a revision of the Cambridge Tripos by adopting the Leibnizian notation for the calculus and abandoning the customary Newtonian theory of ﬂuxions: “the principles of pure D-ism in opposition to the Dot-age of the University” as Babbage wrote in his memoirs (Babbage 1864, 29). It may be assumed that this successful movement triggered oﬀ by a change in notation might have stimulated a new or at least revived interest in operating with symbols. This new research on the calculus had parallels in innovative approaches to algebra which were motivated by the reception of Laplacian analysis.12 In the ﬁrst place, the development of symbolic algebra has to be mentioned. It was codiﬁed by George Peacock in his Treatise on Algebra (1830) and further propagated in his famous report for the British Association for the Advancement of Science (Peacock 1834, especially 198– 207). Peacock started by drawing a distinction between arithmetical and symbolic algebra, which was, however, still based on the common restrictive understanding of arithmetic as the doctrine of quantity. A generalization of Peacock’s concept can be seen in Duncan F. Gregory’s (1813–1844) “calculus of operations.”13 Gregory was most interested in operations with symbols. He deﬁned symbolic algebra as “the science which treats of the combination of operations deﬁned not by their nature, that is by what they are or what they do, but by the laws of combinations to which they are subject” (1840, 208). In his much praised paper “On a General Method in Analysis” (1844), Boole made the calculus of operations the basic methodological tool for analysis.

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However, in following Gregory, he went further, proposing more applications. He cited Gregory, who wrote that a symbol is deﬁned algebraically “when its laws of combination are given; and that a symbol represents a given operation when the laws of combination of the latter are the same as those of the former” (Gregory 1842, 153–154). It is possible that a symbol for an arbitrary operation can be applied to the same operation (ibid., 154). It is thus necessary to distinguish between arithmetical algebra and symbolic algebra, which has to take into account symbolic but nonarithmetical ﬁelds of application. As an example, Gregory mentioned the symbols a and +a. They are isomorphic in arithmetic, but in geometry they need to be interpreted diﬀerently. a can refer to a point marked by a line, whereas the combination of the signs + and a additionally expresses the direction of the line. Therefore symbolic algebra has to distinguish between the symbols a and +a. Gregory deplored the fact that the unequivocity of notation did not prevail as a result of the persistence of mathematical practice. Clear notation was only advantageous, and Gregory thought that our minds would be “more free from prejudice, if we never used in the general science symbols to which deﬁnite meanings had been appropriated in the particular science” (ibid., 158). Boole adopted this criticism almost word for word. In his Mathematical Analysis of Logic he claimed that the reception of symbolic algebra and its principles was delayed by the fact that in most interpretations of mathematical symbols the idea of quantity was involved. He felt that these connotations of quantitative relationships were the result of the context of the emergence of mathematical symbolism, and not of a universal principle of mathematics (Boole 1847, 3–4). Boole read the principle of the permanence of equivalent forms as a principle of independence from interpretation in an “algebra of symbols.” To obtain further aﬃrmation, he tried to free the principle from the idea of quantity by applying the algebra of symbols to another ﬁeld, the ﬁeld of logic. As far as logic is concerned this implied that only the principles of a “true Calculus” should be presupposed. This calculus is characterized as a “method resting upon the employment of Symbols, whose laws of combination are known and general, and whose results admit of a consistent interpretation” (ibid., 4). He stressed (ibid.): It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical Analysis, regardless that in its objects and in its instruments it must at present stand alone. Boole expressed logical propositions in symbols whose laws of combination are based on the mental acts represented by them. Thus he attempted to establish a psychological foundation of logic, mediated, however, by language.14 The central mental act in Boole’s early logic is the act of election used for building classes. Man is able to separate objects from an arbitrary collection which belong to given classes to distinguish them from others. The symbolic representation of these mental operations follows certain laws of combination that

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are similar to those of symbolic algebra. Logical theorems can thus be proven like mathematical theorems. Boole’s opinion has of course consequences for the place of logic in philosophy: “On the principle of a true classiﬁcation, we ought no longer to associate Logic and Metaphysics, but Logic and Mathematics” (ibid., 13). Although Boole’s logical considerations became increasingly philosophical with time, aiming at the psychological and epistemological foundations of logic itself, his initial interest was not to reform logic but to reform mathematics. He wanted to establish an abstract view on mathematical operations without regard to the objects of these operations. When claiming “a place among the acknowledged forms of Mathematical Analysis” (1847, 4) for the calculus of logic, he didn’t simply want to include logic in traditional mathematics. The superordinate discipline was a new mathematics. This is expressed in Boole’s writing: “It is not of the essence of mathematics to be conversant with the ideas of number and quantity” (1854, 12).

2.3. Boole’s Logical System Boole’s early logical system is based on mental operations, namely, acts of selecting individuals from classes. In his notation 1 symbolizes the Universe, comprehending “every conceivable class of objects whether existing or not” (1847, 15). Capital letters stand for all members of a certain class. The small letters are introduced as follows (ibid., 15): The symbol x operating upon any subject comprehending individuals or classes, shall be supposed to select from that subject all the Xs which it contains. In like manner the symbol y, operating upon any subject, shall be supposed to select from it all individuals of the class Y which are comprised in it and so on. Take A as the class of animals, then x might signify the selection of all sheep from these animals, which then can be regarded as a new class X from which we select further objects, and so on. This might be illustrated by the following example: animals A ↓ x sheep

sheep X ↓ y horned

horned sheep Y ↓ z black

black horned sheep Z

This process represents a successive selection which leads to individuals being common to the classes A, X, Y , and Z. xyz stands for animals that are sheep, horned, and black. It can be regarded as the logical product of some common marks or common aspects relevant for the selection. In his major work, An Investigation of the Laws of Thought of 1854, Boole gave up this distinction

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between capital and small letters, thereby getting rid of the complicated consequences of this stipulation. If the symbol 1 denotes the universe, and if the class X is determined by the symbol x, it is consequent that the class not-X has to be denoted by the symbol 1 − x, which forms the supplement to x, thus x(1 − x) = 0. 0 symbolizes nothing or the empty class. Now one can consider Boole’s interpretation of the universal-aﬃrmative judgment. The universal-aﬃrmative judgment “All Xs are Y s” is expressed by the equation xy = x or, by simple arithmetical transformation, x(1 − y) = 0 (p. 22): “As all the Xs which exist are found in the class Y , it is obvious that to select out of the Universe all Y s, and from these to select all Xs, is the same as to select at once from the Universe all Xs.” The universal-negative judgment “No Xs are Y s” asserts that there are no terms common in the classes X and Y . All individuals common would be represented by xy, but they form the empty class. The particular-aﬃrmative judgment “Some Xs are Y s” says that there are some terms common to both classes forming the class V . They are expressed by the elective symbol v. The judgment is thus represented by v = xy. Boole furthermore considers using vx = vy with vx for “some X” and vy for “some Y ,” but observes “that this system does not express quite so much as the single equation” (pp. 22–23). The particular-negative judgment “Some Xs are not Y s” can be reached by simply replacing y in the last formula with 1 − y. Boole’s elective symbols are compatible with the traditional theory of judgment. They blocked, however, the step toward modern quantiﬁcation theory as present in the work of Gottlob Frege, but also in later systems of the algebra of logic like those of C. S. Peirce and Ernst Schröder.15 The basic relation in the Boolean calculus is equality. It is governed by three principles which are themselves derived from elective operations (see ibid., 16–18): 1. The Distributivity of Elections (16–17): it is indiﬀerent whether from of group of objects considered as a whole, we select the class X, or whether we divide the group into two parts, select the Xs from them separately, and then connect the results in one aggregate conception, in symbols: x(u + v) = xu + xv, with u + v representing the undivided group of objects, and u and v standing for its component parts. 2. The Commutativity of Elections: The order of elections is irrelevant: xy = yx. 3. The Index Law: The successive execution of the same elective act does not change the result of the election: xn = x,

for n ≥ 2.

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Boole stressed the importance of the Index Law, which is not generally valid in arithmetic (only in the arithmetic of 0 and 1) and therefore peculiar for elective symbols. It allows one to reduce complex formulas to forms more easily capable of being interpreted. In his Investigation of the Laws of Thought (1854) Boole abandoned the Index Law and replaced it by the Law of Duality (“Boole’s Law”) xx = x, or x2 = x.16 His esteem for this law becomes evident in his claim “that the axiom of the metaphysicians which is termed the principle of contradiction. . . , is a consequence of the fundamental law of thought whose expression is x2 = x” (Boole 1854, 49). Boole referred to the derivation x2 = x x − x2 = 0 x(1 − x) = 0, the last formula saying that a class and its complement have no elements in common. Boole was heavily criticized for this “curious error” (Halsted 1878, 86) of considering the Law of Contradiction a consequence of the Law of Duality, not the other way around (the derivation works, of course, also in the other direction). Boole’s revisions came along with a change in his attitude toward logic. His early logic can be seen as an application of a new mathematical method to logic, thereby showing the eﬃcacy of this method within the broad project of a universal mathematics and so serving foundational goals in mathematics. This foundational aspect diminished in later work, successively being replaced by the idea of a reform of logic. Already in the paper “The Calculus of Logic” (Boole 1848), Boole tried to show that his logical calculus is compatible with traditional philosophical logic. Reasoning is guided by the laws of thought. They are the central topic in Boole’s Investigation of the Laws of Thought, claiming that “there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted” (1854, 6), comparing thereby the laws of thought and the laws of algebra. Logic, in Boole’s understanding, was “a normative theory of the products of mental processes” (Grattan-Guinness 2000a, 51).

2.4. Symbolic Logic within the Old Paradigm: De Morgan Although created by mathematicians, the new logic was widely ignored by fellow mathematicians. Boole was respected by British mathematicians, but his ideas concerning an algebraic representation of the laws of thought received very little published reaction.17 He shared this fate with De Morgan, the second major ﬁgure of symbolic logic at that time.18 Like Boole, the British mathematician De Morgan was inﬂuenced by algebraist George Peacock’s work on symbolic algebra, which motivated him to consider the foundations of algebra in connection with logic. He distinguished,

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for example, algebra as an art associated with what he called “technical algebra” and algebra as science, that is, “logical algebra”: “Technical algebra is the art of using symbols under regulations which . . . are prescribed as the deﬁnition of symbols. Logical algebra is the science which investigates the method of giving meaning to the primary symbols, and of interpreting all subsequent results” (De Morgan 1842, 173–174, reprint p. 338). He used algebraic symbolism in logic, being mainly interested in a reform and extension of syllogistic logic, but ignoring the operational aspect of logic as calculus. He published his main results in a series of papers in the Proceedings of the Cambridge Philosophical Society between 1846 and 1862 (reprinted in De Morgan 1966) and in his book Formal Logic (1847). He has been called “the last great traditional logician” (Hailperin 2004, 346). Among his lasting achievements is the introduction of the technical term of a universe. He spoke, for example, of the “Universe of a proposition, or of a name” that may be limited in any matter expressed or understood” (De Morgan 1846/1966, 2) but continued to distinguish two kinds of the universe of a population, “being either the whole universe of thought, or a given portion of it” (De Morgan 1853/1966, 69). In the ﬁrst of the papers “On the Syllogism,” he introduced an algebraic symbolism for the syllogism, using small letters x, y, z as names contrary to those represented by capitals X, Y , Z (De Morgan 1846/1966, 3). The relations between such names as expressed in standard forms or simple propositions are symbolized as follows (ibid., 4): P )Q P.Q PQ P :Q

signiﬁes ... ... ...

Every P is Q. No P is Q. Some P s are Qs. Some P s are not Qs.

The algebraic symbols thus signify both the quantity of the concepts involved and the copula. For the names X and Y and their contraries x and y, the following equations are valid (ibid.): X)Y = X.y = y)x X:Y = Xy = y:x Y )X = Y.x = x)y

X.Y = X)y = Y )x XY = X:y = Y :x x.y = x)Y = y)X

Y :X = Y x = x:y

xy = x:Y = y:X

De Morgan used this symbolism to reconstruct the theory of syllogism. It served as representation, not as a calculus. Only after having written the 1846 paper, De Morgan found “that the whole theory of the syllogism might be deduced from the consideration of propositions in a form in which deﬁnite quantity of assertion is given both to the subject and the predicate of a proposition,” as he reported in an “Addition,” dated 27 February 1847 (De Morgan 1966, 17). He claimed to have brought

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this idea to paper before he learned of Sir William Hamilton’s quantiﬁcation of the predicate, thereby opening the priority quarrel. De Morgan focused his subsequent logical work on the theory of the copula, following “the hint given by algebra” by separating “the essential from the accidental characteristics of the copula” (1850/1966, 50). The “abstract copula” characterized only by essential features is understood as “a formal mode of joining two terms which carries no meaning, and obeys no law except such as is barely necessary to make the forms of inference follow” (ibid., 51). The abstract copula follows two “copular conditions,” (1) transitiveness X −−− Y −−− Z = X −−− Z (2) convertibility X −−− Y = Y −−− X Aﬃrmative (−−−) and negative (−−) copula are contrary to each other. Of X −−− Y and X −− Y one or the other must be (De Morgan 1850/1966, 51). De Morgan was the ﬁrst to take seriously that traditional syllogistics was incapable of dealing with relational properties like “Smith is smaller than Jones.” His ideas concerning a logic of (two-place) relations can be regarded as his most important contributions (see Merrill 1990, chs. 5–6; Grattan-Guinness 2000a, 32–34). Already in his second paper on the syllogism, he mentioned the role of the copula for expressing the relation between what is connected. He also considered the composition of relations (1850/1966, 55), that is, in modern terms, the relative product. He studied the subject of relations “as a branch of logic” in his fourth paper on the syllogism (De Morgan 1860/1966, 208). De Morgan used capital letters L, M , N for denoting relations, lowercase letters l, m, n for the respective contraries. Additionally, two periods indicate that a relation holds, only one period that the contrary relations holds. Thus, X..LY or X.lY say that X is “some one of the objects of thought which stand to Y in the relation L, or is one of the Ls of Y ” (ibid., 220). X and Y are called “subject” and “predicate,” indicating the mode in which they stand in the relation, thus in both LY.X and X.LY , Y indicates the predicate. If the predicate is itself the subject of a relation, a composition of relations results. “Thus if X..L(M Y ), if X be one of the Ls of one of the M s of Y , we may think of X as an ‘L of M ’ of Y , expressed by X..(LM )Y , or simply by X..LM Y ” (ibid., 221). De Morgan used an accent to signify universal quantity as part of the description of the relation. LM stands for an L of every M , LM X for the same relation to many (ibid.). The converse relation of L, L , is deﬁned as if X..LY , then Y..L−1 X” (ibid., 222). De Morgan then applied this symbolism to his theory of syllogism, introducing “theorem K” as basic for what he called “opponent syllogism,” which is exempliﬁed by the following mathematical syllogism (ibid., 224–225): Every deﬁcient of an external is a coinadequate: external and coinadequate have partient and complement for their contraries,

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and deﬁcient has exient for its converse: hence every exient of a complement is a patient; which is one of the opponent syllogisms of that ﬁrst given. Theorem K says (ibid., 224) that if a compound relation be contained in another relation, by the nature of the relations and not by casualty of the predicate, the same may be said when either component is converted, and the contrary of the other component and of the component change places. One of the examples is that “if, be Z what it may, every L of M of Z be an N of Z, say LM ))N , then L−1 n))m, and nM −1 ))l” (ibid.). The problematic nature of De Morgan’s symbolism becomes obvious in his notation for complex terms. The conjunctive “P and Q” is expressed by P Q, the disjunctive (taken in the inclusive sense) by P, Q. Using this notation he formulated the laws named after him (that can, however, be found already in the work of William of Ockham): “The contrary of P Q is p, q; that of P, Q, is pq” (1847, 118). The equivalent in modern notation is ¬(p ∨ q) = ¬p ∧ ¬q, and ¬(p ∧ q) = ¬p ∨ ¬q, or in the quantiﬁcational version ¬∃x ax = ∀x ¬ax and ¬∀x ax = ∃x ¬ax.

2.5. Reception of the New Logic In 1864, Samuel Neil, the early chronicler of British mid-nineteenth-century logic, expressed his thoughts about the reasons for this negligible reception: “De Morgan is esteemed crotchety, and perhaps formalizes too much. Boole demands high mathematic culture to follow and to proﬁt from” (1864, 161). One should add that the ones who had this culture were usually not interested in logic. The situation changed after Boole’s death in 1864. In the following comments only some ideas concerning the reasons for this new interest are hinted at. In particular the roles of William Stanley Jevons and Alexander Bain are considered. These examples show that a broader reception of symbolic logic commenced only when its relevance for the philosophical discussion of the time came to the fore. 2.5.1. William Stanley Jevons A broader international reception of Boole’s logic began when Jevons (1835– 1882) made it the starting point for his inﬂuential Principles of Science (Jevons 1874). He used his own version of the Boolean calculus introduced in his Pure Logic (Jevons 1864). Among his revisions were the introduction of a simple symbolic representation of negation and the deﬁnition of logical addition as inclusive “or,” thereby creating Boolean algebra (see Hailperin 1981). He also changed the philosophy of symbolism (1864, 5):

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The forms of my system may, in fact, be reached by divesting his [Boole’s] of a mathematical dress, which, to say the least, is not essential to it. The system being restored to its proper simplicity, it may be inferred, not that Logic is a part of Mathematics, as is almost implied in Professor Boole’s writings, but that the Mathematics are rather derivatives of Logic. All the interesting analogies or samenesses of logical and mathematical reasoning which may be pointed out, are surely reversed by making Logic dependent on Mathematics. Jevons’s interesting considerations on the relationship between mathematics and logic representing an early logicistic attitude will not be discussed here. Similar ideas can be found not only in Gottlob Frege’s work, but also in that of Rudolf Hermann Lotze (1817–1881) and Schröder. Most important in the present context is the fact that Jevons abandoned mathematical symbolism in logic, an attitude that was later taken up by John Venn (1834–1923) in his Symbolic Logic (Venn 1894). Jevons attempted to free logic from the semblance of being a special mathematical discipline. He used the symbolic notation only as a means of expressing general truths. Logic became a tool for studying science, a new language providing symbols and structures. The change in notation brought the new logic closer to the philosophical discourse of the time. The reconciliation was supported by the fact that Jevons formulated his Principles of Science as a rejoinder to John Stuart Mill’s (1806–1873) System of Logic of 1843, at that time the dominating work on logic and the philosophy of science in Great Britain. Although Mill had called his logic A System of Logic Ratiocinative and Inductive, the deductive parts played only a minor role, used only to show that all inferences, all proofs, and the discovery of truths consisted of inductions and their interpretations. Mill claimed to have shown “that all our knowledge, not intuitive, comes to us exclusively from that source” (Mill 1843, bk. II, ch. I, §1). Mill concluded that the question as to what induction is, is the most important question of the science of logic, “the question which includes all others.” As a result the logic of induction covers by far the largest part of this work, a subject that we would today regard as belonging to the philosophy of science. Jevons deﬁned induction as a simple inverse application of deduction. He began a direct argument with Mill in a series of papers titled “Mill’s Philosophy Tested” (1877/78). This argument proved that symbolic logic could be of importance not only for mathematics, but also for philosophy. Another eﬀect of the attention caused by Jevons was that British algebra of logic was able to cross the Channel. In 1877, Louis Liard (1846–1917), at that time professor at the Faculté de lettres at Bordeaux and a friend of Jevons, published two papers on the logical systems of Jevons and Boole (Liard 1877a, 1877b). In 1878 he added a booklet titled Les logiciens anglais contemporaines (Liard 1878), which had ﬁve editions until 1907 and was translated into German (Liard 1880). Although Hermann Ulrici (1806–1884)

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had published a ﬁrst German review of Boole’s Laws of Thought as early as 1855 (Ulrici 1855, see Peckhaus 1995), the knowledge of British symbolic logic was conveyed primarily by Alois Riehl (1844–1924), then professor at the University of Graz in Austria. He published a widely read paper, “Die englische Logik der Gegenwart” (“English contemporary logic,” Riehl 1877), which reported mainly Jevons’s logic and utilized it in a current German controversy on the possibility of scientiﬁc philosophy. 2.5.2. Alexander Bain Surprisingly good support for the reception of Boole’s algebra of logic came from the philosophical opposition, namely from the Scottish philosopher Bain (1818–1903) who was an adherent of Mill’s logical theory. Bain’s Logic, ﬁrst published in 1870, had two parts, the ﬁrst on deduction and the second on induction. He made explicit that “Mr Mill’s view of the relation of Deduction and Induction is fully adopted” (1870, I, iii). Obviously he shared the “general conviction that the utility of the purely Formal Logic is but small; and that the rules of Induction should be exempliﬁed even in the most limited course of logical discipline” (ibid., v). The minor role of deduction showed up in Bain’s deﬁnition “Deduction is the application or extension of Induction to new cases” (40). Despite his reservations about deduction, Bain’s Logic became important for the reception of symbolic logic because of a chapter of 30 pages titled “Recent Additions to the Syllogism.” In this chapter the contributions of Hamilton, De Morgan, and Boole were introduced. One can assume that many more people became acquainted with Boole’s algebra of logic through Bain’s report than through Boole’s own writings. One example is Hugh MacColl (1837–1909), the pioneer of the calculus of propositions (statements) and of modal logic.19 He created his ideas independently of Boole, eventually realizing the existence of the Boolean calculus by means of Bain’s report. Even in the early parts of his series of papers “The Calculus of Equivalent Statements,” he quoted from Bain’s presentation when discussing Boole’s logic (MacColl 1877/78). In 1875 Bain’s logic was translated into French, in 1878 into Polish. Tadeusz Batóg and Roman Murawski (1996) have shown that it was Bain’s presentation which motivated the ﬁrst Polish algebraist of logic, Stanisław Piątkiewicz (1848–?) to begin his research on symbolic logic.

3. Schröder’s Algebra of Logic 3.1. Philosophical Background The philosophical discussion on logic after Hegel’s death in Germany was still determined by a Kantian inﬂuence.20 In the preface to the second edition of his Kritik der reinen Vernunft of 1787, Immanuel Kant (1723–1804) had written that logic had followed the safe course of a science since earliest times.

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For Kant, this was evident because of the fact that logic had been prohibited from taking any step backward from the time of Aristotle. But he regarded it as curious that logic hadn’t taken a step forward either (B VIII). Thus, logic seemed to be closed and complete. Formal logic, in Kant’s terminology the analytical part of general logic, did not play a prominent role in Kant’s system of transcendental philosophy. In any case, it was a negative touchstone of truth, as he stressed (B 84). Georg Wilhelm Friedrich Hegel (1770–1831) went further in denying any relevance of formal logic for philosophy (Hegel 1812/13, I, Introduction, XV–XVII). Referring to Kant, he maintained that from the fact that logic hadn’t changed since Aristotle one should infer that it needs to be completely rebuilt (ibid., XV). Hegel created a variant of logic as the foundational science of his philosophical system, deﬁning it as “the science of the pure idea, i.e., the idea in the abstract element of reasoning” (1830, 27). Hegelian logic thus coincides with metaphysics (ibid., 34). This was the situation when after Hegel’s death philosophical discussion on formal logic started again in Germany. This discussion on logic reform stood under the label of “the logical question,” a term created by the neo-Aristotelian Adolf Trendelenburg (1802–1872). In 1842 he published a paper titled “Zur Geschichte von Hegel’s Logik und dialektischer Methode” with the subtitle “Die logische Frage in Hegel’s Systeme.” But what is the logical question according to Trendelenburg? He formulated this question explicitly toward the end of his article: “Is Hegel’s dialectical method of pure reasoning a scientiﬁc procedure?” (1842, 414). In answering this question in the negative, he provided the occasion of rethinking the status of formal logic within a theory of human knowledge without, however, proposing a return to the old (scholastic) formal logic. The term “the logical question” was subsequently used in a less speciﬁc way. Georg Leonard Rabus, the early chronicler of the discussion on logic reform, wrote, for example, that the logical question emerged from doubts concerning the justiﬁcation of formal logic (1880, 1). Although this discussion was clearly connected to formal logic, the socalled reform did not concern formal logic. The reason was provided by the neo-Kantian Wilhelm Windelband who wrote in a brilliant survey on nineteenth-century (philosophical) logic (1904, 164): It is in the nature of things that in this enterprize [i.e., the reform of logic] the lower degree of fruitfulness and developability power was on the side of formal logic. Reﬂection on the rules of the correct progress of thinking, the technique of correct thinking, had indeed been brought to perfection by former philosophy, presupposing a naive world view. What Aristotle had created in a stroke of genius, was decorated with the ﬁnest ﬁligree work in Antiquity and the Middle Ages: an art of proving and disproving which culminated in a theory of reasoning, and after this constructing the doctrines of judgements and concepts. Once one has accepted the foundations, the safely assembled building cannot be shaken:

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it can only be reﬁned here and there and perhaps adapted to new scientiﬁc requirements. Windelband was very critical of English mathematical logic. Its quantiﬁcation of the predicate allows the correct presentation of extensions in judgments, but it “drops hopelessly” the vivid sense of all judgments, which tend to claim or deny a material relationship between subject or predicate. It is “a logic of the conference table,” which cannot be used in the vivid life of science, a “logical sport” which has its merits only in exercising the ﬁnal acumen (ibid., 166–167). The philosophical reform eﬀorts concerned primarily two areas: 1. the problem of a foundation of logic itself. It was dealt with by using psychological and physiological means, thereby leading to new discussion on the question of priority between logic and psychology, and to various forms of psychologism and anti-psychologism (see Rath 1994, Kusch 1995). 2. The problem of the applicability of logic which led to an increased interest in the methodological part of traditional logic. The reform of applied logic attempted to bring philosophy in touch with the stormy development of mathematics and sciences in that time. Both reform procedures had a destructive eﬀect on the shape of logic and philosophy. The struggle with psychologism led to the departure of psychology (especially in its new, experimental form) from the body of philosophy at the beginning of the twentieth century. Psychology became a new, autonomous scientiﬁc discipline. The debate on methodology resulted in the creation of the philosophy of science being ﬁnally separated from the body of logic. The philosopher’s ignorance of the development of formal logic caused a third departure: Part of formal logic was taken from the domain of the competence of philosophy and incorporated into mathematics where it was instrumentalized for foundational tasks. This was the philosophical background of the emergence of symbolic logic in Germany and especially the logical work of the German mathematician Schröder.

3.2. The Mathematical Context in Germany 3.2.1. Logic and Formal Algebra The examination of the British situation in mathematics at the time when the new logic emerged has shown that the creators of the new logic were basically interested in a reform of mathematics by establishing an abstract view of mathematics which focused not on mathematical objects like quantities but on symbolic operations with arbitrary objects. The reform of logic was only secondary. These results can be transferred to the situation in Germany without any problem.

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Schröder was the most important representative of the German algebra of logic.21 He was regarded as having completed the Boolean period in logic (see Bocheński 1956, 314). In his ﬁrst pamphlet on logic, Der Operationskreis des Logikkalkuls (1877), he presented a critical revision of Boole’s logic of classes, stressing the idea of the duality between logical addition and logical multiplication introduced by Jevons in 1864. In 1890, Schröder started the large project of his monumental Vorlesungen über die Algebra der Logik, which remained unﬁnished, although it increased to three volumes with four parts, of which one appeared only posthumously (1890, 1891, 1895, 1905). Contemporaries regarded the ﬁrst volume alone as having completed the algebra of logic (see Wernicke 1891, 196). Nevertheless, Schröder’s logical theory kept, like the one of Boole, close contact to the traditional shape of logic. The introduction of the Vorlesungen is full of references to that time’s philosophical discussion on logic. Schröder even referred to the psychologistic discussion on the foundation of logic, and never really freed his logical theory from the traditional division of logic into the theories of concept, judgment, and inference. Schröder’s opinion concerning the question as to what end logic is to be studied (see Peckhaus 1991, 1994b, 2004a) can be drawn from an autobiographical note (written in the third person), published in the year before his death. It contains his own survey of his scientiﬁc aims and results. Schröder divided his scientiﬁc production into three ﬁelds: 1. A number of papers dealing with some of the current problems of his science. 2. Studies concerned with creating an “absolute algebra,” that is, a general theory of connections. Schröder stressed that these studies represent his “very own object of research” of which only little was published at that time. 3. Work on the reform and development of logic. Schröder wrote (1901) that his aim was to design logic as a calculating discipline, especially making possible an exact handling of relative concepts, and, from then on, by emancipation from the routine claims of spoken language, and also to remove any breeding ground from “cliché” in the ﬁeld of philosophy as well. This should prepare the ground for a scientiﬁc universal language that, widely diﬀering from linguistic eﬀorts like Volapük [a universal language like Esperanto, very popular in Germany at that time], looks more like a sign language than like a sound language. Schröder’s own division of his ﬁelds of research shows that he didn’t consider himself a logician: His “very own object of research” was “absolute algebra,” which was similar to modern abstract or universal algebra in respect to its basic

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problems and fundamental assumptions. What was the connection between logic and algebra in Schröder’s research? From the passages quoted one could assume that they belong to two separate ﬁelds of research, but this is not the case. They were intertwined in the framework of his heuristic idea of a general science. In his autobiographical note he stressed: The disposition for schematizing, and the aspiration to condense practice to theory advised Schröder to prepare physics by perfecting mathematics. This required deepening of mechanics and geometry, but above all of arithmetic, and subsequently he became in time aware of the necessity to reform the source of all these disciplines, logic. Schröder’s universal claim becomes obvious. His scientiﬁc eﬀorts served for providing the requirements to found physics as the science of material nature by “deepening the foundations,” to quote a famous metaphor later used by David Hilbert (1918, 407) to illustrate the objectives of his axiomatic program. Schröder regarded the formal part of logic that can be formed as a “calculating logic,” using a symbolic notation, as a model of formal algebra that is called “absolute” in its last state of development. But what is “formal algebra?” The theory of formal algebra “in the narrowest sense of the word” includes “those investigations on the laws of algebraic operations . . . that refer to nothing but general numbers in an unlimited number ﬁeld without making any presuppositions concerning its nature” (1873, 233). Formal algebra therefore prepares “studies on the most varied number systems and calculating operations that might be invented for particular purposes” (ibid.). It has to be stressed that Schröder wrote his early considerations on formal algebra and logic without any knowledge of the results of his British predecessors. His sources were the textbooks of Martin Ohm, Hermann Günther Graßmann, Hermann Hankel, and Robert Graßmann. These sources show that Schröder was a representative of the tradition of German combinatorial algebra and algebraic analysis (see Peckhaus 1997, ch. 6). 3.2.2. Combinatorial Analysis Schröder developed the programmatic foundations of absolute algebra in his textbook Lehrbuch der Arithmetik und Algebra (1873) and the school program pamphlet Über die formalen Elemente der absoluten Algebra (1874). Among the sources mentioned in the textbook, Martin Ohm’s (1792–1872) Versuch eines vollkommen consequenten Systems der Mathematik (1822) is listed. It stood in the German tradition of the algebraic and combinatorial analysis which started with the work of Carl Friedrich Hindenburg (1741–1808) and his school (see Jahnke 1990, 161–322). Ohm (see Bekemeier 1987) aimed at completing Euclid’s geometrical program for all of mathematics (Ohm 1853, V). He distinguished between number

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(or “undesignated number”) and quantity (or “designated number”) regarding the ﬁrst one as the higher concept. The features of the calculi of arithmetic, algebra, analysis, and so on are not seen as features of quantities but of operations, that is, mental activities (1853, VI–VII). This operational view can also be found in the work of Graßmann, who also stood in the Hindenburg tradition. 3.2.3. General Theory of Forms Graßmann’s Lineale Ausdehnungslehre (1844)22 was of decisive inﬂuence on Schröder, especially Graßmann’s “general theory of forms” (“allgemeine Formenlehre”) opening this pioneering study in vector algebra and vector analysis. The general theory of forms was popularized by Hankel’s Theorie der complexen Zahlensysteme (1867). Graßmann deﬁned the general theory of forms as “the series of truths that is related to all branches of mathematics in the same way, and that therefore only presupposes the general concepts of equality and diﬀerence, connection and division” (1844, 1). Equality is taken as substitutivity in every context. Graßmann chooses as general connecting sign. The result of the connection of two elements a and b is expressed by the term (a b). Using the common rules for brackets we get for three elements ((a b) c) = a b c (§2). Graßmann restricted his considerations to “simple connections,” that is, associative and commutative connections (§4). These connecting operations are synthetic. The reverse operations are called resolving or analytic connections. a b stands for the form which results in a if it is synthetically connected with b: a b b = a (§5). Graßmann introduced furthermore forms in which more than one synthetic operation occur. If the second connection is symbolized with and if there holds distributivity between the synthetic operations, then the equation (a b) c) (b c) is valid. Graßmann called the c = (a second connection a connection on a higher level (§9), a terminology that might have inﬂuenced Schröder’s later “Operationsstufen,” that is, “levels of operations.” Whereas Graßmann applied the general theory of forms in the domain of extensive quantities, especially directed lines, that is, vectors, Hankel later used it to erect on its base his system of hypercomplex numbers (Hankel 1867). If λ(a, b) is a general connection of objects a, b leading to a new object c, that is, λ(a, b) = c, there is a connection Θ which, applied to c and b leads again to a, that is, Θ(c, b) = a or Θ{λ(a, b), b} = a. Hankel called the operation θ “thetic” and its reverse λ “lytic.” The commutativity of these operations is not presupposed (ibid., 18). 3.2.4. “Wissenschaftslehre” and Logic Graßmann had already announced that his Lineale Ausdehnungslehre should be part of a comprehensive reorganization of the system of sciences. His brother,

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Robert Graßmann (1815–1901), attempted to realize this program in a couple of writings published under the series title Wissenschaftslehre oder Philosophie. In its parts on logic and mathematics he anticipated modern lattice theory. He furthermore formulated a logical calculus being in parts similar to that of Boole. His logical theory was obviously independent of the contemporary German philosophical discussion on logic, and he was also not aware of his British precursors.23 Graßmann wrote about the aims of his logic or theory of reasoning (“Denklehre”) that it should teach us strictly scientiﬁc reasoning which is equally valid for all men of any people, any language, equally proving and rigorous. It has therefore to relieve itself from the barriers of a certain language and to treat the forms of reasoning, becoming, thus, a theory of forms or mathematics. Graßmann tried to realize this program in his Formenlehre oder Mathematik, published in six brochures consisting of an introduction (1872a), a general part on “Grösenlehre” (1872b) understood as “science of tying quantities,” and the special parts “Begriﬀslehre oder Logik” (theory of concepts or logic), “Bindelehre oder Combinationslehre” (theory of binding or combinatorics), “Zahlenlehre oder Arithmetik” (theory of numbers or arithmetic), and “Ausenlehre oder Ausdehnungslehre” (theory of the exterior or Ausdehnungslehre). In the general theory of quantities Graßmann introduced the letters a, b, c, . . . as syntactical signs for arbitrary quantities. The letter e represents special quantities: elements, or in Graßmann’s strange terminology “Stifte” (pins), that is, quantities which cannot be derived from other quantities by tying. Besides brackets, which indicate the order of the tying operation, he introduces the equality sign =, the inequality sign Z , and a general sign for a tie ◦. Among special ties he investigates joining or addition (“Fügung oder Addition”) (“+”) and weaving or multiplication (“Webung oder Multiplikation”) (“·”). These ties can occur either as interior ties, if e ◦ e = e, or as exterior exterior ties, if e ◦ e Z e. The special parts of the theory of quantities are distinguished with the help of the combinatorically possible results of tying a pin to itself. The ﬁrst part, “the most simple and, at the same time, the most interior,” as Graßmann called it, is the theory of concepts or logic in which interior joining e + e = e and inner weaving ee = e hold. In the theory of binding or combinatorics interior joining e + e = e and exterior weaving ee Z e hold; in the theory of numbers or arithmetic exterior joining e + e Z e and interior weaving ee = e hold, or 1 × 1 = 1 and 1 × e = e. Finally, in the theory of the exterior or Ausdehnungslehre, the “most complicated and most exterior” part of the theory of forms, exterior joining e + e Z e and exterior weaving ee Z e hold (1872a, 12–13). Graßmann thus formulated Boole’s Law of Duality using his interior weaving ee = e, but he went beyond Boole in allowing interior joining e + e = e, so coming close to Jevons’s system of 1864.

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In the theory of concepts or logic, Graßmann started with interpreting the syntactical elements, which had already been introduced in a general way. Now, everything that can be a deﬁnite object of reasoning is called “quantity.” In this new interpretation, pins are initially set quantities not being derived from other quantities by tying. Equality is interpreted as substitutivity without value change, inequality as impossibility of such a substitution. Joining is read as “and,” standing for adjunction or the logical “or.” Weaving is read as “times,” that is, conjunction or the logical “and.” Graßmann introduced the signs < and > to express sub- and superordination of concepts. The sign expresses that a concept equals or that it is subordinated another concept. This is exactly the sense of Schröder’s later basic connecting relation of subsumption or inclusion. In the theory of concepts, Graßmann expressed this relation in a shorter way with the help of the angle sign ∠. The sign T stands for the All or the totality, the sum of all pins. The following laws hold: a + T = T and aT = a. 0 is interpreted as “the lowest concept, which is subordinate to all concepts.” Its laws are a + 0 = a and a · 0 = 0. Finally Graßmann introduced the “not” (“Nicht”) or negation as complement with the laws a + a = T and a · a = 0.

3.3. Schröder’s Algebra of Logic 3.3.1. Schröder’s Way to Logic In his work on the formal elements of absolute algebra (1874) Schröder investigated operations in a manifold, called domain of numbers (“Zahlengebiet”). “Number” is, however, used as a general concept. Examples for numbers are “proper names, concepts, judgments, algorithms, numbers [of arithmetic], symbols for quantities and operations, points, systems of points, or any geometrical object, quantities of substances, etc.” (Schröder 1874, 3). Logic is, thus, a possible interpretation of the structure dealt with in absolute algebra. Schröder assumed that there are operations with the help of which two objects from a given manifold can be connected to yield a third that also belongs to that manifold (ibid., 4). He chooses from the set of possible operations the noncommutative “symbolic multiplication” c = a . b = ab with two inverse operations measuring (“Messung”)

b . (a : b) = a, a . b = a. and division (“Teilung”) b Schröder called a direct operation together with its inverses “level of operations” (“Operationsstufe”). And again Schröder realized that “the logical addition of concepts (or individuals)” follows the laws of multiplication of real numbers. But there is still another association with logic. In his Lehrbuch, Schröder √ speculated about the relation between an “ambiguous expression” like a

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and its possible values. He determined ﬁve logical relations, introducing his subsumption relations. Be A an expression that can have diﬀerent values a, a , a, . . . . Then the following relations hold (Schröder 1873, 27–29): ⎧ a ⎪ ⎪ ⎪ ⎨ a Superordination A a . ⎪ ⎪ ⎪ .. ⎩ . √ Examples: metal silver; 9 −3. ⎫ a ⎪ ⎪ ⎪ a ⎬ A. Subordination a ⎪ ⎪ ⎪ .. ⎭ . √ Examples: gold metal; 3 9.

Coordination a

a a ··· .

Examples: gold silver [in respect√to the general concept “metal”] or 3 [in respect to the general concept 9 ].

−3

Equality A = B means that the concepts A and B are identical in intension and extension. Correlation A(=)B means that the concepts A and B agree in at least one value. Schröder recognized that if he would now introduce negation, he would have created a complete terminology that allows one to express all relations between concepts (in respect to their extension) with short formulas which can harmonically be embedded into the schema of the apparatus of the mathematical sign language (ibid., 29). Schröder wrote his logical considerations of the introduction of the Lehrbuch without having seen any work of logic in which symbolic methods had been applied. It was while completing a later sheet of his book that he came across Robert Graßmann’s Formenlehre oder Mathematik (1872a). He felt urged to insert a comprehensive footnote running over three pages for hinting at this book (Schröder 1873, note, pp. 145–147). There he reported that Graßmann used the sign + for the “collective comprehension,” “really regarding it as an addition—one could say a ‘logical’ addition—that has besides the features of common (numerical) addition the basic feature a + a = a.” He wrote that he was most interested in the role the author had assigned to multiplication regarded as the product of two concepts which unite the marks being common to both concepts.

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In the Programmschrift of 1874, Schröder also gave credit to Robert Graßmann, but mentioned that he had recently found out that the laws of the logical operations had already been developed before Graßmann “in a classical work” by Boole (Schröder 1874, 7). 3.3.2. Logic as a Model of Absolute Algebra In 1877 Schröder published his Operationskreis des Logikkalkuls, in which he developed the logic of Boole’s Laws of Thought stressing the duality of the logical operations of addition and multiplication.24 An “Operationskreis” (circle of operations) is constituted by more than one direct operation together with their inverses. The “logical calculus” is the set of formulas which can be produced in this circle of operations. Schröder called it a characteristic mark of “mathematical logic or the logical calculus” that these derivations and inferences can be done in form of calculations, namely, in the ﬁrst part of logic as calculation with concepts leading to statements about the objects themselves, that is, categorical judgments, or, in Boole’s terminology, “primary propositions.” In its second part the logical calculus deals with statements about judgments as in conditional sentences, hypothetical or disjunctive judgments, or Boole’s secondary propositions. In this booklet Schröder simpliﬁed Boole’s calculus, stressing, as mentioned, the duality between logical addition and logical multiplication and, thus, the algebraic identity of the structures of these operations. Schöder developed his logic in a systematic way in the Vorlesungen über die Algebra der Logik (1890–1905) designing it as a means for solving logical problems (see Peckhaus 1998, 21–28). Again he separated logic from its structure. The structures are developed and interpreted in several ﬁelds, beginning from the most general ﬁeld of “domains” (“Gebiete”) of manifolds of arbitrary distinct elements, then classes (with and without negation), and ﬁnally proprositions (vol. 2, 1891). The basic operation in the calculi of domains and classes is subsumption, that is, identity or inclusion. Schröder presupposes a, and transitivity “If a b and at the same two principles, reﬂexivity a c, then a c.” Then he deﬁnes “identical zero” (“nothing”) and time b “identical one” (“all”), “identical multiplication” and “identical addition,” and ﬁnally negation. In the sections dealing with statements without negation, he proves one direction of the distributivity law for logical addition and logical multiplication, but shows that the other side cannot be proved; he rather shows its independence by formulating a model in which it does not hold, the “logical calculus with groups, e.g. functional equations, algorithms or calculi.” He thereby found the ﬁrst example of a nondistributive lattice.25 Schröder devoted the second volume of the Vorlesungen to the calculus of propositions. The step from the calculus of classes to the calculus of propositions is taken with the help of an alteration of the basic interpretation of the formulas used. Whereas the calculus of classes was bound to a spatial interpretation especially in terms of the part–whole relation, Schröder used in the calculus of

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propositions a temporal interpretation taking up an idea of Boole from his Laws of Thought (1854, 164–165). This may be illustrated regarding subsumption as the basic connecting relation. In the calculus of classes, a b means that the class a is part of or equal to the class b. In the calculus of propositions, this formula may be interpreted in the following way (Schröder 1891, §28, p. 13):

In the time during which a is true is completely contained in the time during which b is true, i.e., whenever . . . a is valid b is valid as well. In short, we will often say: “If a is valid, then b is valid,” “a entails b” . . . , “from a follows b.” Schröder then introduces symbols, the “sign of products” ,

two new logical and the “sign of sums” . He uses x to express that propositions referring

to a domain x are valid for any domain x in the basic manifold 1, and x to say that the proposition is not necessarily valid for all, but for a certain domain x, or for several certain domains x of our manifold 1, that is, for at least one x (Schröder 1891, 26–27).

§29, For Schröder the use of and in logic is perfectly analogous to arithmetic. The existential quantiﬁer and the universal quantiﬁer are therefore interpreted as possibly indeﬁnite logical addition or disjunction and logical multiplication or conjunction respectively.

This is expressed by the following deﬁnition, which also shows the duality of and (Schröder 1891, §30, 35). λ=n

λ=n aλ = a1 + a2 + a3 + · · · + an−1 + an aλ = a1 a2 a3 · · · an−1 an .

λ=1

λ=1

With this Schröder had all requirements at hand for modern quantiﬁcation theory, which he took, however, not from Frege but from the conceptions as developed by Charles S. Peirce (1839–1914) and his school, especially by Oscar Howard Mitchell (1851–1889).26 3.3.3. Logic of Relatives Schröder devoted the third volume of the Vorlesungen to the “Algebra and Logic of Relatives,” of which only a ﬁrst part dealing with the algebra of relatives could be published (Schröder 1895). The algebra and logic of relatives should serve as an organon for absolute algebra in the sense of pasigraphy, or general script, that could be used to describe most diﬀerent objects as models of algebraic structures. Schröder never claimed any priority for this part of his logic, but always conceded that it was an elaboration of Charles S. Peirce’s work on relatives (see Schröder 1905, XXIV). He illustrated the power of this new tool by applying it to several mathematical topics, such as open problems of G. Cantor’s set theory (e.g., Schröder 1898), thereby proving (not entirely correctly) Cantor’s proposition about the equivalence of sets (“Schröder-Bernstein Theorem”). In translating Richard

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Dedekind’s theory of chains into the language of the algebra of relatives, he even proclaimed the “ﬁnal goal: to come to a strictly logical deﬁnition of the relative concept ‘number of—’ [‘Anzahl von—’] from which all propositions referring to this concept can be deduced purely deductively” (Schröder 1895, 349–350). So Schröder’s system comes close, at least in its objectives, to Frege’s logicism, although it is commonly regarded as an antipode. 3.3.4. The Ideas of Peirce Although Schröder found his way to an algebraic approach to logic independently of Boole, he devoted his early work to a discussion and extension of the Boolean calculus. Main reference point of his mature Vorlesungen, however, was the logical work of the American “polymath” (Grattan-Guinness 2004, 545) Charles S. Peirce. Peirce contributed a great wealth of ideas to modern logic. He approached logic to its full range, interested not only in symbolic logic but also in a reform of traditional syllogistics and applications in the philosophy of science.27 In one of his ﬁrst papers on logic, Peirce improved Boole’s algebra of logic by introducing the inclusive disjunction as Jevons did before him (see Peirce 1868). He introduced “inclusion” as basic logical operator, in an algebraic spirit both for inclusion between classes and implication between propositions (Peirce 1870, WCSP 2, 360). It was later taken up by Ernst Schröder as “subsumption” . Among the ﬁve “icons” for nonrelative logic, “Peirce’s law” {(x y) x} x (see Peirce 1885, WCSP 5, 173) is outstanding. It produces an axiom system for classical propositional logic when being added to an axiom system for intuitionistic logic (see Beth 1962, 18, 128). In the paper “A Boolian Algebra with One Constant” (WCSP 4, 218–221), written around 1880, but not published before 1933, Peirce suggested replacing all logical connectors by only one interpreted as “neither P nor Q,” thereby anticipating the NOR operator, which was independently rediscovered by H. M. Sheﬀer in 1913 (Sheﬀer 1913). In his paper of 1870, Peirce took the ﬁrst step for developing a logic of relatives, thereby elaborating the ideas of De Morgan. He distinguished absolute terms, such as horse, tree, or man, from terms “whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation” (WCSP 2, 365). He discussed simple relative terms, that is, two-place relatives, and conjugate terms, that is, three- or four-place relatives like “giver of — to —” or “buyer of — for — from —” (ibid.). In his 1880 paper “On the Algebra of Logic,” he took up the topic, now speaking of singular reference for nonrelative terms and of dual and plural relatives for two- and more-place relatives. The most elaborated form of his algebra of relatives can be found in his 1885 paper, where he combined it with the theory of quantiﬁcation, the foundation of which had been formulated entirely independently of Frege by Oscar Howard Mitchell in Peirce’s Johns Hopkins logic circle (Mitchell 1883). Whereas Mitchell had developed a system limited

& & &

&

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to a theory of quantiﬁed propositional functions with two prenex quantiﬁers, Peirce developed quantiﬁers as operators on propositional functions over speciﬁc domains.28 In his 1885 paper, Peirce gave credit to Mitchell in the following way (WCSP 5, 178): All attempts to introduce this distinction [of some and all] into the Boolian algebra were more or less complete failures until Mr. Mitchell showed how it was to be eﬀected. His method really consists in making the whole expression of the propositions consist of two parts, a pure Boolian expression referring to an individual and a Quantifying part saying what individual this is. Peirce now used an index notation to express relatives. In the ﬁrst-order part of his logic (ﬁrst-intentional logic), xi yj signiﬁes that x is true of the individual i while y is true of the individual j. The quantiﬁers Σ and Π are used in analogy to their arithmetical meaning. Σi xi means that x is true of some one of the individuals denoted by i, Πi xi means that x is true of all these individuals. Applied to a ordinary language example: Let lij denote that i is a lover of j, and bij that i is a benfactor of j. Then Πi Σj lij bij means that everything is at once a lover and a benefactor of something (WCSP 5, 180). Peirce added considerations on second-intentional logic, that is, secondorder logic (ibid., 185–190) and many valued logic (ibid., 166). In later work he used furthermore “existential graphs” for a graphical representation of quantiﬁcational logic (see CP 4.293–584) which inspired several modern systems for graphical representations of logic (see, e.g., Sowa 1993, 1997). Peirce’s logical considerations were integral part of his triadic category system with ﬁrstness (possibility), secondness (existence), and thirdness (law), his semiotics, and his triadic theory of reasoning with deduction, induction, and abduction (see Hilpinen 2004, 622–628, 644–653). Most of Peirce’s path-breaking thoughts remained unpublished during his lifetime. What he was able to publish, however, excited his contemporary logicians. The best example is Schröder, whose Vorlesungen were deeply inﬂuenced by Peirce, even more, long passages read as critical comments on Peirce’s papers, especially on the seminal papers “On the Algebra of Logic” (Peirce 1880, 1885). In an intermediate word separating the halfs of volume two of the Vorlesungen Schröder wrote that after the completion of the ﬁrst half of volume two in June 1891 he had hoped to publish the second half with the logic of relatives in the autumn of the same year, but (Schröder 1905, XXIV): It is true, seldom in my life an estimation of mine failed to the same extent as then, when I judged the extension and the seriousness of the gaps in my manuscript. This was due to the fact that the only writing that seemed to be useful, Mr. Peirce’s paper on relatives [Peirce 1885], that became indeed the main basis of my volume three, has only a size of 18 pages in print (that could be printed

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on half the number of my pages), and that I thought, that I could get away with a largely reproducing report. I became aware of the enormous signiﬁcance of this paper when I worked at it in detail.

4. Conclusions Like the British tradition, but independent of it, the German algebra of logic was connected to new trends in algebra. It diﬀered from its British counterpart in its combinatorial approach. In both traditions, algebra of logic was invented within the enterprise to reform basic notions of mathematics which led to the emergence of structural abstract mathematics. The algebraists wanted to design algebra as “pan-mathematics,” that is, as a general discipline embracing all mathematical disciplines as special cases. The independent attempts in Great Britain and Germany were combined when Schröder learned about the existence of Boole’s logic in late 1873, early 1874. Finally he enriched the Boolean class logic by adopting Peirce’s theory of quantiﬁcation and adding a logic of relatives according to the model of Peirce and De Morgan. The main interest of the new logicians was to use logic for mathematical and scientiﬁc purposes, and it was only in a second step, but nevertheless an indispensable consequence of the attempted applications, that the reform of logic came into the view. What has been said of the representatives of the algebra of logic also holds for the proponents of competing logical systems such as Gottlob Frege or Giuseppe Peano. They wanted to use logic in their quest for mathematical rigor, something questioned by the stormy development in mathematics. For quite a while, the algebra of logic remained the ﬁrst choice for logical research. Authors like Alfred North Whitehead (1841–1947), and even David Hilbert and his collaborators in the early foundational program (see Peckhaus 1994c) built on this direction of logic, whereas Frege’s mathematical logic was widely ignored. The situation changed only after the publication of Whitehead’s and B. Russell’s Principia Mathematica (1910–1913). But even then important work was done in the algebraic tradition as the contributions of Clarence Irving Lewis (1883–1964), Leopold Löwenheim (1878–1957), Thoralf Skolem (1887–1963), and Alfred Tarski (1901–1983) prove.

Notes 1. Independently of each other, Gregorius Itelson, André Lalande, and Louis Couturat suggested at the 2nd Congress of Philosophy at Geneva in 1904 to use the name “logistic” for, as Itelson said, the modern kind of traditional formal logic. The name should replace designations like “symbolic,” “algorithmic,” “mathematical logic,” and “algebra of logic,” which were used synonymously up to then (see Couturat 1904, 1042). 2. For a book-length biography, see MacHale (1985). See also contemporary obituaries and biographies like Harley (1866), Neil (1865), both reprinted. For a

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comprehensive presentation of Boole’s logic in the context of British mathematics, see Grattan-Guinness (2000a). 3. Whately (1826). Risse (1973) lists 9 editions up to 1848 and 28 further printings to 1908. Van Evra (1984, 2) mentions 64 printings in the United States to 1913. 4. This opinion can be found in a letter of De Morgan’s to Spalding of 26 June 1857 (quoted in Heath 1966, xii) which was, however, not sent. Boole lists Hamilton among the “two greatest authorities in logic, modern and ancient” (1847, 81). The other authority is Aristotle. This reverence to Hamilton might not be without irony because of Hamilton’s disregard of mathematics. 5. See Hamilton 1859–1866, vol. 4 (1866), 287. 6. See his list of symbols in “Logical Notation” in Hamilton 1859–1866, vol. 4 (1866), 469–486. 7. For the priority struggle, see Heath 1966. 8. For diagrammatic methods in logic, see Gardner (1958), Bernhard (2000). 9. See the section “On Expression and Interpretation” in Boole (1847), 20–25, in which Boole gives his reading of the traditional theory of judgment. The section is followed by an application of his notation to the theory of conversion (ibid., 26–30) and of syllogism (ibid., 31–47). 10. This is the subtitle of Boole’s Mathematical Analysis of Logic (1847). 11. For the inﬂuence of Aristotelian logic on Boole’s philosophy of logic, see Nambiar (2000). 12. On the mathematical background of Boole’s Mathematical Analysis of Logic, see Laita (1977), Panteki (2000). 13. On Gregory with focus on his contributions to the foundations of the calculus see Allaire and Bradley (2002). 14. On Boole’s “psychologism,” see Bornet (1997) and Vasallo (2000). 15. For the development of quantiﬁcation theory in the algebra of logic, see Brady (2000). 16. The reason was that already the factorization of x3 = x leads to uninterpretable expressions. On Boole’s Laws of Thought see Van Evra (1977); on the diﬀerences between Boole’s earlier and later logical theory see Grattan-Guinness (2000b). 17. On initial reactions see Grattan-Guinness (2000a), 54–59. 18. For a discussion of De Morgan’s logic see Grattan-Guinness (2000a), 25–37; Merrill (1990); Sánchez Valencia (2004), 408–410, 487–515. 19. On MacColl and his logic see Astroh and Read (1998). 20. See for the following chs. 3 and 4 of Peckhaus (1997), and Vilkko (2002). 21. On Schröder’s biography, see his autobiographical note, Schröder (1901), which became the base of Eugen Lüroth’s widely spread obituary, Lüroth (1903). See also Peckhaus (1997), 234–238; and Peckhaus (2004a). 22. On the various aspects of H. G. Graßmann’s work, see Schubring (1996); Lewis (2004). 23. On Robert Graßmann’s logic and his anticipations of lattice theory see Mehrtens (1979); Peckhaus (1997), 248–250. On the relation zwischen Schröder and the Graßmann brothers see Peckhaus (1996). 24. On Schröder’s algebra of logic see Peckhaus (2004a); Sánchez Valencia (2004), 477–487; Brady (2000). 25. See Schröder (1890), 280. On Peirce’s claim to have proved the second form as well (Peirce 1880, 33) see Houser (1991). On Schröder’s proof see Peckhaus (1994a), 359–374; Mehrtens (1979), 51–56.

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26. See Mitchell (1883), Peirce (1885). On the development of modern quantiﬁcation theory in the algebra of logic see Brady (2000); Peckhaus (2004b). For Mitchell’s biography, see Dipert (1994). 27. For recent work on Peirce’s Logic, see Houser, Van Evra, and Roberts (1997); Brady (2000); Grattan-Guinness (2000a), 140–156; Hilpinen (2004). 28. Brady (2000), 6; see Peirce (1883). For Peirce’s interpretation of Mitchell see also Haaparanta (1993), 112–116.

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Schubring, Gert, ed. 1996. Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Papers from a Sesquicentennial Conference. Dordrecht: Kluwer. (Boston Studies in the Philosophy of Science, vol. 187). Sheﬀer, Henry Maurice. 1913. A set of ﬁve independent postulates for Boolean algebras, with application to logical constants. Transactions of the American Mathematical Society 14: 481–488. Sowa, John F. 1993. Relating diagrams to logic. In Proceedings on Conceptual Graphs for Knowledge Representation, eds. Guy W. Mineau, Bernard Moulin, and John F. Sowa, 1–35. Springer-Verlag: New York. Sowa, John F. 1997. Matching logical structure to linguistic structure. In Houser, Van Evra, and Roberts (eds.) (1997), 418–444. Trendelenburg, Friedrich Adolf. 1842. Zur Geschichte von Hegel’s Logik und dialektischer Methode. Die logische Frage in Hegel’s Systeme. Eine Auﬀoderung zu ihrer wissenschaftlichen Erledigung. Neue Jenaische Allgemeine Literatur-Zeitung 1, no. 97, 23 April 1842, 405–408; no. 98, 25 April 1842, 409–412; no. 99, 26 April 1842, 413–414. Separately published in Trendelenburg Die logische Frage in Hegel’s System. Zwei Streitschriften. Brockhaus: Leipzig 1843. Ueberweg, Friedrich. 1857. System der Logik und Geschichte der logischen Lehren. Adolph Marcus: Bonn. Ueberweg, Friedrich. 1871. System of Logic and History of Logical Doctrines, translated from the German, with notes and appendices by Thomas M. Lindsay. Longmans, Green: London. Reprinted Thoemmes Press: Bristol 1993. Ulrici, Hermann. 1855. Review of Boole (1854). Zeitschrift für Philosophie und philosophische Kritik 27: 273–291. Van Evra, James. 1984. Richard Whately and the rise of modern logic. History and Philosophy of Logic 5: 1–18. Venn, John. 1894. Symbolic Logic, 2nd ed., “revised and rewritten.” Macmillan: London. Reprinted Chelsea Publishing: Bronx, N.Y. 1971. Vasallo, Nicla. 2000. Psychologism in logic: some similarities between Boole and Frege. In Gasser (ed.) (2000), 311–325. Vilkko, Risto. 2002. A Hundred Years of Logical Investigations. Reform Eﬀorts of Logic in Germany 1781–1879. Mentis: Paderborn. Wernicke, Alexander. 1891. Review of Schröder (1891). Deutsche Litteraturzeitung 12: cols. 196–197. Whately, Richard. 1826. Elements of Logic. Comprising the Substance of the Article in the Encyclopaedia Metropolitana: with Additions, &c. J. Mawman: London. Whitehead, Alfred North, Russell, Bertrand. 1910–1913. Principia Mathematica, 3 vols. Cambridge University Press: Cambridge. Windelband, Wilhelm. 1904. Logik. In Die Philosophie im Beginn des zwanzigsten Jahrhunderts. Festschrift für Kuno Fischer, vol. 1, ed. Windelband, 163–186. Carl Winter: Heidelberg.

5

Gottlob Frege and the Interplay between Logic and Mathematics Christian Thiel

Gottlob Frege (1848–1925) has been called the greatest logician since Aristotle, but it is a brute fact that he failed to gain inﬂuence on the mathematical community of his time (although he was not ignored, as some have claimed), and that the depth and pioneering character of his work was—paradoxically— acknowledged only after the collapse of his logicist program due to the Zermelo– Russell antinomy in 1902. Because of this lack of inﬂuence in his time, a leading historian of logic and mathematics has gone so far as to deny Frege a place in the development of mathematical logic. Other historiographers of science, however, are convinced that the history of visible eﬀects of great ideas on science and scientiﬁc communities should be complemented by the recognition even of solitary insights ineﬀective at their time, because the intellectual status of such insights or discoveries will yield most valuable (and otherwise unobtainable) information about the structure and quality of the community that made them possible by providing, as it were, the native soil for their development. Knowledge of this kind is not historically useless. The neglect of Frege by the contemporaneous scientiﬁc community has two very diﬀerent reasons. First, there is little doubt that Frege maneuvered himself out of the mainstream of foundational research (or rather, never succeeded in joining this mainstream) by his insistence on using his newly developed “Begriﬀsschrift,” a logical notation the sophistication and analytical power of which the experts of the nineteenth century (as, in fact, most of those of the twentieth and the early twenty-ﬁrst centuries) failed to recognize. And second, the double disadvantage of working in the no-man’s-land between formal logic and mathematics, and of teaching at the then relatively unimportant small university of Jena gave Frege a low status in the academic world. 196

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The superﬁciality of reception is manifest, for example, in Georg Cantor’s (1885) review of Grundlagen (1884) where Cantor criticizes in condescending manner an allegedly Fregean deﬁnition of (whole) number, whereas the deﬁnition actually found in Grundlagen is quite diﬀerent and would have been worthy of a more careful study. Frege’s correction in his “Erwiderung” (1885) (which he had to publish as a—presumably paid—advertisement) went practically unnoticed. Similarly, already Ernst Schröder in his (1880) review of Frege’s Begriﬀsschrift (1879) had overlooked Frege’s revolutionary technique of quantiﬁcation, claiming (incorrectly) that its eﬀects could have been achieved in a much easier way by Boolean methods. If Frege has been regarded as the founder of modern mathematical logic, this characterization refers to his creation of classical quantiﬁcational logic in his Begriﬀsschrift of 1879 without any predecessor. As to Frege’s motivation, one can only surmise that he felt the urgent need for a logically water tight clariﬁcation of fundamental concepts of analysis like convergence, continuity, uniform continuity, and so on, the precise deﬁnition of which requires nested quantiﬁcation. The mathematical output of the Begriﬀsschrift approach rested on Frege’s replacement of the traditional analysis of elementary propositions into subject and predicate by the general analysis of a proposition into (in our case, propositional) function and argument(s), and its utilization for the expression of the generality of a statement (and of existence statements) by the employment of bound variables and quantiﬁers. For the antecedent part, classical propositional logic, Frege gave a consistent and complete (although not independent) axiom system in terms of negation and conditional, pointing out that other, equivalent axioms and also other connectives could be used, and he managed to get along with the rule of detachment and (not yet suﬃciently precise) substitution rules. In quantiﬁcational logic, he restricted himself to universal quantiﬁcation (which, together with negation, allows the expression of existential statements), and introduced the decisive concepts of the variability domain of a quantiﬁer and the scope of a quantiﬁer and of the quantiﬁed variable. The new devices enabled Frege to precisely deﬁne, for the ﬁrst time, one-one relations, a logical successor and predecessor relation, and a logical heredity relation, in such a way that the arithmetical successor and heredity relations are covered as special cases, and mathematical induction can be formulated, and turns out to have a purely logical foundation. Frege’s Foundations of Arithmetic (Die Grundlagen der Arithmetik, 1884) added, after an elaborate criticism of earlier and contemporary views on the concept of number and on arithmetical statements, a “purely logical” (today dubbed “logicist”) notion of whole number by deﬁning the number n as the extension of the concept “equinumerous to the concept Fn ,” where Fn is a model concept with exactly n objects falling under it, and of a purely logical nature guaranteed by starting with F0 = ¬x = x and constructing Fn+1 recursively from Fn . Frege’s attainment of this notion is somewhat curious because immediately before that he had described and analyzed an attempt at deﬁning number by abstraction directly from equinumerous concepts, but had repudi-

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ated this attempt because of diﬃculties that he considered insurmountable, so that he decided on the explicit deﬁnition just given. Grundlagen also introduced important logical distinctions like that of ﬁrst-order and second-order concepts, with existence and number predicates as examples of the latter. When Frege published Function und Begriﬀ in 1891 and volume 1 of his monumental Grundgesetze der Arithmetik in 1893, he had already realized that extensions of concepts, naively regarded as unproblematic in the explicit deﬁnition of number, must be introduced by an abstraction principle, too. As extensions of concepts have been a main topic of traditional logic at least since the Logic of Port Royal (1662), Frege’s treatment of abstraction in Grundlagen and in Grundgesetze centered around his discovery of the invariance property of statements about “abstract objects,” the logicist deﬁnition of number, and the general abstraction principle (exempliﬁed in Grundgesetze by Frege’s fundamental law V, vide infra) are legitimate and indeed indispensable topics of the history of formal logic. By contrast, the so-called context principle (“The meaning of a word must be asked for in the context of a proposition, not in isolation,” Grundlagen, p. X) and the dichotomy of sense and reference developed in Über Sinn und Bedeutung (1892), often regarded as his most important contribution to philosophy by drawing guidelines for semantics and for a general theory of meaning, have only a negligible role in the history of logic. However, the latter distinction is put to use by Frege in explaining the informative or cognitive value even of judgments that are derived from and therefore based on purely logical premises (as, e.g., according to the logicist thesis, all nongeometrical mathematical theorems), and is of considerable interest for the philosophy of mathematics. To derive the fundamental theorems of arithmetic precisely, that is, within a calculus incorporating strict formation rules for “well-formed formulas” and rules for the logical derivation of conclusions from premises, Frege had to revise and to augment his Begriﬀsschrift. The typically ambiguous “quantiﬁcation axiom” (Begriﬀsschrift, pp. 51 and 62) is now neatly split into a ﬁrst-order and a second-order version (Grundgesetze I, p. 61), but the most momentous change consists in the introduction of new terms of the general form “ Φ(ε),” considered to be names of a new kind of objects called courses-of-values or value-ranges (Wertverläufe) the identity condition for which is given by an abstraction principle accepted by Frege as his fundamental law V, the ﬁfth axiom of his new axiom system:

Φ(ε) = Ψ(α) ⇔ (x)(Φ(x) = Ψ(x)), where Φ(x) and Ψ(x) are functions in Frege’s general sense and the right side of the equivalence expresses the coincidence of their values for every argument, a state of aﬀairs suggesting the identity of the “courses” (or graphs) of the functions in the case of mathematical functions, and thereby the terminology of “courses-of-values.” Frege decided to regard true propositions as names of the “truth value” TRUE and false propositions as names of the “truth value”

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FALSE, respectively, and reconstructed the traditional concepts as one-place functions, the function value of which is one of the two truth values for every argument chosen. So the courses-of-values of such functions are nothing else but the traditional extensions of concepts, or mathematically spoken, the sets or classes determined by the associated function as their deﬁning condition. If concepts are taken as the functions in Frege’s fundamental law V, we get for this special case (in modern notation), { x | F (x) } = { x | G(x) } ⇔ (x)(F (x) ↔ G(x)). In this way, sets have obviously been integrated into the system of Grundgesetze, and since Frege (linking up with the traditional logic of concepts and their extensions) considers abstraction a purely logical operation, set theory becomes (or remains) a proper part of logic. The derivation of arithmetical theorems from the revised and enlarged axiom system of Grundgesetze keeps well within the limits of logic, and in this sense the present set-theoretical foundation of mathematics preserves the intentions and the spirit of Fregean logicism. It was mentioned in the beginning that Frege’s Grundgesetze system foundered at Zermelo’s and Russell’s antinomy, as shown in the appendix of volume 2 of Grundgesetze as well as in Russell’s The Principles of Mathematics, both published in 1903. Though Russell proposed to avoid the antinomy by his type theories, Frege suggested a repair of the axiom system by modifying his fundamental law V; it was shown only much later that this attempt, which has been called “Frege’s way out,” also leads to an impasse by allowing the derivation of other, more complicated antinomies. It is remarkable that the discovery and analysis of Zermelo’s and Russell’s antinomy was made possible only by the extraordinary precision, explicitness, and cogency of Frege’s Grundgesetze system, which in spite of its inconsistency remained a paradigm of a well-designed logical system well into the twentieth century. Among the little-known but precious parts of Grundgesetze, §§90 ﬀ. deserve to be highlighted because of their clear analysis of the nature and the necessary properties of an elementary proof theory and metalogic (“Die formale Arithmetik und die Begriﬀsschrift als Spiele”: Grundgesetze II, p. IX). Attention should also be given to hitherto neglected parts like Frege’s derivation of theorem χ in the appendix to Grundgesetze, where a diagonal argument is used to exhibit a fundamental inconsistency in the (traditional) notion of the extension of a concept (see Thiel 2003). Even the origin of the antinomy has not been located unequivocally up to now. According to the received view in current Frege literature, fundamental law V is responsible for the equivalent of Russell’s antinomy in Grundgesetze. This diagnosis, however, seems a bit rash. It is true that the derivation makes use of fundamental law V, but a careful analysis of it has to inspect not only the logical form of that law but also the structure of the formulae which replace the schematic letters of fundamental law V in every inference that has an instance of it as a premise. Thiel (1975) has tried to show that Frege’s formation rules for function names (which include rules for forming function

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names by the creation of empty places in complex object names) may be too liberal by allowing impredicative function names, and that names of that kind are essentially involved in the derivation of the Zermelo–Russell antinomy in Grundgesetze. A decision on this claim and the questions it raises is still open. A large part of Fregean studies in the past 50 years has been devoted to the investigation of problems that are peculiar to Frege’s systems, without visible impact on the development of the mainstream of mathematical logic invoked in the second paragraph of this chapter. Topics of this kind have been skipped here in spite of their intrinsic interest (as, e.g., the “Julius Caesar problem,” the permutation theorem, and the identiﬁcation thesis of Grundgesetze §10, Frege’s miscarried attempt at a referential completeness proof—which would have implied the consistency of the Grundgesetze system—and last but not least “Hume’s principle” and “Frege’s theorem”). A great thinker’s legacy consists not only in far-reaching insights and eﬃcient methods, it also comprises challenging problems, the solutions of which may sometimes occupy whole generations. Frege, by proving his theorem χ without recourse to Wertverläufe, exhibited an inconsistency (or at least an incoherence) in the traditional notion of the extension of a concept. He prompted our awareness of a situation the future analyses of which will hopefully not only deepen our systematic control of the interplay of concepts and their extensions but also improve our understanding of the historical development of the notion of “extension of a concept” and its historiographical assessment.

Primary Texts (Frege) 1879 Begriﬀsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a.S.: Louis Nebert; repr. in: G. Frege, Begriﬀsschrift und andere Aufsätze, ed. Ignacio Angelelli, Hildesheim: Georg Olms, 1964; Engl. Begriﬀsschrift, a formula language, modeled upon that of arithmetic, for pure thought (transl. Stefan Bauer-Mengelberg), in: Jean van Heijenoort (ed.), From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931 (Cambridge, Mass.: Harvard University Press, 1967), 1–82; Conceptual Notation. A Formula Language of Pure Thought, Modelled upon the Formula Language of Arithmetic, in: G. Frege, Conceptual Notation and Related Articles (trans. and ed. Terrell Ward Bynum, Oxford: Clarendon Press, 1972), 101–203. 1884 Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriﬀ der Zahl, Breslau: Wilhelm Koebner; crit. ed. Christian Thiel, Hamburg: Felix Meiner, 1986; Engl. in: G. Frege, The Foundations of Arithmetic. A logico-mathematical enquiry into the concept of number (bilingual ed., trans. J. L. Austin), Oxford: Basil Blackwell, 1950, rev. 2 1959). 1885 Erwiderung [to Cantor 1885, vide infra], Deutsche Litteraturzeitung 6, no. 28 (11 July 1885), Sp. 1030; Engl. “Reply to Cantor’s Review of Grundlagen der Arithmetik,” in: G. Frege, Collected Papers on Mathematics, Logic, and Philosophy, ed. Brian McGuinness (Basil Blackwell: 1984), 122.

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1891 Function und Begriﬀ. Vortrag gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft für Medicin und Naturwissenschaft. Jena: Hermann Pohle; Engl. in Beaney (1997), The Frege Reader (vide infra), 130–148. 1892 Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik 100: 25–50; Engl. in The Frege Reader (vide infra), 151–171. 1893 Grundgesetze der Arithmetik. Begriﬀsschriftlich abgeleitet, I. Band, Jena: Hermann Pohle. 1903 Grundgesetze der Arithmetik. Begriﬀsschriftlich abgeleitet, II. Band, Jena: Hermann Pohle.

Secondary Literature Beaney, Michael. 1997. The Frege Reader. Oxford/Malden, MA: Blackwell. Cantor, Georg. 1885. [Review of] G. Frege, Die Grundlagen der Arithmetik. . . . Deutsche Litteraturzeitung 6, no. 20 (16 May 1885): 728–729. Russell, Bertrand. The Principles of Mathematics. Vol. I. Cambridge: Cambridge University Press, 1903; London: George Allen & Unwin, 1937 [“Vol. I” omitted]. Schröder, Ernst. 1880. [Review of] G. Frege, Begriﬀsschrift. . . . Zeitschrift für Mathematik und Physik 25 (1880), Historisch-literarische Abtheilung, 81–94; Engl. Review of Frege’s Conceptual Notation. . . , in: G. Frege, Conceptual Notation and Related Articles (vide supra), 218–232. Thiel, Christian. 1975. Zur Inkonsistenz der Fregeschen Mengenlehre. In: idem (ed.), Frege und die moderne Grundlagenforschung. Symposium, gehalten in Bad Homburg im Dezember 1973. Meisenheim am Glan: Anton Hain, 134–159. Thiel, Christian. 2003. The extension of the concept abolished? Reﬂexions on a Fregean dilemma. In Philosophy and Logic. In Search of the Polish Tradition. Essays in Honour of Jan Woleński on the Occasion of his 60th Birthday, eds. Jaakko Hintikka, Tadeusz Czarnecki, Katarzyna Kijania-Placek, Tomasz Placek, and Artur Rojszczak (†). Dordrecht/Boston/London: Kluwer; Synthese Library, vol. 323), 269–273.

Selected Further Readings Angelelli, Ignacio. 1967. Studies on Gottlob Frege and Traditional Philosophy, Dordrecht: D. Reidel. Demopoulos, William, ed. 1995. Frege’s Philosophy of Mathematics. Cambridge, Mass./London: Harvard University Press. Dummett, Michael. 1973. 2 1981. Frege: Philosophy of Language. London: Duckworth. Dummett, Michael. 1981. The Interpretation of Frege’s Philosophy. Cambridge, Mass.: Harvard University Press. Dummett, Michael. 1991. Frege: Philosophy of Mathematics. London: Duckworth. Frege, Gottlob. 1969. Nachgelassene Schriften. Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach (eds.). Hamburg: Felix Meiner, revised and enlarged 2 1983; Engl. Posthumous Writings (trans. Peter Long/Roger White), Oxford: Basil Blackwell, 1979. Haaparanta, Leila and Jaakko Hintikka, eds. 1986. Frege Synthesized. Essays on the Philosophical and Foundational Work of Gottlob Frege. Dordrecht: D. Reidel.

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Kreiser, Lothar. 2001. Gottlob Frege. Leben–Werk–Zeit. Hamburg: Felix Meiner. Schirn, Matthias, ed. 1996. Frege: Importance and Legacy. Berlin/New York: Walter de Gruyter. Sluga, Hans D. 1980. Gottlob Frege. London/Boston/Henley: Routledge & Kegan Paul. Weiner, Joan. 2004. Frege Explained. From Arithmetic to Analytic Philosophy. Chicago/La Salle, Ill.: Open Court.

6

The Logic Question During the First Half of the Nineteenth Century Risto Vilkko

Immanuel Kant wrote, in the preface to the second edition of his Kritik der reinen Vernunft, that since Aristotle it [logic] has not required to retrace a single step, unless, indeed, we care to count as improvements the removal of certain needless subtleties or the clearer exposition of its recognized teaching, features which concern the elegance rather than the certainty of the science. It is remarkable also that to the present day this logic has not been able to advance a single step, and is thus to all appearance a closed and completed body of doctrine. (KrV, B VIII) Kant’s division of logic into its general and transcendental aspects served, during the early nineteenth century, as the basis for the removal of philosophers of logic into, roughly speaking, two opposing camps of the Herbartian formal logicians and the Hegelian idealist metaphysicians. Also it can be assumed that Kant’s disbelief in the possibilities of logic to develop any further from its alleged Aristotelian perfection discouraged many philosophers from trying to improve the logic proper and led most of them, instead, to studying the “applications” of logic, that is, the ﬁelds of study that are nowadays referred to as epistemology, psychology, methodology, and the philosophy of science. However, not all logicians of the early and mid-nineteenth century took Kant’s conception for granted. Herbart saw a promise of further development in Drobisch’s Neue Darstellung der Logik (Herbart 1836, 1267f.). Beneke wrote a few years later that even though Kant’s conception may have felt more or less credible during the 1780s, “since then, the situation has greatly changed” 203

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(Beneke 1842, 1). According to him, “logic has lost its unchangeable character. It has adopted a variety of such aspects of the possibility of which the old logicians, including Kant himself, had no idea” (ibid.). Boole wanted to remark, in 1854, that “syllogism, conversion, &c., are not the ultimate processes of Logic. It will be shown . . . that they are founded upon, and are resolvable into, ulterior and more simple processes which constitute the real elements of method in Logic” (Boole 1854, 10). De Morgan had the courage to write in 1860 that in the ﬁeld of logic “innovations have been listened to in a spirit which seems to admit that Kant’s dictum about the perfection of the Aristotelian logic may possibly be false” (De Morgan 1860, 247). After Hegel’s death in 1831, there arose in the academic circles of Germany a lively discussion concerning the makings of logic both as a philosophical discipline and as a formal and fundamental theory of science which might clarify not only the logical but also the metaphysical foundations of science. In fact, this was perhaps the most popular theme in the philosophical exchange of thoughts in Germany during the mid-nineteenth century. The most characteristic slogans in the discussion were “the logic question” and “reform of logic.” These slogans did not have very speciﬁc meanings. They were used rather loosely to refer to various competing eﬀorts to reform logic. In 1880 Leonhard Rabus (1835–1916) characterized the logic question in his book on nineteenth-century German contributions in the ﬁeld of logic as circling around the fundamental problems of the possibility and justiﬁcation of logic (Rabus 1880, 157; see Vilkko 2002). According to another late nineteenth-century German philosopher, Friedrich Harms (1819–1880), reform of logic could be sought from logic as (1) an organon of sciences, (2) a critique of sciences, or (3) a philosophical science (Harms 1874, 124). As an organon, or as a discipline of the methods of sciences, logic is the science of the forms of thought. As a critique, or as a theory of the necessary preconditions of knowledge and knowing, logic considers such questions as: How are objects given to cognition? What are the basic principles of knowing? What justiﬁes these principles? And how valid are these principles? (Harms 1881, 137). Bacon’s “inductive” reform covered logic as an organon, whereas Locke and Kant treated logic as a critique (ibid., 150). As far as Harms was concerned, this was, however, not enough. Harms wanted to stress that these two aspects of logic must be taken simply as two diﬀerent sides of the one and the same logic. In his view, the most important aspect in this reform concentrated on the form of logic as a philosophical science. He argued as follows: Since logic is a philosophical science due to its content, it must be a philosophical science also due to its form, because the content and the form of a science must coincide. Purely formal logic is, however, originally an empirical science and thus only an instrument for philosophy. When logic is reformed as a philosophical science, it is also reformed as an organon as well as a criterion of correct and consistent thinking (ibid., 121–125, 130). Hermann Ulrici (1806–1884) deﬁned the logic question as “the question about the place, the context, and the working of logic” (Ulrici 1869/70, 1). He began his most important contribution to this debate, titled “Zur logischen

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Frage” (1869/70), by stating that his conception of logic is incompatible not only with that of Hegel but also with every other attempt that denies the purely formal character of logic and tries to identify logic with metaphysics, epistemology, and/or theory of science. According to him, logic “deserves the name of a fundamental science; and it is clearly impossible for such logic to be at the same time also metaphysics and a theory of science” (ibid., 8). In other words, the logic question sprung from a genuine doubt about the justiﬁcation of the formal foundations of logic as the normative foundation of all scientiﬁc activity. On the one hand, most of the participants of the debate opposed Hegel’s attempts to unite logic and metaphysics—on the other, reform was sought to overcome the old scholastic-Aristotelian formal logic. The discussion can thus be characterized as a battle on two fronts. In any case, the need to reform was stimulated by the developments in the ﬁeld of philosophy. As Volker Peckhaus has put it: the reform endeavors that were released through this discussion scarcely considered the formal logic itself, but rather its psychological foundations and its use in theories of science that strove to seize the positive and formal sciences of that time. (Peckhaus 1997, 12) The very slogan “logic question” was used for the ﬁrst time by Adolf Trendelenburg (1802–1872). His writings provoked anew an awareness of the problematic philosophical position of formal logic. What is more, it was from his initiative that the reform discussion of logic really started around the turn of the 1840s. In 1842, he asked in his essay “Zur Geschichte von Hegel’s Logik und dialektischer Methode” whether Hegel’s dialectical method of pure thought should be treated as a scientiﬁc one. His own answer to this question was negative (Trendelenburg 1842, 414). However, of more importance was his criticism of both Herbartian formal logic and Hegelian dialectical logic in his two-volume Logische Untersuchungen (Trendelenburg 1840, I, 4–99). Before going into the details of Trendelenburg’s criticism, let us take a closer look at Herbart’s and Hegel’s conceptions of the nature and the task of logic.

1. Herbart’s Theory of the Structures of Thought Johann Friedrich Herbart (1776–1841) deﬁned philosophy as cultivation and arranging of conceptual material that is given in sense-experience (Herbart 1813, 38f.). His basic division of the ﬁeld of philosophy was the that time usual tripartite one: logic, metaphysics, and aesthetics (the most important part of which was ethics). The task of logic was to take care of the ﬁrst and the foremost duty of philosophy, that is, of conceptual clarity. The task of metaphysics was to justify concepts as objects of thought by analyzing and resolving conceptual contradictions that originate from thought itself. The third constituent, aesthetics, complemented the objects of thought by an analysis of values (Ueberweg 1923, 156f.).

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Herbart remained in many ways faithful to Kant’s conceptions. In his logic writings he himself sketched only the very basics and trusted, when it came to more advanced issues, the textbooks of, for example, Wilhelm Krug and Jakob Fries. Logic meant for him a regulative science which merely establishes the ways of handling concepts as such and lays down the law of contradiction as their highest standard. In his supplement “Hauptpuncte der Logik” to the second edition of his Hauptpuncte der Metaphysik Herbart wrote: Indeed logic is concerned with representations but not with the practice of representing. Hence, it is neither concerned with the mode and the manner of how we get to a representation, nor with the conditions of mind that are given thereby, but only with what becomes represented. (Herbart 1808, 217) In his works, Herbart gives logic such deﬁnitions as, for example, “a general science of understanding” (Herbart 1813, 67) and “a theory of the structures of thought” (Herbart 1808, 222). Logic meant for him a fundamental science that occupies itself ﬁrst of all with separating, classifying, and combining concepts as such; thereafter with making and analyzing judgments; and ﬁnally with revealing the modes of inference. The task of logic was to develop the formal consequences from the given premises (Herbart 1808, 218; 1831, 204). The fundamental point of diﬀerence between Hegel’s and Herbart’s conceptions of logic dealt with the relation between logic and metaphysics. Whereas the former drew an identity between logic and metaphysics, the latter wanted to keep the two strictly separated from each other. Herbart also insisted that for the beneﬁt of pure logic, it is necessary to avoid all psychological considerations. In the second chapter of his Lehrbuch zur Einleitung in der Philosophie, Herbart summarizes his logic conception in ﬁve theses: 1. Logic provides us with the most general regulations of separating, classifying, and combining concepts. 2. Logic presupposes concepts as known and does not distress itself with their speciﬁc contents. 3. Therefore logic is not really an instrument of such an investigation that aims at ﬁnding something novel. It rather gives instructions for revealing what we already know. 4. Nevertheless logic also points out the primary conditions of investigation in general and takes care of the important duty of paying attention to the possibility of committing errors. 5. The term “applied logic” refers to a combination of logic and psychology which, however, falls out as defective on issues where psychology does not already lead the way. (Herbart 1813, 41f.) Even though it is fully justiﬁed to characterize Herbart as an advocate of formal logic and to see his philosophy of logic as an oﬀspring of Kant’s

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empirical realism, it can be asked why the logicians of the early nineteenth century did not seem to be very interested in making an eﬀort toward further development of formal logic. One reasonable answer is based on the fact that was already pointed out in the beginning of this chapter. That is, because during the early nineteenth century logic sprung straight from Kant’s view, according to which the inherited scholastic-Aristotelian logic was to be seen as closed and complete. The only thing there seemed to be left for logicians to deal with was its applications, such as epistemology, methodology, and the philosophy of science.

2. Drobisch’s Formal Philosophy Around the mid-nineteenth century, the perhaps most eminent opponent to the idealist identiﬁcation of logic and metaphysics was the mathematician, astronomer, and philosopher Moritz Wilhelm Drobisch (1802–1896). He was one of the most important and insightful thinkers in the Herbartian school. In the ﬁrst paragraph of the introduction to his Neue Darstellung der Logik (1836), Drobisch introduced philosophy as “the general science” (Drobisch 1836, 1). According to him, it was not the task of philosophy to investigate the auxiliary apparatus of subjective cognition. That was the task of special sciences. For him, like for Herbart, philosophy meant working with purely conceptual material and trying to reach understanding of concepts in themselves (ibid., 2). In Neue Darstellung der Logik, Drobisch focused on what he called “formal philosophy,” that is, logic. He introduced logic as the doctrine of the conditions of correct and consistent thinking (ibid., 5). In his view, logic must not be understood as a description of human thinking. It must not be considered as a descriptive natural history of thinking, but rather as a normative discipline of thought in general or as a “Code of Laws of Thought” (ibid., 6). In this connection Drobisch referred to Kant’s description of logic not as a descriptive but as a demonstrative a priori science, the function of which is to take care of the necessary laws of thought, that is, as the science of the adequate use of understanding and of reason in general (see KrV, B IX, XXII). Drobisch regretted the fact that the law-giving character of logic had been spoiled by the Kantian school. Therefore he saw it necessary to once again impress on philosophers the importance of this normative aspect of logic (Drobisch 1836, 7). Drobisch was concerned about the purity of logic. Already in the preface to the ﬁrst edition of Neue Darstellung der Logik he made it clear that in what follows, logic will be understood as an independent and autonomous formal foundation of all scientiﬁc activity. He wrote that “logic is, in fact, nothing but pure formalism. It is not meant to be, and must not be, anything else” (ibid., VI). Moreover, he did not consider logic as a branch of mathematics (ibid., VIII–X). At the end of the ﬁrst edition of Neue Darstellung der Logik, there is an exceptional and incisive logico-mathematical appendix (ibid., 127–167). In this

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appendix Drobisch concentrates on the problematic connection between logic and mathematics, and introduces his algebraic construction of the simplest forms of judgment and derivation of inferences founded thereupon (ibid., 131– 136). In eﬀect, he develops an extensional algebraic calculus of classes and elementary judgments. Drobisch’s calculation apparatus follows Aristotelian theory of syllogisms. To give just a couple of examples, the classical modi BARBARA and DARAPTI are presented in the following way (ibid., 134f.): BARBARA

M =p S = m (< M = p ) S = p (where p < p)

DARAPTI

M =p M =s s=p

From today’s perspective, Drobisch’s calculus appears as one of the most interesting chapters of Neue Darstellung der Logik. It has been valued as an improvement in comparison with the intensional systems of his famous predecessors Ploucquet, Lambert, Gergonne, and Jacob Bernoulli (Thiel 1982, 763). Unfortunately, Drobisch removed his calculus from the subsequent editions of Neue Darstellung der Logik, which became one of the leading German textbooks of logic during the nineteenth century. He also executed a number of other modiﬁcations to the later editions of his book. In the preface to the second edition, he even admitted that the changes that had been carried out were so extensive that one could almost speak of two altogether diﬀerent books (Drobisch 1851, III). Trendelenburg’s hard criticism was undoubtedly an important catalyst for these changes. But was it the decisive one? Drobisch’s uncompromising attitude toward Trendelenburg in the preface to the second edition suggest that perhaps he just felt that after all his calculus was not quite ripe to be published.

3. Hegel’s Dialectical Logic The ﬁrst one of the two most important sources of Georg Wilhelm Friedrich Hegel’s (1770–1831) dialectical logic is his monumental Wissenschaft der Logik (1812/16). This much debated and thoroughly interpreted work was Hegel’s attempt to provide a comprehensive philosophical synthesis of his union of logic, metaphysics, and epistemology. The second source of Hegel’s logic is Encyclopädie der philosophischen Wissenschaften im Grundrisse, of which Hegel prepared and published three diﬀerent versions—the ﬁrst one in 1817 and the two others in 1827 and 1830. Of particular interest here is the ﬁrst part of the book: “Die Logik, die Wissenschaft der Idee an und für sich” (§§19–244). In the following we pay attention only to the third edition, which was published the year before Hegel’s death. It can be regarded as Hegel’s philosophical

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testament. Reading it this way requires, however, keeping in mind that it does not provide the reader with a thoroughly elaborated and fully developed system but only with a sketch of the foundations of one. And when considering the relations between Wissenschaft der Logik and the Encyclopädie, it is worth knowing that Hegel kept on working intensely with the former one until his death (Nicolin and Pöggeler 1959, xxix, xxxviif.). It is no surprise that Hegel’s and Herbart’s thought with regard to logic did not meet, because the very foundations and the aims of the two diﬀered from each other as greatly as day and night. The following quotation from the introduction to Wissenschaft der Logik gives the reader a hint of the distance between Hegel’s understanding of the term “logic” and that of the Herbartians: logic is to be understood as the system of pure reason, as the realm of pure thought. This realm is truth as it is without veil and in its own absolute nature. It can therefore be said that this content [of pure science] is the exposition of God as he is in his eternal essence before the creation of nature and a ﬁnite mind. (Hegel 1812, 31) Hegel built his dialectical logic on the trivet of Kant’s transcendental logic, Schelling’s identity between the real and the ideal, and Fichte’s Wissenschaftslehre. The resulting theory was designed to serve the needs of his own monumental philosophical system, which divides into the three main constituents of Science of Logic, Philosophy of Nature, and Philosophy of Spirit. Logic was regarded as the foundational science of the system. Hegel’s works provide the reader with the most comprehensive theory of metaphysical philosophy of logic. In his philosophy there is no way of separating logic and metaphysics from one another. In the Encyclopädie Hegel states that “logic therefore coincides with Metaphysics, the science of things set and held in thoughts—thoughts accredited able to express the essential reality of things” (Hegel 1830, 58). If Hegel’s philosophy in toto is a science about the real world of change and development, understood as the collective self-education of humanity about itself, then logic is the construction of the history of thinking. In the Encyclopädie Hegel deﬁnes logic as “the science of the pure Idea; pure, that is, because the Idea is in the abstract medium of Thinking” (ibid., 53). His logic does not consider the categories merely as forms of subjective thinking. They are also seen as the forms of objective Being itself. The Absolute or the Reason—which is the ultimate subject matter of Hegel’s philosophy—is a union of Thinking and Being, and it is the task of logic to develop this unity. Accordingly Hegel divided his logic into two parts: (1) the objective logic concerned with the Being, and (2) the subjective logic concerned with the Thinking. In Wissenschaft der Logik Hegel attacked ﬁercely the “dull and spiritless” (Hegel 1812, 34) attempts of formal logicians to elaborate logic as the most general deductive science of thinking. According to him, the deduction of the “so-called rules and laws, chieﬂy of inference is not much better . . . than a childish game of ﬁtting together the pieces of a colored picture puzzle”

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(ibid.). This “thinking” which formal logic strove to govern constituted for Hegel the mere form of cognition. According to him, formal logic abstracted from all content of cognition and the material constituents of knowledge are, consequently, totally independent of it. Thus, formal logic could provide only “the formal conditions of genuine cognition and cannot in its own self contain any real truth, nor even be the pathway to real truth because just that which is essential in truth, its content, lies outside logic” (ibid., 24). The two most important concepts in Hegel’s critique of the traditional scholastic-Aristotelian conception of logic are “formal” and “abstract.” His accusation of traditional logic as being merely formal thinking is based on a conception according to which logical form and content should correspond to each other. This requirement has several consequences. It explains why logic embraces for him, in addition to the problem of classiﬁcation of propositions and inferences, also the study of the categories on which these classiﬁcations are based. It implies, for example, that logic also deals with the distinctions between diﬀerent levels of knowledge correlative to the various aspects of reality given in the categories. It also has implications for Hegel’s conceptions of judgment and truth (Kakkuri 1983, 41). Hegel’s conclusion of the history of logic until the early nineteenth century was as follows: Before the dead bones of logic can be quickened by spirit, and so become possessed of a substantial, signiﬁcant content, its method must be that which alone can enable it to be pure science. In the present state of logic one can scarcely recognize even a trace of scientiﬁc method. It has roughly the form of an empirical science. (Hegel 1812, 34f.) This leads us to the central topic of Wissenschaft der Logik, that is, the problem of appropriate philosophical method. Hegel’s starting point was the assumption that philosophy had not yet found a method of its own, but merely borrowed bits and pieces from the methodologies of various sciences and, in particular, “regarded with envy the systematic structure of mathematics” (ibid., 35). The connection between logic and philosophy is inextricable because “the exposition of what alone can be the true method of philosophical science falls within the treatment of logic itself; for the method is the consciousness of the form of the inner self-movement of the content of logic” (ibid.). From Kant’s remark of elementary logic having neither lost nor gained any ground since the time of Aristotle, Hegel drew a very radical conclusion. He held the same opinion as Kant in stating that logic had not undergone any positive changes in more than 2000 years. But judging by the logic-compendia of his time the few traceable changes appeared to him as consisting “mainly in omissions” (ibid., 33). Therefore he concluded that it is “necessary to make a completely fresh start with this science” (ibid., 6). Hegel’s logic dealt not only with the traditional Aristotelian laws of thought or Kant’s logic of the understanding but also with metaphysical issues. He

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begun the introduction to the ﬁrst book of Wissenschaft der Logik by regretting the fact that “what is commonly understood by logic is considered without any reference whatever to metaphysical signiﬁcance” (ibid., 28). His at ﬁrst relative and later absolute identiﬁcation of logic and metaphysics is the fundamental point of diﬀerence between Hegel’s philosophy of logic and that of the Herbartian school. There is no undisputable clear-cut answer to the question “What kind of logic is Hegel’s logic?” It is certainly not, for example, a doctrine of the laws of formally impeccable inference. Yvon Gauthier (1984) has suggested an answer that amounts to saying that Hegelian logic is a transcendental logic, which in turn would be the study of the a priori structures of logical thought. According to him, what Hegel calls objective logic is nothing less than metaphysics in the traditional sense, and therefore it is justiﬁed to consider Hegelian logic as transcendental-metaphysical. For Hegel, transcendental-speculative logic, which deals with the most general features of thought, reaches even further than it does for Kant (ibid., 303f.). Whatever the truth, in any case Hegel’s program was one of the most inﬂuential eﬀorts to reform logic during the nineteenth century.

4. Trendelenburg’s Logical Investigations Friedrich Adolf Trendelenburg (1802–1872) was not concerned with logic as mere doctrine of the laws of correct inference. The ﬁrst two chapters of his greatest work, the two-volume Logische Untersuchungen (1840) can well be considered to discuss philosophy of logic in today’s sense of the saying, but the rest of the book—the essence of Trendelenburg’s logical investigations— is perhaps best characterized as fundamental epistemology with a strong metaphysical ﬂavor. Trendelenburg’s intention was to solve what he considered to be the ultimate task of philosophy, namely, the apparent correspondence between “the external reality of Being and the internal reality of Thinking” (Trendelenburg 1840, I, 110). In eﬀect, his Logische Untersuchungen was an attempt ﬁrst to show the defects of both Herbartian and Hegelian logic and then to supplement and reformulate them to achieve a formal and fundamental theory of science and metaphysics.

4.1. Critique of Formal Logic When talking about formal logicians, Trendelenburg meant those philosophers who attempted to explain the pure forms of thought without paying attention to the contents of thought. This tradition rested, according to Trendelenburg, on a strict distinction between thoughts and their objects, that is, between Thinking and Being. Furthermore, because in Trendelenburg’s view these socalled formal logicians took truth as simple correspondence between thoughts and their objects, they also accepted silently the presupposition of harmony

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between forms of thought and things in themselves. In particular Trendelenburg wanted to criticize those philosophers who subscribed to Herbartian conception of logic (Trendelenburg 1840, I, 4–6). Formal theories of logic had traditionally begun with a theory of concepts. Accordingly, Trendelenburg begun his criticism with some critical remarks on traditional theories of concepts and speciﬁed the target of his criticism: “In particular we consider two ingenious and consistent presentations of formal logic, the famous works of A. D. Ch. Twesten [1825] and the University of Leipzig Professor Moritz Wilhelm Drobisch [1836]” (ibid., 7). He was not content with taking concepts as given and understanding them merely as sub- and superordinate combinations of properties. He criticized this view as much too naive for uncovering the secrets of the foundations of human thought. In his view, the traditional subordination of concepts was based on nothing but simple operations of adding and subtracting properties. According to him, every attempt to ﬁnd the essence of thought with the help of such basic operations as these—or with any such alternatives as multiplication and division—remains always futile. Every theory that rested on such theory of concepts became thus “more than dubious” (ibid., 8). Hence, according to Trendelenburg, the whole ediﬁce of formal logic was built on sand. However, for the sake of argument, he assumed that there is nothing wrong with formal theories of concepts and turned to examine the fundamental principle of classical formal logic, that is, the law of identity and contradiction: “A is A and A is not not-A.” Formal logicians had traditionally believed that in the ﬁnal analysis everything else in logic derives from this principle. Trendelenburg wanted to ﬁnd out if this belief really was tenable. Even though he admitted that the principle seemed unassailable at the ﬁrst sight, he wanted the reader to pay closer attention to the latter part of it: A given concept A stands in contradiction with its negation and is logically equivalent with its double negation. According to Trendelenburg, this “blindly accepted” (ibid., 11) interpretation was insuﬃcient for explaining the nuances with regard to contents of concepts. It reduced all of the various conceptual contrasts to the pure formal logical contradiction. Trendelenburg wanted to criticize this inﬂexibility. In his opinion, every purely formal deﬁnition for identity and negation fails to explain them properly (ibid., 11–14). It may be diﬃcult to understand why Trendelenburg wanted to make such an issue about formal logic not paying attention to the contents of judgments if one does not keep in mind that his logical investigations was an attempt to elect the best parts of metaphysics and logic and to reformulate them as a general, formal, and fundamental theory of science. In the introduction to Logische Untersuchungen, he wrote that “the range of these investigations must run through the sphere of logic questions and reach for insight on the whole ﬁeld of science” (ibid., 3). After having scrutinized both formal logic and dialectical method, Trendelenburg announced that the rest of his book will be committed to answering, with regard to the objective foundations of logic, the question about the possibility of knowledge (ibid., 100f.). His project

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was more ambitious than just explicating the laws of correct and consistent deductions. Trendelenburg also probed into diﬀerent types of inference and asked if it was possible to derive all the forms of inference from what the formal logicians regarded as the basic premises of formal logic, that is, the principle of identity and the idea of concepts as combinations of properties. He found nothing to complain about the classical forms of deductive inference. However, the problematic cases were logic of induction and inferences based on analogy. “What a great shame it is,” he wrote, “if there is no ability to understand the logic of induction and analogy expressed by science; and in case it is generality and not necessity that follows, then the principle of identity and contradiction is not the [basic] principle of logic” (ibid., 18). Indeed, according to Trendelenburg, inadequate understanding of logic of induction was one of the most alarming shortcomings of early nineteenth-century formal logic. Trendelenburg dedicated the end of the ﬁrst chapter to his favorite subject, Aristotelian philosophy. He had noticed that formal logicians often appealed to Aristotle and willingly called themselves Aristotelians. Trendelenburg, however, had found a number of reasons why they should not be regarded Aristotelian. According to him (ibid., 18–21): 1. Aristotle did not propose that the forms of thought should be understood purely in themselves; 2. Understanding concepts simply as given combinations of properties does not correspond to Aristotle’s reﬁned theory of concepts; 3. The nineteenth-century formulation for the principle of identity and contradiction, “A is A and A is not not-A,” diﬀers signiﬁcantly from Aristotle’s original formulation: “The same attribute cannot at the same time belong and not belong to the same subject” (Met. 1005b 18–20); 4. Aristotle did not regard aﬃrmation and negation as purely logical forms; 5. Aristotle considered modal judgments of necessity and possibility as rooted in the nature of things; 6. Aristotle did not postulate syllogisms as merely formal relations between judgments. This was roughly what Trendelenburg left in the hands of the public for deciding whether formal logic could be taken seriously with regard to the logic question.

4.2. Critique of Metaphysical Logic If it was the most serious defect of Herbartian formal logic to strictly separate Thinking from Being and to concentrate only on the former one, then Hegelian metaphysical logic was guilty of exactly the opposite. According to Hegel, Thinking and Being could not be separated from each other. In his system

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knowledge of Being derives straight from Thinking in itself. Trendelenburg, however, could not see how this could be possible. Shortly, in his opinion the biggest fault with Hegel’s conception of logic was the attempt to completely and unjustly neglect the decisive intermediate role of the Aristotelian concept of motion in obtaining knowledge (Harms 1881, 236). Trendelenburg’s criticism of Hegel’s dialectical method ranges over a broad ﬁeld of philosophical topics. In the following we shall, however, concentrate only on those aspects and arguments that can be regarded as belonging to the ﬁeld of philosophy of logic. Trendelenburg summarized the basic situation with dialectical method in the following way: The dialectical method strives for the greatest possible. It wants to develop and create the pure Idea as if in a divine intellect—solely out of itself. Content and form are supposed to arise simultaneously. Because the pure Idea brings forth only what lies deep in itself, it must create such a world where nothing exists in itself and every thought is a genuine part of the totality. . . . We must, however, resign [from the dialectical method] at once. The means are too frail for executing the plan of such a titanic project. (Trendelenburg 1840, I, 94) Trendelenburg’s ﬁrst argument concerned the alleged presuppositionless of Hegel’s logic. According to Hegel, pure Thought needed no support from perception or sense experience. The pure Idea was the stone foundation of his logic and vice versa. Therefore, according to Hegel, logic was both quite easy and extremely hard: From diﬀerent points of view, Logic is either the hardest or the easiest of sciences. Logic is hard, because it has to deal neither with perceptions nor, like geometry, with abstract representations of the senses, but with the pure abstractions; and because it demands a force and facility of withdrawing into pure thought, of keeping ﬁrm hold on it, and of moving in such an element. Logic is easy, because its facts are nothing but our own thought and its familiar forms or terms: and these are the acme of simplicity, the ABC of everything else. (Hegel 1830, §19) Trendelenburg could not accept this view. In his opinion Hegel’s “pure Thought” did not deserve its name because it could not escape from tacitly presupposing the fundamental principle of all knowledge, that is, the Aristotelian idea of motion. According to Trendelenburg, even the most elementary dialectical steps were impossible without support from this concealed principle: “Wherever we turn to, motion remains the presupposed vehicle of the dialectically breeding Thought. . . . This spatial motion is hereupon the ﬁrst assumption of this presuppositionless logic” (Trendelenburg 1840, I, 24–29; see also Petersen 1913, 156). Trendelenburg’s second argument against the dialectical method concerned the two seemingly logical relations of negation and identity. However, Trende-

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lenburg wanted to point out that with a closer look it can be seen that it is not the logical negation that works in Hegel’s system: What is the nature of this dialectical negation? It can have a twofold character. Either it is understood in a pure logical way, so that it simply denies what the ﬁrst concept aﬃrms without replacing it with something new, or it can be understood in a real way, so that the aﬃrmative concept is denied by a new aﬃrmative concept, in what way both of the two must be replaced with each other. We call the ﬁrst instance logical negation, and the second one real opposition. (Trendelenburg 1840, I, 31) Is it now possible for the logical negation, Trendelenburg asked further, to stipulate such progress that from a given denial a new positive concept arises which exclusively unites in itself both the aﬃrmation and the negation? According to Trendelenburg’s deﬁnition for logical negation, this was totally out of question. In other words, it would be a mistake to treat the dialectical negation as logical contradiction. Hence, it must be regarded as a real opposition. However, if it is a real opposition, then it is unattainable from the logical point of view and Hegel’s dialectic is not the dialectic of pure Thought. Hence the one who takes a closer look on the so-called negations of Hegel’s dialectical logic shall in most cases discover ambiguities (ibid., 30–45). According to the rules of Hegel’s dialectical logic, identity creates a new concept of a higher level out of a given concept and its opposite. This dialectical product is the truth of its “ingredients.” Hence, dialectical identity appears to be a real unit, even though it is, in the ﬁnal analysis, only a kind of shallow similarity of abstraction. Trendelenburg could not see how it could be possible for two distinct concepts to mutate into a third, new one. He wrote that “dialectical identity oﬀers more than it has” (ibid., 55). If the dialectical identity was supposed to be some kind of an impetus of the concrete reality, then it surely could not be an identity of abstraction. According to Trendelenburg, there is an obvious contradiction between the origin of the dialectical concept of identity and its alleged eﬀect (ibid., 45–56). Trendelenburg’s third point of criticism concerned Hegel’s conception of immediacy. In Aristotle’s philosophy, Trendelenburg clariﬁed, every such element of thought is immediate that does not reduce to any other element, for example, the basic elements of representation in general or certain particulars sensed in such a manner that nothing whatsoever comes in between the sensuous representation and its object. In the nineteenth-century philosophy it was, according to Trendelenburg, more customary to use the term “immediate” in the latter sense of the word. Since the whole dialectic was in the ﬁnal analysis nothing but a chain of mediation, immediacy was out of question in this sense. However, in Hegel’s system, the concept of immediacy is prominent everywhere in the dialectical process of mediation. Now it seemed to be, in Trendelenburg’s opinion, that in this context immediacy can only mean selfsubsistence, that is, Being-for-self. Hegel himself expressed this quite clearly

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in his Encyclopädie: “Being-for-self, as reference to itself, is immediacy, and as reference of the negative to itself, is a self-subsistent, the One. This unit, being without distinction in itself, thus excludes the other from itself” (Hegel 1830, §96). Later in the same volume he exclaims that “the immediate judgment is the judgment of deﬁnite Being. The subject is invested with a universality as its predicate, which is immediate, and therefore a sensible quality” (Hegel 1830, §172; emphasis added). In Trendelenburg’s opinion, this explanation left no room for misunderstanding: In Hegel’s logic the term “immediate” refers to something foreign to his system, that is, to something sensuous. And above we just saw how in Hegel’s dialectic the function of immediacy was by no means supposed to lead the way from pure thoughts to something sensuous. Thus, Hegel’s use of the term “immediate” remains ambiguous (Trendelenburg 1840, I, 56–59). Trendelenburg closed the ﬁrst chapter of Logische Untersuchungen by estimating whether it is right to regard the Herbartian projects of formal logic as latest extensions to the Aristotelian tradition of logic. Accordingly, at the end of the second chapter, he paid attention to dialectical method having sought for its original from Plato’s Parmenides dialogue (ibid., 89). The latter part of the Parmenides dialogue (137c–166c), where Socrates and Parmenides discuss the intertwined concepts of the one and many and the problematic relations between parts and wholes, has been interpreted in many diﬀerent ways since time immemorial. Hegel recognized a resemblance between Parmenides’s holistic concept of One and his own Absolute. Admittedly there are certain similarities. However, there are also other ways of understanding the passage. Another possibility is to read Parmenides simply as Plato’s reply to his critics. According to a number of scholars, the safest principle of interpretation is to excavate the hints that Plato himself gives to the reader for understanding the dialogue. Trendelenburg subscribed to this strategy. One of these hints is also the heart of Trendelenburg’s last argument against Hegel: In the beginning of the latter part of the dialogue Parmenides suggests that Socrates could use some training in the art of dialectic so that he might be more successful in searching for solutions to Parmenides’s philosophical dilemmas (135c–136a). Thus, the arguments and proofs of the latter part can be regarded as merely heuristic. It can be read as just an evaluation of various juxtaposed philosophical arguments and theses, some of which reappear in other dialogues by Plato—others being mere formal supplements to Parmenides’s arguments. Trendelenburg also held the opinion that it is hard to ﬁnd a credible uniform philosophical doctrine hidden beneath Parmenides’s lesson to Socrates (Trendelenburg 1840, I, 89). The task of Hegel’s dialectic was to show how a closed system could seize the whole reality. The outcome, however, failed to convince Trendelenburg. It seemed to him evident that the role of perception is silently assumed everywhere in Hegel’s theory and that the concepts of the pure Thought are, in the ﬁnal analysis, nothing but diluted representations. “Intuition is,” he wrote in the closing pages to the second chapter of Logische Untersuchungen, “vital for

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human thought and it starves to death if it must try to live on its own entrails” (ibid., 96). At the end of the whole work, Trendelenburg glanced back to his work and wrote: As we have seen, formal logic is essential but not suﬃcient for accomplishing the logical task. Hegel’s dialectic, in its turn, gives a promise of more—as a matter of fact of the greatest that can be imagined—but falls out as impossible. (Trendelenburg 1840, II, 363) As the discussion on the reform of logic moved on, there appeared certain general points of agreement concerning the basic nature and the task of logic. It became common to accept that the possible reform of logic must go hand in hand with the reform of philosophy. The Kantian appreciation of mathematics against its Hegelian devaluation became rehabilitated even though the question about the relationship between logic and mathematics remained diﬃcult. On the one hand formal logic became almost resistant to philosophical criticism, but on the other hand it lost at least part of its prestige as the foremost constituent of philosophy proper as it gradually was transformed into a subdiscipline of mathematics.

5. Herbartian and Hegelian Reactions to the Criticism Trendelenburg’s Logische Untersuchungen had a devastating impact in both the Herbartian and the Hegelian camps. Academic public expected the leading representatives from both sides to formulate and present counterarguments. This they also did. The leading Herbartian philosophers were, however, a little slower in defending themselves than their Hegelian colleagues. Even though Trendelenburg explicitly aimed his censure at Drobisch and August Twesten (1789–1876), apparently most of those who took part in the evaluation discussion of Logische Untersuchungen read the ﬁrst chapter of the book as censure of Herbartian logic and metaphysics. Drobisch wrote and published several articles in defense of Herbart. Twesten did not reply on Logische Untersuchungen. It took more than 10 years before Drobisch was ready to step forth with counterarguments. The ﬁrst set of his answers was published in 1851 in the preface to the second edition of his Neue Darstellung der Logik (1851). The second one came out the year after, in the form of a journal article (Drobisch 1852). Before these two contributions, only Hermann Kern had dared to defend Herbartian philosophy against Trendelenburg’s authority. Kern published an essay for justifying Herbartian metaphysics nine years after Logische Untersuchungen (Kern 1849). In addition to Drobisch and Kern, Ludwig Strümpell appears to be the only eminent Herbartian who had the courage to defend Herbartian philosophy in public against Trendelenburg with his essay “Einige Worte über Herbart’s Metaphysik in Rücksicht auf die Beurtheilung derselben durch Herrn Professor Trendelenburg” (1855). Even

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Trendelenburg himself was astonished to ﬁnd out how long it took for his opponents to prepare any answers to his criticism (Trendelenburg 1855, 317). In the second edition of Neue Darstellung der Logik (1851), Drobisch still subscribed to formal logic. In general, Drobisch accused Trendelenburg of making formal logic appear as if it was philosophically much less sophisticated than it really was. In particular he emphasized that Trendelenburg’s statement about formal logic totally separating thoughts from their objects is incorrect: Formal logic does not presuppose pure thought and does not attempt to analyze or explain the forms of thought in abstracto. . . . Formal logic does not recognize forms without content. It only recognizes such forms that are independent of particular contents which they might fulﬁl. Contents, which they cannot completely do without, remain thus indeterminate and accidental. (Drobisch 1851, IV) Drobisch also still held that there is no insurmountable dividing wall between Aristotelian logic and formal logic (ibid., III–XIV). A year later Drobisch admitted, in a journal article “Ueber einige Einwürfe Trendelenburg’s gegen Herbart’sche Metaphysik” (1852), that it might have been a good idea to reply on Logische Untersuchungen a little sooner. However, he thought that it still was not too late to break that silence. This article was, above all, an act in defense of Herbartian metaphysics. When it comes to Trendelenburg’s arguments against Herbartian philosophy of logic, this time Drobisch only referred brieﬂy to the preface to the second edition of his Neue Darstellung der Logik (ibid., 11–12) Evidently Trendelenburg had expected more vivid reactions to the “twoedged critique” of his Logische Untersuchungen. This is at least what the critique itself (Trendelenburg 1840, 4–99), his reply to Drobisch (Trendelenburg 1855), and its extension (Trendelenburg 1867) suggest. Perhaps he had even planned the ﬁrst two chapters of Logische Untersuchungen rather as an opening of a polemic than as a coup de grâce. At least Hegelian philosophers reacted a little faster. Hegelians were naturally very sore with Trendelenburg’s criticism. Diﬀerences between their published reactions were largely due to diﬀerent personal temperaments. For instance, the leading Hegelian of the 1840s, Karl Rosenkranz (1805–1879), could not quite understand how “a man, who is so throughly familiar with Aristotle’s philosophy, [could] have sunk so deep that he denies νοησιζ τηζ νοησεως from νους” (Rosenkranz 1844, xviif.). There were, however, other Hegelians who did not manage to keep themselves as dispassionate as Rosenkranz. Karl Michelet characterized Trendelenburg’s philosophy as “jumble” (Michelet 1861, 126), and Arnold Ruge wrote, in Deutsche Jahrbücher für Wissenschaft und Kunst, that those who are dull enough to be unable to recognize the progress Hegel has stimulated have no scientiﬁc importance—and even less do they possess positive political credibility. Their work is still-born,

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a matter of deepest ignorance and complete lack of strength. Those who cannot digest Hegel, cannot digest either the heroes of the German Spirit: Luther, Leibniz and Kant! (cited in Petersen 1913, 158) Trendelenburg’s victory over both of his opponents seems to have been undisputable. In 1859 Rosenkranz conﬁrmed, in his Wissenschaft der logischen Idee (1858/59), that as a consequence of Trendelenburg’s censure in his Logische Untersuchungen the whole discussion around Hegelian dialectic had come to durable stagnation—the advance of Hegelian philosophy had ceased. Some 30 years after Logische Untersuchungen, Hermann Bonitz wrote, in his memorial essay to Trendelenburg, that “in any case it is true that now, after three decades, the substantial inﬂuence of Hegelian philosophy has been conﬁned to a very modest group of faithful adherents and that Trendelenburg has had a considerable eﬀect on this change with his criticism” (Bonitz 1872, 23). Forty years after Logische Untersuchungen, Friedrich Harms valued Trendelenburg’s contribution to the nineteenth-century philosophy of logic as easily the most signiﬁcant. Either neglecting or not knowing what for instance Gottlob Frege and Ernst Schröder had recently accomplished, he wrote that Trendelenburg’s Logische Untersuchungen is the latest signiﬁcant attempt to reform logic. . . . we are living in a time of fragmentary eﬀorts to reform logic. These attempts do not have any accurate continuity. They attempt to remodel logic from greatly varying starting-points and with greatly varying results. The future will make the best out of what Lotze, Ulrici, Ueberweg, Chr. Sigwart, and others have accomplished. (Harms 1881, 238)

References Aristotle. 1928. Metaphysica. The Works of Aristotle. Translated into English, vol. 8, ed. W. D. Ross. Oxford: Clarendon Press. Beneke, Friedrich E. 1842. System der Logik als Kunstlehre des Denkens. Berlin: Dümmler. Bonitz, Hermann. 1872. Zur Erinnerung an Friedrich Adolf Trendelenburg. Abhandlungen der königlichen Akademie der Wissenschaften zu Berlin aus dem Jahre 1872. Berlin: Buchdruckerei der königlichen Akademie der Wissenschaften, 1–39. Boole, George. 1854. An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities. London: Walton & Maberly; Cambridge: Macmillan. De Morgan, Augustus. [1860] 1966. Logic. In De Morgan, On the Syllogism and Other Logical Writings, ed. Peter Heath, 247–270. Reprint, London: Routledge & Kegan Paul. Drobisch, Moritz W. 1836. Neue Darstellung der Logik nach ihren einfachsten Verhältnissen. Nebst einem logisch-mathematischen Anhange. Leipzig: Voß. Drobisch, Moritz W. 1851. Neue Darstellung der Logik nach ihren einfachsten Verhältnissen, mit Rücksicht auf Mathematik und Naturwissenschaft, Zweite, völlig umgearbeitete Auﬂage. Leipzig: Voß.

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Drobisch, Moritz W. 1852. Ueber einige Einwürfe Trendelenburg’s gegen Herbart’sche Metaphysik. Zeitschrift für Philosophie und philosophische Kritik (Neue Folge) 21: 11–41. Gauthier, Yvon. 1984. Hegel’s Logic from a Logical Point of View. In Logic, Methodology and Philosophy of Science VI. Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science, Hanover, 1979. Studies in Logic and Foundations of Mathematics, no. 104, eds. L. Jonathan Cohen, Jerzy Łoś, Helmut Pfeiﬀer, and Klaus-Peter Podewski, 303–310. Amsterdam, New York, Oxford: North-Holland. Harms, Friedrich. 1874. Die Reform der Logik (aus den Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin). Berlin: Vogt. Harms, Friedrich. 1881. Philosophie in ihrer Geschichte, Zweiter Theil, Geschichte der Logik. Berlin: Hofmann. Hegel, Georg W. [1812] 1923. Wissenschaft der Logik. Erster Band: Die objective Logik. Ed. Georg Lasson. Reprint, Leipzig: Meiner. Hegel, Georg W. [1830] 1911. Encyclopädie der philosophischen Wissenschaften im Grundrisse. Zum Gebrauch seiner Vorlesungen. Dritte Ausgabe. Ed. Georg Lasson. Reprint, Leipzig: Meiner. Herbart, Johann F. [1808] 1887. Hauptpuncte der Logik. In the 2nd edition of Hauptpuncte der Metaphysik, ed. Karl Kehrbach, 217–226. Reprint, Langensalza: Beyer & Söhne. Herbart, Johann F. [1813] 1891. Lehrbuch zur Einleitung in die Philosophie, ed. Karl Kehrbach. Reprint, Langensalza: Beyer & Söhne. Herbart, Johann F. [1831] 1897. Kurze Encyklopädie der Philosophie aus praktischen Gesichtspuncten entworfen, ed. Karl Kehrbach. Reprint, Langensalza: Beyer & Söhne. Herbart, Johann F. 1836. Rezension von Neue Darstellung der Logik nach ihren einfachsten Verhältnissen. Nebst einem logisch-mathematischen Anhange. Von M. W. Drobisch. Göttingische gelehrte Anzeigen 10: 1267–1274. Kakkuri, Marja-Liisa. 1983. Abstract and concrete: Hegel’s logic as logic of intensions. Ajatus 39: 40–106. Kant, Immanuel. [1787] 1904. Kritik der reinen Vernunft. Zweite Auﬂage, ed. by Königlich Preußische Akademie der Wissenschaften. Reprint, Berlin: Reimer. Kern, Hermann. 1849. Ein Beitrag zur Rechtfertigung der herbartschen Metaphysik. Coburg: Gymnasium in Coburg. Michelet, Karl L. 1861. Die dialektische Methode und der Empirismus. In Sachen Trendelenburgs gegen Hegel. Der Gedanke 1: 111–126, 187–201. Nicolin, Friedhelm and Otto Pöggeler. 1959. Zur Einführung. In Hegel, Enzyklopädie der philosophischen Wissenschaften im Grundrisse, IX–LII. Hamburg: Meiner. Peckhaus, Volker. 1997. Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert. Berlin and New York: Akademie Verlag. Petersen, Peter. 1913. Die Philosophie Friedrich Adolf Trendelenburgs. Ein Beitrag zur Geschichte des Aristoteles im 19. Jahrhundert. Hamburg: Boysen. Plato. 1953. Parmenides. In The Works of Plato, Translated into English with Analyses and Introductions by B. Jowett, vol. 2, 669–718. Oxford: Clarendon Press. Rabus, Georg L. 1880. Die neuesten Bestrebungen auf dem Gebiete der Logik bei den Deutschen und Die logische Frage. Erlangen: Deichert.

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Rosenkranz, Karl. 1844. Georg Wilhelm Friedrich Hegel’s Leben. Supplement zu Hegel’s Werken. Berlin: Duncker und Humblot. Rosenkranz, Karl. 1858/59. Wissenschaft der logischen Idee. Königsberg: Bornträger. Strümpell, Ludwig. 1855. Einige Worte über Herbart’s Metaphysik in Rücksicht auf die Beurtheilung derselben durch Herrn Professor Trendelenburg. Zeitschrift für Philosophie und philosophische Kritik (Neue Folge) 27: 1–34, 161–192. Thiel, Christian. 1982. From Leibniz to Frege: Mathematical Logic between 1679 and 1879. In Logic, Methodology and Philosophy of Science VI. Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science, Hanover, 1979. Studies in Logic and Foundations of Mathematics, no. 104, eds. L. Jonathan Cohen, Jerzy Łoś, Helmut Pfeiﬀer, and Klaus-Peter Podewski, 755–770. Amsterdam, New York, Oxford: North-Holland. Trendelenburg, Adolf. 1840. Logische Untersuchungen. Berlin: Bethge. Trendelenburg, Adolf. 1842. Zur Geschichte von Hegel’s Logik und dialektischer Methode. Die logische Frage in Hegel’s Systeme. Eine Auﬀoderung [sic] zu ihrer wissenschaftlichen Erledigung. Neue Jenaische Allgemeine Literatur-Zeitung 1(97): 405–408; 1(98): 409–412; 1(99): 413–414. Trendelenburg, Adolf. [1853] 1855. Ueber Herbart’s Metaphysik und eine neue Auﬀassung Derselben. In Historische Beiträge zur Philosophie, Zweiter Band, vermischte Abhandlungen, 313–351. Berlin: Bethge. Trendelenburg, Adolf. 1867. Ueber Herbart’s Metaphysik und neue Auﬀassung Derselben, Zweiter Artikel. In Historische Beiträge zur Philosophie, Dritter Band, vermischte Abhandlungen, 63–96. Berlin: Bethge. Twesten, August. 1825. Die Logik, insbesondere der Analytik. Schleswig: Königlichen Taubstummen-Institut. Ueberweg, Friedrich. 1923. Grundriss der Geschichte der Philosophie, Vierter Theil, Die deutsche Philosophie des neunzehnten Jahrhunderts und der Gegenwart. Berlin: Mittler und Sohn. Ulrici, Hermann. 1869/70. Zur logischen Frage. (Mit Beziehung auf die Schriften von A. Trendelenburg, L. George, Kuno Fischer und F. Ueberweg.) I. Formale oder materiale Logik? Verhältniß der Logik zur Metaphysik. Zeitschrift für Philosophie und philosophische Kritik (Neue Folge) 55: 1–63; II. Die logischen Gesetze, ibid.: 185–237; III. Die Kategorieen, 56: 1–46; IV. Begriﬀ, Urtheil, Schluß, ibid.: 193–250. Vilkko, Risto. 2002. A Hundred Years of Logical Investigations. Reform Eﬀorts of Logic in Germany, 1781–1879. Paderborn: Mentis.

7

The Relations between Logic and Philosophy, 1874–1931 Leila Haaparanta

One who seeks to discuss the relations between logic and philosophy in the nineteenth century and the early twentieth century has to pay special attention to his or her use of the term “logic.” In the context of nineteenth-century and early twentieth-century philosophy, that term occasionally refers to similar activities to those we now call logic. In those days, logic could mean what we nowadays tend to call logic proper, that is, working with formal systems that resemble those of mathematics. However, it could also mean activities that we would now wish to label as “epistemology,” “philosophy of science,” “philosophy of language,” or “philosophy of logic.” Therefore, it may sound strange to promise to discuss the relations between nineteenth-century and early twentieth-century logic and philosophy. It is more to the point to claim that this chapter gives a survey of the ﬁeld of philosophy where (1) the philosophical foundations of modern logic were discussed and (2) where such themes of logic were discussed that were on the borderline between logic and other branches of the philosophical enterprise, such as metaphysics and epistemology. What will be excluded in this chapter are the formal developments on the borderline between logic and mathematics, hence, contributions made by such logicians as Augustus De Morgan (1806–1871), George Boole (1815–1864), Ernst Schröder (1841–1902), and Giuseppe Peano (1858–1932), for example (see chapters 4 and 9 in this volume). Gottlob Frege (1848–1925) and Charles Peirce (1839–1914) are included, since their work in logic is closely related to and also strongly motivated by their philosophical views and interests. In addition, this chapter pays attention to a few philosophers to whom logic amounted to traditional Aristotelian logic and to those who commented on the nature of logic from a philosophical perspective without making any signiﬁcant contribution to the development of formal logic. 222

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The choice of the years 1874 and 1931 has its reasons. In 1874 Franz Brentano (1838–1917) held his inaugural lecture “Über die Gründe der Entmutigung auf philosophischem Gebiete” in Vienna. The lecture showed the way to philosophers who made a sharp distinction between subjective psychological acts studied by the empirical sciences and the objective contents of those acts represented by means of logic (Brentano 1929, 96). Basically the same distinction had already been made by Bernard Bolzano (1781–1848) in his Wissenschaftslehre (1837). In 1931, which is the last year that is taken into account in this chapter, a volume of Erkenntnis was published which contained Rudolf Carnap’s (1891–1970) criticism of Martin Heidegger (1889–1976) titled “Überwindung der Metaphysik durch logische Analyse der Sprache” and Arendt Heyting’s (1898–1980) “Die intuitionistische Grundlegung der Mathematik,” which was inspired by Edmund Husserl’s (1859–1938) and Heidegger’s thoughts. Those articles were important in view of the division of philosophical schools in the twentieth century. This chapter is far from being the whole story of the relations between logic and philosophy 1874–1931. Instead, it consists of a number of themes and opens up a few perspectives on the period. There is slight emphasis on German philosophy. The chapter focuses on Frege, Husserl, and Peirce. Frege and Peirce are chosen because of their central role in the development of modern logic. Husserl is chosen because he wrote a great deal on the philosophical problems related to the logical enterprise. If we use the labels of our time, we would say that Husserl was one of the most important philosophers of logic of his own time.

1. The Historical Setting, 1874–1931 Even if Kant thought that no signiﬁcant changes are possible in logic, his own transcendental logic raised several new themes that we could now call philosophy of logic. Transcendental logic was philosophy of certain logical categories, especially of their metaphysical limits and epistemological import. After Kant, the role of those categories was discussed in various ways. There were philosophers such as Johann Gottlieb Fichte (1764–1814), Friedrich Wilhelm Schelling (1775–1854), and G. W. F. Hegel (1770–1831) who anchored logical categories to the world, who argued that logical categories are categories of being, of what there is (see chapter 6). In the second half of the century, the situation changed. Philosophers started to debate on the relation between logic and psychology. That debate increased interest in the epistemological questions related to logic, but it also brought about a new formulation of metaphysical or ontological problems. The basic question was no longer what the most general structure of the world is. Instead, philosophers pondered on whether there was a speciﬁc abstract realm that had thoughts as its denizens and that logic could represent. In his Lehrbuch der Logik (1920), Theodor Ziehen listed and characterized various groups of nineteenth-century logic (Ziehen 1920, 155–216). The main

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opposition in the last decades of the century was that between psychologists and antipsychologists. In the ﬁrst half of the century, there were a number of psychologists such as Friedrich Beneke (1798–1854), Otto Friedrich Gruppe (1804–1876), William Whewell (1794–1866), and August Comte (1798–1857). In Germany several schools arose in the late nineteenth century that attacked psychologism, such as neo-Kantians like Hermann Cohen (1842–1918) and Paul Natorp (1854–1924) and logicists like Frege and Husserl. Husserl’s early views are usually considered psychologistic. Among logicists there were philosophers whom Ziehen called value-theoretical logicists, such as Wilhelm Windelband (1848–1915) and Heinrich Rickert (1863–1936), and moderate logicists like Hermann Lotze (1817–1881) and Gustav Teichmüller (1832–1888). Moderate or weak psychologists included several thinkers, for example, Christoph Sigwart (1830–1904), Wilhelm Wundt (1832–1920), Benno Erdmann (1851–1922), and Theodor Lipps (1851–1914).1 The contents of these doctrines will be clariﬁed later in this chapter. The opposition between logic and metaphysics became important in a new way in the beginning of the twentieth century, when Carnap raised criticism against Heidegger’s views. Researchers who have tried to trace the origin of the distinction between the analytic and the phenomenological, more generally, the Continental tradition, have paid attention to the debate between Carnap and Heidegger concerning the relation between logic and metaphysics. Various interpretations can be proposed concerning the core of Carnap’s criticism. It is not clear how Heidegger would have defended his view or attacked Carnap’s position. Michael Friedman (1996, 2000), among others, has studied the theme by taking the historical context into account. He has argued that the roots of Carnap’s thought were in the neo-Kantianism of the Marburg school, while Heidegger’s philosophy ensued from the Southwest school. According to Friedman, that diﬀerence largely explains the fact that Carnap emphasized the role of logic, while Heidegger stressed the centrality of questions concerning human beings and their values. There are various ways of making the distinction between the analytic and the phenomenological tradition. Several criteria have been suggested, such as their attitudes toward the history of philosophy, toward their own history, toward science, and toward the idea of scientiﬁc philosophy; their views on what are the central problems of philosophy, the objects of philosophical research, and the methods of philosophy; and their attitudes toward the ideal of clarity in philosophy. It has been suggested that views on the relation between logic and metaphysics are an important criterion if we wish to divide philosophers into the two camps. The diﬀerent criteria turn out to be problematic in closer scrutiny.2 One who seeks to locate the diﬀerences between the two traditions cannot ignore the fact that there were at least two ideas that early phenomenology, especially Husserl’s thought, and most of early analytic philosophy shared. First, there was the idea of pure philosophy, which presupposed a belief in the sharp distinction between knowledge a priori and knowledge a posteriori. That

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belief had one of its origins in late nineteenth-century antipsychologism. Second, there was the belief in the method of analysis as the method of philosophy. That method, though in diﬀerent versions, was used by Frege, the “godfather” of analytic philosophy, and Husserl, the pioneer of phenomenology. These two common features were intertwined in various ways.3 Carnap’s and Heidegger’s debate had its background in lively discussion concerning the philosophy of logic and mathematics that was going on particularly in German philosophy at the end of the nineteenth century and at the beginning of the twentieth century. Hermann Lotze (1817–1881), who was professor at the University of Göttingen, inﬂuenced a number of those who took part in the discussion. On Lotze’s view, objectivity is not the same as that actuality (Wirklichkeit) which belongs to concrete beings. Lotze also regarded abstract objects like thoughts and values as objective in the sense that they are valid. Frege was one of Lotze’s students, and so was Bruno Bauch, Frege’s colleague in Jena, a neo-Kantian philosopher and the founder of the society for German idealism. Frege also belonged to that society. Heinrich Rickert, who was professor in Freiburg, was also inﬂuenced by Lotze’s philosophy. Carnap was a student of Bauch’s, while Heidegger was a student of Rickert’s. Rickert and Windelband were central ﬁgures of the neo-Kantianism of Southwest Germany.4 Frege’s philosophical environment was not the Southwest school but rather the Marburg school. Frege was most likely to receive his concept of truth value from Windelband, who was one of the so-called value-theoretical logicists; they were philosophers who thought that besides the moral values the realm of values includes the truth values studied by logic.5 Like Frege, Husserl criticized psychologists and held the view that logic and mathematics study abstract objects, such as numbers and thoughts, that is, the structure of thoughts and the inferential relations between thoughts. At the beginning of the twentieth century, Husserl was professor in Göttingen, until he moved to Freiburg in 1916, to follow Rickert in the professorship. Husserl’s follower in Freiburg was Heidegger. Frege was a logicist in two meanings, Husserl in one meaning of the word. A competing doctrine, namely formalism, was represented by David Hilbert (1862–1943), who was Husserl’s colleague and friend in Göttingen. In the philosophy of mathematics, logicism meant two things; on the one hand, it was the view that arithmetic or even the whole of mathematics can be reduced to logic; on the other hand, it was the view that numbers are abstract objects that are independent of the human mind. On this latter view, mathematical knowledge has to do with these very objects. Frege’s logicist program had to do with arithmetic, and it included deﬁning the concept of number by means the concepts of “extension of a concept” and “equinumerous.” Frege took extensions of concepts to be logical objects.6 Husserl’s studies in the foundations of logic and mathematics were closely connected to the rise of phenomenology. Logicism in the latter meaning was a natural starting point of Husserl’s phenomenology; namely, if the objects

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of logic and mathematics have their origins in consciousness, even if not in empirical consciousness studied by psychology, as Husserl argued, one has to ﬁnd out how to make the distinction between empirical and non-empirical consciousness. If psychology is interested in empirical consciousness, and if philosophy, including the philosophy that studies the relations between the subject and logic and mathematics, is interested in pure consciousness, how are these consciousnesses distinguished from each other?7 Are we dealing with ontologically two diﬀerent consciousnesses, or are we talking about two diﬀerent points of view to the same consciousness? If the latter holds, what do we mean by saying that in the last analysis it is the same consciousness we are talking about, and if it is one and the same, how can we justify the claim that it is one and the same consciousness? Several diﬃcult ontological and epistemological questions arise. Therefore, it is not surprising that Husserl moved from studies in the philosophical foundations of logic to studies of consciousness. Husserl asked how logic as science is possible. He wished to justify the ﬁeld of knowledge called logic, but it often seems that he also wished to justify a certain logic, namely, classical logic, by studying its origin in consciousness. There has been a debate on whether Husserl wished to take a position on the correctness of logical systems, and if he did, whether he was a conservative or a revisionist in logic. One has raised the question whether Husserl would have suggested giving up the law of excluded middle or any other law of logic, if we cannot ﬁnd philosophical justiﬁcation for those laws. Phenomenologists often emphasize the incommensurability of philosophy and the sciences. We could think that Husserl’s philosophical studies and logicians’ debate on the acceptability of various logics are incommensurable. Dieter Lohmar has, for his part, sought to show that Husserl was a moderate revisionist. In his view, Husserl thought that it is possible that we cannot ﬁnd justiﬁcation for all laws of classical logic.8 Radical revisionists in logic were Oskar Becker (1889–1964) and Heyting, who sought to change logic on the basis of Husserl’s and Heidegger’s thought. They tried to develop intuitionistic logic by using Husserl’s concepts of meaning intention and meaning fulﬁllment or disappointment.9 As noted, one of Heyting’s papers, inspired by Husserl and Heidegger, came out in the same volume of Erkenntnis where Carnap had his criticism of Heidegger, the criticism where Carnap sought to show by means of logic that Heidegger’s sentences are meaningless. In this chapter, I proceed as follows. First, I consider logic as a category theory, hence, as a doctrine that is interested in the categories of thought and being. I discuss those views in which logic is understood as the ideal language that mirrors reality in the right way. Husserl’s formal ontology is related to these doctrines. This consideration brings us to Heidegger’s view of metaphysics. I then take up the ideas of one world and of the plurality of worlds and consider the theories of modalities. The doctrine of three realms was much discussed in the late nineteenth century. I will consider the acknowledgment of the third realm and the reasons for such an acknowledgment. This question

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is tied to the debate on psychologism and antipsychologism and the problem of the objectivity of the realm of thoughts that logic speaks about. The important point made by antipsychologists was epistemological rather than ontological. That point brings us to the question of the possibility of logical knowledge and the various ways of answering the question given by Kant, Frege, and Husserl. I then continue with epistemological considerations and discuss Frege’s idea that in logical inference no intuitive gaps are allowed. If a logical theorem is justiﬁed, there is no reference to intuition in the inferential chain. In addition, philosophers raised the question concerning the justiﬁcation of traditional logical laws and a speciﬁc logical language. Husserl was one of those who raised such questions. That theme also brings us to intuitionistic logic and to the relations between logic and experience. Finally, Frege’s and Peirce’s methodologies of logic are discussed, and Frege’s semantic views are presented.

2. The Relations between Logic, Metaphysics, and Ontology In nineteenth-century logic and philosophy, logic was often understood contentually or materially (inhaltlich). The idea that logic has content received various meanings. (1) Logic was regarded as contentual in the sense that it was assumed to speak about the objects of the world. Kant’s transcendental logic was contentual in this sense in a peculiar way; it showed us the form of the phenomenal world. Hegelian logic was contentual, because it sought to mirror the historical development of reality. (2) Logic was taken to be contentual in the sense of being transcendental, that is, being a picture of the a priori conditions of all thought. (3) Logic was thought to have content in the sense that it was assumed to speak about the objects of the abstract realm, that is, to convey thoughts, which were considered objective. (4) Logic was thought to have content in the sense that it was assumed to mirror the structure of the psychological realm. Philosophers who regarded logical categories as categories of being or as categories of objects of knowledge and experience represented the ﬁrst or the second position. Leibniz and Kant belonged to that tradition of logic, even if their views otherwise diﬀered radically. Frege also thought that an ideal language can be discovered that is the correct mirror of the universe. However, Frege is also famous for his writings about objective thoughts and of his view of logic as a representative of the realm of abstract objects. Hence, besides being a mirror of all that there is, for Frege logic was a mirror of a speciﬁc realm; the problem for interpreters has been whether Frege considered that realm in the framework of epistemology only, or whether he regarded its objects as having an ontological status. This doctrine, whether in its epistemological version or both its epistemological and its ontological version, was in opposition with the fourth doctrine listed, which was called psychologism.

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2.1. The Leibnizian Starting Point: Logic as the Mirror of Reality Leibniz was the most prominent of the pre-Fregean thinkers who maintained that the terms of our natural language do not correspond to the things of the world in a proper way and that we should therefore construct a new language which mirrors correctly the whole universe. One important characteristic of modern logic was that unlike traditional logic, it proposed a new language— mathesis universalis, lingua characteristica, begriﬀsschrift, or whatever it was called by various authors. Modern logicians, primarily Frege, wished to establish a new language that mirrors the world and replaced the grammatical subjectpredicate analysis of sentences by the argument-function analysis. Therefore, the term “linguistic turn,” as applied to Frege, may lead us astray, if we do not remember that Frege also turned away from language. That is, unlike traditional logicians, he paid little attention to grammatical concepts like those of subject and predicate in his logical studies. In addition to the dream of ideal language, there was the idea of calculus strongly emphasized by Boole and his followers. It meant the eﬀort to formulate the rules of logical inference explicitly by presenting logical and non-logical vocabulary, formation rules, and transformation rules. Boole stated as follows: We might justly assign it as the deﬁnitive character of a true Calculus, that it is a method resting upon the employment of symbols, whose laws of combination are known and general, and whose results admit of a consistent interpretation. (Boole 1965, 4) The nineteenth century saw a breakthrough of the two ideas, even if emphases varied among logicians. Frege stressed that he did not want to put forward, in Leibniz’s terms, only a calculus ratiocinator, by which he primarily meant the rules of logical inference. He argued that his conceptual notation was to be a lingua characterica, which was the term that he used for Leibniz’s lingua characteristica. That is, his notation was to be a proper language which speaks about all that there is.10 Frege raised criticism against Boole, because in his view Boole merely focused on developing a Leibnizian calculus in his logical works. However, this was not exactly what Boole himself thought of his project, because he included the idea of logic as a mental or philosophical language in his philosophical remarks on logic (Boole 1958, 11, and Boole 1965, 5). It has been argued in the literature that since diﬀerent logicians emphasized diﬀerent sides of the Leibnizian project, they ﬁnally came to advocate conﬂicting views of the basic nature of logic. It has been claimed that Boole, Peirce, and Schröder, for example, were inclined to stress the importance of developing a calculus, whereas Frege and the early Russell were among those who laid more emphasis on the idea of logic as a universal language. The systematic consequences of the two views have been studied by a number of authors, especially Jean van Heijenoort (1967), Warren D. Goldfarb (1979), and Jaakko Hintikka (1979, 1981a, 1981b). According to these studies, those

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who stressed the idea of logic as language thought that language speaks about one single world. This was the position to which Frege was committed. He thought that there is one single domain of discourse for all quantiﬁers, as he assumed that any object can be the value of an individual variable and any function must be deﬁned for all objects. This is what was stated by his principle of completeness (Grundsatz der Vollständigkeit) (GGA II, §§56–65). On the other hand, those who supported the view that logic is a calculus were ready to give various interpretations or models for their formal systems. This appears to have been Boole’s standpoint. Boole wrote: Every system of interpretation which does not aﬀect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. (Boole 1965, 3) However, it is not clear how this passage ought to be interpreted. It is noteworthy that Boole’s statement is not far from what Frege thought. Frege also wished to construct such a language as can be applied to various ﬁelds like arithmetic and geometry (BS, 1964, “Vorwort,” XII). However, ﬁelds of application are not what is meant by the distinction between the one-world and the many-world view. Moreover, Frege wanted to develop both a language and a calculus; if he wanted to develop them as they are understood by contemporary scholars, he could not consistently support both of the implications stressed by those scholars, that is, he could not preach for the one-world view and for the plurality of worlds at the same time. The twentieth-century perspective has also given more content to the two views. It has been claimed that those who support the idea of logic as language tend to think that they cannot step beyond the limits of language and that this prevents them from developing a proper semantic theory for their language. On the other hand, it has been argued that those who endorse the view of logic as calculus are inclined to think that it is possible to look at a formal system, as it were, from the outside and develop a semantic theory for it. For example, even if Frege introduced his doctrine of senses (Sinne) and references (Bedeutungen), which is a semantic doctrine, he did not believe that he could propose a proper semantic theory for a formal or a natural language. He repeatedly pointed out that he can only give suggestions and clues concerning his basic semantic concepts and the semantic properties of his conceptual notation.11 Frege made the distinction between language and calculus on the basis of his interpretation of Leibniz’s project, but he was not conscious of all the implications of the two views of logic which have been detected in the literature. Hence, there are at least three diﬀerent (though closely connected) stories to be told, as far as the ideas of a universal language and calculus are concerned. There is the story of the content which Leibniz gave to his idea, the story Frege and Boole told about their projects, and the story told from the

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twentieth-century perspective that tries to capture the far-reaching systematic implications of the two extreme positions. In Frege’s hands, the dream of a universal language was tied to the task of philosophy. Otherwise Frege did not write much about the task of philosophy. In the beginning of his Begriﬀsschrift (1879) he writes that if one task of philosophy is to free the human mind from the power of word by revealing the mistakes that are often almost unavoidably caused by the use of language, then his conceptual notation, which has been constructed for this purpose, will be a useful tool for a philosopher (Frege, BS, 1964, XII–XIII). Frege often complains that natural language leads us astray. However, he nowhere states that it would be the only task of philosophy to clarify language. There is one story to be told concerning the relations between Kant and Frege which illuminates Frege’s position among the opponents and the supporters of metaphysics. In the preface of his Begriﬀsschrift, Frege states that he tries to realize Leibniz’s idea of lingua characterica. The term was most likely to come from the Leibniz edition by J. E. Erdmann from the years 1839 and 1840, as the word characterica is used there instead of the word characteristica used by Leibniz (see Haaparanta 1985, 102–117). Adolf Trendelenburg also used the same word in his writing “Über Leibnizens Entwurf einer allgemeinen Charakteristik” (1867). According to Trendelenburg, philosophers ought to construct a Leibnizian universal language, Begriﬀsschrift, by taking Kant’s theory of knowledge into account. In his view, Kant’s contribution was that he distinguished the conceptual and the empirical component of thought and stressed the importance of studying the conceptual component. Trendelenburg also tells us about Ludwig Benedict Trede, who in his article “Vorschläge zu einer nothwendigen Sprachlehre” in 1811 tried to create a universal language by following Leibniz and Kant. Frege also called his language conceptual notation, which he, it is true, took to be a less successful name for it. He also used the expression “the formula language of pure thought” in the subtitle of his book Begriﬀsschrift and the expression “the intuitive representation of the forms of thought” in his article “Über die wissenschaftliche Berechtigung einer Begriﬀsschrift” (1882) (Frege, BS, 1964, 113–114). The above-mentioned connections have been noticed and also stressed by a few scholars several years ago (see Sluga 1980; Haaparanta 1985). Even if there were no similarities whatsoever between Trede’s notation and Frege’s language, we can say that by his reference to Trendelenburg Frege told us something about the philosophical background of his conceptual notation. On the basis of what has been said, we may argue that Frege’s conceptual notation was itself a philosophical position taking. It was not in favor of psychologistic transcendentalism, according to which the necessary conceptual conditions which make knowledge and experience possible are typical of the human mind. Nor was it in favor of transcendental idealism, if we think that a transcendental idealist is one who acknowledges a transcendental subject. We can say, however, that Frege was a transcendentalist in a very weak sense; he tried to write down the forms of thought, which Kant would have called

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the necessary conditions of knowledge and experience. It is, of course, obvious that Frege’s conceptual notation was not a codiﬁcation of those forms that we ﬁnd in Kant’s table of categories. The Vienna Circle gave a special treatment to the new logic that Frege had developed. The manifest of the Vienna Circle was directed against metaphysics, and the same spirit can be found in many other writings of the members of the circle. In the manifest, the new logic was described as a neutral system of formulas, a symbolism which is free from the slag of the historical languages, as a tool by means of which it is possible to show that the statements made by metaphysicians and theologians are pseudo-statements, that they express feeling of life, which would be properly expressed by art. The Vienna Circle regarded the close relation with traditional languages as the main problem of metaphysics. They also blamed metaphysics for assuming that thought can know itself without empirical material; that kind of knowing was sought by transcendental philosophy. The Vienna Circle declared that it is not possible to develop metaphysics from “pure thought” (Der Wiener Kreis 1973, 308). They believed that logical analysis overcomes not only scholastic metaphysics but also Kantian and modern apriorism. That position taken by the Vienna Circle meant the rejection of synthetic judgments a priori and hence the rejection of transcendental knowledge. Hence, if we draw a line from Kant to Frege and further to the Vienna Circle, there is a crucial change in how the relations between being and the pure forms are understood. It is as early as in his Allgemeine Erkenntnistheorie (1918) that Schlick raised the question of whether there are any pure forms of thought and answered that thought with its judgments and concepts does not press any form on reality (Schlick 1918, 304–305). For Schlick, that means the repudiation of Kant’s philosophy (ibid., 306). In his article “Die Wende der Philosophie” (1930) he argued that the greatest change is due to a new insight concerning the nature of the logical, which was made by Frege, Russell, and particularly Wittgenstein. According to that new understanding, the pure form is merely the form of an expression, but that form cannot be presented (Schlick 1938, 33–34). It is true that Frege did not present the system of signs called conceptual notation, if presenting it had meant giving a semantic theory for the system in a metalanguage. If Frege thought that forms of thought are proper objects of knowledge, that knowledge was for him a kind of immediate recognition. Recognition of the correct forms, the result of which is the creation of conceptual notation, can be called immediate intellectual seeing or intuition. In his late writings in 1924 and 1925, Frege stressed that we see correctly, if natural language does not disturb our intellectual seeing. Moreover, when Frege discussed certain important features of his language, such as the distinctions between the diﬀerent meanings of “is,” which are existence, predication, identity, and class inclusion, he gave lengthy arguments for the distinctions. One of the most central reasons he put forth was that his new language takes care of the diﬀerence between individuals and concepts, which is missing both in Aristotelian logic and in Boole’s logic, and that the

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diﬀerence is mirrored by the distinction between identity and predication as well as by the distinction between predication and class inclusion. For example, to preserve the distinction between objects and concepts, Frege considered it necessary to realize that the “is” of identity diﬀers from the “is” of predication and, moreover, that this distinction reﬂects how things really are (“Über Begriﬀ und Gegenstand,” 1892, KS, 168). Moreover, the motivation for denying that existence is a ﬁrst-order predicate came from Kant’s thought. Frege also gave a positive contribution by trying to tell what existence is, namely, that it is a second-order concept. We can say that Frege had not only a view of the word “being” but also a view of the forms of being, which are forms of thought, and those forms were meant to be codiﬁed as his ideal language.12 There was a well-known controversy between logic and metaphysics in the early days of the analytic tradition and the phenomenological movement, to which I already referred. The Vienna Circle declared in 1929 that the new logic, the ideal language developed by Frege, Russell, and Whitehead, frees philosophy from considering the true nature of reality. It was believed that by means of the new formula language, it was possible to show that metaphysical statements are meaningless. It was not thought that the very ideal language would have a metaphysical content. For a logical empiricist, Heidegger’s philosophy was an example of the meaninglessness of metaphysics. In 1931 Carnap published his article “Überwindung der Metaphysik durch logische Analyse der Sprache,” in which he studied Heidegger’s sentences and stated that the sentences of a metaphysician cannot be combined with the ways in which logic and science proceed. In his Was ist Metaphysik? (1929) as well as in the afterwords of its later editions, Heidegger discussed the criticism that had been raised against the way he used the word “nichts.” According to Heidegger, nothing is the origin of negation, not the other way round. His message was that logic has its origin in the being of Dasein (Heidegger 1992, 37) and philosophy can never be measured by means of the standards of the idea of science (ibid., 41). Hence, for Heidegger the origin of the logical concept of being was the being of Dasein. There thus seemed to be a sharp contrast between Heidegger, who spoke about the meaning of being and a linguistic philosopher who spoke about the diﬀerent meanings of the word “is.” It was Frege who distinguished the diﬀerent meanings of “is” in his conceptual notation, and therefore it may seem that Frege was clearly among those who wished to limit the talk about being to the word “is.” This is not the case, as Frege was not an opponent of metaphysics. It is more to the point to say that Frege’s thought lay somewhere between the philosophy of the Vienna Circle and Heidegger’s fundamental ontology. The view of philosophy held by the Vienna Circle was characterized by the fact that philosophy was taken to be an art of using a tool and the good tool was Frege’s, Russell’s, and Whitehead’s formula language. However, the language lost the metaphysical content that it had for Frege. The pure forms were interpreted as the forms of a system of signs; the system of signs was no

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more “an intuitive representation of the forms of thought,” as Frege wrote. In its manifest the Vienna Circle declared that there are no depths in science but there is surface everywhere (Der Wiener Kreis 1973, 306). In that sense, the circle also wanted philosophy to be like science. Both Frege and Heidegger were interested in the philosophical basis of logic, Frege mainly in the epistemological basis and Heidegger in the origin of logic in the being of Dasein. Both thought that there is something under the “surface.” The Vienna Circle thought that philosophy is activity; that was especially emphasized by Schlick in “The Future of Philosophy” (1931). Schlick referred to Wittgenstein, for whom philosophy was not a theory but a certain kind of activity, that is, of clarifying meanings and writing formulas which do the job of clariﬁcation (Schlick 1938, 132). It is true, the incentive for that kind of philosophizing was given by Frege, but it would be far from the truth to argue that Frege held that view. Edmund Husserl touched on the relations between logic and being in several connections, for example, when he distinguished between formal ontology and material ontologies in his Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie I (1913). It was already in the ﬁrst volume of the Logische Untersuchungen (1900) that he divided logic into two parts according to the two tasks that he believed logic to have. One of the tasks was to give the formal categories of meaning, whereas the other was to put forward the formal categories of objects. Husserl listed the basic concepts of pure logic or analytical categories both in the Logische Untersuchungen I (A 244–245/B 243–244) and in the Ideen I (Husserl 1950b, 26–32). His categories of meaning include such concepts as belong to the essence of the proposition or apophansis, such as subject and predicate, conjunctive, disjunctive, and hypothetical connections, that is, what we would call logical connectives, and the concepts of concept, proposition, and truth. In addition to the categories of meaning, he gave a list of pure formal objective categories, such as object, property, relation, state of aﬀairs, identity, whole and part, number, and genus and species. Husserl called these categories of objects substrate-categories. In the Ideen I, Husserl states that “formal ontology contains the forms of all ontologies . . . and prescribes for material ontologies a formal structure common to them all” (Husserl 1950b, 27; Kersten’s translation, 21), and then goes on with treating formal ontology and pure logic as synonymous terms. He also claims that pure truths of meaning can be converted into pure truths of objects (ibid., 28). In the Logische Untersuchungen, Husserl pays attention to the distinction between formal and empirical (or material) concepts, as well as to the distinction between formal or analytic propositions and laws and material propositions and laws. He states that concepts like something, one, object, quality, relation, association, plurality, number, order, ordinal number, whole, part, magnitude, and so on, have a basically diﬀerent character from concepts like house, tree, color, tone, space, sensation, feeling, and so on, which for their part express genuine contents. It is not clear how we should make the

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distinction between form and content. What is clear, however, is that Husserl and his contemporaries took the very distinction between form and matter, or form and content, to be essential to logical studies, whether logicians were interested in concepts or in inferences in which the concepts were used.

2.2. The Absence of the Metaphysics of Modalities Our contemporary modal logic is usually considered as an extension of the two-valued predicate calculus that was developed in the late nineteenth century. However, the roots of our modal theory reach far back to Aristotelian logic, which regarded modal logic as a legitimate branch of logical studies. Interest in modal notions is a new phenomenon among logicians only when it is considered in the framework of the developments of those late nineteenth-century logicians who are honored as the pioneers of modern logic. In the beginning of the twentieth century, logicians were not willing to discuss modal concepts. They were mainly inspired by the extensionalist program which was preached by Frege, among others, and codiﬁed in the Principia Mathematica. Modal notions seemed to escape all treatments that are interested only in references. Later in the twentieth century, logicians proposed axiomatic systems for modal logic, which, however, ﬁrst avoided all systematic semantic considerations. Since the late 1950s, they introduced and developed interpretations for the axioms of modal systems. These interpretations are useful for clarifying which systems of axioms most naturally correspond to our intuitions concerning modal notions and their relations. Leibniz is an important ﬁgure behind our contemporary modal logic. It is also known that he is an important ﬁgure behind Frege’s logical work. Nonetheless, given that Frege set out to realize what Leibniz had dreamt of, it is surprising that he was reluctant to develop modal logic in the early twentieth century. Even if he started from Leibniz’s program in arguing that we must construct a proper language that represents the world, he was not true to Leibniz’s view that there could be alternative worlds to which our ideal language would be related. Frege’s conceptual notation was meant to represent only one world. As already noted, this doctrine of Frege’s is most clearly visible in his requirement that all of the predicates of the language must be deﬁned for all objects. Frege’s formula language was thus meant to speak about all that there is, and its quantiﬁers were meant to range over all individuals. Modal logic, as we understand it nowadays, was thus blocked out in the very beginning. Frege gave another reason for his unwillingness to discuss the concepts of necessity and possibility within the limits of his logic. The reason was that those concepts do not concern logic at all but that they have to do with the nature of the grounds of our judgments (BS, §4). For Frege, logic is interested in the objective realm of thoughts.13 Frege regarded the act of judging as a psychological phenomenon, which belongs to the realm of our private minds.14 Hence, even if Frege severely criticized all eﬀorts to reduce logical laws to

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psychological laws, he restricted modal notions to the realm of psychology, thus agreeing with psychologists. He did not believe that thoughts are necessary or possible as such, but he insisted that they are necessary or possible for our private minds. Like psychologists, he connected modal concepts with the concept of certainty and took them to modify our acts of thinking, which are units of the subjective realm. Signs for modal concepts did not play any role in his ideal language (for Frege’s views in more detail, see chapter 12 in this volume).

3. The Relations between Logic, Epistemology, and Psychology 3.1. Logical Psychologism In the late nineteenth century, the question of what logic mirrors, if it mirrors something, was mostly discussed in a way that was determined by the debate on the relation between logic and psychology. Contemporary naturalism has its roots in late nineteenth-century psychologism. The word “naturalism” was also used in the late nineteenth century. Like contemporary naturalism, late nineteenth-century naturalism and its version called psychologism had various contents(see Haaparanta 1995, 1999b; Kusch 1995). In her book Philosophy of Logics (1978) Susan Haack distinguishes between strong and weak logical psychologism. According to the strong view, logic describes our thought and may also tell us how we ought to think (Haack 1978, 238). In his book Husserl and Frege (1982), J. N. Mohanty describes strong logical psychologism as a doctrine according to which logic is a branch of psychology, the laws of logic describe actual human thought, and psychological study is therefore both suﬃcient and necessary for studying the foundations of logic (Mohanty 1982, 20). In Haack’s terminology, weak logical psychologism is the view that logic determines how we ought to think (Haack 1978, 38). Mohanty, for his part, characterizes the weak version as a thesis that it is necessary but not suﬃcient to study human thinking processes if we want to clarify the theoretical foundations of logic (Mohanty 1982, 20). Many logicians who are regarded as antipsychologists (Frege, for example) might accept what Haack calls weak logical psychologism. However, they would not say that determining the norms of thought would be the only or the basic task of logic (GGA I, “Einleitung,” XV). Logical psychologism had two diﬀerent roots in nineteenth-century philosophy. First, there was an interpretation of transcendentalism which regarded the transcendental conditions of experience as the conditions determined by the mental structure of the human race. Second, there was the tradition of empiricism, which attempted to base all knowledge on experience. German, French, and British logical psychologism in the nineteenth century was so complicated a doctrine that many ways of classifying it are possible. It could

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be a doctrine concerning the basic concepts of logic, the basic laws of traditional logic, or the nature of logical inference. If logical psychologism was understood as a theory that primarily concerns the conceptual tools of logic, it either claimed that such concepts as unity, plurality, negation, and possibility are structural features of the human mind, or it argued that those concepts are abstracted from sense perception. The latter position is linked with the empiricist tradition of the modern times. The former position followed if one interpreted transcendentalism by saying that the transcendental conditions of experience are determined by the structure of our factual minds. If logical psychologism was a doctrine concerning such laws as the law of excluded middle or the law of noncontradiction, it either maintained that those laws are structural features of the human mind or claimed that those laws have their origin in sense perception. There were a number of philosophers who stressed that the laws of logic have an empirical basis in sense perception, but who did not call themselves psychologists. J. S. Mill, for example, did not want to take that label (Mill 1906, Book II). If a psychologist claimed that the basic laws of logic represent the constant and innate structures of the human mind, he regarded the laws of logic as factual in the sense that he took them to be research objects of the science called psychology. Hence, both the empiricist and the transcendentalist version of psychologism were epistemological theories that tried to reveal the natural origin of logic and to justify certain logical concepts, logical laws, and logically valid inferences by means of the revealed origin. The foregoing classiﬁcation contained two basic forms of psychologism. Husserl also hinted at a similar division, when he distinguished between empirical and transcendental psychology as two diﬀerent bases of psychologism (LU I, A 123/B 123). One of the versions abstracts such laws as the law of noncontradiction from the objects of experience, whereas the other version pushes the structure of the mental realm into the objects of experience. In his Philosophie als strenge Wissenschaft (1910–1911) and in his lecture notes “Logik als Theorie der Erkenntnis” (1910–1911) Husserl characterized naturalism in various ways.15 He stated that naturalism is a phenomenon consequent on the discovery of nature, which is to say, nature considered as a unity of spatiotemporal being subject to exact laws of nature (PsW, 79). He also remarked that psychology is concerned with “empirical consciousness,” with consciousness from the empirical point of view, whereas phenomenology is concerned with “pure consciousness,” which is consciousness from the phenomenological point of view (PsW, 91). For Husserl, the phenomenological point of view was the philosophical point of view. Moreover, he continued that any psychologistic theory “naturalizes” pure consciousness (PsW, 92). Naturalizing pure consciousness amounts to identifying it with empirical consciousness. If the realm of pure consciousness had been the realm of norms for Husserl, his criticism would have been that naturalism deduces norms from facts. However, the core of the distinction between pure and empirical consciousness was not at that point.

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In his lectures on logic as the theory of knowledge (1910–1911), Husserl distinguished between laws of logic, laws of natural sciences, and norms created by human beings. The distinction corresponds to that made by Frege in his preface to the ﬁrst volume of the Grundgesetze. Husserl admitted that there are major reasons which speak in favor of logical psychologism. However, he explained by means of an analogy that logical psychologism is not true. In his “Göttinger Vorlesungen über Urteilstheorie” in the summer term of 1905 Husserl talked about the analogy between geometry and logic. There he points out that it is common to draw a false analogy; the psychologistic view is that the art of logical reasoning is related to psychology as geodesy is related to geometry or as technical physics is related to theoretical physics (19b). In his lectures in 1910 and 1911, Husserl explained what he thought is the right analogy (20b). Just as geodesy is related to ideal geometry, normative logic is related to logic as a theoretical discipline. Moreover, just as behind geodesy there is a natural science or several of them, likewise behind normative logic there is psychology. In Husserl’s view, the norms of logic are inferred from the facts of pure or theoretical logic, not from the facts given by psychology; the facts given by pure logic have to do with the structures of propositions and with the inferential relations between propositions. Husserl also stated in his lecture notes that naturalistic philosophy is characterized by the fact that it acknowledges only one ﬁeld of possible knowledge, which is nature (17a). Moreover, he stated that naturalism recognizes only one method of giving foundations for knowledge; it argues that all knowledge is based on experience (17b). In Husserl’s view, the essential diﬀerence between naturalism and antinaturalism was that naturalism does not acknowledge the ideal realm. Husserl characterized the ideal realm as eternal, self-identical, timeless, spaceless, unmovable, and unchangeable; he did not state that it is something that is expressed by normative propositions. He also remarked that there is no mysticism in such a view (28a, 28b). As we will see in the next section, in his later writings he expressed his view in constructivist terms and stressed the diﬀerence between two attitudes more than the diﬀerence between the two realms.

3.2. Antipsychologism and the Doctrine of the Third Realm In the passages quoted, Husserl acknowledged what is called “the third realm” by Frege. The doctrine of the three realms can be found in Lotze. According to Lotze, the being of abstract objects is not like the being of concrete objects. Instead, abstract objects are valid, geltend. Lotze took it to be important to distinguish between what is valid and what is (was gilt and was ist) (Lotze 1874, 16 and 507). Frege presented a doctrine of three realms, by means of which he expressed his view on the being and the being known of logical categories and of thoughts that are constituted by those categories. In the ﬁrst volume of his Grundgesetze der Arithmetik (1893) and in his article “Der Gedanke” (1918) he made a

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distinction between the subjective realm of ideas (Vorstellungen), the objective realm of actual (wirklich) objects, and the realm of objects that do not act on our senses but are objective, that is, the realm of such abstract objects as numbers, truth values, and thoughts (GGA I, XVIII–XXIV; KS, 353). His conceptual notation, which he called the formula language of pure thought, was meant to mirror parts of the third realm, as it was meant to present the structure of thoughts and the inferential relations between thoughts. It is usually assumed that Frege’s acknowledgment of the third realm was a Platonic doctrine. Some interpreters have challenged the received view, but others, most notably Tyler Burge (1992), have given strong arguments for the view that Frege held a Platonic ontology; Burge also emphasizes that Frege did not seek to defend his position, except for showing problems in competing views, and that he did not make any eﬀort whatsoever to develop a sophisticated version of his ontology. In spite of Burge’s carefully documented argumentation, other interpretations remain serious candidates. When Frege discussed his third realm in his “Der Gedanke,” he remarked that he must use metaphorical language. In other words, such expressions as “the content of consciousness” and “grasping the thought” must not be understood literally (KS 359, n. 6). As Frege expressed his worry about the fact that natural language leads us astray as early as in the preface of his Begriﬀsschrift, the interpretation that Frege did not take numbers or thoughts to have being in the proper sense of the word “being” is at least worth considering. Frege did think that the objects of the third realm are objective, hence, independent of subjective minds. That is not yet an ontological position. On Frege’s view, thoughts and their constitutive logical categories are denizens of the third realm, but their being is not like the being of the denizens of the objective and actual realm. Thomas Seebohm has argued that Frege presented a transcendental argument to the eﬀect that the existence of mathematical objects and logical categories is a necessary condition of the meaningfulness of mathematical and logical practice (Seebohm 1989, 348). If that argument holds, Frege’s acknowledgment of the third realm would have ensued from his epistemological views. Husserl argued that we must acknowledge an ideal realm of abstract objects to avoid psychologism. He pointed out that there is an unbridgeable diﬀerence between the sciences of the real and the sciences of the ideal, as the former are empirical, while the latter are a priori. Husserl realized that if we acknowledge the ideal realm, we must face an epistemological problem concerning our access to this realm. Most of Husserl’s logical studies after his Logische Untersuchungen are an eﬀort to answer this question by means of phenomenology. In his last logical works, titled Formale und transzendentale Logik (1929) and Erfahrung und Urteil (1939), which was published posthumously, he sought to show that we have an access to the denizens of the ideal realm, because we have set the structure of transcendental consciousness to those denizens, hence, we have maker’s knowledge of that realm.

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Husserl did not think that we could be mistaken about what the correct logical categories are. His problem was how to give a justiﬁcation for what he regarded as our true beliefs concerning those categories. In his sixth logical investigation (LU II, 1901, 1921) Husserl studies the components of meaning which determine the form of a proposition and calls them categorial meaningforms (Bedeutungsformen). In his view, those forms are expressed in natural language in several ways, for example, by deﬁnite and indeﬁnite articles, numerals, and by such words as “some,” “many,” “few,” “is,” “and,” “if—then,” and “every” (LU II, A 601/B2 129; LU II, A 611/B2 139). Husserl asked what the origin of logical forms is, when nothing in the realm of real objects seems to correspond to them (LU II, A 611/B2 139). He took it to be a problem how the logical words originally get their meaningfulness, hence, what kind of activity of a subject is required so that the logical words become meaningful. In his last works on logic, Husserl sought to show that that activity is precisely the activity of transcendental consciousness. Kant interpreted logical categories as the pure concepts of understanding, which correspond to certain types of judgments and which give form to the objects of experience. In his Begriﬀsschrift, Frege, for his part, introduced eight signs as the basic signs of his formula language of pure thought; those signs expressed the basic logical categories and made it possible for Frege to present most types of judgments listed in Kant’s table. As was noted, in Frege’s doctrine of the three realms the logical categories were regarded as constitutive for the denizens of the third realm called thoughts.

3.3. On the Possibility of Logical Knowledge 3.3.1. What Is Logical Knowledge? Kant is famous for his eﬀort to answer the so-called transcendental questions, such as “How is pure mathematics possible?”, “How is pure natural science possible?”, and “How is metaphysics as science possible?” This type of questions have two readings. One either wants to know whether x is possible and wishes to have a justiﬁcation for its possibility, or one assumes that x is possible and tries to ﬁnd out the conditions of its possibility.16 If one raises the question concerning the possibility of logical knowledge, one may think of two questions, ﬁrst, whether logical knowledge is possible at all, and second, if it is, under what conditions it is possible. This section is a short study of a few late nineteenth-century and early twentieth-century logicians’ and philosophers’ views of that possibility. Frege’s and Husserl’s views will again be in focus. By logical knowledge, one may mean knowledge which is reached by means of logical inference, hence, knowledge based only on logical truths. For example, knowing that p & q → p

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would be an example of logical knowledge. By logical knowledge, one may also mean knowledge concerning the basic concepts of logic, the logical forms of propositions, the basic laws of logic, or the rules of logical inference. For example, statements like Existence is a logical concept. The logical form of the sentence “Man is an animal” is “∀x(F (x) → G(x))”. The law of noncontradiction holds. would express logical knowledge in the intended meanings. In his Grundlagen der Arithmetik (1884, §3) Frege stated that the distinctions between a priori and a posteriori, synthetic and analytic, concern not the content of the judgment but the justiﬁcation for making the judgment. He excluded the naturalistic interpretation of his claim and stated that by his distinctions he intends to refer to the ultimate ground on which rests the justiﬁcation for holding a proposition to be true. He continued that the problem becomes that of ﬁnding the proof of the proposition. By his characterizations of analytic and synthetic truths and truths a priori and a posteriori, he expressed the view that the justiﬁcation of analytic truths a priori comes from general logical laws and deﬁnitions. In his view, logical laws neither need nor admit of justiﬁcation. However, the question remains what Frege would have named as the source of knowledge if he had thought that we can know the structure of the ideal logical language in the proper sense of knowing. Did he think that we know that existence is a logical concept? If he thought that way, what would he have labeled as the source of knowledge, hence, what would have been a justiﬁcation for such a claim? As already stated in section 3.2, Husserl studied the components of meaning which determine the form of a proposition and called them categorial meaningforms (Bedeutungsformen). He took it to be a problem how the logical words originally get their meaningfulness, hence, what kind of activity of a subject is required so that the logical words become meaningful. In Husserl’s thought, the questions of origin were linked with the questions of justiﬁcation. 3.3.2. Can We Have Logical Knowledge? Emil Lask on Kant’s View Kant thought that categories, hence, logical concepts, have their origin in the logical forms of propositions. However, he took the list of the logical forms of propositions for granted. If Kant’s transcendental deduction was a justiﬁcation of certain logical concepts, the idea of that justiﬁcation was to show the role of those concepts in cognition and experience; it was to show how the pure concepts of understanding contribute to making objects of knowledge possible and how they are linked with the forms of intuition. Expressed in contemporary terminology, Kant sought to give us the epistemological foundation of logic by showing how the pure forms of thought

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are applied to sensuous experience. Moreover, by that project Kant also tried to show us how logical knowledge in general is possible. He argued that logical concepts are hidden in objects of experience, they have their origin in those objects, and we can have knowledge of them precisely via their link to what is given in intuition. Emil Lask, a student of Rickert, whose thought inﬂuenced Heidegger’s early philosophy, praised Kant’s Copernican revolution in his work on logic and the doctrine of categories. According to his writing published in 1911, Kant had shown that certain questions concerning objects belong to logic, hence not to metaphysics (LP, 31).17 However, Lask argued that Kant’s critique of knowledge could not touch on the questions concerning logic or the logical forms of objects in a proper manner (LP, 260–262). In his view, that followed because Kant was committed to a two-world doctrine in which a distinction was made between the world of sensory objects and the transcendent world. Lask argued that as Kant neither regarded logic as sensory nor took it to be metaphysical, he made it homeless (heimatlos; LP, 263). We may disagree on Lask’s two-world interpretation of Kant’s thought. However, it is interesting to ﬁnd out how Lask solved the problem concerning the homelessness of logic which he thought to have found in Kant’s philosophy. His starting point was to give up the two-world doctrine and replace it by an epistemological doctrine concerning the concept of objectivity. That doctrine came from Lotze. As was noted, according to Lotze the being of abstract objects is not like the being of concrete objects. Lotze took them to be valid, and he considered it to be important to distinguish between what is valid and what is (Lotze 1874, 16 and 507). Lask supported that kind of division between two worlds (LP, 6), but he did not consider it an ontological distinction. Instead, for him that was a distinction between two diﬀerent attitudes or points of view, which we can take toward our sensory experience (LP, 48–49, and 88–91). Lask thought that the logical attitude considers psychological, physical, and cultural beings in a way that diﬀers from the attitude of everyday experience and scientiﬁc activity; it is interested in what is valid for those beings. On Frege’s View In his “Über die wissenschaftliche Berechtigung einer Begriﬀsschrift” (1882) Frege writes: “a perspicuous representation of the forms of thought (eine anschauliche Darstellung der Denkformen) has . . . signiﬁcance extending beyond mathematics. May philosophers, then, give some attention to the matter!” (BS, 1964, 114). The forms of thought Frege talked about were not meant to be the forms which the human mind happens to have. However, Frege thought that we (or he) can have an access to those forms and they can be written down as a language, as a conceptual notation (begriﬀsschrift), as he thought to have done in his book titled Begriﬀsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. These forms were not tied to human psychology, but they were pure, hence, not naturalistically characterizable forms. If Frege thought that logical forms can be known in the proper sense of knowing, he must have meant by “knowing” some kind of immediate

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recognition of the presence of the forms of thought. That recognition, the result of which is Frege’s conceptual notation, could be called intuition, in the sense of immediate intellectual seeing. How is this intellectual seeing possible at all? In view of Frege’s conceptual notation, which is meant to be a genuine language that speaks about the world and carries ﬁxed meanings, intellectual seeing presupposes grasping the correct structure of thoughts. Frege did not think that meanings could be given syntactically, hence, for him, giving meanings to logical constants did not amount to giving inferential rules, say, rules of introduction and elimination. For him, meanings of logical function names were found by means of grasping thoughts and by analyzing them. Still, Frege assumed that meanings are present in the syntax of the conceptual notation and there is no way of giving a semantic theory for that notation. He thought that our knowledge concerning the structure of the ideal logical language, hence, the basic logical concepts and the logical forms of propositions, and concerning the basic laws of logic carries its own justiﬁcation, which has to do with “immediate seeing,” which is not disturbed by sensory data. However, even if Frege did not seek to present any theory of logical knowledge, that did not mean that his ideal language would not have been motivated by epistemological considerations.18 On Husserl’s View Husserl’s doctrine of categorial perception in the Logische Untersuchungen was meant to be a solution to the problem concerning the origin of logical knowledge. Husserl introduced a new concept of perception that was not sensuous perception. In his view, categorial meanings are originally related to objects of sense perception but in a peculiar manner; logical forms are in the objects of sensuous acts but hidden in them as it were. In categorial perception, which was the term Husserl used, the subject sees the sensuous object diﬀerently; he or she perceives the object via logical forms, hence, the object is for him or her in these forms (LU II, A 615/B2 143). Sensuous objects are objects of sensuous acts, whereas ideal objects are objects which arise in that kind of “seeing diﬀerently” (LU II, A 617/B2 146). In Husserl’s view, such acts as the act of conjunction, disjunction, and generalization need sensuous acts which are their foundation (LU II, A 618/B2 146). In the later edition of the sixth logical investigation in 1921, Husserl remarked that these acts that are not founding acts are in relation to what appears in the sensuous founding acts (LU II, B2 146).19 Lask and Husserl thus shared the idea that the philosophical nature of logic must be studied by studying the attitudes or the points of view which the subject of knowledge has to the objects of knowledge. Hence, in Husserl’s view the origin of logical concepts is in sense perception; logical forms can become ideal objects studied by the science called logic, because there is a subject who sees the objects of perception in an explicating manner (LU II, A 623–625/B2 151–153). Logical concepts, like the concepts of whole and part, are possibilities in objects which become articulated in categorial acts (LU II, A 627/B2 155). Logical forms are not in themselves

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but the subject makes them exist. Husserl thus found the origin of logical concepts in the activity of the subject. For Husserl, the forms of thought have been set into sensuous objects and as objective they can be known by us. To know them is to construct new categorial objects, and that constructing is categorial intuition. On this reading of Husserl’s text, logical knowledge is possible because it is knowledge concerning our own constructions. Frege thought that we cannot take distance from logical categories, we can only write them down when we see them correctly; we cannot present an epistemological theory for them. Nonetheless, Frege gave us several epistemological arguments which aimed at supporting his choice of a certain kind of logical language. Unlike Frege, Husserl thought that logical categories and even logical laws need and can be given justiﬁcation, which means giving an epistemological theory for logic.20 A somewhat surprising conclusion can be drawn if we pay attention to the connection between the views of the possibility of logical knowledge just discussed and of the nature of philosophy. The philosophers of the Vienna Circle thought that Frege’s, Russell’s, and Whitehead’s logic was a neutral system of formulas and a useful tool for clarifying thoughts, hence, philosophy was for them a certain kind of activity, namely, the activity of clarifying thoughts by means of the new tool. They did not suggest that philosophers ought to present theories of anything, not even of logical knowledge. Later, it has been typical of the analytic tradition to put forward philosophical, formal, and semiformal theories of various kinds, including theories of logic, or logics, and of natural language. From that perspective, Husserl’s way of thinking of the possibility of logical knowledge and his search for a theory of that kind of knowledge is a more natural background for the analytic tradition than Frege’s approach.

4. Discovery, Justiﬁcation, and Intuition 4.1. The Rejection of Intuition The problem of justiﬁcation became a central theme in the philosophy of logic during the ﬁrst decades of the twentieth century. The role of intuition as a justiﬁer was discussed by logicians and philosophers. From what has been said, it seems that Husserl opposed reference to intuition in cases where Frege was ready to rely on intuitive knowledge. If we argue that way, we suggest that Husserl’s demand for justiﬁcation goes further than that of Frege’s. It is true Husserl thought that even if propositions that are taken to be basic in a formal system are not in need of justiﬁcation in terms of logical inference, they need another kind of justiﬁcation, namely, a philosophical justiﬁcation. Unlike Frege, Husserl thought that logical categories and logical laws need and can be given philosophical justiﬁcation in the sense of giving a philosophical or an epistemological theory for logic. In section 4.2.1, I trace Husserl’s view back to its Kantian origins.

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In a postumous work published in 1991, J. Alberto Coﬀa considers the semantic tradition from Bolzano to Carnap, hence from the early nineteenth century until the early twentieth century. According to Coﬀa, the semantic tradition reacts against Kant’s philosophy. He claims that that tradition tried to get rid of all references to intuition, which it took to be Kant’s great problem. Coﬀa points out that the semantic tradition can be deﬁned by means of its problem, its enemy, its goal, and its strategy. According to Coﬀa, its problem was a priori, its enemy was pure intuition, on which Kant relied when he studied the possibility of mathematics, its aim was to develop a concept of a priori in which pure intuition played no role, and its strategy was to base that theory on the development of semantics (Coﬀa 1991, 22). Coﬀa also argues that in geometry it is particularly necessary to refer to constructions which are seen immediately but that even calculus, which was the strongest branch of eighteenth-century mathematics, had the same practice (ibid., 23). Coﬀa remarks that by the end of the nineteenth century Bolzano, Helmholz, Frege, Dedekind, and many others helped settle that Kant was not right, that concepts without intuition were not empty (ibid., 140). The pioneers of logic at the end of the nineteenth century stressed that in the ﬁeld of logic one is not allowed to refer to intuition; each inferential step must be written down. Particularly, Frege’s conceptual notation was meant to be a tool by means of which each step in the process of inference can be written down exactly without any resort to intuition. However, even if Frege and Peirce, among others and maybe most prominently, were creating a new logic and Frege tried to carry out a program which aimed at reducing arithmetic to logic, they did not, and they did not even want to, get rid of intuition altogether. Of course the very concept of intuition was problematic for them. If we look at the pages of Frege’s Begriﬀsschrift, we notice that he appeals to what we would nowadays call our pattern recognition abilities both in his analysis of sentences and in his ways of presenting inferences. Peirce laid even more emphasis on the role of intuition. For example, in 1898 he praised Kant for understanding the role of constructions or diagrams in mathematical inference. He wrote that mathematical inference proceeds by means of observation and experiment and that the necessary nature of this inference is merely caused by the fact that a mathematician observes and tests a diagram which is his own creation (“The Logic of Mathematics in Relation to Education,” 1898, CP, 3.560). In 1896, Peirce noted that logic has to do with observing facts concerning mental constructions (NE 4, 267). He very often stressed the value of ﬁgures in inference and states in 1902 that all knowledge has its origin in observation (NE 4, 47–48). Of course these pioneers of modern logic did not assume that a logician is able to see all the consequences of given premises, and they did not give a logician a permission to refer to seeing the conclusion of given premises immediately. Nevertheless, they thought that when taking the shortest steps in an inferential process, a logician does something that can be naturally called perceiving or seeing.

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4.2. Husserl’s Problem of Justiﬁcation and Frege’s and Peirce’s Discoveries Even if the idea of axiomatic science in logic is not discussed in this section, the methods of logical discovery and justiﬁcation deserve attention. Frege claimed that all great scientiﬁc improvements of modern times have their origin in the improvement of method. He wrote: I would console myself on this point with the realization that a development of method, too, furthers science. Bacon, after all, thought it better to invent a means by which everything could easily be discovered than to discover particular truths, and all great steps of scientiﬁc progress in recent times have had their origin in an improvement of method. (Frege, BS, 1964, XI; Frege 1972, 105) The method Frege proposed for science was his begriﬀsschrift, the new logic, but there was even a deeper truth in his statement. A better method was also needed if one wished to improve logic. In his paper “Explanation of Curiosity the First” (1908) Peirce described Euclid’s procedure in proving theorems. Euclid ﬁrst presented his theorem in general terms and then translated it into singular terms. Peirce paid attention to the fact that the generality of the statement was not lost by that move. The next step was construction, which was followed by demonstration. Finally, the ergo-sentence repeated the original general proposition. Peirce laid much emphasis on the distinction between corollarial and theorematic reasoning in geometry. He took an argument to be corollarial if no auxiliary construction was needed. For Peirce, construction was “the principal theoric step” of the demonstration (CP, 4.616). Peirce also stressed that it is the observation of diagrams that is essential to all reasoning and that even if no auxiliary constructions are made, there is always the step from a general to a singular statement in deductive reasoning; that means introducing a kind of diagram to reasoning. Peirce’s methodological interests are well known. For example, in 1882 he stated in his “Introductory Lecture on the Study of Logic”: “This is the age of methods; and the university which is to be the exponent of the living condition of the human mind, must be the university of methods” (W 4, 379). Moreover, in his “Introductory Lecture on Logic” (1883) he made an interesting remark on methodology. He wrote: But modern logicians generally, particularly in Germany, do not regard Logic as an art but as a science. They do not conceive the logician as occupied in the study of methods of research, but only as describing what they call the normative laws of thought, or the essential maxims of all thinking. Now I have not a high respect for the Germans as logicians. I think them very unclear and obtuse. But I must admit that there is much to be said in favor of distinguishing

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Logic from Methodology. . . . Let us say then that Logic is not the art of method but the science which analyzes method. (W 4, 509–510) As Peirce thus regarded logic as science, it is no surprise that he was also interested in the methodological commitments and choices of the one who works in the science of logic. There is an interesting history of method from Kant to Frege and Husserl. What was especially important is that all the way the task is twofold. On the one hand, Kant considered transcendental forms, that is, logical concepts, to be our method or tools for reaching the phenomenal world, as he considered them to be our tools for constructing that world. On the other hand, he regarded it as necessary to have a proper method, which is transcendental analysis, for knowing those very tools. Frege’s task was also twofold. Frege set out to ﬁnd a new method for science, which would be his begriﬀsschrift, but he also needed a new method of discovering that very method. In Husserl’s philosophy the method of ﬁnding the method came to be a method of knowing the ideal world. That happened because Husserl considered the logical tools to have being as structures of that world. According to Husserl, logic tries to claim something about the structure of the realm of ideal objects, which is strange for us in the sense that it is independent of our subjective mental realms. Husserl’s question brings us back to the question of method, as Husserl assumed that we have knowledge of the ideal realm only if there is a reliable route from our subjective minds to the objective realm, that is, only if we have proper tools for reaching that realm. Therefore, for him the foundational task was to know and describe these very tools. 4.2.1. Husserl and the Justiﬁcation of Logic Husserl’s question “How is logic as science possible?” amounted to the question “What were the methods of discovery and justiﬁcation that justiﬁed modern logic as science?” Husserl also proposed this problem for those who are interested in the foundations of logic. He compared the activities of a practicing artist with those of a scientist. He argued that both of them are in an equally bad shape if we think of how conscious they are of the principles of their creation or their evaluation. Husserl even claimed that mathematics has no special position on this issue. A mathematician is often unable to inform us of his steps of discovery or to give us a proper theoretical evaluation, that is, a justiﬁcation, of his results (LU I, A 9–10/B 9–10). Husserl proposed that all discovery and testing rest on regularities of form and that regularities of form also make the theory of science, that is, logic, possible (LU I, A 22/B 22). Husserl’s thought lends itself easily to the framework of the philosophical tradition introduced by Kant. His main works in the ﬁeld of logic bear Kantian labels in their very titles. His trilogy of logic consisted of the book titled Logische Untersuchungen I–II (1900–1901), the ﬁrst volume of which he calls Prolegomena zur reinen Logik, Formale und transzendentale Logik (1929), and Erfahrung und Urteil (1939), which was posthumously completed and

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published by Ludwig Landgrebe. Even if Husserl attached his philosophy to Kantian themes, he was convinced that he had to raise heavy criticism against Kant’s ideas. He blamed Kant for having failed to achieve a “pure” theory of knowledge, which would be free from all naturalistic elements, such as psychological and anthropological assumptions. No more was he satisﬁed with neo-Kantians’ developments, which he called transcendental psychology (LU I, A 92–97/B 92–97).21 He admitted, though, that Kant’s philosophy also had features that go beyond psychologism (LU I, A 94/B 94, note). In his early writings, Husserl seemed to speak in favor of psychologism, for example, in his book Philosophie der Arithmetik (1891), which Frege, the devoted antipsychologist, heavily attacked in 1894 (“Rezension von: E. Husserl, Philosophie der Arithmetik, Erster Band, Leipzig, 1891,” KS, 179–192). Some scholars, for example Mohanty (1982), have disputed that Husserl was a psychologist in the sense that Frege gave to the term. Mohanty stresses that Frege’s criticism led Husserl to revise some parts of his theory of number and it may have made him pay more attention to distinguishing between act, content, and object. However, Mohanty points out that it could not lead Husserl to reject such a version of psychologism which Frege attacked simply because Husserl never subscribed to that version (Mohanty 1982, 22–26). However that may be, it was at the very end of the nineteenth century that Husserl clearly joined the antipsychologistic camp, which his Logische Untersuchungen testiﬁed. It may be noted that in that work he also pointed out that he does not want to reject everything that he has done in his Philosophie der Arithmetik (LU II, B1 283, note). In the Logische Untersuchungen, the main starting points for Husserl were Bolzano, Lotze, and Brentano, to whom Husserl paid homage in those two logical works (LU I, A 219–227/B 219–227, and LU II, A 344–350/B1 364–370). Bolzano (1837) had introduced Sätze an sich and Vorstellungen an sich, which he regarded as neither existing in space and time nor depending on our mental acts (Bolzano 1929, §19). Hence, Bolzano distinguished the proposition itself from our thinking of it and acknowledged a speciﬁc realm of ideal objects, for which he did not admit proper existence, though. As was noted, Lotze, for his part, considered being and validity to be two senses of actuality (Wirklichkeit) and distinguished between the being of concrete things and the validity of abstract objects. For him, validity was a way of being independent of subjective mental acts (Lotze 1874, 507). Even if Brentano was not a defender of abstract entities, he distinguished between mental acts and their objects, which have intentional inexistence in those acts but need not have any real existence (Brentano 1924, 124–125). Husserl was inﬂuenced by Brentano already from the middle of the 1880s, when he was Brentano’s student in Vienna.21 As was noted in section 3.2, Husserl approved of those ideas and made a distinction between the real and the ideal. He stated: There is an essential, quite unbridgeable diﬀerence between sciences of the ideal and sciences of the real. The former are a priori, the latter

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empirical. The former set forth ideal general laws grounded with intuitive certainty in certain general concepts; the latter establish real general laws, relating to a sphere of fact, with probabilities into which we have insight. (LU I, A 178/B 178; LI I, 185) Husserl observed that once the distinction between the ideal and the real realm is acknowledged, we quite naturally come to realize one crucial problem. This problem constituted an important part of Husserl’s criticism against Kant. In 1929 Husserl maintained that because Kant did not make the distinction between the ideal and the real, he failed to ask one important question. Because Kant did not assume any world of ideal objects of thought, he could not ask how we can have an access to these objects (FTL, 233–235). In the Formale und transzendentale Logik, Husserl was explicit in stressing the importance of Kant’s theories concerning the Humean problem, which include his doctrine of transcendental synthesis and of transcendental abilities in general. Husserl praised Kant’s questions concerning our knowledge and its presuppositions. However, he blamed Kant for not asking transcendental questions about formal logic (FTL, 228–230). Kant took Aristotelian logic to be a complete system, which needs no major corrections. All we can do for what he called general logic was to make it more elegant; the proper task of that logic, which is to expose and prove the formal rules of all thought, had already been accomplished, in Kant’s view (KRV, B viii–ix). Kant asked how pure mathematics is possible, how pure natural science is possible, and how metaphysics as natural disposition and as science is possible (KRV, B 20–22), but he did not ask how logic as science is possible. Husserl believed that if Kant had distinguished between the ideal and the real realm, it would have occurred to him to ask such an epistemological question. Husserl concluded that both Hume and Kant realized the transcendental problem of the constitution of what he called the real realm. He thought that they failed to see the corresponding problem concerning the constitution of the ideal objects, such as the judgments and the categories which belong to the sphere of reason and which logic is interested in. In other words, Kant did not make his analytic a priori a problem (FTL, 229–230). Husserl’s question in his logical works can thus be formulated in three ways: (1) How can we have knowledge of the realm of ideal objects? (2) How can we rely on what logic claims? (3) How can we justify the analytic truths a priori? These formulations have close connections. The ideal realm consists of abstract objects like numbers and thoughts, and it is precisely logic that tries to say something about the structure of thoughts and about the inferential links between thoughts. Therefore, because Kant did not ask how we can know anything about the ideal realm, he did not ask how logic as science is possible, either. Moreover, since logical laws are analytic a priori, Husserl asked how we can rely on the analytic a priori claims which logic oﬀers to us.22 Husserl thus blamed Kant for not being able to ask how we can have knowledge of the ideal realm. We could certainly defend Kant by the argument

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that because he did not postulate any such problematic realm as the realm of abstract objects, he did not need to face such epistemological problems as Husserl. We may also say that even if Kant did not ask Husserl’s question, his Kritik der reinen Vernunft served as an answer to that question. However, the point in Husserl’s argument might be construed as the statement that we cannot know anything that is not made objective, hence that the proper deﬁnition of the concept of knowledge implies that the object of knowledge is distinguished from the knowing subject. On this construal of his argument, Husserl required that logical concepts and laws are something that can be known in the proper sense of knowledge. If this is what he meant, the point of his criticism was that Kant did not set the conceptual tools of logic outside consciousness to study those tools.23 Husserl thus asked the question which Kant did not ask and tried to do what Kant did not do, namely, lay the epistemological foundations of logic. But what was actually the philosophical incentive of the question concerning how logic as science is possible? From Galilei and Descartes to Kant, philosophers had sought for a ﬁrm foundation for modern natural science, for mathematics and even for metaphysics. If we believe that the history of logic can be reconstructed as a Kuhnian science, hence, that the question of foundations arises in logic when the received framework is threatened, we quite naturally see the nineteenth century as a revolutionary period in logic. Aristotelian logic was losing ground in those days, and new formal developments arose. What this period needed, then, was an epistemological justiﬁcation for either the old logic or for those new suggestions. Hence, on this construal, Husserl’s question was necessitated by the new developments of logic in the nineteenth century. Husserl remarked: “how could such a logic [scientiﬁc logic] become possible while the themes belonging to it originally remained confused?” (FTL, 158; Husserl 1969, 178). The foundational crisis was not the most perspicuous reason for the question concerning the possibility of logic as science. The question arose as a natural consequence of the various confrontations within logic and philosophy of logic in the nineteenth century. As we saw in Husserl’s case, it arose from a philosophical position that postulated a speciﬁc realm of abstract objects like thoughts which logic speaks about. If that kind of realm is assumed and acknowledged, it is quite natural to ask how we can have knowledge of it, that is, how we can rely on logic which is supposed to speak about it. But why does anyone want to assume such an objective realm? I already suggested one answer that had to do with the proper concept of knowledge. Other guesses can also be made. Husserl’s argumentation suggests that historically the objectivity of the ﬁeld of interest of logic was probably necessitated by a proper criticism against a psychological or anthropological interpretation of Kant’s transcendentalism, which was represented by such logicians as Jakob Fries and Benno Erdmann, for example. Fries thought that logical concepts must be understood as the ways in which the human species organizes experience, and the logical laws must be construed as anthropological laws.24

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On Husserl’s judgment, the philosophy of logic of his own day was strongly anthropologistic; he even argued that it was rare to ﬁnd a thinker who would be free from the inﬂuence of that doctrine (LU I, A 116/B 116). In addition to the empiricist tradition, psychologism in logic had a natural connection with Kant’s transcendentalism, for the transcendental structure of human thought was easily construed as a psychological structure, which is typical of the human race. If one wanted to save transcendental logic from that kind of reading, one had better regard the transcendental structure as the structure of some objective realm.25 4.2.2. The Role of Judgments in Frege’s and Peirce’s Logical Discoveries Frege and Peirce discovered quantiﬁcation theory independently of each other. They both introduced a new formula language in which arguments or indices were distinguished from functions or relative terms. In his paper “Über den Zweck der Begriﬀsschrift” (1883) Frege remarked: In fact, it is one of the most important diﬀerences between my way of thinking and the Boolean way—and indeed I can add the Aristotelian way—that I do not proceed from concepts but from judgements. (BS 1964, 101) That Frege opposed Aristotle and Boole has been noticed by all interpreters, but it was about 30 years ago that Frege’s way of thinking was taken under more extensive historical consideration. Interpreters such as Hans Sluga (1980) linked Frege’s view with Kant’s idea that judgments have priority over their constitutive concepts. Kant was also one of Peirce’s philosophical heroes. Murray Murphey (1961) already noted that Peirce’s logical discovery brought him closer to Kant, as Peirce distinguished between indices and relative terms, hence, as it were, wrote down Kant’s distinction between intuitions and concepts. In his paper “Booles rechnende Logik und die Begriﬀsschrift” (1880/81), Frege clariﬁed the diﬀerence between his conceptual notation and Boolean logic. He stated that the real diﬀerence is that in logic he avoids a division into two parts, of which the ﬁrst is dedicated to the relation of concepts, that is, to primary propositions, and the second to the relation of judgments, that is, to secondary propositions, by construing judgments as prior to concept formation (ibid., 14 and 52). He continued that unlike Boole, he reduces his primary propositions to the secondary ones, which comes up in that he construes the subordination of two concepts as a hypothetical judgment (ibid., 17–18). This result came out when Frege broke up the judgment which contained subordination, which is a relation between two concepts. Before Frege was able to do this, he had to realize the distinction between individuals and concepts. This is what he also emphasized. In the article he remarked that his view does justice to that distinction. In Frege’s view, the problem with Boole’s notation lay in that Boole’s letters never meant individuals but always extensions of concepts. The distinction between individuals and concepts, or

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more generally functions, hence, between proper names and function names, was a crucial part in Frege’s discovery. It seems on the basis of Frege’s remarks in the Grundlagen that even if Frege criticized Kant’s concept of intuition, he viewed the distinction between intuitions and concepts as a precursor of his own distinction (GLA, §27, n.). The same methodological change from the Boolean method to the analysis of judgments was essential to Peirce’s discovery. I already mentioned that Kant was an important ﬁgure behind Peirce’s philosophy. As Murphey remarked, it was the manner in which Kant discovered his categories that interested Peirce most of all (Murphey 1961, 33). In the 1870s, Peirce discovered his logic of relatives, which was inspired by De Morgan’s ideas and Boole’s algebra of logic. Peirce’s articles titled “The Logic of Relatives” (1883) and “On the Algebra of Logic: A Contribution to the Philosophy of Notation” (1885) contained the ﬁrst presentation of his quantiﬁcation theory, which he himself called his general algebra of logic and which, as he wrote, he developed on the basis of O. H. Mitchell’s, his student’s, ideas (CP, 3.363 and 3.393). The ﬁrst important change from Boole’s logical algebra was that Peirce added indices to relations. Indices referred directly to individuals. Second, he introduced the quantiﬁers “some” and “every.” When he introduced his two improvements of logic, Peirce referred to Mitchell’s article “On a New Algebra of Logic” (1883). He expressed his indebtedness to Mitchell regarding both indices and quantiﬁers. However, when he described Mitchell’s way of using indices, he deviated from what Mitchell said. Peirce interpreted Mitchell’s formula “F1 ” as “the proposition F is true of every object in the universe” and formula “Fu ” as “the proposition F is true of some object in the universe.” For Mitchell, the symbol F was any logical polynomial involving class terms and their negatives. He did not take it to be a proposition, but rather called it a predicate or a description of every or some part of the universe (Mitchell 1883, 75 and 96). Moreover, Peirce used the concept of individual, which Mitchell did not use. Otherwise, it is true that Mitchell had both indices and quantiﬁers, as Peirce declared. Mitchell supported the view that objects of thought, in which logic is interested, are either class terms or propositions, but that every proposition expresses a relation among class terms (Mitchell 1883, 73). Because Mitchell thought that, basically, every proposition expresses a relation among class terms, he relied on the Boolean method, which started from concepts and came up with propositions by combining concepts. It is precisely this way of thinking which Frege attacked, as we noted. Hence, even if Mitchell did suggest indices and quantiﬁers, the new logical language cannot be encountered in his treatment. Peirce’s contribution was to take propositions as the starting point of analysis and generate a distinction between relative terms and the names of individuals. In his article “On a New List of Categories” (1867), which was meant to improve Kant’s doctrine of categories, Peirce relied on the subject-predicate form of propositions and assumed that in the aggregate of a subject and a predicate the subject represented what he calls substance (CP, 1.547 and 1.548).

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For Peirce, substance was the present in general, hence not an individual. It is not until the 1880s that individuals in the sense of Kant’s intuitions appeared in Peirce’s logical notation. These observations suggest that between Peirce’s “New List” (1867) and his discovery of the new notation (1883, 1885) there was a methodological change, which contributed to his logical discovery. Hence, the decisive insight both for Frege and Peirce was that a judgment is not an aggregate of terms that represent concepts or classes but that its elements have diﬀerent kinds of roles in their contexts. Two of those basic roles are that of representing relations and that of denoting individuals.

5. Origins of Twentieth-Century Semantics: Frege’s Distinction between Sinne and Bedeutungen Even if Frege did not have any semantic theory, he expressed views of semantic concepts and had considerations in his works that can be called semantic. For Frege, the Sinne, senses, of sentences are thoughts and the Bedeutungen, references, of sentences are truth values, the True and the False. Sentences are compounded out of proper names, which refer to objects, and function names, which refer to functions. The Sinne of function names are simply parts of thoughts.26 But what are the Sinne expressed by proper names? In “Über Sinn und Bedeutung” (1892), Frege remarked that the sense of a proper name is a way the object to which this expression refers is presented, or a way of “looking at” this object. Furthermore, he stated that the sense expressed by a proper name belongs to the object to which the proper name refers. In other words, for Frege, senses were not primarily senses of names but senses of references. Hence, it is more advisable to speak about senses expressed by names than senses of names. Frege also gave examples of senses, like “the Evening Star” and “the Morning Star” as senses of Venus, and “the teacher of Alexander the Great” and “the pupil of Plato” as senses of Aristotle (“Über Sinn und Bedeutung,” KS, 144). Nonetheless, Frege admitted that we speak meaningfully about entities which do not exist. In his view, a sentence lacks only a truth value—but not a sense—if it contains a name that has no reference (“Über Sinn und Bedeutung,” KS, 148). Russell adopted a critical standpoint against this idea, according to which an expression can have a sense although it lacked a reference. In his article “On Denoting” (1905) he argued that a sentence like “The present King of France is bald” should be construed as the sentence “One and only one being has the property of being the present King of France, and that being is bald.” The property of being the present king of France does not belong to any being, and therefore the sentence is false. Moreover, according to Russell, the sentence “The present King of France is not bald” is false if it means that there is an entity which is now king of France and is not bald. Russell, however, suggested another analysis for the latter sentence which says that it

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is false that there is an entity that is now king of France and is bald. On this interpretation, the sentence turns out to be true (Russell 1956, 53). Frege regarded it as possible for an object to be given to us in a number of diﬀerent ways. He observed that it is common in our natural language that one single proper name expresses many of those senses which belong to an object. For Frege, to each way in which an object is presented there corresponds a special sense of the sentence that contains the name of that object. The diﬀerent thoughts that we get from the same sentence have the same truth value. In Frege’s view, we must sometimes stipulate that for every proper name there is just one associated manner of presentation of the object denoted by the proper name (“Der Gedanke,” 1918, KS, 350). However, he believed that diﬀerent names for the same object are unavoidable, because one can be led to the object in a variety of ways (“Über den Begriﬀ der Zahl,” 1891/92, NS, 95). For Frege, our knowledge of an object determines what sense, or what senses, the name of the object expresses to us. One sense or a number of senses provides us only with one-sided knowledge (einseitige Erkenntnis) of an object. Frege argued: “Complete knowledge [allseitige Erkenntnis] of the reference would require us to be able to say immediately whether any given sense belongs to it. To such knowledge we never attain” (“Über Sinn und Bedeutung,” KS, 144; Frege 1952, 58).27 On the basis of Frege’s hints, we may conclude that his concept of Sinn is thoroughly cognitive. Many of his formulations suggest that Sinne are complexes of individual properties of objects, hence, something knowable. If this interpretation of the concept of Sinn were correct, it would have been Frege’s view that we know an object completely only if we know all its properties, which is not possible for a ﬁnite human being. It would also follow that according to Frege, each object could in principle have an inﬁnite number of names which would correspond to the modes of presentation of the object. Frege did not hold the position that knowing some arbitrary property or complex of properties of an object constitutes knowing the object completely since, for him, a necessary condition for knowing an object would be knowing all the properties of that object. Nevertheless, on the suggested interpretation he thought that in a weaker sense we know an object precisely by knowing some properties of that object. It is true Frege’s weaker sense of knowing an object is not free from problems, either, even if it is more natural than the stronger sense. This is because Frege does not explain which properties of an object one must know to know the object. In Frege’s view, we are not able to speak about the senses of proper names as senses, for if we start speaking about them, they turn into objects, which, again, have their own senses. But what are these objects in case we speak about the senses expressed by proper names? Frege said that senses can be named (“Über Sinn und Bedeutung,” KS, 144–145) and proposed such examples as “the teacher of Alexander the Great” and “the pupil of Plato.” But if senses were complexes of the properties that belong to objects, as suggested, their names ought to be such as “being the teacher of Alexander the Great” or

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“being the pupil of Plato.” Frege’s examples suggest that when we name a sense of an object, we do not name any new object which would be a complex of individual properties of that object, but we name the original object in a new way. Hence, it follows from these examples that we do not succeed in naming a sense of an object as any new object, after all. Instead, we only name the object itself as considered under the description with which the sense provides us. There has been much discussion on what Frege’s motivation for adopting the distinction between senses and references might have been. When he introduced the distinction, he primarily referred to identity statements. It seems as if the distinction between Sinn and Bedeutung had, above all, been meant to give an adequate account of the symbol of identity, which Frege wanted to preserve in his language. By making the distinction between Sinn and Bedeutung, he sought to give a natural reading for identity statements. When introducing the concepts of sense and reference, Frege tried to solve the problems that what we now call intensional contexts caused for what we now call his idea of extensional language. The principle of functionality, which we may call the principle of compositionality in the case of references, is the core of that idea.28 Everything worked well according to what we would call truth tables when Frege constructed complex sentences out of simple sentences by means of conditionality (BS, §5). The trouble for Frege was caused by what became later called intensional contexts. Frege tried to deal with those contexts by introducing the concepts of indirect sense and indirect reference, the latter being the same as the normal sense of an expression. Frege claimed that in certain indirect contexts our words automatically switch their references to what normally are their senses. In a letter to Russell, he even recognized the need for using special signs for words in indirect speech (BW, 236). For example, in the complex sentences “A believes that a is P ” and “A believes that b is P ,” “that a is P ” and “that b is P ” name two diﬀerent thoughts, since “a” and “b” have diﬀerent senses. Let us assume that a and b have the same normal reference. Given that the truth value of the complex sentence is considered to be the value of a function whose arguments are the references of the components of the sentence, it does no harm to what we call the principle of functionality even if the complex sentences have diﬀerent truth values. Since the arguments of the function diﬀer from each other, that is, because a and b have diﬀerent indirect references, the references of the complex expressions may quite well be diﬀerent, and the principle of functionality is thus saved. Frege’s theory of Sinn and Bedeutung was not only a solution oﬀered to the problems that indirect contexts caused to the idea of extensional language, but it was also a direct consequence of his idea of a universal language. As noted, Frege’s begriﬀsschrift, conceptual notation, was meant to be a realization of Leibniz’s great idea. Leibniz thought that the terms of our natural language do not correspond to the things of the world in a proper way, and therefore we ought to construct a new language which mirrors correctly the whole universe.29 He dreamed of a language that speaks about the actual world in the sense of mirroring the individual concepts instantiated in this world. Frege’s

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world diﬀered from that of Leibniz in the sense that for him the actual world was the only world. For Frege, Sinne were something that we cannot avoid when we try to reach the world by means of our language. Frege’s belief in the inescapability of Sinne can thus be considered a special form of the Kantian belief that we must always consider objects through our conceptual systems. In “Ausführungen über Sinn und Bedeutung” (1982–1985) he remarked: “Thus it is via a sense and only via a sense that a proper name is related to an object” (NS, 135; Frege 1979, 124). Hence, the distinction between senses and references was something that Frege would have accepted in any case because of his belief in the role of conceptual machinery in reaching the world. That observation brings us back to where we started, namely, to how Frege understood the nature of his conceptual notation.30

Notes I have used extracts from my article “Analysis as the Method of Logical Discovery: Some Remarks on Frege and Husserl,” Synthese 77 (1988), 73–97, with the kind permission of Springer Science+Business Media. The chapter also contains passages from my article “Existence and Propositional Attitudes: A Fregean Analysis,” Logical Analysis and History of Philosophy 4 (2001), 75–86, which appear here with the kind permission of Mentis, and from my article “Finnish Studies in Phenomenology and Phenomenological Studies in Finland,” in Leila Haaparanta and Ilkka Niiniluoto (eds.), Analytic Philosophy in Finland, Poznań Studies in the Philosophy of the Sciences and the Humanities, vol. 80 (Rodopi, Amsterdam, 2003), 491–509, which appear here with the kind permission of Rodopi. I have used the manuscripts “Göttinger Vorlesungen über Urteilstheorie” (1905) and “Logik als Theorie der Erkenntnis” (1910–1911) with the kind permission of the Husserl Archives at the University of Leuven. 1. For the debate between psychologists and antipsychologists, see, for example, Kusch (1995). 2. See Friedman (1996, 2000) and Haaparanta (1999a, 2003). 3. See Beaney (2002) and Haaparanta (2007). 4. See Haaparanta (1985, 1999a) and Friedman (1996, 2000). 5. See Gabriel (1986). Cf. Ziehen (1920), 132–240. 6. See, for example, Haaparanta (1985) and Mancosu (1998). Also see Detlefsen (1992) and chapters 9 and 14 in this volume. 7. See Haaparanta (1988, 1999b). 8. See Lohmar (2002a, 2002b). 9. See Becker (1927) and Heyting (1930a, 1930b, 1931). 10. See, for example, Leibniz (1961a), 84 and 192, and Leibniz (1961b), 29, 152, and 283. See, for example, Frege, “Booles rechnende Logik und die Begriﬀsscrift” (1880/1881), NS, 9–52, “Über den Zweck der Begriﬀsschrift” (1883), BS (1964), 98, “Über die Begriﬀsschrift des Herrn Peano und meine eigene” (1896), KS, 227, GGA II, §§56–65, and “Anmerkungen Freges zu: Philip E. B. Jourdain, The development of the theories of mathematical logic and the principles of mathematics” (1912), KS, 341. For the terminological diﬀerence between Leibniz and Frege, see Haaparanta (1985), 11, and its references.

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11. The idea that Jaakko Hintikka (1979, 1981a, 1981b) has labeled as the idea of the ineﬀability of semantics and to which Hugly (1973) has also paid attention in Frege’s logic is visible at various points in Frege’s writings. For example, see Frege’s remarks on senses in “Über Sinn und Bedeutung” (1892), KS, 144–145, on functions in “Über Begriﬀ und Gegenstand” (1892), KS, 170, on the concept of identity in “Rezension von: E. G. Husserl, Philosophie der Arithmetik I” (1894), KS, 184, and on the concept of truth in “Der Gedanke” (1918), KS, 344. Also see his informal explanations of the semantics of his conceptual notation, “Darlegung der Begriﬀsschrift,” in GGA I. See Haaparanta (1985), 33, 41–43, 61–62, and 66. 12. See, for example, “Dialog mit Pünjer über Existenz” (before 1884), in NS, GLA, §53, “Über Begriﬀ und Gegenstand,” (1891), KS, 173, and Frege’s letter to Hilbert 6.1.1900, BW, 75. See also Haaparanta (1985). 13. See Frege’s “Vorwort” to GGA I. Also see his article “Der Gedanke” (1918), KS, 342–362. 14. See “Der Gedanke,” KS, 351, where Frege discusses the realm of representations (Vorstellungen). In Frege’s view, representations like sense impressions and feelings need someone who has them. Obviously, acknowledging the truth of a thought, that is, judging, needs someone who acknowledges. Frege’s terminology thus suggests that he takes the acts of judging to belong to the realm of our minds. 15. References are to the manuscripts “Göttinger Vorlesungen über Urteilstheorie” (1905) and “Logik als Theorie der Erkenntnis” (1910–1911). 16. See, for example, Kemp Smith (1962), 43–45. 17. See Haaparanta and Korhonen (1996), 40–41. See Crowell (1992) and Friedman (1996), 58–59. 18. See Haaparanta (1985). 19. See Haaparanta and Korhonen (1996), 42. 20. See Haaparanta (1988). 21. See Husserl’s biography in Schuhmann (1977). Even if the doctrine of “propositions in themselves” was popular among a number of Husserl’s predecessors and contemporaries, Husserl’s view can also be interpreted as ensuing from certain internal motives of his philosophy. This kind of reading is suggested by Cooper-Wiele (1989), who emphasizes the role of the idea of a totalizing act in Husserl’s thought. See Cooper-Wiele (1989), 11 and 90–108. 22. For Husserl’s concept of formal or analytical law, see LU II, A 246–251/B1 252–256. For Husserl’s discussion concerning the relationship between logical laws and the analytic a priori, see, for example, Husserl (1950b), 28. 23. The same problem had also been tackled by Hegel from an opposite point of view. In Hegel’s view, Kant’s problem was that his critical philosophy tried to study the faculty of knowledge before the act of knowing. Hegel argued that other tools can be studied before they are used, but the use and study of logical tools is one and the same process (Hegel, 1970, §10 and §41, Zusatz 1). 24. See Fries (1819), 8. Also see Fries (1827), 4. For Erdmann’s psychologistic interpretation of transcendentalism, see Erdmann (1923), 472–477. For Frege’s criticism of Erdmann, see GGA I, “Vorwort,” xv–xvi. 25. Kusch (1995) has studied the sociological aspects of the debate on psychologism. My presentation is restricted to those aspects that are internal to the philosophical discussion. For Husserl’s criticism of psychologism, also see Willard (1984), 143–166. 26. See Haaparanta (1985). Also see chapter 13 in this volume.

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27. The word “complete” is not a good translation for allseitig, but it is in any case not so misleading as the word “comprehensive” chosen by Geach and Black. A better expression would, perhaps, be “knowledge from every angle.” 28. See Frege’s argumentation in “Über Sinn und Bedeutung.” 29. See note 10. See also Leibniz (1969), sec. 8. 30. There are a great number of studies in late nineteenth-century and early twentieth-century philosophy, especially Frege and Husserl, that one could recommend for further reading, for example, Beaney (1996), Bilezki and Matar (1998), Dummett (1993), Floyd and Shieh (2001), Glock (1999), Hill (1991), Hill and Rosado Haddock (2000), Kreiser (2001), Macbeth (2005), Mendelsohn (2005), Reck (2002), Schumann (1977), Tieszen (1989, 2004), Tragesser (1977), and Weiner (2004).

References Beaney, Michael. 1996. Frege: Making Sense. London: Duckworth. Beaney, Michael. 2002. Decompositions and Transformations: Conceptions of Analysis in the Early Analytic and Phenomenological Traditions. Southern Journal of Philosophy 43: 53–99. Becker, Oskar. 1927. Mathematische Existenz: Untersuchungen zur Logik und Ontologie mathematischer Phänomäne. Halle: Max Niemeyer. Bilezki, Anat, and Anat Matar, eds. 1998. The Story of Analytic Philosophy: Plot and Heroes. London: Routledge. Bolzano, Bernard. [1837] 1929–1931. Wissenschaftslehre in vier Bänden, hrsg. von W. Schultz. Leipzig: Verlag von Felix Meiner. Boole, George. [1847] 1965. The Mathematical Analysis of Logic, being an essay towards a calculus of deductive reasoning. Oxford: Basil Blackwell. Boole, George. [1854] 1958. An Investigation of the Laws of Thought, on which are founded the mathematical theories of logic and probabilities. New York: Dover. Brentano, Franz. 1929. Über die Gründe der Entmutigung auf philosophischem Gebiete. In Über die Zukunft der Philosophie, ed. O. Kraus, 83–100. Leipzig: Verlag von Felix Meiner. Burge, Tyler. 1992. Frege on Knowing the Third Realm. Mind 101: 633–650. Carnap, Rudolf, 1931. Überwindung der Metaphysik durch logische Analyse der Sprache. Erkenntnis 2: 219–241. Coﬀa, J. Alberto. 1991. The Semantic Tradition from Kant to Carnap. To the Vienna Station, ed. Linda Wessels. Cambridge: Cambridge University Press. Cooper-Wiele, J. K. 1989. The Totalizing Act: Key to Husserl’s Early Philosophy. Dordrecht: Kluwer. Crowell, Stephen G. 1992. Lask, Heidegger, and the Homelessness of Logic. Journal of the British Society for Phenomenology 23: 222–239. Detlefsen, Michael. 1992. Brouwerian Intuitionism. In Proof and Knowledge in Mathematics, ed. Michael Detlefsen, 208–250. London: Routledge. Dummett, Michael. 1993. Origins of Analytical Philosophy. London: Duckworth. Erdmann, Benno. [1892] 1923. Logische Elementarlehre, dritte vom Verfasser umgearbeitete Auﬂage, hrsg. von E. Becher. Berlin: Walter de Gruyter. Floyd, Juliet, and Sanford Shieh, eds. 2001. Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy. New York: Oxford University Press.

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Frege, Gottlob. [1879] 1964. Begriﬀsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. In Frege (1964), 1–88. Frege, Gottlob. [1883] 1964. Über den Zweck der Begriﬀsschrift. In Frege (1964), 97–106. Frege, Gottlob. [1884] 1968. Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriﬀ der Zahl. In The Foundations of Arithmetic/Die Grundlagen der Arithmetik, repr. and trans. by John. L. Austin. Oxford: Basil Blackwell. (Referred to as GLA.) Frege, Gottlob. 1893. Grundgesetze der Arithmetik, begriﬀsschriftlich abgeleitet, Erster Band. Jena: Verlag von H. Pohle. (Referred to as GGA I.) Frege, Gottlob. 1903. Grundgesetze der Arithmetik, begriﬀsschriftlich abgeleitet, Zweiter Band. Jena: Verlag von H. Pohle. (Referred to as GGA II.) Frege, Gottlob. 1952. Translations from the Philosophical Writings of Gottlob Frege, ed. Peter Geach and Max Black. Oxford: Basil Blackwell. Frege, Gottlob. 1964. Begriﬀsschrift und andere Aufsätze, hrsg. Ignacio Angelelli. Hildesheim: Georg Olms. (Referred to as BS.) Frege, Gottlob. 1967. Kleine Schriften, hrsg. Ignacio Angelelli. Darmstadt: Wissenschaftliche Buchgesellschaft, und Hildesheim: Georg Olms. (Referred to as KS.) Frege, Gottlob. 1969. Nachgelassene Schriften, hrsg. Hans Hermes, Friedrich Kambartel, und Friedrich Kaulbach. Hamburg: Felix Meiner. (Referred to as NS.) Frege, G. 1972. Conceptual Notation and Related Articles, trans. and ed. by T. W. Bynum. Oxford: Clarendon Press. Frege, Gottlob. 1976. Wissenschaftlicher Briefwechsel, hrsg. Gottﬁred Gabriel, Hans Hermes, Friedrich Kambartel, Christian Thiel, und Albert Veraart. Hamburg: Felix Meiner. (Referred to as BW.) Frege, Gottlob. 1979. Posthumous Writings, trans. Peter Long and Roger White. Oxford: Basil Blackwell. Friedman, Michael. 1996, Overcoming Metaphysics: Carnap and Heidegger. In Origins of Logical Empiricism, Minnesota Studies in the Philosophy of Science, Vol. XVI, ed. Richard N. Giere and Alan W. Richardson, 45–79. Minneapolis: University of Minnesota Press. Friedman, Michael. 2000. A Parting of the Ways: Carnap, Cassirer and Heidegger. Chicago and La Salle: Open Court. Fries, Jakob F. [1811a] 1819. System der Logik, zweite verbesserte Auﬂage. Heidelberg: Mohr und Winter. Fries, Jakob F. [1811b] 1827. Grundriss der Logik, dritte Auﬂage. Heidelberg: Christian Friedrich Winter. Gabriel, Gottfried. 1986. Frege als Neukantianer. Kantstudien 77: 84–101. Glock, Hans-Johann. 1999. The Rise of Analytic Philosophy. Oxford: Blackwell. Goldfarb, Warren D. 1979. Logic in the Twenties: The Nature of the Quantiﬁer. Journal of Symbolic Logic 44: 351–368. Haack, Susan. 1978. Philosophy of Logics, Cambridge: Cambridge University Press. Haaparanta, Leila. 1985. Frege’s Doctrine of Being. Acta Philosophica Fennica 39. Haaparanta, Leila. 1988. Analysis as the Method of Discovery: Some Remarks on Frege and Husserl. Synthese 77: 73–97.

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Haaparanta, Leila. 1994. Intentionality, Intuition and the Computational Theory of Mind. In Mind, Meaning and Mathematics: Essays on the Philosophical Views of Husserl and Frege, ed. Leila Haaparanta, 211–233. Dordrecht: Kluwer. Haaparanta, Leila. 1995. On the Possibility of Pure Epistemology: A Husserlian Point of View. In Mind and Cognition: Philosophical Perspectives on Cognitive Science and Artiﬁcial Intelligence, ed. Leila Haaparanta and Sara Heinämaa, 151–167. Acta Philosophica Fennica 58: 151–167. Haaparanta, Leila. 1999a. On the Relations between Logic and Metaphysics: Frege between Heidegger and the Vienna Circle. In Metaphysics in the Post-Metaphysical Age, Papers of the 22nd International Wittgenstein Symposium, Contributions of the Austrian Ludwig Wittgenstein Society, vol. 7 (1), ed. Uwe Meixner and Peter Simons, 243–248. Haaparanta, Leila. 1999b. On the Possibility of Naturalistic and of Pure Epistemology. Synthese 118: 31–47. Haaparanta, Leila. 2003. Finnish Studies in Phenomenology and Phenomenological Studies in Finland. In Analytic Philosophy in Finland, Poznań Studies in the Philosophy of the Sciences and the Humanities, vol. 80, ed. Leila Haaparanta and Ilkka Niiniluoto, 491–509. Amsterdam: Rodopi. Haaparanta, Leila. 2007. The Method of Analysis and the Idea of Pure Philosophy in Husserl’s Transcendental Phenomenology. In The Analytic Turn: Analysis in Early Analytic Philosophy and Phenomenology, ed. Michael Beaney, 262–274. London: Routledge. Haaparanta, Leila and Anssi Korhonen. 1996. Kolmannen valtakunnan vieraina— huomautuksia loogisten objektien olemassaolosta. In Tieto, totuus ja todellisuus, ed. Ilkka A. Kieseppä, Sami Pihlström, and Panu Raatikainen, 38–44. Helsinki: Gaudeamus. Hegel, Georg Wilhelm Friedrich. [1830] 1970. Enzyklopädie der philosophischen Wissenschaften. In G. W. F. Hegel, Werke in zwanzig Bänden: Band 8, ed. E. Moldenhauer und K. M. Michel. Frankfurt: Suhrkamp. Heidegger, Martin. [1929] 1992. Was ist Metaphysik?. Frankfurt: Vittorio Klostermann. van Heijenoort, Jean. 1967. Logic as Calculus and Logic as Language. Synthese 17: 324–330. Heyting, Arend. 1930a. Sur la logique intuitionniste. Académie Royale de Belgique, Bulletin 16: 957–963; the English translation “On Intuitionistic Logic,” in Mancosu (1998), 306–310. Heyting, Arend. 1930b. Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der Preussischen Akademie der Wissenschaften, 42–56; English translation “The Formal Rules of Intuitionistic Logic,” in Mancosu (1998), 311–327. Heyting, Arend. 1931. Die intuitionistische Grundlegung der Mathematik. Erkenntnis 2: 106–115. Hill, Claire Ortiz. 1991. Word and Object in Husserl, Frege, and Russell: The Roots of Twentieth-Century Philosophy. Athens: Ohio University Press. Hill, Claire Ortiz and Guillermo E. Rosado Haddock. 2000. Husserl or Frege? Meaning, Objectivity, and Mathematics. Chicago: Open Court. Hintikka, Jaakko. 1979. Frege’s Hidden Semantics. Revue Internationale de Philosophie 33: 716–722. Hintikka, Jaakko. 1981a. Semantics: A Revolt against Frege. In Contemporary Philosophy, vol. 1, ed. Guttorm Fløistad, 57–82. The Hague: Martinus Nijhoﬀ.

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Hintikka, Jaakko. 1981b. Wittgenstein’s Semantical Kantianism. In Ethics, Proceegings of the Fifth International Wittgenstein Symposium, ed. E. Morscher and R. Stranzinger, 375–390. Vienna: Hilder–Pichler–Tempsky. Hugly, Peter. 1973. Ineﬀability in Frege’s Logic. Philosophical Studies 24: 227–244. Husserl, Edmund [1891] 1970. Philosophie der Arithmetik, mit ergänzenden Texten (1890–1901), Husserliana XII, hrsg. L. Eley. The Hague: Martinus Nijhoﬀ. Husserl, Edmund. [1900, 1913] 1950a. Logische Untersuchungen I, Husserliana XVIII, Text der 1. (1900) und der 2. (1913) Auﬂage, hrsg. E. Holenstein. The Hague: Martinus Nijhoﬀ. (Referred to as LU I, A/B.) Husserl, Edmund. [1901, 1913, 1921] 1984. Logische Untersuchungen II, Husserliana XIX/1–2, Text der 1. (1901) und der 2. (1913, 1. Teil; 1921, 2. Teil) Auﬂage, hrsg. U. Panzer. The Hague: Martinus Nijhoﬀ. (Referred to as LU II, A/B1 and LU II, A/B2 .) Husserl, Edmund. [1913, 1921] 1970. Logical Investigations I–II, trans. J. N. Findlay. New York: Humanities Press. (Referred to as LI.) Husserl, Edmund. [1910–1911] 1965. Philosophie als strenge Wissenschaft, hrsg. R. Berlinger. Frankfurt: Vittorio Klostermann. (Referred to as PsW.) Husserl, Edmund. [1913] 1950b. Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie I (1913), Husserliana III, hrsg. Walter Biemel. Den Haag: Martinus Nijhoﬀ; trans. W. R. Boyce Gibson, Allen & Unwin, London, and Macmillan, New York, 1931, and F. Kersten, Martinus Nijhoﬀ, The Hague, 1982. Husserl, Edmund. 1929. Formale und transzendentale Logik: Versuch einer Kritik der logischen Vernunft. Halle: Verlag von Max Niemeyer. (Referred to as FTL.) Husserl, Edmund. [1939] 1964. Erfahrung und Urteil: Untersuchungen zur Genealogie der Logik, red. und hrsg. L. Landgrebe. Hamburg: Claassen Verlag. (Referred to as EU.) Husserl, Edmund. 1969. Formal and Transcendental Logic, trans. D. Cairns. The Hague: Martinus Nijhoﬀ. Kant, Immanuel. [1781, 1987] 1904. Kritik der reinen Vernunft, 1781 (A), 2nd ed. 1787 (B). In Kant’s gesammelte Schriften, Band III, Berlin: G. Reimer; trans. Norman Kemp Smith (1929). Macmillan, London, 1929. (Referred to as KRV.) Kemp Smith, Norman. 1962. A Commentary to Kant’s ‘Critique of Pure Reason,’ 2nd ed. New York: Humanities Press. Kreiser, Lothar. 2001. Gottlob Frege: Leben – Werk – Zeit. Hamburg: Felix Meiner Verlag. Kusch, Martin. 1995. Psychologism: A Case Study in the Sociology of Philosophical Knowledge. London and New York: Routledge. Lask, Emil. [1911] 1923. Die Logik der Philosophie und die Kategorienlehre. In E. Lask, Gesammelte Schriften II, hrsg. Eugen Herrigel, 1–282. Tübingen: J. C. B. Mohr. (Referred to as LP.) Leibniz, Gottfried Wilhelm. [1890] 1961a. Die philosophischen Schriften von Gottfried Wilhelm Leibniz, Siebenter Band, hrsg. von C. I. Gerhardt. Hildesheim: Georg Olms. Leibniz, Gottfried Wilhelm. [1903] 1961b. Opuscules et fragments inédits de Leibniz, ed. Louis Couturat. Hildesheim: Georg Olms. Leibniz, Gottfried Wilhelm. 1969. Discourse on Metaphysics. In G. W. Leibniz, Philosophical Papers and Letters, trans. and ed. L. E. Loemker, 2nd ed., 303–330. Dordrecht: Reidel.

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Lohmar, Dieter. 2002a. Elements of a Phenomenological Justiﬁcation of Logical Principles, including an Appendix with Mathematical Doubts concerning some Proofs of Cantor on the Transﬁniteness of the Set of Real Numbers. Philosophia Mathematica 10: 227–250. Lohmar, Dieter. 2002b. The Transition of the Principle of Excluded Middle from a Principle of Logic to an Axiom: Husserl’s Hesitant Revisionism in Logic. In New Yearbook of Phenomenology and Phenomenological Philosophy, ed. Burt Hopkins and Steven Crowell, 53–68. Madison: University of Wisconsin Press. Lotze, Hermann. 1874. System der Philosophie, Erster Teil: Drei Bücher der Logik. Leipzig: Verlag von G. Hirzel. Macbeth, Danielle. 2005. Frege’s Logic. Cambridge, Mass.: Harvard University Press. Mancosu, Paolo, ed. 1998. From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. New York: Oxford University Press. Mendelsohn, Richard L. 2005. The Philosophy of Gottlob Frege. Cambridge: Cambridge University Press. Mill, John Stuart. [1843] 1906. A System of Logic: Ratiocinative and Inductive. New York: Longmans, Green. Mitchell, O. H. 1883. On a New Algebra of Logic. In Studies in Logic by Members of the Johns Hopkins University, ed. Charles Peirce, 72–106. Boston: Little, Brown. Mohanty, J. N. 1982. Husserl and Frege. Bloomington: Indiana University Press. Murphey, Murray G. 1961. The Development of Peirce’s Philosophy. Cambridge, Mass.: Harvard University Press. Peirce, Charles Sanders. 1931–1935. Collected Papers of Charles Sanders Peirce (CP), vols. I–VI, ed. Charles Hartshorne and Paul Weiss. Cambridge, Mass.: Harvard University Press. Peirce, Charles Sanders. 1958. Collected Papers of Charles Sanders Peirce (CP), vols. VII–VII, ed. Arthur Burke. Cambridge, Mass.: Harvard University Press. Peirce, Charles Sanders. 1976. The New Elements of Mathematics by Charles S. Peirce (NE), vols. 1–4, ed. Carolyn Eisele. The Hague: Mouton. Peirce, Charles Sanders. 1984. Writings of Charles S. Peirce: A Chronological Edition (W), vol. 4, ed. E. Moore et al. Bloomington: Indiana University Press. Reck, Erich H., ed. 2002. From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy. New York: Oxford University Press. Russell, Bertrand. 1956. Logic and Knowledge: Essays 1901–1950, ed. Robert Charles Marsh. London and New York: Routledge. Schlick, Moritz. 1918. Allgemeine Erkenntnislehre. Wien: Verlag von Julius Springer. Schlick, Moritz. 1938. Gesammelte Aufsätze 1926–1936, Wien: Gerold. Schlick, Moritz. 1986. Die Probleme der Philosophie in ihrem Zusammenhang, Vorlesung aus dem Wintersemester 1933/34, hrsg. von H. Mulder, A. J. Kox, und R. Hegselmann. Frankfurt am Main: Suhrkamp. Schuhmann, Karl. 1977. Husserl-Chronik. Denk- und Lebensweg Edmund Husserls. The Hague: Martinus Nijhoﬀ. Seebohm, Thomas. 1989. Transcendental Phenomenology. In Husserl’s Phenomenology: A Textbook, ed. J. N. Mohanty and W. R. McKenna, 345–385. Washington, D.C.: Center for Advanced Research in Phenomenology & University Press of America. Sluga, Hans D. 1980. Gottlob Frege. London: Routledge. Tieszen, Richard. 1989. Mathematical Intuition: Phenomenology and Mathematical Knowledge. Dordrecht: Kluwer.

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Tieszen, Richard. 2004. Husserl’s Logic. In Handbook of the History of Logic, vol. 3: From Leibniz to Frege, ed. Dov M. Gabbay and John Woods, 207–321. Amsterdam: Elsevier. Tragesser, Robert S. 1977. Phenomenology and Logic. Ithaca, N.Y.: Cornell University Press. Trendelenburg, Adolf. 1867. Über Leibnizens Entwurf einer allgemeinen Charakteristik. In Adolf Trendelenburg, Historische Beiträge zur Philosophie, Dritter Band: Vermischte Abhandlungen, 1–47. Berlin: Verlag von G. Bethge. Weiner, Joan. 2004. Frege Explained: From Arithmetic to Analytic Philosophy. La Salle, Ill.: Open Court. Willard, Dallas. 1984. Logic and the Objectivity of Knowledge: A Study in Husserl’s Early Philosophy. Athens: Ohio University Press. Der Wiener Kreis. [1929] 1973. The Vienna Circle of the Scientiﬁc Conception of the World (Wissenschaftliche Weltauﬀassung, Der Wiener Kreis). In Otto Neurath: Empiricism and Sociology, ed. Martha Neurath and Robert S. Cohen, 301–318. Dordrecht: Reidel. Ziehen, Theodor. 1920. Lehrbuch der Logik auf positivistischer Grundlage mit Berücksichtigung der Geschichte der Logik. Bonn: A. Marcus & E. Webers Verlag.

8

A Century of Judgment and Inference, 1837–1936: Some Strands in the Development of Logic Göran Sundholm Dedicated to Per Martin-Löf on the occasion of his 60th birthday. “O judgement! thou art ﬂed to brutish beasts, And men have lost their reason.” —Julius Caesar, Shakespeare My oﬃce in the present chapter is to tell how, within a century, the notions of judgment and inference were driven out of logical theory and replaced by propositions and (logical) consequence. Systematic considerations guide the treatment. My history is unashamedly Whiggish: A current position will be shown as the outcome, or even culmination, of a historical development. No apology is oﬀered, nor, in my opinion, is one needed. Philosophy in general, and the philosophy of logic in particular, treats of conceptual architecture. The logical ediﬁce is an old one and its supporting concepts have a venerable pedigree. Many parts of the building are buried in the past. Thus, the study of conceptual architecture has to be aided by conceptual archaeology. In the light The present chapter is based on lectures that I have given to second-year philosophy students at Leyden since 1990, and also draws on my inaugural lecture (1988). Per MartinLöf’s (1983) Siena lectures were an important source of inspiration, as were innumerable subsequent conversations with him on the history and philosophy of logic. In recent years conversations with my colleagues Maria S. van der Schaar and E. P. Bos have also been helpful. I am also indebted to Dr. Björn Jespersen and Dr. van der Schaar for valuable comments on the penultimate draft and to Dott.ssa Arianna Betti for much appreciated help with word processing. The material has been treated in invited lecture-courses, at the ESSLI Summerschool in Saarbrücken 1991, and at the universities of Siena 1992, Campinas and Rio de Janeiro 1993, Turku 1998, and Amsterdam 1999, as well as in a complete semester-course at Stockholm 1994. I am indebted to hosts and participants alike. My Cracow 1999 LMPS 11 lecture, now published as (2002), brieﬂy tells the inference half of the tale. Translations into English are in general my own.

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of the many changes that logic underwent during my chosen period, this may even seem rather apposite. Within the philosophy of logic, to understand what a present-day position is, it is often essential to understand how it became what it is. Furthermore, the systematic philosophical underpinning of presentday logic is not ﬁxed; the balance is not ready to be drawn up. Accordingly, a survey of the historical development that led to the various options is a required aid for an informed choice among contemporary alternatives. In one essential respect, though, mine diﬀers from a Whig history. The ﬁnal outcome is not necessarily seen as an improvement on earlier but now largely abandoned views. My own preferences go in the direction of anti-realism, but a deliberate attempt has been made to adopt a neutral stance when describing the various positions. The heroes and villains of my plot, in rough chronological order, are John of St. Thomas, Bernard Bolzano, Franz Brentano, Gottlob Frege, the Ludwig Wittgenstein of the Tractatus, Arend Heyting, and Gerhard Gentzen. Minor roles will be played by Immanuel Kant, Johann Gottlieb Fichte, David Hilbert, Bertrand Russell and G. E. Moore, Harold Joachim, and L. E. J. Brouwer. The treatment will not be exhaustive. In particular, many eminent logicians will not be treated, even though they do belong to the period under consideration, for the simple reason that their contributions did not touch the systematic theme that uniﬁes my exposition. The criteria for inclusion and exposition are based also on systematic considerations. It is my conviction, with respect to our present stage of logical knowledge, both systematic and historical, that this deserves preference above a mere recording of chronological facts. The systematic framework in which such facts are ﬁtted confers coherence and memorability on the unfolding tale. Such a procedure is not without its dangers. They have been faced with great lucidity by Jonathan Barnes: On the one hand, no discussion of the ancient theories will have any value unless it is conducted in moderately precise and rigorous terms; and on the other, a rigorous and precise terminology was unknown in the ancient world. If I insist on precision I shall be guilty of anachronism. If I stick to the ancient formulations, I shall be guilty of incoherence. I prefer anachronism.1 Barnes’s point is well taken and applies with equal force to the nineteenth century. Taking my cue from him, if methodological demands force me into anachronism, I would rather be coherent than (chronologically) right. The (Oxford English) dictionary explains logic as the art and science of valid reasoning. In my chosen century, the central notion of logic is that of judgment. Its form and function in inference will play a crucial role in the sequel. Changes in the conception of judgment and, concomitantly, of inference, are central here. Other topics, such as the position of the law of the excluded third, its function as a criterion for signiﬁcance, and its relation to the knowability of truth, also serve to structure the chapter. The (un)deﬁnability of truth, as well as the nature of the formal calculus used (if any), will also so serve.

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1. L’ancien Régime: The Logic That Was to Be Overturned The preface to early editions of Quine’s Methods of Logic opened with the terse observation that “Logic is an old subject, and since 1879 it has been a great one.”2 One would be hard put not to agree with the ﬁrst part of Quine’s quip, but a number of us have taken issue with the second. Surely, logic was great also prior to 1879, the year in which Frege published his Begriﬀsschrift. George Boolos and Hilary Putnam have respectively dated the inauguration of logical greatness to 1847 and 1854 on the strength of the appearance of George Boole’s logical works.3 Contrary to the received Massachusetts wisdom of Harvard and MIT, it seems obvious to me that the year 1837 deserves pride of place within the history of logic as the proper counterpart to 1879.4 To grasp the substance and magnitude of the logical revolution, we have to consider in outline the kind of logic that was superseded. To a large extent it was nothing but a latter-day version of traditional logic, with the typical methodological accretions that became common after the Port Royal Logic.5 We do well to remember that traditionally logic was conceived of as more wideranging than what is today the case. As a matter of fact, the “sweet Analytics of Aristotle”—Prior and Posterior—are not addressed to the same problematic. The Analytica Priora is devoted to the theory of consequence, that is, an answer is oﬀered to the question: What follows from what? The Analytica Posteriora, on the other hand, treats of the theory of demonstration, where the crucial question is: How does one obtain further knowledge from known premises? Present-day logic restricts itself to the theory of consequence and relegates the theory of demonstration to epistemology. In the nineteenth century, on the other hand, these epistemological concerns constituted a part of logical theory. At the beginning of my chosen period, the traditional patrimony is still very much in charge. The following familiar square oﬀers a convenient starting point for (my description of) the successive revolutions in logic: The Traditional Structure of Logic: Operation of the Intellect

(Mental) Product

(External) Sign

1

Simple Apprehension

Concept, Idea, (Mental) Term

(Written/spoken) Term

2

Judging, Composition/Division of two terms

Judgment, (Mental) Proposition: S is P .

Assertion, (Written/spoken) Proposition

3

Reasoning, Inferring

(Mental) Inference

(Written/spoken) Inference, Reasoning

The diagram6 employs a conceptual order of priority from left to right, from acts, via products, to signs: Acts of various kinds have mental products that

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(may) have (outward) linguistic signs, be they spoken or written. But for this “horizontal” order of conceptual priority there is also a “vertical” order of priority among the (act-)kinds, that is, the operations of the mind. The proper subject matter of logic is reasoning, that is, the third operation of the mind. Nevertheless, the two other operations have to be included in the domain of logic, since inferences are built from judgments. Judgments, in turn, are formed through the composition, or division, of two concepts (“terms”). In logic, the conceptual order starts with terms, and proceeds via judgment to inference. The traditional diagram exhibits a characteristic tripartite act

object ←→ sign

structure. Indeed, Johann Gottlieb Fichte went so far as to claim that essentially there are only two philosophical positions with respect to its epistemological components act object.7 Either you give the object through the act, in which case—with Fichte—you are an idealist, or you direct the act toward the prior object, in which case you are a dogmatist.8 Under this act/object structure, concepts are objects of acts of grasping (“apprehending”), and similarly the judgments made (“mental propositions”) are products of the acts of judging. With respect to the third operation, though, the traditional position is not consequent. To sort this out we note a basic ambiguity in the term inference. On the one hand, inference may be taken in the sense of an inference pattern (German Schlussweise). Such a pattern, or mode, of inference can be given by means of a schema I: J1 J2 . . . Jk , J where I deliberately have allowed more than the customary two premises of traditional syllogisms. The mode I of inference corresponds to a rule of inference according to which you have the right to make, that is, to know, the judgment J, provided that you have already made, that is, provided that you already know, the judgments J1 , J2 , . . . , Jk . On the other hand, inference can also pertain to an act of inference, say, for instance, one made according to the mode I. Such an act has, or perhaps better, proceeds according to, the structure | | | J1 J2 . . . J3 9 . J The product of an act of inference, though, is not an act of inference—the act, clearly, does not have itself as product—nor is it the mode of inference I, according to which the act was carried out; on the contrary, it is the judgment made J. The traditional diagram is accordingly in error when it puts (mental)

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inference in the product place of the act of inference (reasoning). What could such a “mental inference” be? No suitable entity seems available for service in the role. The inference mode is not an act of inference, nor a product of such an act; it is a blueprint, or manual, for inference acts that have products. An inference act is a mediate act of judgment, in which one judgment, the conclusion, is known on the basis of certain other judgments, the premises, being known. Thus, an act of inference is a particular kind of judging, whence its (mental) product is a judgment made. Already Kant famously reversed one of the above orders of priority, namely, that between rows 1 and 2. Concepts are no longer held to be prior to judgments: “We can reduce all actions of reason to judgements, so that reason generally can be regarded as a capacity for judgement.”10 This reversal, in one form or other, we shall encounter in most of the thinkers here considered. Also other paradigm shifts in philosophy can be accounted for in terms of the traditional diagram. The most original contribution of twentieth-century philosophy, namely, the abolition of the primacy of the inner mental life that was eﬀected by Wittgenstein,11 can be seen as nothing but a reversal of the priorities between the second and third columns. The outward sign is no longer conceptually posterior to the inner product.

2. Speech Act Intermezzo: A Uniﬁed Linguistic Account for Some Nineteenth-Century Changes Traditionally, the linguistic counterpart to the mental judgment made is the assertion. This term, in common with other English -ion words, exhibits a process/product ambiguity.12 It may concern the act of asserting (judging) or the product of such an act, the assertion (judgment) made. The appropriate linguistic tool for assertion is the declarative sentence. In general, when S is a declarative sentence the question Is it true that S? may legitimately be put. An assertion that snow is white is readily eﬀected by means of a single utterance of the declarative sentence Snow is white.13 By convention, in the absence of counterindications that it should not be so held, a single utterance of a declarative is an assertion. For instance, the declarative S is not used assertorically in Consider the example: S. or He claimed that S, but I don’t know whether it really constitutes so.14

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Not every use of a declarative sentence is assertoric, but assertoric uses can be recognized as such since the counterquestions: How do you know that S is true? What are your grounds?15 are a legitimate response to an assertoric use of S. The content of the assertion eﬀected by means of an assertoric utterance of the declarative Snow is white, that is, the assertion that snow is white, is given by means of a nominalized that clause, that snow is white. In general, a single utterance of this clause alone will not serve to eﬀect an assertion that snow is white.16 To get back to a declarative, a single utterance of which will so serve, one must either append is true or preﬁx it is true to the clause in question. Then we obtain, respectively, that snow is white is true and it is true that snow is white, single utterances, either of which do suﬃce for asserting that snow is white. Note that the ﬁrst of these two formulations admits of the preﬁx the content. It then yields a yet fuller but still equivalent formulation of the judgment made: The content that snow is white is true. The second formulation, though, resists the corresponding interpolation, which results in ungrammatical nonsense: It is true the content that snow is white. These considerations suggest that judgable content A is true is the proper form of judgment, when one prefers a unary form of judgment that makes explicit the content judged in the judgment made.17 The content in question will be given by a that-clause formed from a declarative S. The judgment made in or by the act of judging that is made public through the assertoric utterance of the declarative S accordingly, takes the form that S is true.18

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It must be stressed, though, that this path to this unary, content-explicit form of judgment is manifestly not language independent, because it draws extensively on linguistic considerations, albeit very simple ones. As such, it would be rejected out of hand by most major ﬁgures considered here, even though the unary form of judgment itself is accepted. Diﬀerent analyses oﬀered by various logicians reach the same result, but diﬀerent routes are taken. Nevertheless the speech act theory route to the unary form of judgment constitutes as good an example as any of the characteristic—twentieth-century— linguistic turn in philosophy that was inaugurated by Frege (1891): ontological and epistemological questions are now answered (while recast in linguistic form) via a detour through language.19 However, drawing on the traditional conceptual link between judgment and assertion, namely, that between mental object and exterior sign, the above exposition, in spite of its anachronistic (twentieth-century) ﬂavor, explains why the (nineteenth-century) unary form of judgment has to take the form it has.

3. Revolution: Bolzano’s Annus Mirabilis I postulated that 1837 was a crucial year for logic, no reason being given. However, in this year the four hefty tomes of Bernard Bolzano’s Wissenschaftslehre made their weighty appearance.20 This event constitutes the greatest revolution in logical theory since Aristotle, even though the Wissenschaftslehre fell stillborn from the press, as far as near-time inﬂuence is concerned, owing to clerical and political censorship. Indeed, in the preface to the second edition of his main work (the ﬁrst edition of which appeared in the year of Bolzano’s birth) a very distinguished professor of philosophy could still write: “Since Aristotle, [Logic] has not had to retreat a single step. Also remarkable is that it has not been able to take a single step forward, and thus to all appearance is closed and perfect,”21 which state of aﬀairs continued until the coming of the second nineteenth-century revolution in logic. Within logical theory, 1879, the year of Quine’s choosing, is the counterpart to the second revolutionary year 1848. Traditional logic was ﬁrst and foremost a term logic, rather than a propositional logic. In spite of the medieval scholastic achievements concerning the theory of consequentiae, and the insights of the—much earlier—Stoic logic, the syllogism, in one version or other, still ruled supreme, which circumstance renders Kant’s opinion considerably less farfetched than it might seem today. For instance, his own conception of logic as set out in the Jäsche Logik (whether it be truly Kantian or not) is cast entirely in the customary traditional mold.22 Bolzano’s revolution with respect to the traditional picture is threefold. First, the middle (“product”) column of the traditional schema is objectiﬁed. The mental links are severed, and thus, in particular, the traditional notions mental term (concept, idea) and mental proposition (judgment) are turned into their ideal, or Platonist, counterparts idea-in-itself (Vorstellung an sich)

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and proposition-in-itself (Satz an sich).23 Second, the pivotal middle square of the diagram is altered: The judgment made no longer takes the traditional (S is P ) form. Logic is no longer term logic. Instead Bolzano uses the propositional, unary form of judgment that was canvassed above, with his Sätze an sich taking the role of judgable contents: The Satz an sich S is true.24 Third, Bolzano bases his logical theory, not on inference (from judgments known to judgment made), but on (logical) consequence between propositions.25 Judgment is dethroned and its content now holds pride of place in logical theory. Needless to say, Bolzano, a priest steeped in the tradition, does not jettison everything traditional: A Satz an sich, that is, the judgable content, rather than the judgment made, has (or can brought to) the canonical form V1 has V2 , which is very close to the Aristotelian form S is P . Instead of the Aristotelian judgment Man is mortal we ﬁnd the Bolzanian content Man has mortality. The precise reasons for this shift from the concrete mortal to the abstractum mortality need not detain us here; in essence, Bolzano takes the Aristotelian form of judgment and turns it into a form of content, where the contents are objectiﬁed denizens of the ideal—Platonic—third realm.26 Bolzano’s key notion is that of proposition-in-itself: The idea-in-itself is explained as a part of a proposition-in-itself that is not a proposition-in-itself. Bolzano’s logical objectivism is a Platonism: As already noted, his crucial an sich notions are all ideal. We are not told very much about what ideal means here. Instead, his manner of proceeding is that of a via negativa: a list of nonapplicable attributes is oﬀered. Thus, the ideal realm is characterized as atemporal, aspatial, inert, nonlinguistic, nonmental, unchangeable, nongenerated. . . . Furthermore, the propositions-in-themselves serve in various logical roles, in particular as contents of mental acts and declarative sentences.27 However, not only propositions and their parts are ideal an sich notions: The truth of a true proposition-in-itself is truth-in-itself. Bolzano’s explanation of truth is an interesting one. According to him, all propositions have or can be brought to the logical form (V1 has V2 ), and so truth only has to be explained

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for propositions of this form: The proposition (V1 has V2 ) is true if and only if V1 really (German wirklich) has V2 , for instance, the proposition-in-itself that snow has whiteness is true if and only if snow really has whiteness.28 This, virtually “disquotationalist,” rendering is compatible with currently fashionable “minimalist” positions concerning truth. Bolzano, however, was on the road toward a more substantial notion of truth when he noted that the following proportion holds concerning truth and a certain kind of existence, namely, that of instantiation (German Gegenständlichkeit), that is, the higherorder property of an idea-in-itself of being instantiated: the similarity between this relation among propositions and . . . that among ideas is obvious. Namely, what holds, concerning ideas, for the circumstance whether indeed a certain object falls under them or not, holds, concerning propositions, for the circumstance whether truth pertains to them or not.29 In the form of a proportion: proposition-in-itself idea-in-itself = . truth instantiation Thus, what it is for a proposition-in-itself to be true is what it is for an idea-in-itself to have something falling under it. In other words, applied to my (snow-bound) stock example: the proposition-in-itself that snow is white is true (or has truth, in the terminology preferred by Bolzano) precisely when the idea-initself the whiteness of snow has nonemptiness, that is, when some entity falls under the whiteness of snow.30 Bolzano here anticipates something of considerable importance for the analysis of truth, and we shall have occasion to return to his comparison in the sequel. Bolzano’s apparatus for logical analysis, comprising propositions, ideas, and instantiation, is highly versatile.31 Thus, for instance, as Leibniz knew, the four categorical Aristotelian judgments are readily cast in the required form. For instance, an E judgment, No V1 are V2 , is rendered the Idea (in-itself) of a V1 that is V2 does not have existence (Gegenständlichkeit).32

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The notion of truth in itself for propositions-in-themselves is bivalent: For every proposition-in-itself A, A has truth or A has falsity, also in-itself. The an sich character of the truth of true propositions-in-themselves is one of the pillars on which Bolzano’s logical realism rests.33 Another is the reduction of epistemological matters to the Platonist an sich notions. The ﬁrst instance of this reduction concerns judgment: a judgment of the novel form, that is, proposition-in-itself A is true is correct (richtig) if A really is a truth-in-itself.34 This reduces the epistemic notion of the correctness for judgments to the Platonist an sich notion of truth for propositional contents. Here Bolzano pays a price—in my opinion too high a price—for his iron-hard realism in logic and epistemology. Under the Bolzano reduction, a blind judgment, a mere guess, without any trace of justiﬁcation, is a piece of knowledge (an Erkenntnis).35 The only thing that matters is the an sich truth, whether knowable or not, of the proposition-in-itself that serves as content of the judgment in question.36 Thus, for instance, according to Bolzano, if, independently of any counting, it happens to hit bull’s eye, my unfounded claim that the City Hall at Leyden has 1234 window panes, is simply a piece of knowledge. In this I, for one, cannot follow him. Bolzano deserves high praise for his lucid and uncompromising realism. Also antirealists proﬁt from reading him: His version of realism is one of the very best on oﬀer.37 Admitting blind judgments as pieces of knowledge, however, is not just realism but realism run rampant. Bolzano’s transformation of the third and ﬁnal notion in the traditional picture, namely, that of inference, makes an unmistakably modern impression. The changed form of judgment transforms the inference schema I into I : A1 is true A2 is true . . . Ak is true . C is true An inference according to I is valid if the proposition-in-itself C is a logical consequence of the propositions-in-themselves A1 , A2 , . . . , Ak . Such a logische Ableitbarkeit—Bolzano’s terminology—holds between the A’s and C when each uniform variation V of all nonlogical ideas that makes all the A’s true also makes C true.38 In other—more modern—words, C is a logical consequence of A1 , A2 , . . . , Ak when the proposition-in-itself (A1 & A2 & · · · & Ak ) ⊃ C is not just true but logically true, that is, true under all uniform variations of its nonlogical parts.39 The notion of an Ableitbarkeit provides yet another Bolzano reduction of an epistemic notion to Platonist an sich notions. In the same fashion that

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Bolzano reduced the (epistemic) correctness (Richtigkeit) of the judgment made to the an sich truth of its an sich content, the validity of an inference is also reduced to, or in this case perhaps better, replaced by something on the level of the Platonist contents of judgments. Indeed, the fourth chapter, §§223–268, of the Wissenschaftslehre bears the title “Von den Schlüssen,” but deals with Ableitbarkeiten among propositions-in-themselves, rather than with judgments that are made on the basis of certain other judgments already having been made.40 Thus the inference is valid or not, irrespective of whether it transmits knowledge from premise judgments to the conclusion judgment, solely depending on the an sich truth-behavior of the propositions-in-themselves that serve as contents of the judgments in question, under all variations with respect to suitable in-themselves parts of the relevant propositions. Bolzano’s position is accordingly threatened not just by the phenomenon of blind knowledge. Under his account also inference can be blindly valid, irrespective of whether it preserves knowability from premise(s) to conclusion. Logical consequence (logische Ableitbarkeit) is a relation that may obtain between any propositions whatsoever, be they true or false. Bolzano also studies another consequence relation among propositions, but now restricted to the ﬁeld of truths-in-themselves only, that he calls Abfolge (grounding). The theory of Bolzano’s grounding relation is diﬃcult and as yet not very well explored; it can be seen as yet another reduction of epistemic notions to Platonist ones. Consider the inference I : A is true . B is true When I is valid, that is, preserves knowledge from premise to conclusion, and the premise is known, the judgment A is true serves to ground the judgment B is true. Then a certain relation obtains between the propositions A and B that serve as contents of the judgments in question. Abfolge can be seen as a “propositionalization” Abf (A, B) of that relation: The relation of grounding, which holds in the ﬁrst instance between pieces of knowledge, that is, between judgments known, is turned into a propositional relation (“connective”) between propositions, that is, contents of judgments. Every truth has a grounding tree that is partially ordered according to the Abfolge relation.41 It can be seen as an ideal proof that shows why the true proposition is true, somewhat along the lines of Aristotelian demonstrations διοτι.42 In the light of Bolzano’s innovations and ensuing reductions, it is important to distinguish between the holding of a consequence, that is, the preservation of truth from antecedent propositions to consequent proposition, and the validity of an inference ﬁgure, that is, the preservation of knowability from premise judgments to conclusion judgment.43 This insight is lost to modern philosophy of logic that largely accepts the Bolzano reduction to such an extent that (validity of) inference and (logical holding of) consequence are identiﬁed.

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4. Revisionism; the “Novel” Contributions of Brentano Franz Brentano, in lectures given at the Universities of Würzburg and Vienna, from the early 1870s onward, proposed another revision of traditional doctrine. Because of his distaste for all Platonist notions in logic, such as Bolzano’s proposition-in-itself, Brentano rejected the single unary form of judgment that ascribes truth to a Platonist content.44 Instead, he canvassed the use of two unary forms of judgments, namely, α IS (exists), in symbols α+, and α IS NOT (does not exist), in symbols α−, where α is a (general) concept. Brentano, however, was not the ﬁrst to note this. Already Bolzano explicitly considered these forms, under the respective guises of α has Non-Emptiness (Gegenständlichkeit) and α has Emptiness, and determined their most important properties. In particular, we already noted, Bolzano knew that the four Aristotelian categorical judgments can be dealt with using these two forms.45 Credibility might not be stretched to the point of credulity if we surmise that this anticipation provides one of the reasons for Brentano’s staggering lack of generosity toward the Great Bohemian: When . . . I drew attention to Bolzano, this . . . in no way, was intended to recommend Bolzano as a teacher and leader to the young people. What they could learn from him, I dare say, they could learn better from me. . . . And . . . as I myself never took a single thesis from Bolzano, so I was never able to convince my pupils that they would ﬁnd there a true enrichment of their philosophical knowledge.46 Under the circumstances, “methinks the learned Gentleman doth protest too much!” However, it is not unlikely that also Bolzano’s logical objectivism disqualiﬁed him as a “teacher and leader” in the eyes of Brentano, who distrusted all kinds of logical Platonism. Of more lasting value than Brentano’s employment—and alleged rediscovery—of the Leibniz–Bolzano reductions are his views on the blind judgment.47 These have profound consequences for his formulation of the traditional laws of thought, such as noncontradiction and excluded third, as well as for the relation between truth and evidence. Young man Brentano construed evidence as “experience of truth” (German Erlebnis der Wahrheit—Husserl’s terminology), whence the order of dependence goes from truth to evidence.48

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Later, under the pressure from the phenomenon of blind judgment, he reversed this order of priority and held that truth (correctness, German Richtigkeit) should be seen as possibility for evident judgment: Truth pertains to the judgments of he who judges rightly, that is, to the judgements of him who judges what someone would judge who judged with evidence; that is, he who asserts what would be asserted also by someone judging with evidence.49 Similarly he is led to a negative formulation of the law of excluded middle: It is impossible that someone, who rejects something that is wrongly accepted by someone else, rejects it wrongly, as well as that someone who accepts something, that is wrongly accepted by someone, accepts it wrongly, presupposed . . . that both judge with the same mode of representation and with the same mode of judgement.50 From an antirealist point of view, Brentano is certainly on the right track; he refrains from asserting that a content must be either true or false, in entire independence of whether it is known to be so. His formulation, though, is not entirely correct. Brentano, the great crusader against the blind judgment, here forgets to take it into account. Of course, it is possible that the object A is wrongly accepted by P1 , as well as wrongly rejected by P2 , namely, when P1 and P2 both judge blindly, that is, without evidence. On the other hand, the corresponding formulation of Noncontradiction is correct: It is impossible that someone rightly rejects what is rightly accepted by someone else.

5. Functions Triumphant: Frege’s Account of Judgment and Inference Frege, pace Quine, is generally held to have inaugurated the revolution in logic. From the present perspective though, his contribution is remarkably slender. Logical objectivism, with its novel unary judgment, is present wholesale already in Bolzano, where it is cast in a more perspicuous form. Frege, furthermore, does not treat of logical consequence among propositions, or Thoughts, as he called them. For better or worse, Bolzano, with his insistence on replacing inference with the notion of consequence, makes a much more modern impression than Frege, whose traditional views on inference have come in for much criticism. We must not forget, however, that Frege was a mathematician and from the outset his aims were those of a mathematician rather than of a philosopher. His contributions to my topic are all subservient to the aim of providing a secure foundation for mathematical analysis, very much in the style of traditional Aristotelian foundationalism: One seeks a small number of primitive concepts, and basic truths concerning those primitives, in terms of which, at least a very sizable part and preferably all, of mathematics can be formulated, while its truths can be derived by means of primitive inference steps, where the basic

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axioms and primitive inference steps are made evident from the concepts they contain.51 In the Begriﬀschrift booklet from 1879 (what turns out to be) a preliminary version of the formal language is given and the basic notions explained. In the Grundlagen der Arithmetik from 1884 the program of securing the mathematical theorems by means of reducing the mathematical axioms to logical theorems is spelled out informally. However, Frege was aware of the fact that he had only made plausible the reduction of arithmetic to logic, since, possibly at the instigation of the Brentanist Carl Stumpf, the Grundlagen development was informal and not carried out in the begriﬀsschrift. Thus Frege could not guarantee that his demonstration were really gap-free. The means of demonstration, whether logical or arithmetical, were not explicitly listed. Accordingly, his inferences have not been made evident solely from the concepts employed in them, and so the arithmetical ediﬁce remains shaky. The (considerable) changes in the begriﬀsschrift that were put into eﬀect around 1890 served to make the formal execution of the logicist program feasible; unfortunately, the project failed owing to the emergence of the Zermelo–Russell paradox in Frege’s system. Thus, when compared to Bolzano, Frege’s most important contribution is his begriﬀsschrift.52 By creating this formal language, Frege provides a partial realization of the Leibnizian calculus ratiocinator project. That an inference step, or axiom, is valid depends on contentual aspects pertaining to the notions from which the step, or axiom, in question has been built.53 However, once such a step has been explicitly formulated and validated in terms of contents, it is mechanically recognizable as such. No further contentual, “intuitive” considerations are required to determine whether the inference in question is valid; being of the appropriate syntactic form suﬃces and that form is mechanically, or “blindly,” recognizable. As far as the theoretical framework is concerned, Frege’s one step over and beyond Bolzano is minute but with enormous consequences. In both early and mature formulations of his theory of judgment, Bolzano’s unary form of judgment is retained: The circumstance that S is a fact and a judgment is not the mere grasping of a Thought, but the acknowledgment of its truth.54 Frege, however, by training and profession was a mathematician. His teaching activity was mainly devoted to analytical geometry. Through his mentor Ernst Abbe, one of Riemann’s few students, he also gained access to the latest developments in the then emerging function theory, that is, that branch of mathematical analysis that deals with analytic functions in the complex plane. His logical revolution draws heavily on the notion of function: Instead of Bolzano’s clumsy form of content “A has b”, Frege carves up his contents using the versatile form P (a),

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that is, function P applied to argument a. Frege’s logic is mathematized from the outset. It is especially well suited for coping with Weierstraß’s rigorous treatment of analysis; indeed, the notation could have been (and probably was) invented for the very purpose.55 The familiar concepts of pointwise continuity, and its reﬁnement into uniform continuity, illustrate this: (∀x ∈ I)(∀ε > 0)(∃δ > 0)(∀y ∈ I)(|x − y| < δ ⊃ |f (x) − f (y)| < ε) and (∀ε > 0)(∃δ > 0)(∀x ∈ I)(∀y ∈ I)(|x − y| < δ ⊃ |f (x) − f (y)| < ε). These succinct formulations show how admirably the Fregean quantiﬁer is geared to expressing distinctions involving multiple generality.56 A verbal, natural language treatment would be much harder to take in. Frege’s function-theoretic conception of logic imposed an interesting bifurcation on his views on truth. Mathematicians speak of the value of a function for a certain argument. For instance, 2 + 2 is the value of the function x + 2 for the argument 2. In the ﬁrst instance, the plus-two function takes numbers into numbers, but owing to Frege’s doctrine of universality, it has to be extended into one deﬁned for all objects. One then makes use of what Quine has called a “don’t care” argument, for instance, r+2 if r is a number; ξ + 2 =def the Moon otherwise. Adopting the same perspectives also at the level of sentences, from the complete sentence Caesar conquered Gaul, we get the function ξ conquered Gaul, which must also be deﬁned for all objects, including me, the Moon, and Louis XIV, as well as the number of those grains of sand at Syracuse beach that were not counted by Archimedes when writing the Sandreckoner, and plutonium, an element unknown at the time of Frege. A value of the conquering Gaul function will have to be something close to a judgable content, or Thought. It will not, however, be a judgable content, because it is not invariant under diﬀerent descriptions of the argument. Frege’s by now notorious example concerning the planet Venus makes this clear: Venus = Venus, The Morning Star = Venus, and the Evening Star = Venus are three values of the function ξ = Venus.

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Since in all three cases the argument—Venus = The Morning Star = The Evening Star—is the same, the value has to be the same. The Thought expressed is diﬀerent in all three cases.57 Accordingly, the value of the function for the argument Venus (under any description) is not a Thought. Instead Frege avails himself of certain ideal objects, the True and the False, that are known as “truth values,” and serve as appropriate function-values. By the truth value of a sentence, Frege understands the circumstance that it is true or that it is false.58 Thus, the common function-value in the three cases above is the truth value the True. Sentences are then seen as truth value names.59 In his elucidation of the revised begriﬀsschrift Frege lays down, for each regular sentence, under what condition it is a name of the True. The sentence then expresses, or has as its sense (Sinn), the Thought that this truth condition is fulﬁlled.60 Frege’s theory of meaning is a bipartite mediation theory, very much along the lines of early medieval theories of signiﬁcation: The sign expresses its sense that refers to an entity (called Bedeutung by Frege). In Frege’s theory a number of themes are dealt with that were touched on in the section 2. Frege deemed it necessary to include in his begriﬀsschrift a speciﬁc symbol, that makes explicit the assertoric force that the Kundgabe of a judgment made carries. Frege’s view of inference has come in for much criticism; an inference is an “act of judgement, which is made, according to logical laws, on the basis of judgements already made.”61 On the symbolic level, this is reﬂected in the omnipresence of the judgment stroke, both on premises and conclusion, in Frege’s formal inference-ﬁgures in the Gg. Modus ponens, for instance, takes the form A⊃B A 62 . B The sign “” has changed its meaning and in the logic of today it is an ordinary (meta)mathematical predicate applicable to certain (meta)mathematical objects, namely wﬀ’s, that is, elements of a free algebra of “expressions” generated over a certain “alphabet.” When ϕ ∈ wﬀ, “ ϕ” has the meaning there exists an inductively deﬁned derivation-tree of wﬀ’s with ϕ as end formula; in particular, the Frege sign does no longer function as a force indicator, but can be negated and occur in an antecedent of an implication.63 In Frege, however, it is clear that it expresses assertoric force. Thus, both premises and conclusion of inferences are known, since, as we remarked, assertions made do contain claims to knowledge. The practice of drawing inferences from mere hypothesis, however, in particular as embodied in the works of Gerhard Gentzen, is held to refute Frege at this point.64 Frege was ﬁrmly committed to realism: Being true (Wahrsein) is something diﬀerent from being held true (Fürwahrhalten), be it by one, be it by many, be it by all, and

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can in no way be reduced to it. It would be no contradiction that something is true that is held false by all.65 This is a clear statement of one of the central roles of truth, namely, its metaphysical role. By this I understand the task of truth to hold open the possibility of making mistakes. It is a minimum requirement on any viable epistemological position that it must allow for the possibility of mistaken acts of knowledge: “What is true is independent of our acknowledgement. We can make mistakes.”66 The opposite, “Protagorean” position would make man the measure of all things and would equate truth with truth-for-us. It would constitute an epistemological nihilism, where anything goes, along the lines of moral nihilism within ethics: “If God is dead, everything is permitted.” Mistaken deeds, be they logico-epistemic or ethical, presuppose a norm. Frege avails himself of the required norm via the notion of truth for judgable contents. He then reduces the rightness (Latin rectitudo) of epistemic acts, that is, the notion that is needed, strictly speaking, to uphold metaphysical realism, to that of the correctness of the judgment made in such an act, and that correctness ﬁnally to the truth of the Thought that serves as content of the judgment made. Truth for Thoughts, ﬁnally, is bivalent: Any Thought is either true or false, come what may. Frege secures this via his doctrine of sharp concepts. Thoughts are the result of applying concepts, that is, functions from objects to truth values, to objects. Concepts have to be sharply deﬁned on all objects: The Law of Excluded Third is really the requirement that concepts be sharply delineated in another guise. An arbitrary object Δ either falls under the concept Φ, or it does not fall under it: tertium non datur.67 Thus Frege, and before him Bolzano, secures the metaphysical role of truth, namely, that of providing the notion of rightness for epistemic acts, via the bivalence of truth for judgmental contents. This is not the only way to secure the notion of rightness for acts; Brentano, for instance, rejected the notion of proposition (Thought) and instead used product-correctness as the basic, absolute notion. Wittgenstein, on the other hand, did not take propositional truth as the basic notion, the way Frege and Bolzano did, but reduced it to the ontological notion of obtaining with respect to states of aﬀairs. One can also take the notion of rightness as a primitive notion sui generis, which is my own preferred option.68 Frege throughout his career held the view that truth (for propositions) is sui generis and indeﬁnable. Since the Thought that S is the same as the Thought that it is true that S, every Thought contains (the notion of) truth and so there is no neutral ground left from which to formulate a deﬁnition: Every putative deﬁniens irreducibly contains the deﬁniendum in question. Frege’s realism, just like Bolzano’s, is a logical one: There is no attempt at a further ontological reduction of propositional truth. For Frege, a fact is nothing but

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a true Thought and correspondence theories of truth are ﬁrmly rejected.69 His ontology is very sparse: objects, functions, and that is all—no facts, no states of aﬀairs, no tropes, or what have you. Frege held the wheel at the ﬁrst bend of the linguistic turn. His only category distinction is that between saturated and unsaturated entities, and this ontological distinction draws on the linguistic distinction between expressions with and without gaps into which other expression may be ﬁtted. In spite of his thoroughgoing realism, Frege appears committed to the view that every true proposition can be known as such: The most secure demonstration is obviously the purely logical, which, abstracting from the particular character of the things, rests only on the laws on which all knowledge depends. We then divide all truths that require a justiﬁcation into two kinds, in that for the one, the demonstration can proceed purely logically, for the other has to be based on facts of experience.70 Truths are then divided into those that need justiﬁcations and those that do not; the former are split into those that have purely logical demonstrations and those whose demonstrations rest on experiential facts. Thus, in either case, it appears that if the truth is one that stands in need of justiﬁcation, then there is a demonstration. Thus all truths can be known: If it needs no justiﬁcation, it can be known from itself, whereas truths that do need justiﬁcation can be known through a demonstration, be it logical or empirical.

6. Truth Made: The Correspondence Theory Strikes Back Half a decade after Frege’s Hochleistungen, G. E. Moore inaugurated his realist apostasy from the Hegelianism of his philosophical apprenticeship by adopting something very much like Bolzano’s theory of propositions with an an sich notion of simple truth. In this he was soon followed by Bertrand Russell.71 Russell and Moore were not crystal clear (to put it mildly). The best formulation of their novel theory was oﬀered by a staunch upholder of the old order, the idealist H. H. Joachim, whose aptly titled (1906) book The Nature of Truth has a chapter Truth as a Quality of Independent Entities. His characterization of the an sich theory of truth is a powerful one: “Truth” and “Falsity,” in the only strict sense of the terms, are characteristics of “Propositions.” Every Proposition, in itself in an entire independence of mind, is true or false; and only Propositions can be true or false. The truth or falsity of a Proposition is, so to say, its ﬂavor, which we must recognize, if we recognize it at all, immediately: much as we appreciate the ﬂavor of pineapple or the taste of gorgonzola.72

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Joachim also articulated clearly the possibility of unknowable truths on the an sich reading of truth: “The independent truth will be and remain entirely in itself, unknown and unknowable.”73 In an oblique way Russell had already admitted of the possibility of unknowable truths: Now, for my part, I see no possible way of deciding whether propositions of inﬁnite complexity are possible or not; but this at least is clear, that all the propositions known to us (and it would seem, all propositions that we can know) are of ﬁnite complexity.74 In philosophy, claims that something cannot be done are dangerous and invariably tend to provoke attempts to achieve what has been denied. Frege’s view that truth is sui generis and cannot be deﬁned was challenged even before it had been published:75 After yet another decade of logico-semantical soul-searching Moore and Russell were veering toward the correspondence theory of truth.76 Both gave reductions of truth in ontological terms by means of a truth-maker 77 analysis in the form proposition A is true = there exists a truth-maker for A. In a truth-maker analysis, to each proposition there is related a suitable notion of truth-maker and also a suitable notion of existence with respect to such truth-makers. Moore chose “facts” as his truth-makers and Russell used “complexes.” For Moore, a proposition is true if it corresponds to an existing fact, and for Russell it is true if the complex to which it corresponds exists. The intricacies of their respective ontologies of facts and complexes need not detain us here; both were superseded by Wittgenstein’s Tractatus and are now merely of historical interest. The Tractatus rests on three main pillars, to wit (i) Wittgenstein’s famous picture theory of linguistic representation; (ii) the doctrine of logical atomism, according to which every proposition is a truth-function of elementary propositions; and (iii) the Saying/showing doctrine. Of these the picture theory serves to structure the work.78 In a brief attempt at an exposition, I treat the proposition (∗)

Peter is the father of John

as if it were a Tractarian elementary proposition.79 Thus, our example (∗) is an elementary proposition of the form aRb. Hence, it must (?) immediately (?) strike us as a picture and indeed even one that obviously resembles its subject matter (4.12). How can we make sense of this? On the ontological side, in the world, we have the state of aﬀairs that Peter and John stand in the father-son relation. We now have to construe the propositional sign used to express the proposition (∗) as a fact that

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serves to present this state of aﬀairs. The two structures—linguistic and ontological—have to be, in mathematical parlance, isomorphic. Language

World

“Peter” Peter “John” John Q(a, b) father-son relation Q(“Peter”, “John”) Peter and John’s standing in the father-son relation Our task to ensure isomorphism between language and world amounts to ﬁnding an appropriate Q-relation. Obviously the ﬁeld of such a relation must consist of expressions and this is the key to Wittgenstein’s solution: Q(α, β) =def the expressions α and β stand, respectively, immediately to the left and to the right of the sign-array “is the father of.”80 Hence, “that ‘Peter’ stands in a certain relation, namely the Q-relation, to ‘John’, says that Peter and John stand in the father-son relation” (3.1432). Using the Q-relation, the sentence-sign (∗) is (or can be viewed as) a fact, since the two proper names do stand in the Q-relation. This syntactic fact in turn serves to present the state of aﬀairs that Peter is the father of John. When this state of aﬀairs exists (or obtains), it is a fact, and the proposition is true. In this case the proposition is a picture of the fact. According to the picture theory, every atomic, or elementary, proposition E presents a state of aﬀairs (Sachverhalt) SE that may or may not obtain (4.21).81 Accordingly, if the presented state of aﬀairs SE obtains the elementary proposition is true and depicts (what is then) the fact SE (4.25, 2). States of aﬀairs are logically independent of each other; from the obtaining of one nothing can be concluded about the obtaining of another (2.062). A point (Wahrheitsmöglichkeit) v in logical space LS is an assignment of + (obtains) and − (does not obtain) to each state of aﬀairs (4.3); in other words, a point in logical space is a function v from states of aﬀairs to {+, −}. Thus, LS = {+, −}SV , that is, the collection of functions from the collection SV of Sachverhalte to {+, −}. A situation (Sachlage) σ in logical space is a partition of LS into two parts σ + and σ − (2.11). Points in the positive part σ + are compatible and those in the negative part σ − are incompatible with σ. A proposition A is a truth-functional combination of elementary propositions (5).82 The truth-functional composition of the proposition A determines whether A is true or false with respect to or at a point v in LS. A point v ∈ LS induces a {T(rue), F(alse)}-valuation v on truth-functional propositions in the following way: For an elementary proposition E, v(E) = T

if v(SE ) = +;

v(E) = F

if v(SE ) = −.

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Thus, an elementary proposition is true at a point v if v is compatible with the state of aﬀairs that the elementary proposition presents. For the proposition A = N(ξ), v(A) = T if v(B) = F for every proposition B in the range ξ;83 v(A) = F otherwise. Thus, a generalized (joint) negation is true only if all the negated propositions are true (6, 5.5ﬀ.). A proposition C is a logical consequence of a class Γ of propositions if for every v ∈ LS such that v(A) = T for every A ∈ Γ also v(C) = T (5.11, 512). Because every proposition is obtained through repeated applications of the N-operation to (suitably presented) ranges of propositions, the explanation determines fully whether a proposition is true or false at a point in logical space (5.501–3). The sense (Sinn) of the proposition A is a certain Sachlage σA in LS (4.021, 4.2).84 The positive part of the sense of A is given by { v ∈ LS | v(A) = T }, and similarly for the negative part, of course. The thesis of truth-functionality then ensures that the Sachlage σA , that is, the sense of A can be “computed” from the symbol A. From this epitome it should be clear that the Tractarian logical theory is a realism of the kind that was inaugurated by Bolzano.85 However, Wittgenstein carries the logical realism of Bolzano and Frege to a ﬁtting conclusion: The logical realism of Bolzano is here replaced by an ontological realism. Propositional truth, the primitive an sich notion of logical realism, is reduced one step further to a prior ontological notion, namely, the obtaining of states of aﬀairs. Neither Bolzano nor Frege ignored epistemological issues; in fact, they were of an all-encompassing importance for Frege’s logicist project. Wittgenstein, on the other hand, deliberately eschews epistemic concerns in logic, for instance, the Frege–Russell assertion sign (4.442). Also the epistemic notion of inference is eliminated in favor of logical consequence by means of the Bolzano reduction (5.132). Nevertheless, concerning the deployment of logic, Wittgenstein held that it must be possible to compute mechanically from the symbols alone whether one proposition follows from another (5.13, 6.126, 6.1262). He was wrong in this. In general, the “computation” cannot be executed, owing to its inﬁnitary character. When he wrote the Tractatus, Wittgenstein was not aware of the unsolvability of the general Entscheidungsproblem for the predicate calculus. It was discovered—by Church and Turing—only in 1936, and poses an insuperable technical obstacle for the Tractarian philosophy of logic and language. Thus, Wittgenstein’s vision that everything important concerning logic could be read oﬀ mechanically am Symbol allein was rendered illusory. Wittgenstein was certainly aware of the fact that reasoning presupposes a correctness norm, because otherwise correct (right) and correct-for-me coincide, in which case there is no possibility for mistakes anymore. However, rather than taking rightness (rectitudo) of acts as a primitive notion, he adopts an

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ontological reduction of rightness. The order of explanation runs as follows: The rightness of the act of inference is reduced to the correctness of the product of the act, that is the judgment made, or knowledge obtained, which notion, in its turn, is reduced to the truth of the content. The truth of the content, ﬁnally, is reduced to an ontological notion, namely, that of the obtaining of states of aﬀairs.86 If objectivity is guaranteed at that level, say, in the form of bivalence for states of aﬀairs—a state of aﬀairs either obtains or does not obtain—it can be exported back to other levels, whence the possibility for mistakes is held open. In diagram form: The truth-maker reduction in Wittgenstein’s TRACTATUS (4) (2) {content of object} ← act of knowledge ↓ SC obtains ↔ [{Proposition C} is true] ↑ ↑ (1) state of aﬀairs [object of the act] (3) = [asserted statement, statement known]. From an epistemological point of view, the rightness notion for acts of knowledge is the most crucial one. It is enough to uphold the diﬀerence between appearance and reality, and, as such, constitutes the minimum requirement on a viable epistemology.87 The need for an ultimate correctness-norm for acts of knowledge, Wittgenstein certainly knew and accepted. Whereas I prefer to take it as primitive, Wittgenstein in the Tractatus reduces the rightness of the act to the correctness of the assertion made, and that in turn to the truth of the propositional content, which, ﬁnally, is reduced to the obtaining (and nonobtaining) of the corresponding state of aﬀairs. Committed realists, when challenged, often reduce the norm of rightness one step further, from the notion of obtaining for states of aﬀairs, to “reality itself,” which accordingly has to provide for the obtaining and nonobtaining of states of aﬀairs. When this reduction is coupled with the idea that “reality itself” is the sum total of all (material) objects and the wish to treat also reality itself as a material object, conceptual confusion results. However, without being a transcendent notion, reality cannot fulﬁll its required role as norm. It certainly cannot be subject to contingent facts the way material objects are, because such facts are responsible to the norm, whence it cannot be a material object. Wittgenstein had thought harder about these issues than most and such confusion is certainly avoided in the Tractatus: “Reality is the obtaining and non-obtaining of states of aﬀairs” (2.06). On such a view, the notion of obtaining (and nonobtaining) of states of aﬀairs can (pleaonastically) be reduced to reality itself. On the other had, “reality itself” thus construed is in no way less transcendent a notion than that of the obtaining of states of aﬀairs.

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7. Constructive Proofs of Propositions, the Traditional Form of Judgment Resurfaces Moore and Russell continue, or perhaps rediscover, the realist stance in logic that had been advocated by Bolzano. Classical, bivalent logic is upheld for an sich bearers of truth—the “propositions”—either by means of a primitive sui generis notion of an sich truth (Bolzano, Frege) or by means of an ontological reduction of truth via a truth-maker analysis (Moore, Russell, Wittgenstein). One would not expect such metaphysical generosity concerning truth to come cheap. The currency in which the price has to be paid is, however, epistemological rather than metaphysical: Unknowable truths cannot be ruled out. The issue is by now a familiar one, owing to the works of Michael Dummett, who has challenged realist accounts of truth on meaning-theoretical grounds: Bivalent truth cannot serve as a key concept in an adequate theory of meaning, owing to the occurrence of propositions with undecidable truth-conditions.88 However, Dummett was not the ﬁrst to challenge unreﬂective realism. Already in the 1880s, the Berlin mathematician Leopold Kronecker and his pupils, among whom was Jules Molk, challenged the automatic use of realist logic: Deﬁnitions should be algebraic and not merely logical. It is not enough just to say: “something either is or is not.” Being and nonbeing have to be set forth with respect to the particular domain within which we operate. Only in this way do we take a step forward. If we deﬁne, for instance, an irreducible function as a function that is not reducible, that is to say, that is not decomposable into other functions of a ﬁxed kind, we do not give an algebraic deﬁnition at all, we only enunciate what is but a simple logical truth. In Algebra, for it to be rightful to give this deﬁnition, it must be preceded by the indication of a method that permits one, with the aid of ﬁnitely many rational operations, to obtain the factors of a reducible function. Such a method only confers an algebraic sense on the words reducible and irreducible.89 In other words, the following “deﬁnition” is not a permissible one: 1 if the Riemann hypothesis is true; f (x) =def 0 if the Riemann hypothesis is false. When the deﬁnition is read classically (or “logically”), the function f is constant and therefore, trivially, a computable function. However, at the moment of writing, we are unable to compute the “computable” function in question. On the “logical” view, f (14), say, is a natural number, but its numerical value cannot be ascertained. Deﬁnitions by means of undecided cases do not admit the eﬀective substitution of deﬁniens for deﬁniendum. They contravene the canon for deﬁnitions that has been with us for three centuries, ever since

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Pascal.90 This is the price that a mathematician has to pay for unrestricted use of classical logic. His language will then contain nonprimitive terms that cannot be eliminated in favor of primitive vocabulary: Accordingly, there is no guarantee that meaning has been conferred on the terms in question. The Kronecker criticism, in my opinion rightly, rules out deﬁnition of functions by means of undecidable separation of cases. Possibly a classical mathematician could live happily without these contrived functions. However, Dirichlet’s famous deﬁnition of the function that is 1 on rational real numbers and 0 on irrational real numbers provoked a change in the conception of what a function is and can hardly be dismissed for want of mathematical interest. It also proceeds by an undecided separation of cases. Many proofs in classical analysis make use of this method. For instance, the standard “bisection of intervals” proof of the Bolzano–Weierstraß theorem that every bounded inﬁnite set of real numbers has an accumulation point proceeds in exactly this fashion.91 Again, these are mathematical matters and perhaps the classical logician, rather than the classical mathematician, need not be worried. Alas, this hope turns out to be forlorn: We only have to notice that Frege’s explanation of the classical quantiﬁer is cast in the form of an undecided separation of cases for matters to become more serious. Quantiﬁer(phrase)s are function(expression)s that take (expressions for) propositional functions and yield (expressions for) propositions. Propositions, for the mature Frege, are ways of specifying truth values, and it seems advisable to make explicit also the relevant domain of quantiﬁcation.92 Accordingly, we consider a truth value valued function A[x] ∈ {The True, The False}, provided that x ∈ D. Frege then deﬁnes the universal quantiﬁer by means of the following explanation: The True, if A[a/x] = The True, provided a ∈ D; (∀x ∈ D)A[x] =def The False, otherwise. However, when the domain D is inﬁnite, unsharp, or otherwise undecidable, the separation of cases cannot be carried out and the deﬁned quantiﬁer cannot be eliminated. Uncharitably put “the classical logician literally does not know what he is talking about.” To my mind, this is the strongest way to marshal undecidability considerations against classical logic. The law of excluded middle is not the real issue.93 Already the classical rules of quantiﬁer formation are unsound: They do not guarantee that “propositions” formed accordingly actually do have content. Until 1930, content was a very live issue. Work on the foundations of mathematics was dominated by the wish to secure a foundation for the practice of mathematical analysis after the ε-δ fashion of Weierstraß that satisﬁes the following conditions:

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(Ai) a formal system is given, in a syntactically precise way, (Aii) with meaning explanations that endow the well-formed expression of its formal language with content, (Aiii) in such a way that its axioms and rules of inference are made evident, and (B) classical logic is validated. Frege’s GGA was the ﬁrst substantial attempt to meet the double desiderata of contentual formalization (A) and classical logic (B), but it foundered on the Zermelo–Russell paradox: Somewhere in Frege’s §§29–31 there is an error, since otherwise every regular expression would have a Bedeutung and every derivable expression would be a name of the True. Whitehead and Russell also failed in their attempted Principia Mathematica execution of the foundationalist program: Their meaning explanations do not suﬃce to make evident the “Axioms” of Inﬁnity, Choice, and Reducibility. Similarly, Wittgenstein’s Tractatus provides (an attempt at) a semantic superstructure for the formal languages designed by Frege and Peano (as modiﬁed by Whitehead and Russell), as does the work of Frank Ramsey (1926). By 1930, faith in the project is waning: Carnap (1931) represents logicism’s last stand. The metamathematical Hilbert program (1926) was an attempt to secure the unlimited use of classical logic, at the price of giving up content, by means of an application of positivist philosophy of science to mathematics. The use of classical logic and impredicative methods are all ﬁne as long as “the veriﬁable consequences,” that is, those theorems that do have content, actually “check out.”94 Passing content by, this means that every free-variable equation between numerical terms that is derivable using also ideal axioms without content has to be correct, when read with content. Hilbert discovered that this holds if the ideal system is consistent, that is, does not derive, say, the formula 0 = 1. In a way, this would have been an ideal approach to the foundations of mathematics for the working mathematician. The conceptual analysis required for foundational work, at which a mathematician does not necessarily excel, is replaced by a clear-cut mathematical issue, to be resolved by a (meta)mathematical proof, just like any other mathematical problem. Alas, it was too good to be true: With the appearance of Gödel (1931) all hope ended here, but the mathematical study of languages without content, which Hilbert had introduced in pursuit of a certain philosophical program, stayed on as an mathematical research program even when the philosophical position had collapsed. Shortly after 1930, the ﬁrst wave of (meta)mathematical results come in: Tarski and Lukasiewicz (1930), the already mentioned Gödel (1931), and Tarski (1933a, 1933b). Under the inﬂuence of these (meta)mathematical successes, even Carnap, the last logicist diehard, jettisons content and anything goes: Up to now, in constructing a language the procedure has usually been, ﬁrst assign a meaning to the fundamental mathematico-logical

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symbols, and then to consider what sentences and inferences are seen to be logically correct in accordance with this meaning. . . . The connection will only become clear when approached from the opposite direction: let any postulates and rules be chosen arbitrarily; then this choice, whatever it may be, will determine what meaning is to be assigned to the fundamental logical symbols.95 The weakness of the position to which Carnap converted is obvious: If anything goes, what guarantee is there that content can be assigned? After all, there had been a few attempts at securing analysis already, meaning explanations and all, that had foundered on inconsistencies in the underlying formalisms. In such a calculus, demonstrably, no content can be had. Carnap’s novel gospel is an extremely liberal one: Principle of Tolerance. It is not our business to set up prohibitions, but to arrive at conventions. . . . In logic, there are no morals. Everyone is at liberty to build up his own logic . . . as he wishes.96 A quarter of a century earlier, at the same time when, in Cambridge, Russell and Moore bit the bullet of unknowable truths, Carnapian licentiousness was rejected on the other side of the North Sea in the (1907) doctoral dissertation of a young Amsterdam mathematician who took over the torch of mathematical constructivism from Kronecker. L. E. J. (“Bertus”) Brouwer (1881–1966) claimed that language use was responsible to the mathematical deed of construction and not the other way round: In the ediﬁce of mathematical thought thus erected, language plays no part other than that of an eﬃcient, but never infallible or exact, technique for memorizing mathematical constructions, and for communicating them to others so that mathematical language by itself can never create new mathematical systems. But because of the highly logical nature of mathematical language the following question naturally presents itself. Suppose that, in mathematical language, trying to deal with an intuitionist mathematical operation, the ﬁgure of an application of one of the principles of classical logic is, for once, blindly formulated. Does this ﬁgure of language then accompany an actual languageless mathematical procedure in the actual mathematical system concerned? 97 In particular, laws of whatever theoretical logic have no validity on their own, but have to be applied in such a fashion that they do ensure proper content. A year after his thesis, Brouwer reaches the conclusion that the law of excluded middle cannot guarantee that the required deed of construction can be executed, whence it has to be rejected not as false but as unfounded.98 Thus he refrains from asserting that “A ∨ ¬A is true.”99 Also the method of proofs by means of nonconstructive dilemma that proceeds by obtaining

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the conclusion that C is true from both the assumption that A is true as well as from the opposite assumption that A is false and concludes that C is true, is rejected, as is the method of indirect (or apagogic) proof, when one assumes a negative claim, obtains a contradiction and concludes a positive claim from this contradiction. Reductio ad absurdum proofs, on the other hand, are perfectly acceptable to constructivists: In these one proves a negative claim from a positive assumption that yields a contradiction. Here a method is provided for obtaining a contradiction from an assumption that constitutes a construction for the negation. Mere formulation or postulation does not automatically confer validity on the rules in question. Formulation alone is not enough to secure preservation of content at the level of the mathematical deed of construction. In this, surprisingly enough, Brouwer resembles Frege who, at roughly the same time, severely criticized formalist accounts of mathematics for their lack of content.100 For Frege, however, it was a commonplace that the contents expressed by declaratives have to be bivalent propositions, tertium non datur. Frege hoped to secure this by making the bond between propositions and truth values a tight one: A proposition is a means of presenting a truth value. Owing to lack of eﬀectiveness in some of the chosen means of presentation, for example, quantiﬁcation with respect to an inﬁnite domain via an undecidable separation of cases, an operational want of content is the result. Accordingly, Brouwer, as well as other mathematical constructivists who insist on the constructional deed in mathematics, will have to provide for another notion of proposition than that of (a mode of presentation of) a truth value, if the formal logical calculi shall not be void of content. This Brouwer did only by precept in his mathematical work: With a lifelong love-hate relationship to language, he never took to formalization and the emerging symbolic calculi of logic.101 It was left to others, to wit Hermann Weyl, one of few ﬁrst-rate mathematicians with a sympathy for intuitionism, and Brouwer’s pupil Arend Heyting, to formulate the required notions explicitly. Brouwer’s style of exposition in his intuitionistic writings was not to everybody’s taste and Weyl, who deftly wielded a polemical pen, took over the early propaganda work, at which he excelled. From his study at Göttingen, Weyl had ﬁrsthand knowledge of Husserl’s phenomenology, and this inﬂuence can be seen in his writings around 1920.102 It was left to him, possibly drawing on work of Schlick and Pfänder, to formulate explicitly the required notion of constructive existence to be applied in a constructive truth-maker analysis: An existential proposition—for instance, “there is an even number” —is not at all a proper judgement that expresses a state of aﬀairs; existential states of aﬀairs are an empty invention of logicians. “2 is an even number”: that is a real judgement that expresses a state of aﬀairs; “there is an even number” is only a judgement-abstract that has been obtained from this judgement.103

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Here we have a novel form of judgment, namely, α exists, where α is a general concept. Its assertion condition is given by the rule a is an α , α exists whence one is entitled to assert that α exists only if one already knows an α.104 The contribution of Heyting is twofold. First, he gave an explicit formulation of the proper intuitionistic rules of logic.105 Second, he intervened decisively in a confused debate whether logic according to intuitionists would need a third truth value: true, false, and undeﬁned, thereby leading to a law of the “excluded fourth,” and so on.106 In his intervention Heyting formulated explicitly a constructivist notion of proposition that admits of a truth-maker analysis: A proposition p, for example, “Euler’s constant is rational” expresses a problem, or better still, a certain expectation (that of ﬁnding two integers a and b such that C = a/b) that may be realized or disappointed.107 Here the intuitionistic novelty is introduced: proofs of propositions, that is, judgable contents, rather than judgments. All previous proving in the history of logic and mathematics had been at the level of judgment and not at that of their contents. These proofs of propositions are not epistemic but ontological in character; inspection of the examples given by Heyting and Brouwer reveals that they are common or garden mathematical objects: functions, ordered pairs, and so on. A proposition A is given by a certain set Proof(A) of proofobjects for the proposition in question. Many alternative formulations have been oﬀered: Proposition Intention Expectation Problem Type Set Speciﬁcation

Proof

Heyting (1934)

Fulﬁllment

Heyting (1930), (1931)

Solution Object Element Program

Heyting (1930), Kolmogorov (1932) Howard (1980) Martin-Löf (1982) Martin-Löf (1982)

The explanation of the standard logical constants then take the following form: ⊥

There are no proofs for ⊥.

& When a is a proof for A and b is a proof for B, a, b is a proof for A & B. ∨

When a is a proof for A, i(a) is a proof for A ∨ B. When b is a proof for B, j(a) is a proof for A ∨ B.

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⊃ When b is a proof for B, provided x is a proof for A, λx.b is a proof for A ⊃ B. ∀ When D is a set, P is a proposition, when x ∈ D, and b is a proof for P , when x ∈ D, λx.b is a proof for (∀x ∈ D)P . ∃

When D is a set, a ∈ D, P is a proposition, when x ∈ D, and b is a proof for P [a/x], a, b is a proof for (∃x ∈ D)P .108

The constructivist truth-maker analysis then takes the form proposition A is true = Proof(A) exists, where the notion of existence is the constructive (Brouwer–)Weyl existence already explained.109 The wheel has come full circle: A judgment made that ascribes truth to a proposition is elliptic for another judgment in the fully explicit form: a is a Proof(A), which is nothing but a judgment of the traditional form: S is P .110 Transformation of the form of judgment Traditional binary form S is P

Existential unary forms Brentano ± Concept α IS (exists)

Bolzano 1837 unary form Prop. A is true

Frege 1879 Prop. P (A) is true

Russell, Moore, 1910 Truth-maker analysis Prop. A is true = The concept Truth-maker [A] exists Realist:

Constructive:

Tractatus 1921 Elementary prop. A is true = Sachverhalt SA obtains

Heyting 1930 Prop. A is true = Proof(A) exists Weyl 1921 Constructive existence

p is a Proof(A)

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8. Inference versus Consequence: How Gentzen Had It Both Ways The interpreted formal systems of Frege, Whitehead and Russell, and Heyting were all axiomatic. These systems (are meant to) have an intended interpretation in terms of the respective meaning explanations. In such systems, a formal derivation is or can be read as a proof that shows that its conclusion formula, when read according to its interpretation, does express a truth. In the modern metamathematical systems of propositional and predicate logic, on the other hand, the end-formula has no intended interpretation, but has to be true under any truth value assignment or set-theoretic interpretation, respectively. Frege, furthermore, explicitly held that one can only draw inferences from known premises. This claim has been controverted, most famously by Gentzen, who created another kind of formalism in his 1933 Göttingen dissertation.111 The derivable objects are still formulae, but may depend on assumptions, and several rules serve to discharge open assumptions. A derivation takes the general form: A1 , A2 , . . . , Ak . . . . (D) . C where A1 , . . . , Ak are the undischarged assumption on which the end-formula C depends. The rules of inference are divided into two groups of introduction and elimination rules. The conjunction introduction rule (&I), say, allows you to proceed to the conclusion A & B, given two derivations of A and B, respectively, that depend on open assumptions in the lists Γ and Δ, respectively. The derivation of A & B depends on open assumptions in the joint list Γ, Δ. The rule (&E) of conjunction elimination, on the other hand, allows you to obtain the conclusion A from the premise A & B, and also the conclusion B from the same premise, while the open assumptions remain unchanged. The rule (⊃I) of implication introduction allows one to proceed to A ⊃ B from the premise B that has been derived from assumption formulae in the list Γ, while discharging as many premises of the form A as one wants—one, many, or none. The derivation of A ⊃ B depends on assumptions in the list Γ1 , where Γ1 coincides with Γ, except possibly for some deleted occurrences of the assumption formula A. The system is convenient to work with when one actually has to ﬁnd the derivations in question. Michael Dummett put the case for Gentzen’s natural deduction as follows: Frege’s account of inference allows no place for a[n] . . . act of supposition. Gentzen later had the highly successful idea of formalizing inference so as to leave a place for the introduction of hypotheses.

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Indeed, “it can be said of Gentzen that it was he who showed how proof theory should be done.”112 However, Dummett’s comparison between Frege and Gentzen is not entirely fair, since it does not take the metamathematical paradigm shift into account. For Frege, the formal system was a tool in the epistemological analysis of mathematics: it was actually used for for proving theorems. For Gentzen, (meta)mathematician, or Beweistheoretiker, the formal system was Hilbertian, that is, an object of study, without content, about which one proves (meta)mathematical theorems, such as, for instance, his famous (1936, 1938) consistency theorem by means of ε0 -induction. For a fair comparison, the respective formal systems of Frege and Gentzen accordingly have to be placed on an equal footing: We either divest Frege systems of their content and treat them as if they were metamathematical, or we supply meaning explanations for the key notions in Gentzen’s systems, so as to endow its object “language” with content. The present chapter is devoted to the notion of judgment, and an inference is nothing but a judgment of a particular (mediate) kind. However, without content no judgment, so it is to the second of these alternatives that we have to turn. Our task is to give a reformulation, call it Gentzen, of Gentzen, at the same level of interpretation as that provided by Frege. The early stages of the conversion present no diﬃculties: It is clear that the wﬀ’s in the formal language, say, of ﬁrst-order arithmetic, can be interpreted as propositions. The syntactic terms are readily turned into numerical expressions, and the predicates < and = obviously lend themselves for interpretation as the computable numerical relations less than and identity, respectively. So far so good; with respect to elementary syntax and semantics, Frege and Gentzen march in step. The diﬃculties arise when we turn to the pragmatic dimension that is involved in Frege’s use of the turnstile as an assertion sign, that is, as an explicit force indicator. Gentzen does not use a turnstile, but if he had it would undoubtedly have been used as a Kleene–Rosser theorem predicate; Gentzen was a (meta)mathematician. Here we see a ﬁrst diﬃculty for Gentzen: Gentzen (and with him other metamathematicans) used his wﬀ’s in two roles. Wﬀ’s are fed to connectives, that is, Frege’s Gedankengefüge, to build other, more complex wﬀ’s: Accordingly, for Gentzen they are propositions. On the other hand, Gentzen also used wﬀ’s as end formulae of derivation trees: Accordingly, for Gentzen, the wﬀ’s also have to be turned into theorems, that is, assertions (judgments made) that propositions are true. Here Gentzen confronts a potentially damaging ambiguity. However, we must allow him the same leeway as that oﬀered to Frege: He can make use of the turnstile as an assertion sign, and also other force indicators, should he want to do so. The obvious option for Gentzen is to use two force indicators, one for assertion () and another for assumption (). Finally, Gentzen also has to interpret the derivation trees of Gentzen. The Gentzen derivation D will be interpreted by means of the following procedure:

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i. Append “is true” to each wﬀ that stands on its own at a node in D (rather than as part of another wﬀ); ii. preﬁx transformed wﬀ’s “A is true” of assumption formulae by the assumption sign “” and preﬁx transformed “A is true” of conclusion formulae by the assertion sign “”; iii. interpret all wﬀ’s in D as propositions. The result is the tree D :

(D )

A1 is true, .

A2 is true, .

. . . , Ak is true, . .

. C is true. We may have some hope that D will serve as a ﬂow-chart for a proof-act that yields the knowledge that proposition C is true. However, this simpleminded approach does not work: The interaction of the two kinds of force—assumption and assertion—is more involved. Consideration of an example, in which prooftheoretical experts will recognize one of Dag Prawitz’s reduction ﬁgures, makes this clear:113 [A] | (d) B | A⊃B A B This tree is dressed according to the procedure and transformed into the tree

(d )

A is true (1) | | B is true (2) A ⊃ B is true (3) A is true B is true (5)

(4)

The force apparatus is almost equal to its task: the proposition A occurs as part of an assertion (4), of an assumption (1), and as an unasserted part of an assertion (3). The notation is rich enough to distinguish these cases clearly. With respect to the proposition B matters are less fortunate, though. For the proposition B, assertion (5) and unasserted part (3) are coped with, in the same way as for the proposition A. The premise that B is true of the (⊃I) rule (2), however, is neither assumed nor asserted, and its force cannot be expressed with the two force indicators at hand. There one asserts that the proposition B is true, provided that the proposition A is true.114 One must note, though, that it is not the assertion that is hypothetical or conditional; the assertion

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is categorical, whereas the notion of truth has been made conditional. We no longer ascribe outright truth to the proposition B, but only the constrained notion . . . is true, provided that A is true. Thus we have an unconditional, categorical assertion that conditional truth pertains to the proposition B. Strictly speaking, this is a novel form of judgment. The derivation tree D above, where the assumptions A1 is true, A2 is true, . . . , Ak is true are still open, or undischarged, does not allow for the ascription of outright truth to the proposition C, but only of truth on condition that A1 is true, A2 is true, . . . , Ak is true. The general case of the weakened, conditional truth in question will then be: . . . is true (A1 is true, A2 is true, . . . , Ak is true). Accordingly nodes in derivation trees are not covered with statements of the form A is true, but with statements of the conditional form. Eﬀecting this transformation, the derivation tree D ultimately takes the form D : A1 is true (A1 is true), A2 is true (A2 is true), . . . , Ak is true (Ak is true) . . . . (D ) . C is true (A1 is true, A2 is true, . . . , Ak is true). The relevant notion of assertion is still categorical, but the truth that is asserted of various proposition may be weakened. We must distinguish between the two statements: i. proposition A ⊃ B is true, ii. B is true, provided that A is true, or its (synonymous) variant, ii . if A is true, then B is true. The statement (i) is explained classically via truth-making of atomic propositions and then inductively via the truth tables, say, and constructively in terms of an assertion-condition demanding a (canonical) proof-object, as in section 7. From the constructive point of view, an assertion of the ﬁnal statement in D , that is, (∗)

C is true (A1 is true, . . . , Ak is true),

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demands a dependent proof-object: (∗∗)

c is a proof of C, provided that x1 is a proof of A1 , . . . , xk is a proof of Ak .

Accordingly, the conditional statement (∗) represents a novel form of judgment, with the assertion condition (∗∗). This suggests how natural deduction derivations should be interpreted: They are notations for dependent proof-objects. Gentzen did not have only one format for natural deduction derivations but two. Sometimes they are considered as mere notational variants.115 In the present context their diﬀerences are signiﬁcant. In 1936 he used a sequential format for the derivations.116 The derivable objects are no longer well-formed formulae, but sequents. A sequent A1 , A2 , . . . , Ak ⇒ C lists all the open assumptions on which C depends. Derivations have no assumptions, but axioms only of the form A ⇒ A, with the Gentzen interpretation A is true, provided that A is true, indeed, something undeniably correct, albeit not very enlightening. Consideration of the tree D shows that its top formulae are axioms of this kind and that the conditional statements at the nodes in the tree are nothing but sequents in another notation. Because there are no acts of assumption, no discharge of assumptions takes place, but antecedent (assumption-)formulae can get struck out; for instance, the rule (⊃I) takes the form A, Γ ⇒ B . Γ⇒A⊃B Conjunction introduction (&I) will be Γ⇒A Δ⇒B . Γ, Δ ⇒ A & B On the Gentzen interpretation the sequent A1 , . . . , Ak ⇒ C is interpreted as C is true, on condition that A1 is true, . . . , Ak is true, and the Gentzen sequent should properly include the truth ascriptions: A1 is true, . . . , Ak is true ⇒ C is true.

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Derivations in the sequential format of natural deduction describe, or are, blueprints for proof-acts that certain propositions are conditionally true. Properly speaking, we have here a treatment of consequence relations among propositions: The consequent proposition is true when the antecedents are all true. One should also note that the statement proposition C is true is a special case of the sequent, when the number k of antecedent propositions = 0. The sequents can also be read as closed sequents (A1 , . . . , Ak ) ⇒ C.117 Just as the Gentzen sequents represent a novel form of judgment, so do these closed sequents, and their Gentzen interpretation should be the sequent (A1 , . . . , Ak ) ⇒ C holds. To have the right to assert that a closed sequent holds we must give a verifying object. This is a function f that takes proofs a1 , . . . , ak of the antecedent propositions into a proof f (a1 , . . . , ak ) of the consequent proposition C. We must distinguish between three equiassertible statements: the proposition A ⊃ B is true (demands a proof of A ⊃ B); the conditional statement (open sequent) A true ⇒ B true (demands a dependent proof b of B, provided that x is a proof of A); the closed sequent (A) ⇒ B holds (demands a function from Proof(A) into Proof(B)). The assertion condition is diﬀerent in all three cases, but one can be met only if the other two can also be met. Furthermore, one reason that these notions are not always kept apart is that all three are refutable by the same counterinstance, namely, a proof-object a of A and a dependent proof-object c for the open sequent B true ⇒ ⊥ true. With this distinction, my treatment of the sequent calculus comes to an end. At the level of assertion, there is apparently little to choose between Gentzen and Frege. It only seems fair to let Frege have the last word: The repeated Frege conditional ... C Ak .. . A1

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is commonly read as the implication (A1 ⊃ (A2 ⊃ (· · · ⊃ (Ak ⊃ C) · · · ))). However, rotating the Frege conditional 90° clockwise, while altering the notation only slightly, produces another familiar result as the late Pavel Tichý (1988, pp. 248–252) observed, namely A1 , A2 , . . . , Ak ⇒ C.118 The correspondence between the calculi of Frege and Gentzen operates even with respect to the ﬁne structure of the rules, sometimes even exhibiting a surprising(?) resemblance of terminology. Thus, the following question acquires some urgency: Did Gentzen read Frege’s Grundgesetze prior to 1933, the year in which his dissertation was composed?119 Be that as it may. Bolzano gave us a coherent theory of (logical) consequence between propositions. Frege was more right about inference from judgments made to judgment than he is given credit for. However, only in Gentzen’s sequential natural deduction do we have a theory that treats of both consequence as well as inference.

Brief Biographical Notes 1. Bernard Bolzano, 1781–1848120 A Bohemian priest of Italian origin, who held the chair of Philosophy of Religion at the Charles University in Prague from 1805 until 1820, when he was summarily dismissed, as well as barred from public teaching and preaching, for holding too liberal views concerning matters both spiritual and temporal, gave fundamental contributions to mathematical analysis (“Bolzano–Weierstraß theorem”). A wholly admirable man, he led a retiring life with friends in the Bohemian countryside, devoting himself to logical and mathematical researches. The magnitude of Bolzano’s contribution to logical theory, as well as to philosophy in general, can hardly be overestimated. Being censored, it was left unrecognised, thereby retarding logical progress by half a century. Appreciation is mounting with the growing volume of the Gesamtausgabe, and Bolzano might yet receive the credit that is so amply his due: “the greatest philosopher of the nineteenth century, bar none.”121 2. Franz Brentano, 1838–1917 Brentano belonged to a prominent German cultural family. Ordained a priest, he held, after impressive Aristotle studies, a (Catholic) extraordinary chair in Philosophy at Würzburg. Misgivings over the deﬁnition of Papal Infallibility in 1870 led him to renounce the priesthood and change his chair for one in Vienna, where his lectures aquired cult status as society happenings. The Concordat between Austria and the Vatican allegedly forbade Austrian ex-priests to marry and led him to resign also this chair and his Austrian citizenship in 1880; having taken Saxon citizenship he then married, conﬁdently expecting reappoinment. This never happened, reputedly

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at the personal instigation of the emperor; after 15 years as Privatdozent, the former Professor Brentano left Austria and settled in Florence. Blindness darkened his last decade, to which belong important brief essays on truth as well as a wide-ranging correspondence with utterly devoted pupils. Italy’s entry into World War I forced a ﬁnal move to Switzerland, where Brentano died in 1917. Brentano, an outstandingly successful lecturer and supervisor, though not devoid of dictatorial leanings, had highly able doctoral students, who bitterly disappointed him by not speaking in unison with their master’s voice. Nevertheless, his inﬂuence ranges wide, not only among devoted Brentanists, but also in two major schools of twentieth-century philosophy, to wit, the LvowWarsaw school under Twardowski, the ﬁrst to introduce analytical techniques, and Husserl’s phenomenology. 3. Gottlob Frege, 1848–1925 A German mathematician at Jena who taught (mainly) analytical geometry several hours a week, never reached the rank of Ordinarius but gave fundamental contributions to the foundations of logic and mathematics. In the Begriﬀsschrift and Die Grundlagen der Arithmetik the program of reducing arithmetic to logic, as well as the logic to which it was to be reduced, are set out with great lucidity. His attempt at a fully rigorous execution of his foundationalist program, in the Grundgesetze der Arithmetik, proved to be irredeemably ﬂawed owing to the emergence of the Zermelo–Russell paradox within the system. Three important essays from the early 1890s provide a philosophical underpinning for the Grundgesetze. Of these, Über Sinn und Bedeutung is commonly regarded as the origin of modern philosophy of language. Frege founded no school, and, for a long time was only known through and for his inﬂuence on major ﬁgures such as Russell, Carnap, and Wittgenstein. A deeply conservative man in matters cultural and political, Frege died forgotten in the Weimar Republic to which he could not relate. His contributions to logic, its philosophy, and the philosophies of mathematics and language are now recognized in their own right, and not only as an inﬂuence on others, whereby Frege rightly emerges as a major thinker of the nineteenth century. 4. Ludwig Wittgenstein, 1889–1951 The youngest son of Karl Wittgenstein, a main architect of the Industrial Revolution in Austria, as well as one the wealthiest men in Europe, was educated at the Oberrealshule in Linz, where Adolf Hitler was a fellow pupil, and subsequently at the Technische Hochschule, Berlin-Charlottenburg, and Manchester University, prior to settling at Cambridge, where his work on the foundations of logic ripened in close contact with Bertrand Russell, in relation to whom Wittgenstein went through the stages of pupil, co-worker, and implacable critic. At the outbreak of World War I, Wittgenstein volunteered for the Austrian army, and during the war he reﬁned and deepened his views on logic that were published in the aphoristic Logisch-Philosophische Abhandlung, now universally known as Tractatus (Logico-Philosophicus). After the Great War, Wittgenstein gave away his in-

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herited fortune and became a schoolteacher in the hinterlands of lower Austria. A Vienna lecture by L. E. J. Brouwer, March 1928, rekindled his interest in philosophy, and led him to return to England. Keynes wrote to his wife: “God is in England. I met him at the 5.15 train.” Elected a fellow of Trinity, and from 1939 professor in succession to G. E. Moore, Wittgenstein developed an entirely novel conception of philosophy, on which he published nothing. A stern man, more unsparing of himself than of others, Wittgenstein died in 1951, his last words being: “Tell them I have had a wonderful life.”

Notes 1. Barnes (1988, p. 48). I am indebted to my Leyden colleague Dr. J. van Ophuijsen for drawing my attention to this marvelous passage. 2. (1950), p. vii. 3. Putnam (1982), Boolos (1994). Boolos also canvasses 1858, the year in which Dedekind cut the rationals, as a candidate. 4. I have argued as much, in and out of print, since 1988 (p. 4). 5. Thus texts would comprise three main parts, or “books”: Of Terms, Of Judgement, and Of Reasoning, and possibly a ﬁnal part treating Of Method. Kant’s Jäsche Logik is a good case in point. 6. The diagram draws on a similar one in Maritain (1946, p. 6) but is reasonably standard. Maritain’s source, and also that of virtually all other neo-Thomists, is the splendid Ars Logica by John of St. Thomas. 7. Fichte (1797) (which bears the title Wissenschaftslehre). The convenient representation of the act/object distinction was introduced in Martin-Löf (1987). 8. A committed anti-antirealist, or the unbiased reader, might prefer the less pejorative realist for the other alternative. 9. Here the vertical bars above the judgments J1 , J2 , . . . , Jk represent acts that yield, respectively, the judgment in question. 10. KdrV, A69. “Wir können alle Handlungen des Verstandes auf Urteile zurückführen, so daß der Ve r s t a n d überhaupt als ein Ve r m ö g e n z u u r t e i l e n vorgestellt werden kann.” 11. And by Heidegger, or so I have been told. 12. As does judgment: act of judging versus judgment made. 13. Recent scholarly tradition associates this familiar example with Tarski (1944). It is, nevertheless, considerably older than so. We ﬁnd it in Boole (1854, p. 52), as well as in Hilbert and Ackermann (1928, p. 4) (who undoubtedly have it from Boole). The latter is cited by Tarski in Der Wahrheitsbegriﬀ. However, the ultimate source for the present-day logical obsession with arctic meteorology might well be Aristotle’s Prior Analytics, Book A, ch. iv, where we ﬁnd a discussion of things—among them snow—that admit the predication of “white.” 14. Here the letter S serves as a schematic letter for declaratives. 15. For lack of space, in what is after all an inquiry into the recent history of logic, I must here leave well-known (spät) Wittgensteinian claims to the contrary without due consideration. I simply register my conviction that they do not present an insurmountable obstacle, and that my view as given in the text is essentially correct.

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16. Sometimes a single utterance of the nominalization will, nevertheless, eﬀect the required assertion, for example, when responding to the question: “Which of the two alternatives is the true one?” 17. The apt term judgable content we owe to Frege’s Bs. His beurtheilbarer Inhalt has been variously rendered into English as (i) possible content of judgment (Geach), (ii) content that can be judged (Van Heijenoort), (iii) judicable content (Jourdain), and (iv) judgable content (Dummett). Of these, the last deserves preference over the second for the sake of brevity, while the ﬁrst is likely to cause serious confusion, owing to its pointing in the direction of modal logic. 18. This is not the full story. The right to ask for grounds, when faced with an assertion made, shows that there is an implicit claim to knowledge contained in the assertoric force with which the sentence has been uttered, and which sometimes comes to the fore, for instance, in Moore’s paradoxical assertion of It is raining but I do not believe it. Thus, I know that snow is white, which is in the performative ﬁrst person, might have been a more felicitous form to use for the assertion made by means of my assertoric utterance of the declarative “Snow is white,” were it not for the fact that it is prone to be conﬂated with the third-person use of the ﬁrst person, which, when applied to me, is synonymous with Göran Sundholm knows that snow is white. 19. This linguistic turn in philosophy was so named by Gustav Bergmann (1964, p. 177). The term gained wide currency after Richard Rorty (1967) chose it for his title. 20. Weighty in every sense of the word; its four volumes add up to a total of close to 2500 pages. 21. Kant, KdrV, B VIII (my translation). 22. The Kantian authority for Jäsche’s text is not undisputed, see Boswell (1988). 23. The English rendering of Bolzano’s Satz an sich is a matter of some delicacy. The modern, Moore-Russell notion of proposition, being an English counterpart of the Fregean Thought (German Gedanke), really is an an sich notion, and, for our purposes, essentially the same as Bolzano’s Satz an sich. Thus, proposition-initself is pleonastic: The in-itself component is already included in the proposition. Furthermore, the mental propositions and their linguistic signs, that is, written or spoken propositions, as explained, carry assertoric force, whereas Bolzano’s Sätze an sich manifestly do not, serving, as they do, in the role of judgmental content. Accordingly, it might be better to use Sentence in itself, which does not seem to carry the presumption of assertoric force. However, as Ockham and other medieval thinkers noted, the propositio mentalis, and its matching exterior signs, can be further analyzed into propositio judicationis, which does carry assertoric force, and propositio apprehensionis, which does not. Ockham has Quodlibetal Questions with congenial titles: Questio V:vi “Is an act of apprehending really distinct from an act of judging?” and Q iv:16 “Does every act of assenting presuppose an act of apprehending with respect to the same object?” So from this point of view, Bolzano’s proposition-in-itself is obtained by severing the (mental) links that tie the propositio apprehensionis to its mental origin and its linguistic signs.

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24. WL, §34. 25. Occasionally I shall permit myself to drop the “in-itself” idiom in the interest of perspicuity and readability and speak just of “propositions.” 26. Contrary to a common misapprehension, Frege, who employed the term third realm in 1918, is not its progenitor. It was in general use in neo-Kantian circles. Gottfried Gabriel’s lemma Reich, drittes in the Historisches Wörterbuch der Philosophie tells the full story. 27. In some theories, for instance, that of Frege, propositions (thoughts) are explained as the meanings of declarative sentences. This is not Bolzano’s way of proceeding, the sui generis, absolutely mind- and language-independent propositionsin-themselves, are there in their own right, so to say, and they are capable of fulﬁlling various logical oﬃces, among them that of serving as sentence meanings. 28. The reader will please note long shadows being cast forward toward Tarski (1944). Bolzano (§28) has a discussion of whether really is really necessary in the right side of this explanation. 29. WL, §154 (4). “Auch leuchtete jedem die Ähnlichkeit ein, die zwischen diesem Verhältnisse unter den Sätzen und zwischen jenem, welches . . . unter Vorstellungen . . . obwaltet. Was nähmlich bei Vorstellungen den Umstand gilt, ob ein gewisser Gegenstand durch sie in der Tat vorgestellt werde, das gilt bei Sätzen der Umstand, ob ihnen Wahrheit zukomme oder nicht.” 30. When snow is white, the idea-in-itself the whiteness of snow is instantiated and the idea the blackness of snow is not. 31. Sebestik (1992) oﬀers a beautiful précis of Bolzano’s framework. 32. WL, §138. The categorical judgments of the A and I forms and O are treated of, respectively, at §225 Anmerkung, and §171. The treatment of an O judgment (Some α is not β) follows the pattern of the I judgments: The idea-in-itself of an α that is not β has Gegenständlichkeit. 33. The falsity of false propositions-in-themselves is also an sich. 34. WL, §34. The status of my chosen form of judgment [A is true] is very delicate with respect to Bolzano’s system. On the one hand, it has to be a proposition-in-itself, since the iteration of . . . is true is the key step in Bolzano’s non-apagogic “proof” that there are inﬁnitely many true propositions-in-themselves (WL, §32). On the other hand, propositions-in-themselves are supposed to be sentence meanings, as well as the bearers of truth and falsity, as is clearly documented by the following passage (cited from Mark Textor [1996], p. 10) in Bolzano’s Von der mathematischen Lehrart (my translation): “not what grammarians call a proposition, namely the linguistic expression, but rather the sense of this expression, that must always be only one of true or false, is for me a proposition in itself or an objective proposition.” The sense (Sinn) of the declarative sentence (grammatical proposition) “Snow is white” is that snow is white. Furthermore, that-clauses are what yield grammatical declarative sentences when saturated with “is true” or “is false.” Accordingly, that-clauses seem to be the appropriate linguistic counterparts to propositions-in-themselves. But then, the ascription of truth to a proposition-in-itself is not a proposition-in-itself, since the declarative “the proposition-in-itself A is true” isn’t a that-clause, that is, does not have the required form for being (the linguistic counterpart of) a proposition-in-itself. This note attempts to answer Wolfgang Künne, who objected, after my Cracow lecture, that for Bolzano, [A is true] is just another proposition-in-itself, but not a judgment. In spite of the considerations, on balance, I am inclined to think that

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this might be an instance where I let my systematic preferences override historical subtleties, and that some injustice is done to Bolzano. 35. Franz Brentano, for whom the problem of the blind judgment became a major issue, and whose views will be considered in the sequel, propagated this apt terminology. As Per Martin-Löf has pointed out to me, the notion and term might ultimately derive back to Plato, The Republic, 506c: “opinions divorced from knowledge, are ugly things[.] The best of them are blind. Or do you think that those who hold some correct opinion without evidence diﬀer appreciably from blind men who go the right way?” 36. WL, §36. Fairness bids me to report that Bolzano was aware of a certain awkwardness in his doctrine at this point. In WL, §314, with the telling title “Are there Deﬁnite Limits to Our Capacity for Knowledge?,” he notes: For since every judgment that agrees with the truth is a piece of knowledge, even if that agreement is only accidental and had come about only by way of previous errors, it can very well be seen that the limits of our capacity for knowledge, if we were able to abide by such a broad deﬁnition, would ﬂuctuate everywhere, since mere chance and even a mistake could contribute to its enlargement. In this connection we should further note that according to Bolzano, every truth is knowable, since God knows the truth of every true proposition-in-itself, whence it can be known: ab esse ad posse valet illatio. 37. In much the same way, Bolzano proﬁted immeasurably from having Kant, the foremost idealist of the age, as his main target. That can be seen by comparing the pristine clarity of Bolzano’s work with the murkiness of early Moore and Russell 60 years later. Their realism was the result of an apostasy from and battle with a much inferior version of idealism, namely the British Hegelianism of Bradley, Green, and Bosanquet. Another example of the same phenomenon is provided by Wittgenstein, who, according to Geach (1977, vi), held Frege’s Der Gedanke in low esteem: “it attacked idealism on its weak side, whereas a worthwhile criticism of idealism would attack it just where it was strongest.” The destructive side of a philosophical position seems to gain in quality with the target it attacks. 38. Apparently Bolzano was unable to give a material criterion for what it is to be a (non)logical idea, but then so were his successors, who oﬀered virtually identical accounts of logical truth and consequence a century later. 39. The foregoing brief formulations do not do perfect justice to Bolzano on a number of scores. (i) Logical consequence (in the modern sense) is a two-place relation between antecedent and succedent propositions, whereas Ableitbarkeit, be it logical or not, is a three-place relation between antecedent proposition(s), consequent proposition(s), and idea(s) (that occur in at least one antecedent or consequent proposition), where the ideas indicate the places where the variation takes place. Logical Ableitbarkeit considers variation with respect to the collection of all nonlogical ideas that occur in the antecedent and consequent propositions. As a limiting case (possibly one rejected by Bolzano) one might consider merely material Ableitbarkeit that consists in the preservation of truth under variation with respect to no ideas, and which holds between A1 , A2 , . . . , Ak and C, when the implication (A1 & A2 & · · · & Ak ) ⊃ C

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is just true, but not necessarily logically true. (ii) With respect to the consequence (or sequent in the terminology of Hertz and Gentzen) A1 , A2 , . . . , Ak ⇒ B1 , B2 , . . . , Bm , Bolzano demands that the A’s and the B’s be compatible, that is, there must be some suitable variation that makes them all true. Furthermore (iii), modern theory holds that the multiple-succedent sequent should be read as A1 & A2 & · · · & Ak ⊃ B1 ∨ B2 ∨ · · · ∨ Bm ; the sequent is valid when this implication is a logical truth. Bolzano, however, uses another meaning for the sequent, namely, A1 & A2 & · · · & Ak ⊃ B1 & B2 & · · · & Bm ; according to Bolzano an Ableitbarkeit with many succedent propositions holds when every variation that makes all antecedent propositions true also makes every (and not just at least one) succedent proposition true. In Siebel (1996) Bolzano’s theory of Ableitbarkeit is studied in depth and related (with due consideration for signiﬁcant diﬀerences) to a number of well-known modern topics, such as Russell’s theory of propositional functions, the Quine-AjdukiewiczTarski account of logical truth and consequence, and the relevance logic of Anderson and Belnap. Nevertheless, in spite of the sometimes considerable diﬀerences, it is proper to regard Bolzano as the founder of the modern theory of (logical) consequence among propositions; he is the ﬁrst to reduce the validity of inference (from judgment to judgment) to a matching relation among propositions (-in-themselves) that serve as contents of the relevant premise and conclusion judgments, respectively. 40. In WL, part III (“Erkenntnislehre”), ch. II (“Von den Urtheilen”), §300 (“mediation of a judgement through other judgements”), Bolzano considers also inferences proper, that is, mediate acts of judgments, and not only their Platonist simulacra, namely, consequence relations (Ableitbarkeiten) among the respective judgmental contents. Lack of space prevents me from developing this theme any further. 41. A true proposition-in-itself can stand in the relation of Abfolge to more than one grounding proposition. 42. WL, §220. See Aristotle, An. Post., I:13. 43. Validity of an inference ﬁgure must be distinguished from that of validity (rightness) of an act of inference. An act of inference, that is, a mediate act of judgment, is valid (right, real, or true) if its axioms, that is, according to Frege’s GLA, §3, p. 4, characterization, judgments neither capable of nor in need of demonstration, really are correct, and the inference-ﬁgures employed therein really are valid, that is, do preserve knowability. 44. Compare, for instance, the fragments reprinted in part III and appendix 2 of Brentano (1930), with titles such as “Against so-called Judgmental Contents” and “On the Origin of the Erroneous Doctrine of the entia irrealia.” 45. Furthermore, these reductions were well known already to Leibniz, for instance in the General Inquisitions, §§146–151. Franz Schmidt provides a list of 28 (!) diﬀerent Leibnizian reductions of the four categorical judgments in Leibniz (1960, pp. 524– 529). For instance, the singular aﬃrmative judgment “Some A are B” is rendered alternatively as “AB is,” “AB is a thing,” and “AB has existence,” by reductions 17,

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22, and 27. Here Bolzano’s German hat Gegenständlichkeit is matched by Leibniz’s Latin est or est Ens. 46. Letter from Brentano to Hugo Bergmann, June 1, 1909, quoted from Bergmann (1968, pp. 307–308) (my translation). Wenn ich . . . auf Bolzano aufmerksam machte, so geschah dies, . . . , keineswegs, um den jungen Leuten Bolzano als Lehrer und Führer zu empfehlen. Was sie von ihm, das dürfte ich mich sagen, konnten sie besser von mir lernen. . . . Und wie gesagt, wie ich selbst von Bolzano nie auch nur einen einzigen Satz entnommen habe, so habe ich auch niemals meinen Schülern glaubhaft gemacht, daß sie dort eine wahre Bereicherung ihrer philosophischen Erkenntnis gewinnen würden. 47. Brentano (1889, Anm. 27, pp. 64–72) and the fragments in Brentano (1930, part IV) are important here. Also relevant is his Versuch über die Erkenntnis, that is, Brentano’s (1903) attempt at an Essay on human knowledge after the fashion of Locke and Leibniz. For instance, its ﬁrst part bears the grandiloquent title: Destroy prejudice! An appeal to the contemporary age, that it free itself, in the spirit of Bacon and Descartes, from all blind Apriori. 48. Note that this use of the term evidence is diﬀerent from its use within current analytical philosophy of science and the Anglo-Saxon common-law legal systems. (“My lord, I beg leave to enter exhibit 4 into evidence.”) There one is concerned with supporting evidence for a claim. Brentano’s use is concerned with that which is evident (known). Evidence is the quality that pertains to what is evident. 49. Brentano (1930, p. 139). 50. Brentano (1956, p. 175, §39). According to the editor, this negative formulation of the Law of Excluded Third derives from an unpublished fragment “Über unsere Axiome” from 16.2.1916. 51. Scholz (1930) remains the standard treatment of Aristotelian foundationalism. 52. Frege (1879) is a book with the title Begriﬀsschrift, whereas “begriﬀsschrift” is an English (loan-)word for the eponymous formal language developed in that work. This is not an ideal solution to the title/notion ambiguity of the German term. Using either of the two standard English renderings—ideography and concept(ual ) notation—seems a worse option, though. 53. This is how it ought to be; regarding Frege, his GGA axiom 5 concerning Werthverläufe and the use of classical second-order impredicative quantiﬁcation remain unjustiﬁed. In place of notions we could speak of terms or concepts here. Either choice runs the risk of being taken in too narrow a sense, though. Today a term is a syntactic entity only, often associated with the formation rules of ﬁrst-order predicate calculus, whereas for Frege a concept is conﬁned to a certain kind of function. 54. The ﬁrst formulation combines passages from (1879, pp. 2, 4): “Der Umstand, dass. . .” and “. . .ist eine Thatsache.” Apparently the form of judgment is (. . .ist eine Thatsache), where the blank has to be ﬁlled with an “Umstand.” Accordingly the form of the judgment made through an assertoric utterance of “Snow is white” is: The circumstance that snow is white is a fact

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In place of circumstance, Frege also allows for “Satz.” In (1918, p. 74, n. 8) he identiﬁes fact with true Thought, which yields the ﬁnal reformulation: The proposition that snow is white is true. The second formulation is taken from 1892 (p. 34, n. 7): Ein Urteil ist mir nicht das bloße Fassen eines Gedankens, sondern die Anerkennung seiner Wahrheit. 55. Frege (1880/81, pp. 36 ﬀ.) attempts to sell his begriﬀsschrift to the mathematicians by treating of the standard notions pertaining to continuity, but to no avail, alas. The mathematicians, and among them to his shame Felix Klein, did not rise to the occasion and the paper was rejected by, for instance, the Mathematische Annalen. In desperation, Frege sought refuge with the philosophers at the Zeitschrift für Philosophie und philosophische Kritik, but they proved equally cold-hearted. In spite of it being unpublished, Frege’s piece must be give full evidentiary value since it was written for publication and repeatedly submitted. Later Frege established very good relations with the Zeitschrift where some of this very best papers appeared, in 1882 and 1892. 56. That is, iterated combinations of “for all/there is” and “there is/for all.” The passage from continuity to uniform continuity provides a clear example of the shift from “∀∃” to “∃∀”. The (linear) logical notation employed here is reasonably standard, using inverted A and E for Alle (all) and Es gibt (there is). It is due to Gerhard Gentzen (1934–35), but derives in essence from Peano, via mediation through Whitehead. Frege’s own begriﬀsschrift is two-dimensional and has great versatility, as well as a strange beauty of its own. It was never able, however, to gain proselytes, and so it perished with its progenitor in the early stages of the mounting metamathematical revolution in the late 1920s. 57. Frege’s checkered struggle toward an identity criterion for propositions (his Thoughts) is long and fascinating; see Sundholm (1994c). 58. Frege (1892, p. 34): “Ich verstehe unter dem Wahrheitswerte eines Satzes den Umstand, daß er wahr ist oder daß er falsch ist.” 59. Frege’s notion of a proper name (Eigenname), following the German translation of John Stuart Mill’s System of Logic, comprises not just grammatical proper names but singular terms in general. 60. GGA, I, §32. Note that this formulation admit the equation of proposition (Thought) with truth-condition, the Thought that snow is white = the Thought expressed in “snow is white” = the truth-condition of “snow is white,” whence for a declarative sentence S: S = that S is true = the truth-condition of “S” is fulﬁlled = the proposition expressed in “S” is true. Thus also, the Thought that S = the thought that the truth-condition of “S” is fulﬁlled.

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61. Frege (1906b, II, p. 387): “eine Urteilsfällung, die auf Grund schon früher gefällter Urteile nach logischen Gesetzen vollzogen wird.” 62. Bs, §6, and GGA, I, §14. The change from Frege’s two-dimensional notation to a one-dimensional Gentzen notation is not always anodyne, but here, where the concern is assertoric force, rather than the speciﬁc contents, it seems innocent enough. 63. The ins and outs of Frege’s assertion sign are treated very well in Stepanians (1998, chs. 1–5). Concerning the origin of its use as a theorem predicate, see Kleene (1952, p. 88, p. 526). 64. Dummett (1973, p. 309, p. 435). I beg to diﬀer and will return to the issue in section 8. 65. GGA, I, preface, pp. xv–xvi: Wahrsein ist etwas anderes als Fürwahrgehalten werden, sei es von Einem, sei es von Vielen, sei es von Allen, und es ist in keiner Weise darauf zurückzuführen. Es ist kein Widerspruch, dass etwas wahr ist, was von Allen für falsch gehalten wird. This marvelous credo is embedded in a passage pp. xv–xvii that is highly germane to the realism issue. 66. Nachlass, p. 2. (The Logik of the 80s): “Was wahr ist, ist unabhängig von unser Anerkennung. Wir können irren.” It is not required that there be mistaken acts of knowledge, but only that their possibility is not ruled out. 67. GGA, II, p. 69: Das Gesetz des ausgeschlossenen Dritten ist ja eigentlich nur in anderer Form die Forderung, dass der Begriﬀ scharf begrenzt sei. Ein beliebiger Gegenstand Δ fällt entweder unter den Begriﬀ Φ, oder er fällt nich unter ihn: tertium non datur. 68. The crucial primacy of the sui generis notion of rightness was noted by Martin-Löf (1987, 1991). In the light of this, Sundholm (2004) spells out various interrelations between diﬀerent roles of truth. 69. Locus classicus: “Der Gedanke” (1918). 70. Bs, preface, p. IX: Die festeste Beweisführung ist oﬀenbar der rein logische, welche, von der besonderen Beschaﬀenheit der Dinge absehend, sich allein auf die Gesetze gründet, auf denen alle Erkenntnis beruht. Wir theilen danach alle Wahrheiten, die einer Begründung bedürfen, in zwei Arten, idem der Beweis bei den einen rein logisch vorgehen kann, bei den andern sich auf Erfahrungsthatsachen stützen muss. GLA, §§3–4, contains a further elaboration of this theme into an account of the distinctions analytic/synthetic, a priori/a posteriori. Frege’s considerations here, successively stepping from a known truth to its grounds seeking the ultimate laws of justiﬁcations, are strongly reminiscent of Bolzano’s use of his grounding trees with respect to Abfolge. 71. Moore (1898, 1902) and Russell (1903, appendix A, §477, 1904). Cartwright (1987) treats of their early theory in some depth. The notion of proposition is here essentially the same as in Bolzano, and, to some extent, Frege. The grave responsibility for mistranslating the Fregean Gedanke (Thought) into proposition rests on Moore and Russell: Moore (1898, p. 179) introduced the terminology: “We have approached

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the nature of the proposition or judgment. A proposition is composed not of words, nor yet of thoughts, but of concepts.” Russell (1903, appendix A, §477) completes the error by coupling Frege’s Gedanke with his own proposition. Through PM this misidentiﬁcation eventually became standard throughout all of modern logic. In its original sense from the tradition, a proposition was either the (mental) judgment made, or its outward announcement in language, whereas after Russell and Moore it is turned into the content of a proposition in the original sense. 72. Joachim (1906, p. 37), who apparently wrote this marvelous passage in ignorance of Bolzano, drawing only on what he could ﬁnd in Russell and Moore, for instance: [There] is no problem at all in truth or falsehood; that some propositions are true and some false, just as some roses are red and some white; that belief is a certain attitude towards propositions, which is called knowledge when they are true, error when they are false. (Russell 1904, p. 523) Note how Russell adheres to the Bolzano reduction of knowledge to the mere truth of its content. Wittgenstein (Tractatus 6.111) also took notice of this passage from Russell. 73. Joachim (1906, p. 39). Russell and Moore both responded to Joachim’s book in Mind. Moore’s response is particularly interesting: “That some facts are facts, and some truths true, which never have been, are not now, and never will be experiences at all, and which are not timelessly expressed either” (1907, p. 231). What Moore countenances here are propositions that will remain unknown at all times; that, though, does not make them unknowable. The opposite view presupposes what Lovejoy (1936) called the principle of Plenitude, namely, that all potentialities will eventually become actual. (Martin-Löf 1991 rejects the application of Plenitude to knowability: what is knowable need never be known.) Only a year later did Moore commit himself in a review of William James: “It seems to me, then, that very often we have true ideas which we cannot verify; true ideas, which in all probability no man will ever be able to verify” (Moore 1907–08, p. 103). 74. Russell (1903, p. 145). (I am indebted to Prof. Peter Hylton for drawing my attention to this passage.) Since inﬁnitely complex propositions have to be unknowable, one way of deciding the issue concerning their existence is to deny that there are unknowable propositions. Because Russell is unable to pronounce on the issue, this means that he does not want to rule out unknowable truths. Also this passage was noticed by Wittgenstein, see Tractatus 4.2211. Russell’s is the earliest position (known to me) that allows for unknowable truths. Frege, as we saw, rejects them; every truth either is knowable in itself or has a Begründung, that is, a proof. 75. The undeﬁnability of truth was claimed in print only in Der Gedanke (1918, p. 60). In Nachlass, p. 140 (Logik 1897), Frege had made the same points almost verbatim. They in turn go back in nuce to the Logik of the 80s. 76. Moore in the lecture course from 1910–11 that was published later (1953). Russell in a number of places, for example, 1912 (p. 74) and PM, p. 43. 77. The notion was explicitly formulated by Mulligan, Simons, and Smith (1984). 78. The classic Stenius (1960) remains eminently readable. Hacker (1981) oﬀers the best presentation of the theory and its diﬃculties. 79. It most certainly is not; Peter and John, assuming they are empirical subjects, are complexes (5.541–5.421) composed of thoughts (3), that is, picture-facts (2.16),

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and will be analyzed in terms of the propositions that describe the complexes in question (3.24). The transcendent subject (5.63–5.641), on the other hand, “thinks out” the sentence-senses, which constitute the method of projection to the world (3.11–3.13), whereas the empirical subject is composed of sentential signs, that is, thought-facts. 80. The deﬁnition of the Q-relation reminds one of the ways that Frege formed unsaturated expressions. It is clear, I think, that this is one of very many places where the inﬂuence of “the great works” of Frege (see the preface to the Tractatus) can be felt. 81. In the next few paragraphs I use expository devices from the metatheory of the propostional calculus to survey the logico-semantical doctrines of the Tractatus. References to the Tractatus are by thesis number. Enderton (1972) contains the relevant model theory. 82. Wittgenstein’s notion of proposition (Satz) is not that of Bolzano–Frege– Russell (Satz an sich/Gedanke/proposition). In the Tractatus a proposition is a meaningful sentence in use and the Sätze might well better be rendered sentences in English translation. 83. Here N is Wittgenstein’s generalized Sheﬀer-stroke that negates every member of the range ξ of propositions. 84. The Fregean proposition is a sense, whereas the Tractarian proposition (sentence) has sense. 85. Jan Sebestik (1990) suggests that Robert Zimmerman’s Gymnasium textbook Philosophische Propädeutik, which is replete to the point of plagiarism with material taken from Bolzano’s Wissenschaftslehre, might be the missing link between Bolzano and Wittgenstein. 86. After the metamathematical revolution around 1930, Wittgenstein’s ontological notion, obtaining of the states of aﬀairs makes the elementary proposition true, is transformed into the model-theoretic: A |= ϕ, that is, the set-theoretical structure A satisﬁes the wﬀ ϕ (Tarski and Vaught 1957). See also Sundholm (1994b). 87. The crucial epistemological role of rightness in upholding the distinction between appearance and reality was noted and stressed by Martin-Löf (1987). 88. Dummett (1976) is the locus classicus, while Dummett (1991) oﬀers a booklength treatment. The secondary literature on Dummett’s argument has reached the proportions of an avalanche. Sundholm (1986) is an early survey, and Sundholm (1994a) approaches Dummett’s position from a more severely constructivist standpoint. 89. Molk (1885, p. 8): Les déﬁnitions devront être algébraiques et non pas logiques seulement. Il ne suﬃt pas de dire: “Une chose est ou et non pas.” Il faut montrer ce ques veut dire être et ne pas être, dans le domaine particulier dans lequel nous nous mouvons. Alors, seulement nouns faisons un pas en avant. Si nous déﬁnissions, par exemple, une fonction irréductible comme une fonction qui n’est pas réductible, c’est a dire quie n’est pas décomposable en d’autres fonctions d’une nature déterminée, nous ne donnons point de déﬁnition algébraique, nous n’énonçons qu’une simple vérité logique.

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Pour qu’en Algèbre, nous soyons en droit de donner cette déﬁnition, il faut qu’elle soit précédé de l’exposé d’une méthode nous permettant d’obtenir a l’aide d’un nombre ﬁni d’operations rationelles, les facteurs d’une fonction réductible. Seule cette méthode donne aux mots réductible et irréductible. 90. See the Port-Royal Logic, Arnauld and Nicole (1662, part IV, ch. III). 91. The ∞ set D ⊆ , being bounded, is contained in a closed real interval I. Deﬁne I0 =def I =def [a0 , b0 ] Ik+1 =def [ak , ak + bk /2] Ik+1 =def [ak + bk /2, bk ]

if this left half of Ik contains ∞ many points from D; otherwise. (NB. Here we cannot decide whether a half has ∞ many points from D.)

Hence, each of the nested intervals Ik contains ∞ many points from D, and length (Ik ) → 0, when k → ∞. Thus, k Ik contains exactly one point that is the required accumulation point for D. 92. Frege does not include the set D, the domain of quantiﬁcation, since he quantiﬁes over all individual objects. 93. Martin-Löf (1983, p. 33) hints at this way of understanding Brouwer’s criticism. It was noted explicitly by Aarne Ranta (1994, p. 38). See also Sundholm (1998), where also Poincaré’s criticism of impredicability is cast in the same mold. The law of excluded middle does not only serve as a principle of reasoning. It is also used meaning-theoretically to delimit the notion of proposition. Thus, for Frege, a proposition is a method for determining one of the truth values True and False. Similarly, every proposition implies itself and something which is not a proposition implies nothing, Russell (1903, §16) notes, and goes on to use “P ⊃ P ” as an explanation of what it is for something P to be a proposition. But an assertion that P ⊃ Q is true is equivalent to is an assertion that P is false or Q is true. Thus an assertion that P is a proposition amounts to an assertion that P is false or P is true. The issue resurfaces in the Cambridge Letter R 12 from Wittgenstein to Russell, June 1913, where “ ‘aRb.v.∼aRb’ must follow directly without the use of any other premiss.” Also Cantor’s explanation of a well-deﬁned set (1882, p. 114) makes meaning-theoretical use of the law of excluded middle. 94. Appropriately enough, free-variable equations between computable terms, with only true numerical substitution instances, are called veriﬁzierbar (veriﬁable) in the canonical exposition Hilbert and Bernays (1934, p. 237). 95. Carnap (1934, p. xv). 96. Carnap (1934, pp. 51–52). 97. Brouwer (1981, p. 5). This formulation, albeit late, expresses Brouwer’s lifelong view. 98. Brouwer (1908). 99. One does not, of course, claim that A ∨ ¬A is false, that is, that ¬(A ∨ ¬A) is true, because the latter claim is refutable outright: Assume that ¬(A ∨ ¬A) is true. Assume further that A is true. Under this assumption, A ∨ ¬A is also true. Therefore, the assumption that A is true leads to a contradiction. Therefore, A is false, now only under the sole assumption that ¬(A ∨ ¬A) is true. Hence ¬A is true,

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still under the same assumption. But then, under the same assumption, also A ∨ ¬A is true. Thus the assumption that ¬(A ∨ ¬A) is true leads to the conclusion that also A ∨ ¬A is true, which is a contradiction. Therefore the assumption is wrong and ¬(A ∨ ¬A) is false. Thus, ¬¬(A ∨ ¬A) is true. 100. GGA, II, §§87–147, as well as his undigniﬁed diatribes (1906a) and (1906b), (1908), against Korselt and Thomae, respectively. The need for content in mathematical sign-languages is a theme that Frege pursued from his earliest writings; see for instance, the long Nachlass paper on Boole’s logic and his own begriﬀsschrift (1880/81) and above all (1882). 101. Even after World War II—Brouwer lectured regularly at Cambridge from 1946 to 1951—he would proclaim, apparently with a deadpan face, that “Absurdity of absurdity of absurdity is equivalent to absurdity,” rather than use the pellucid ¬¬¬A ↔ ¬A. See Brouwer (1981, p. 12). 102. Most clearly perhaps in the introduction to Weyl (1918a), but also in the treatment of logic in (1918b). 103. Weyl (1921, p. 54): Ein Existentialsatz—etwa “es gibt eine gerade Zahl”—ist überhaupt kein Urteil im eigentlichen Sinne, das einen Sachverhalt behauptet; ExistentialSachverhalte sind eine leere Erﬁndung der Logiker. “2 ist eine gerade Zahl”: das ist ein wirkliches, einem Sachverhalt Ausdruck gebendes Urteil; “es gibt eine gerade Zahl” ist nur ein aus diesem Urteil gewonnenes Urteilsabstrakt. 104. The novel form of judgment and the explicit formulation of the rule that provides its assertion-condition are both due to Per Martin-Löf (1994). 105. Heyting (1930a). 106. The debate in question is treated in Thiel (1988) and Franchella (1994). 107. Heyting (1930b, p. 958): Une proposition p, comme, par example, “la constante d’Euler est rationelle”; exprime un problème, ou mieux encore une certaine attente (celle de trouver deux entiers a et b tels que C = a/b), qui pourra être réalisée ou déçue. 108. This table is based on a streamlined formulation oﬀered by Per MartinLöf (1984), and, in each case, lays down what a canonical proof-object is for the proposition in question. For the signiﬁcance of canonical in this context, see Sundholm (1997), where a full exposition of the intuitionistic meaning explanations is oﬀered. 109. It should be stressed that these meaning explanations for the logical constants, and the ensuing truth-deﬁnition, are neutral with respect to the underlying logic; in fact the framework can be viewed as a Tarskian truth deﬁnition—another neutral account. If we allow nonconstructive existence claims, also classical logic holds under the proof-object semantics. We have to show, reasoning nonconstructively, that Proof(A ∨ ¬A) = ∅. Assume that Proof(A) = ∅. Let a ∈ Proof(A); then i(a) ∈ Proof(A ∨ ¬A). Assume that Proof(A) = ∅. Then λx.x ∈ Proof(A) → Proof(⊥) = Proof(¬A), and so j(λx.x) ∈ Proof(A ∨ ¬A). Hence, in either case, Proof(A ∨ ¬A) = ∅, so the proposition A ∨ ¬A is true. Q.E.D.

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110. In Martin-Löf’s constructive type theory (1982, 1984) the elliptic form of judgment A is true is replaced by the explicit p is a Proof(A); Martin-Löf (1983) makes clear that this constitutes a return to the traditional S is P form of judgment. 111. Published as Gentzen (1934–35). 112. Dummett (1973, p. 309, and p. 435, respectively). 113. Prawitz (1965, p. 37). Prawitz prefers the opposite order between the two premises of the (⊃I) rule, but this is of no importance for the present point. 114. Provided that, given that, on condition that, under the assumption that, under the hypothesis that. . . . Many variations in the wording are possible here. 115. For instance by Prawitz (1971, remark 1.6, p. 243), Dummett (1977, pp. 121– 122, and 1991, p. 248), as well as Sundholm (1983). 116. The sequential form of natural deduction uses both introduction rules and elimination rules. It must not be confused with the sequent calculus of Gentzen (1934–35) that uses no elimination rules but has both left and right introduction rules, on both sides of the sequent arrow. 117. Gentzen did not consider closed sequents; the exploration of their theory is due to Peter Schroeder-Heister (1981, 1984, 1987), half a century after Gentzen. 118. Kutschera (1996) and Schroeder-Heister (1999) both discuss the matter in apparent unawareness of Tichý’s explicit treatment. Tichý’s remarkable chapter 13— Inference—merits attention, as does his paper “On Inference” (1999). 119. The Übersicht (1934–35, p. 176) does mention Frege, Russell, and Hilbert as particularly important for the formalization of logical inference, but the remark does not presuppose familiarity with the details of Frege’s formalization. The introduction to Hilbert and Ackermann (1928, p. 2), which Gentzen did know, makes similar mention of the same authors. 120. Full biographies are available for a number of authors treated of in the present chapter: Frege (Kreiser 2001), Wittgenstein (McGuinness 1988; Monk 1990), Brouwer (Van Dalen 1999), and Gentzen (Menzler-Trott 2001). 121. Simons (1999, p. 115).

References Arnauld, Antoine, and Pierre Nicole. 1662. La Logique ou L’Art de Penser. Barnes, Jonathan. 1988. Meaning, Saying, and Thinking. In Dialektiker und Stoiker, eds. Klaus Döring and Theodor Ebert, 47–61. Stuttgart: Franz Steiner Verlag. Bergmann, Gustav. 1964. Logic and Reality. Madison: University of Wisconsin Press. Bergmann, S. H. 1968. Bolzano and Bertano. Archief für Geschichte der Philosophie 48: 306–311. Bolzano, Bernard. [1837] 1929–1931. Wissenschaftslehre in vier Bänden, hrsg. von W. Schultz. Leipzig: Verlag von Felix Meiner. (WL) Boole, George. 1854. The Laws of Thought. London: Macmillan. Boolos, George. 1994. 1879? In Reading Putnam, eds. Bob Hale and Peter Clark, 31–48. Oxford: Basil Blackwell. Boswell, Terry. 1988. On the textual authenticity of Kant’s logic. History and Philosophy of Logic 9: 193–203. Brentano, Franz. 1889. Vom Ursprung sittlichher Erkenntnis (Philosophische Bibliothek 55). Hamburg: Felix Meiner, 1955.

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Brentano, Franz. 1903. Versuch über die Erkenntnis (Philosophische Bibliothek 194). Leipzig: Felix Meiner, 1925. Brentano, Franz. 1930. Wahrheit und Evidenz (Philosophische Bibliothek 201). Hamburg: Felix Meiner, 1974. Brentano, Franz. 1956. Lehre vom richtigen Urteil, ed. F. Mayer-Hillebrand. Bern: Francke Verlag. Brouwer, L. E. J. 1981. Cambridge Lectures on Intuitionism, ed. Dirk van Dalen. Cambridge: Cambridge University Press. Brouwer, Luitzen Egbertus Jan. 1908. De onbetrouwbaarheid der logische pricipes. Tijdschrift voor wijsbegeerte 2: 152–158. Cantor, Georg. 1882. Über unendliche lineare Punktmannigfaltigkeiten, III. Mathematische Annalen 20: 113–121. Carnap, Rudolf. 1931. Die logizistische Grundlegung der Mathematik. Erkenntnis 2: 91–105. English translation, Logicist Foundations of Mathematics. In Philosophy of Mathematics, eds. Paul Benacerraf and Hilary Putnam, 31–41. Englewood Cliﬀs, N.J.: Prentice Hall, 1964. Carnap, Rudolf. 1934. Logische Syntax der Sprache. Wien: Springer. Cited after the English translation by the Countess von Zeppelin, Logical Syntax of Language, London: Routledge and Kegan Paul, 1937. Cartwright, Richard. 1987. A Neglected Theory of Truth. In Philosophical Essays, 71–93. Cambridge, Mass.: MIT Press. Dummett, M. A. E. 1977. Elements of Intuitionism. Oxford: Oxford University Press. Dummett, Michael. 1973. Frege. Philosophy of Language. London: Duckworth. Dummett, Michael. 1976. What Is a Theory of Meaning? (II). In Truth and Meaning, eds. Gareth Evans and John McDowell, 67–137. Oxford: Clarendon Press. Dummett, Michael. 1991. The Logical Basis of Metaphysics. London: Duckworth. Enderton, H. B. 1972. A Mathematical Introduction to Logic. New York: Academic Press. Fichte, Johann Gottlieb. 1797. Erste Einleitung in die Wissenschaftslehre. Philosophisches Journal 5: 1–47. Franchella, M. 1994. Heyting’s contribution to the change in research into the foundations of mathematics. History and Philosophy of Logic 15: 149–172. Frege, Gottlob. 1879. Begriﬀsschrift. Halle: Louis Nebert. (Bs) Frege, Gottlob. 1880/81. Boole’s rechnende Logik und die Begriﬀsschrift. In Frege (1983), 9–52. Frege, Gottlob. 1882. Über die wissenschaftliche Berechtigung einer Begriﬀsschrift. Zeitschrift für Philosophie und philosophische Kritik 81: 48–56. Frege, Gottlob. 1884. Die Grundlagen der Arithmetik. Breslau: Koebner. (GLA) Frege, Gottlob. 1891. Funktion und Begriﬀ. Jena: Hermann Pohle. Frege, Gottlob. 1892. Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik 100: 25–50. Frege, Gottlob. 1906a. Antwort auf die Ferienplauderei des Herrn Thomae. Jahresbericht der deutschen Mathematiker-Vereinigung 15: 586–590. Frege, Gottlob. 1906b. Grundlagen der Geometrie I, II, III. Jahresbericht der deutschen Mathematiker-Vereinigung 15: 293–309, 377–403, 423–430. Frege, Gottlob. 1908. Die Unmöglichkeit der Thomaeschen formalen Arithmetik aufs neue nachgewiesen. Jahresbericht der deutschen Mathematiker-Vereinigung 17: 52–55. And Schlußbemerkung on p. 56.

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Frege, Gottlob. 1918. Der Gedanke. Beiträge zur Philosophie des deut/-schen Idealismus 1: 58–77. Frege, Gottlob. 1983. Nachgelassene Schriften, eds. Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach. Hamburg: Felix Meiner. Geach, Peter. 1977. Preface. In Gottlob Frege, Logical Investigations, vii–ix. Oxford: Basil Blackwell. Gentzen, Gerhard. 1934–35. Untersuchungen über das logische Schliessen. Mathematische Zeitschrift 39: 176–210, 405–431. Gentzen, Gerhard. 1936. Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen 112: 493–565. Gentzen, Gerhard. 1938. Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie. In Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften (N.F.) 4, 19–44. Leipzig: Felix Meiner. Gödel, Kurt. 1931. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38: 173–198. Hacker, P. M. S. 1981. The Rise and Fall of the Picture Theory. In Perspectives on the Philosophy of Wittgenstein, ed. Irving Block, 85–109. Oxford: Blackwell. Heyting, Arend. 1930a. Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der preussischen Akademie von Wissenschaften Phys.–math. Klasse, 42–56. English translation in Mancosu (1998), 311–327. Heyting, Arend. 1930b. Sur la logique intuitionniste. Acad. Roy. Belgique, Bull. Cl. Sci. 5, 16: 957–963. English translation in Mancosu (1998), 306–310. Heyting, Arend. 1931. Die intuitionistische Grundlegung der Mathematik. Erkenntnis 2: 106–115. English translation in P. Benacerraf and H. Putnam, Philosophy of Mathematics (2nd ed.), Cambridge: Cambridge University Press, 1983, 52–61. Heyting, Arend. 1934. Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie (Ergebnisse der Mathematik, vol. 3:4). Berlin: Springer. Hilbert, David. 1926. Über das Unendliche. Mathematische Annalen 95: 161–190. Hilbert, David and Paul Bernays. 1934. Grundlagen der Mathematik, vol. I. Berlin: Springer. Hilbert, David, with W. Ackermann. 1928. Grundzüge der theoretischen Logik. Berlin: Springer. Howard, W. A. 1980. The Formulae-as-Types Notion of Construction. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, eds. J. Seldin and R. Hindley, 479–490. London: Academic Press. Joachim, H. H. 1906. The Nature of Truth. Oxford: Clarendon Press. Kant, Immanuel. [1781, 1987] 1904. Kritik der reinen Vernunft, 1781 (A), 2nd ed. 1787 (B). In Kant’s gesammelte Schriften, Band III, Berlin: G. Reimer; trans. Norman Kemp Smith (1929). Macmillan, London, 1929. (KdrV ) Kleene, Stephen Cole. 1952. Introduction to Metamathematics. Amsterdam: NorthHolland. Kolmogoroﬀ, A. N. 1932. Zur Deutung der intuitionistischen Logik. Mathematische Zeitschrift 35: 58–65. English translation in Mancosu (1998), 328–334. Kreiser, Lothar. 2001. Gottlob Frege. Leben–Werk–Zeit. Hamburg: Felix Meiner. Kutschera, Franz von. 1996. Frege and Natural Deduction. In Frege: Importance and Legacy, ed. M. Schirn, 301–304. Berlin: De Gruyter. Leibniz, Gottfried Wilhelm. 1960. Fragmente zur Logik. Berlin: Akademie-Verlag. Edited, selected, and translated by Franz Schmidt.

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Lovejoy, Arthur. 1936. The Great Chain of Being. Cambridge, Mass.: Harvard University Press. Mancosu, Paolo, ed. 1998. From Brouwer to Hilbert. Oxford: Oxford University Press. Maritain, Jaques. 1946. An Introduction to Logic. London: Sheed and Ward. Martin-Löf, Per. 1982. Constructive Mathematics and Computer Programming. In Logic, Methodology and Philosophy of Science VI, Hannover 1979, ed. L. J. Cohen et al., 153–175. Amsterdam: North-Holland. Martin-Löf, Per. 1983. On the meanings of the logical constants and the justiﬁcations of the logical laws, lectures delivered in Sienna, ﬁrst distributed in 1985, and printed in Nordic Journal of Philosophical Logic I (1996): 11–60. Available at http://www.hf.uio.no/ﬁlosoﬁ/njpl. Martin-Löf, Per. 1984. Intuitionistic Type Theory. Naples: Bibliopolis. Martin-Löf, Per. 1987. Truth of a proposition, evidence of judgement, validity of a proof. Synthese 73: 191–212. Martin-Löf, Per. 1991. A Path from Logic to Metaphysics. In Atti del Congresso Nuovi Problemi della Logica e della Filosoﬁa della Scienza, Viareggio, 8–13 gennaio, 1990, eds. G. Corsi and G. Sambin, 141–149. Bologna: CLUEB. Martin-Löf, Per. 1994. Analytic and Synthetic Judgements in Type Theory. In Kant and Contemporary Epistemology, ed. P. Parrini, 87–99. Dordrecht: Kluwer. McGuinness, Brian. 1988. Wittgenstein, a Life: Young Ludwig, 1889–1921. London: Duckworth. Menzler-Trott, E. 2001. Gentzens Problem: Mathematishe Logik im nationalsozialistischen Deutschland. Basel: Birkhäuser. Molk, Jules. 1885. Sur une notion qui comprend celle de la dvisibilité et sur la théorie générale de l’élimination. Acta Mathematica 6: 1–166. Monk, Ray. 1990. Wittgenstein: The Duty of Genius. London: Jonathan Cape. Moore, G. E. 1898. The nature of judgement. Mind 8: 176–193. Moore, G. E. 1902. Truth. In Encyclopaedia of Philosophy and Psychology, ed. J. M. Baldwin, vol. 2, 716–718. London: Macmillan. Moore, G. E. 1907. Mr. Joachim’s Nature of Truth. Mind 16: 229–235. Moore, G. E. 1907–08. Professor James’ “Pragmatism”. Proc. Arist. Soc. 8. Cited from the reprint in Some Problems of Philosophy. London: Routledge and Kegan Paul, 1922, 97–146. Moore, G. E. 1953. Some Main Problems of Philosophy. London: George Allen and Unwin. Mulligan, Kevin, Peter Simons, and Barry Smith. 1984. Truth-makers. Philosophy and Phenomenological Research 44: 287–321. Prawitz, Dag. 1965. Natural Deduction. Uppsala: Almqvist & Wiksell. Prawitz, Dag. 1971. Ideas and Results in Proof Theory. In Proceedings of the Second Scandinavian Logic Symposium, ed. J. E. Fenstad, 235–307. Amsterdam: NorthHolland. Putnam, Hilary. 1982. Peirce the logician. Historia Mathematica 9: 290–301. Quine, Willard Van Orman. 1950. Methods of Logic. New York: Holt. Ramsey, Frank Plumpton. 1926. The foundations of mathematics. Proceedings of the London Mathematical Society 25(2): 338–384. Ranta, Aarne. 1994. Type-Theoretical Grammar. Oxford: Clarendon Press. Rorty, Richard. 1967. The Linguistic Turn. Chicago: University of Chicago Press. Russell, Bertrand. 1903. The Principles of Mathematics. Cambridge: Cambridge University Press.

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Russell, Bertrand. 1904. Meinong’s theory of complexes and assumptions. Mind 13: 204–219, 336–354, 509–524. Russell, Bertrand. 1912. The Problems of Philosophy. Oxford: Oxford University Press. 1982. Scholz, Heinrich. 1930. Die Axiomatik der Alten. Blätter für deutsche Philosophie 4: 259–278. Cited from Scholz (1961), 27–45. English translation by Jonathan Barnes, Ancient Axiomatic Theory. In Articles on Aristotle 1: Science, eds. J. Barnes, M. Schoﬁeld, and R. Sorabji, 1975, 50–60. London: Duckworth. Scholz, Heinrich. 1961. Mathesis Universalis, eds. Hans Hermes, Friedrich Kambartel, and Joachim Ritter. Basel: Benno Schwabe Verlag. Schroeder-Heister, Peter. 1981. Untersuchungen zur regellogischen Deutung von Aussagenverknüpfungen. Diss., Bonn. Schroeder-Heister, Peter. 1984. A natural extension of natural deduction. Journal of Symbolic Logic 49: 1284–1300. Schroeder-Heister, Peter. 1987. Structural Frameworks with Higher-Level Rules. Habilitationsschrift. University of Konstanz, Department of Philosophy. Schroeder-Heister, Peter. 1999. Gentzen-Style Features in Frege. In Abstracts of the 11th International Congress of Logic, Methodology, and Philosophy of Science (Cracow, August 1999), 499. Cracow. Sebestik, Jan. 1990. The Archeology of the Tractatus: Bolzano and Wittgenstein. In Wittgenstein—Towards a Re-Evaluation, Proc. 14th Int. Wittgenstein Symp., Kirchberg am Wechsel, 13–20 August 1989, Vol. I, eds. Rudolf Haller and J. Brandl, 112–128. Wien: Verlag Hölder-Pichler-Tempsky. Sebestik, Jan. 1992. The Construction of Bolzano’s Logical System. In Bolzano’s Wissenschafstlehre 1837–1987. International Workshop Firenze, 16–19 settembre 1987, 163–177. Firenze: Leo S. Olschki. Siebel, Mark. 1996. Der Begriﬀ der Ableitbarkeit bei Bolzano. St Augustin: Academia Verlag. Simons, Peter. 1999. Bolzano, Brentano, and Meinong: Three Austrian Realists. In German Philosophy since Kant (Royal Institute of Philosophy Supplement 44), ed. Anthony O’Hear, 109–136. Cambridge: Cambridge University Press. Stenius, Erik. 1960. Wittgenstein’s Tractatus: A Critical Expostion of Its Main Line of Thought. Oxford: Basil Blackwell. Stepanians, Markus. 1998. Frege und Husserl über Urteilen und Denken. Paderborn: Schöningh. Sundholm, B. G. 1983. Systems of Deduction. In Handbook of Philosophical Logic I, eds. D. Gabbay and F. Guenthner, 133–188. Dordrecht: Reidel. Sundholm, B. G. 1986. Proof Theory and Meaning. In Handbook of Philosophical Logic, eds. D. Gabbay and F. Guenthner, vol. III, 471–506. Dordrecht: Reidel. Sundholm, B. G. 1988. Oordeel en Gevolgtrekking: Bedereigde Species? (Judgement and Inference. Endangered Species?). Inauguaral Lecture, Leyden University, September 9, 1988. Published in pamphlet form by the university. Sundholm, B. G. 1994a. Vestiges of Realism. In The Philosophy of Michael Dummett, eds. Brian McGuinness and G. Oliveri, 137–165. Dordrecht: Kluwer. Sundholm, B. G. 1994b. Ontologic versus Epistemologic: Some Strands in the Development of Logic, 1837–1957. In Logic and Philosophy of Science in Uppsala, eds. Dag Prawitz and Dag Westerståhl, 373–384. Dordrecht: Kluwer. Sundholm, B. G. 1994c. Proof-theoretical semantics and Fregean identity-criteria for propositions. Monist 77: 294–314.

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Sundholm, B. G. 1997. Implicit epistemic aspects of constructive logic. Journal of Logic, Language and Information 6: 191–212. Sundholm, B. G. 1998. Intuitionism and Logical Tolerance. In Alfred Tarski and the Vienna Circle (Vienna Circle Institute Yearbook), eds. Jan Wolenski and E. Köhler, 135–149. Dordrecht: Kluwer. Sundholm, B. G. 2002. A Century of Inference: 1837–1936. In Logic, Methodology and Philosophy of Science 11, Cracow 1999, eds. Jan Wolenski and K. Placek. Dordrecht: Kluwer. Sundholm, B. G. 2004. Antirealism and the Roles of Truth. In Handbook of Epistemology, eds. I. Niiniluoto, M. Sintonen, and J. Wolenski, 437–466. Dordrecht: Kluwer. Tarski, Alfred. 1933a. Pojece prawdy wjezykach deukcyjnynch, Travaux de la Société des Sciences et des Letttres de Varsovie, Classe III Sciences Mathématiques et Physiques, vol. 34. (German translation by Leopold Blaustein: Tarski 1935). Tarski, Alfred. 1933b. Einige Betrachtungen über die Begriﬀe der ω-Widerspruchsfreiheit und der ω-Vollständigkeit. Monatshefte für Mathematik und Physik 40: 97–112. Tarski, Alfred. 1935. Der Wahrheitsbegriﬀ in den formalisierten Sprachen. In Studia Philosophica I (1936), 261–405. Lemberg: Polish Philosophical Society. Oﬀprints in monograph form dated 1935. Tarski, Alfred. 1944. The semantic conception of truth and the foundations of semantics. Philosophy and Phenomenological Research 4: 341–375. Tarski, Alfred. 1956. Logic, Semantics, Metamathematics. Oxford: Clarendon Press. Tarski, Alfred, with Jan Lukasiewicz. 1930. Untersuchungen über den Aussagenkalkül. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III 23: 30–50. English translation in Tarski (1956, pp. 38–59). Tarski, Alfred, with Robert Vaught. 1957. Arithmetical extensions of relational systems. Compositio Mathematicae 13: 81–102. Textor, Mark. 1996. Bolzanos Propositionalismus. Berlin: De Gruyter. Thiel, Christian. 1988. Die Kontroverse um die intuitionistische Logik vor ihre Axiomatisierung durch Heyting im Jahre 1930. History and Philosophy of Logic 9: 67–75. Tichý, Pavel. 1988. Frege’s Foundations of Logic. Berlin: De Gruyter. Tichý, Pavel and Jindra Tichý. 1999. On Inference. In The LOGICA Yearbook 1998, Philosophical Institute, Czech Academy of Science, 73–85. Prague: FILOSOFIA Publishers. Van Dalen, Dirk. 1999. Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer, vol. 1. The Dawning Revolution. Oxford: Clarendon Press. Weyl, Hermann. 1918a. Raum–Zeit–Materie. Berlin: Julius Springer. Weyl, Hermann. 1918b. Das Kontinuum. Leipzig: Veit. Weyl, Hermann. 1921. Über die neue Grundlagenkrise der Mathematik. Mathematische Zeitschrift 10: 39–79. English translation in Mancosu (1998), pp. 86–118. Wittgenstein, Ludwig. 1921. Logisch-philosophische Abhandlung. Annalen für Naturphilosophie 14: 198–262. Reprinted with English translation in Wittgenstein (1922). Wittgenstein, Ludwig. 1922. Tractatus logico-philosophicus. London: Kegan Paul.

9

The Development of Mathematical Logic from Russell to Tarski, 1900–1935 Paolo Mancosu, Richard Zach, and Calixto Badesa

The following nine itineraries in the history of mathematical logic do not aim at a complete account of the history of mathematical logic during the period 1900–1935. For one thing, we had to limit our ambition to the technical developments without attempting a detailed discussion of issues such as what conceptions of logic were being held during the period. This also means that we have not engaged in detail with historiographical debates which are quite lively today, such as those on the universality of logic, conceptions of truth, the nature of logic itself, and so on. While of extreme interest, these themes cannot be properly dealt with in a short space, as they often require extensive exegetical work. We therefore merely point out in the text or in appropriate notes how the reader can pursue the connection between the material we treat and the secondary literature on these debates. Second, we have not treated some important developments. While we have not aimed at completeness, our hope has been that by focusing on a narrower range of topics our treatment will improve on the existing literature on the history of logic. There are excellent accounts of the history of mathematical logic available, such as, to name a few, Kneale and Kneale (1962), Dumitriu (1977), and Mangione and Bozzi (1993). We have kept the secondary literature quite present in that we also wanted to write an essay that would strike a balance between covering material that was adequately discussed in the secondary literature and presenting new lines of investigation. This explains, for instance, why the reader will ﬁnd a long and precise exposition of Löwenheim’s (1915) theorem but only a short one on Gödel’s incompleteness theorem: Whereas there is hitherto no precise presentation of the ﬁrst result, accounts of the second result abound. Finally, the treatment of the foundations of mathematics is quite restricted, and it is 318

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ancillary to the exposition of the history of mathematical logic. Thus, it is not meant to be the main focus of our exposition.1 Page references in citations are to the English translations, if available, or to the reprint edition, if listed in the bibliography. All translations are the authors’, unless an English translation is listed in the references.

1. Itinerary I. Metatheoretical Properties of Axiomatic Systems 1.1. Introduction The two most important meetings in philosophy and mathematics in 1900 took place in Paris. The First International Congress of Philosophy met in August and so did, soon after, the Second International Congress of Mathematicians. As symbolic, or mathematical, logic has traditionally been part of both mathematics and philosophy, a glimpse at the contributions in mathematical logic at these two events will give us a representative selection of the state of mathematical logic at the beginning of the twentieth century. At the International Congress of Mathematicians, Hilbert presented his famous list of problems (Hilbert 1900a), some of which became central to mathematical logic, such as the continuum problem, the consistency proof for the system of real numbers, and the decision problem for Diophantine equations (Hilbert’s tenth problem). However, despite the attendance of remarkable logicians like Schröder, Peano, and Whitehead in the audience, the only other contributions that could be classiﬁed as pertaining to mathematical logic were two talks given by Alessandro Padoa on the axiomatizations of the integers and of geometry, respectively. The third section of the International Congress of Philosophy was devoted to logic and history of the sciences (Lovett 1900–1901). Among the contributors of papers in logic we ﬁnd Russell, MacColl, Peano, Burali-Forti, Padoa, Pieri, Poretsky, Schröder, and Johnson. Of these, MacColl, Poretsky, Schröder, and Johnson read papers that belong squarely to the algebra of logic tradition. Russell read a paper on the application of the theory of relations to the problem of order and absolute position in space and time. Finally, the Italian school of Peano and his disciples—Burali-Forti, Padoa, and Pieri—contributed papers on the logical analysis of mathematics. Peano and Burali-Forti spoke on deﬁnitions, Padoa read his famous essay containing the “logical introduction to any theory whatever,” and Pieri spoke on geometry considered as a purely logical system. Although there are certainly points of contact between the ﬁrst group of logicians and the second group, already at that time it was obvious that two diﬀerent approaches to mathematical logic were at play. Whereas the algebra of logic tradition was considered to be mainly an application of mathematics to logic, the other tradition was concerned more with an analysis of mathematics by logical means. In a course given in 1908 in

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Göttingen, Zermelo captured the double meaning of mathematical logic in the period by reference to the two schools: The word “mathematical logic” can be used with two diﬀerent meanings. On the one hand one can treat logic mathematically, as it was done for instance by Schröder in his Algebra of Logic; on the other hand, one can also investigate scientiﬁcally the logical components of mathematics. (Zermelo 1908a, 1)2 The ﬁrst approach is tied to the names of Boole and Schröder, the second was represented by Frege, Peano, and Russell.3 We will begin by focusing on mathematical logic as the logical analysis of mathematical theories, but we will return later (see itinerary IV) to the other tradition.

1.2. Peano’s School on the Logical Structure of Theories We have mentioned the importance of the logical analysis of mathematics as one of the central motivating factors in the work of Peano and his school on mathematical logic. First of all, Peano was instrumental in emphasizing the importance of mathematical logic as an artiﬁcial language that would remove the ambiguities of natural language, thereby allowing a precise analysis of mathematics. In the words of Pieri, an appropriate ideographical algorithm is useful as “an instrument appropriate to guide and discipline thought, to exclude ambiguities, implicit assumptions, mental restrictions, insinuations and other shortcomings, almost inseparable from ordinary language, written as well as spoken, which are so damaging to speculative research” (Pieri 1901, 381). Moreover, he compared mathematical logic to “a microscope which is appropriate for observing the smallest diﬀerence of ideas, diﬀerences that are made imperceptible by the defects of ordinary language in the absence of some instrument that magniﬁes them” (382). It was by using this “microscope” that Peano was able, for instance, to clarify the distinction between an element and a class containing only that element and the related distinction between membership and inclusion.4 The clariﬁcation of mathematics, however, also meant accounting for what was emerging as a central ﬁeld for mathematical logic: the formal analysis of mathematical theories. The previous two decades had in fact seen much activity in the axiomatization of particular branches of mathematics, including arithmetic, algebra of logic, plane geometry, and projective geometry. This culminated in the explicit characterization of a number of formal conditions for which axiomatized mathematical theories should strive. Let us consider ﬁrst Pieri’s description of his work on the axiomatization of geometry, which had been carried out independently of Hilbert’s famous Foundations of Geometry (1899). In his presentation to the International Congress of Philosophy in 1900, Pieri emphasized that the study of geometry is following arithmetic in becoming more and more “the study of a certain order of logical relations; in freeing itself little by little from the bonds which still keep

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it tied (although weakly) to intuition, and in displaying consequently the form and quality of purely deductive, abstract and ideal science” (Pieri 1901, 368). Pieri saw in this abstraction from concrete interpretations a unifying thread running through the development of arithmetic, analysis, and geometry in the nineteenth century. This led him to a conception of geometry as a hypothetical discipline (he coined the term “hypothetico-deductive”). In fact he goes on to assert that the primitive notions of any deductive system whatsoever “must be capable of arbitrary interpretations in certain limits assigned by the primitive propositions,” subject only to the restriction that the primitive propositions must be satisﬁed by the particular interpretation. The analysis of a hypothetico-deductive system begins then with the distinction between primitive notions and primitive propositions. In the logical analysis of a hypothetico-deductive system it is important not only to distinguish the derived theorems from the basic propositions (deﬁnitions and axioms) but also to isolate the primitive notions, from which all the others are deﬁned. An ideal to strive for is that of a system whose primitive ideas are irreducible, that is, such that none of the primitive ideas can be deﬁned by means of the others through logical operations. Logic is here taken to include notions such as, among others, “individual,” “class,” “membership,” “inclusion,” “representation,” and “negation” (383). Moreover, the postulates, or axioms, of the system must be independent, that is, none of the postulates can be derived from the others. According to Pieri, there are two main advantages to proceeding in such an orderly way. First of all, keeping a distinction between primitive notions and derived notions makes it possible to compare diﬀerent hypothetico-deductive systems as to logical equivalence. Two systems turn out to be equivalent if for every primitive notion of one we can ﬁnd an explicit deﬁnition in the second one such that all primitive propositions of the ﬁrst system become theorems of the second system, and vice versa. The second advantage consists in the possibility of abstracting from the meaning of the primitive notions and thus operate symbolically on expressions which admit of diﬀerent interpretations, thereby encompassing in a general and abstract system several concrete and speciﬁc instances satisfying the relations stated by the postulates. Pieri is well known for his clever application of these methodological principles to geometrical systems (see Freguglia 1985; Marchisotto 1995). Pieri refers to Padoa’s articles for a more detailed analysis of the properties connected to axiomatic systems. Alessandro Padoa was another member of the group around Peano. Indeed, of that group, he is the only one whose name has remained attached to a speciﬁc result in mathematical logic, that is, Padoa’s method for proving indeﬁnability (see the following). The result was stated in the talks Padoa gave in 1900 at the two meetings mentioned at the outset (Padoa 1901, 1902). We will follow the “Essai d’une théorie algébrique des nombre entiers, précédé d’une introduction logique a une théorie déductive quelconque.” In the Avant-Propos (not translated in van Heijenoort 1967a) Padoa lists a number of notions that

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he considers as belonging to general logic such as class (“which corresponds to the words: terminus of the scholastics, set of the mathematicians, common noun of ordinary language”). The notion of class is not deﬁned but assumed with its informal meaning. Extensionality for classes is also assumed: “a class is completely known when one knows which individuals belong to it.” However, the notion of ordered class he considers as lying outside of general logic. Padoa then states that all symbolic deﬁnitions have the form of an equality y = b where y is the new symbol and b is a combination of symbols already known. This is illustrated with the property of being a class with one element. Disjunction and negation are given with their class interpretation. The notions “there is” and “there is not” are also claimed to be reducible to the notions already previously introduced. For instance, Padoa explains that given a class a to say “there is no a” means that the class not-a contains everything, that is, not-a = (a or not-a). Consequently, “there are a[’s]” means: not-a = (a or not-a). The notion of transformation is also taken as belonging to logic. If a and b are classes and if for any x in a, ux is in b, then u is a transformation from a into b. An obvious principle for transformations u is: if x = y then ux = uy. The converse, Padoa points out, does not follow. This much was a preliminary to the section of Padoa’s paper titled “Introduction logique a une théorie déductive quelconque.” Padoa makes a distinction between general logic and speciﬁc deductive theories. General logic is presupposed in the development of any speciﬁc deductive theory. What characterizes a speciﬁc deductive theory is its set of primitive symbols and primitive propositions. By means of these, one deﬁnes new notions and proves theorems of the system. Thus, when one speaks of indeﬁnability or unprovability, one must always keep in mind that these notions are relative to a speciﬁc system and make no sense independently of a speciﬁc system. Restating his notion of deﬁnition he also claims that deﬁnitions are eliminable and thus inessential. Just like Pieri, Padoa also speaks of systems of postulates as a pure formal system on which one can reason without being anchored to a speciﬁc interpretation, “for what is necessary to the logical development of a deductive theory is not the empirical knowledge of the properties of things, but the formal knowledge of relations between symbols” (1901, 121). It is possible, Padoa continues, that there are several, possibly inﬁnite, interpretations of the system of undeﬁned symbols which verify the system of basic propositions and thus all the theorems of a theory. He then adds: The system of undeﬁned symbols can then be regarded as the abstraction obtained from all these interpretations, and the generic theory can then be regarded as the abstraction obtained from the specialized theories that result when in the generic theory the system of undeﬁned symbols is successively replaced by each of the interpretations of this theory. Thus, by means of just one argument that proves a proposition of the generic theory we prove implicitly a proposition in each of the specialized theories. (1901, 121)5

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In contemporary model theory, we think of an interpretation as specifying a domain of individuals with relations on them satisfying the propositions of the system, by means of an appropriate function sending individual constants to objects and relation symbols to subsets of the domain (or Cartesian products of the same). It is important to remark that in Padoa’s notion of interpretation something else is going on. An interpretation of a generic system is given by a concrete set of propositions with meaning. In this sense the abstract theory captures all of the individual theories, just as the expression x + y = y + x captures all the particular expressions of the form 2 + 3 = 3 + 2, 5 + 7 = 7 + 5, and so on. Moving now to deﬁnitions, Padoa states that when we deﬁne a notion in an abstract system we give conditions which the deﬁned notion must satisfy. In each particular interpretation the deﬁned notion becomes individualized, that is, it obtains a meaning that depends on the particular interpretation. At this point Padoa states a general result about deﬁnability. Assume that we have a general deductive system in which all the basic propositions are stated by means of undeﬁned symbols: We say that the system of undeﬁned symbols is irreducible with respect to the system of unproved propositions when no symbolic deﬁnition of any undeﬁned symbol can be deduced from the system of unproved propositions, that is, when we cannot deduce from the system a relation of the form x = a, where x is one of the undeﬁned symbols and a is a sequence of other such symbols (and logical symbols). (1901, 122) How can such a result be established? Clearly one cannot adduce the failure of repeated attempts at deﬁning the symbol; for such a task, a method for demonstrating the irreducibility is required. The result is stated by Padoa as follows: To prove that the system of undeﬁned symbols is irreducible with respect to the system of unproved propositions it is necessary and suﬃcient to ﬁnd, for any undeﬁned symbol, an interpretation of the system of undeﬁned symbols that veriﬁes the system of unproved propositions and that continues to do so if we suitably change the meaning of only the symbol considered. (1901, 122)6 Padoa (1902) covers the same ground more concisely but also adds the criterion of compatibility for a set of postulates: “To prove the compatibility of a set of postulates one needs to ﬁnd an interpretation of the undeﬁned symbols which veriﬁes simultaneously all the postulates” (1902, 249). Padoa applied his criteria to showing that his axiomatization of the theory of integers satisﬁed the condition of compatibility and irreducibility for the primitive symbols and postulates. We thus see that for Padoa the study of the formal structure of an arbitrary deductive theory was seen as a task of general logic. What can be said about

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these metatheoretical results in comparison to the later developments? We have already pointed out the diﬀerent notion of interpretation which informs the treatment. Moreover, the system of logic in the background is never fully spelled out, and in any case it would be a logic containing a good amount of set-theoretic notions. For this reason, some results are taken as obvious that would actually need to be justiﬁed. For instance, Padoa claims that if an interpretation satisﬁes the postulates of an abstract theory, then the theorems obtained from the postulates are also satisﬁed in the interpretation. This is a soundness principle, which nowadays must be shown to hold for the system of derivation and the semantics speciﬁed for the system. For similar reasons the main result by Padoa on the indeﬁnability of primitive notions does not satisfy current standards of rigor. Thus, a formal proof of Padoa’s deﬁnability theorem had to wait until the works of Tarski (1934–1935) for the theory of types and Beth (1953) for ﬁrst-order logic (see van Heijenoort 1967a, 118–119, for further details).

1.3. Hilbert on Axiomatization In light of the importance of the work of Peano and his school on the foundations of geometry, it is quite surprising that Hilbert did not acknowledge their work in the Foundations of Geometry. Although it is not quite clear to what extent Hilbert was familiar with the work of the Italian school in the last decade of the nineteenth century (Toepell 1986), he certainly could not ignore their work after the 1900 International Congress in Mathematics. In many ways Hilbert’s work on axiomatization resembles the level of abstractness also emphasized by Peano, Padoa, and Pieri. The goal of Foundations of Geometry (1899) is to investigate geometry axiomatically.7 At the outset we are asked to give up the intuitive understanding of notions like point, line, or plane and consider any three system of objects and three sorts of relations between these objects (lies on, between, congruent). The axioms only state how these properties relate the objects in question. They are divided into ﬁve groups: axioms of incidence, axioms of order, axioms of congruence, axiom of parallels, and axioms of continuity. Hilbert emphasizes that an axiomatization of geometry must be complete and as simple as possible.8 He does not make explicit what he means by completeness, but the most likely interpretation of the condition is that the axiomatic system must be able to capture the extent of the ordinary body of geometry. The requirement of simplicity includes, among other things, reducing the number of axioms to a ﬁnite set and showing their independence. Another important requirement for axiomatics is showing the consistency of the axioms of the system. This was unnecessary for the old axiomatic approaches to geometry (such as Euclid’s) because one always began with the assumption that the axioms were true of some reality and thus consistency was not an issue. But in the new conception of axiomatics, the axioms do not express truths but only postulates whose consistency must be investigated.

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Hilbert shows that the basic axioms of his axiomatization are independent by displaying interpretations in which all of the axioms except one are true.9 Here we must point to a small diﬀerence with the notion of interpretation we have seen in Pieri and Padoa. Hilbert deﬁnes an interpretation by ﬁrst specifying what the set of objects consists in. Then a set of relations among the objects is speciﬁed in such a way that consistency or independence is shown. For instance, for showing the consistency of his axioms, √ he considers a domain given by the subset of algebraic numbers of the form 1 + ω 2 and then speciﬁes the relations as being sets of ordered pairs and ordered triples of the domain. The consistency of the geometrical system is thus discharged on the new arithmetical system: “From these considerations it follows that every contradiction resulting from our system of axioms must also appear in the arithmetic deﬁned above” (29). Hilbert had already applied the axiomatic approach to the arithmetic of real numbers. Just as in the case of geometry, the axiomatic approach to the real numbers is conceived in terms of “a framework of concepts to which we are led of course only by means of intuition; we can nonetheless operate with this framework without having recourse to intuition.” The consistency problem for the system of real numbers was one of the problems that Hilbert stated at the International Congress in 1900: But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a ﬁnite number of logical steps based upon them can never lead to contradictory results. (1900a, 1104) In the case of geometry, consistency is obtained by “constructing an appropriate domain of numbers such that to the geometrical axioms correspond analogous relations among the objects of this domain.” For the axioms of arithmetic, however, Hilbert required a direct proof, which he conjectured could be obtained by a modiﬁcation of the arguments already used in “the theory of irrational numbers.”10 We do not know what Hilbert had in mind, but in any case, in his new approach to the problem (1905b), Hilbert made considerable progress in conceiving how a direct proof of consistency for arithmetic might proceed. We will postpone treatment of this issue to later (see itinerary VI) and go back to specify what other metatheoretical properties of axiomatic systems were being discussed in these years. By way of introduction to the next section, something should be said here about one of the axioms, which Hilbert in his Paris lecture calls axiom of integrity and later completeness axiom. The axiom says that the (real) numbers form a system of objects which cannot be extended (Hilbert 1900b, 1094). This axiom is in eﬀect a metatheoretical statement about the possible interpretations of the axiom system.11 In the second and later editions of the Foundations of Geometry, the same axiom is also stated for points, straight lines and planes:

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(Axiom of completeness) It is not possible to add new elements to a system of points, straight lines, and planes in such a way that the system thus generalized will form a new geometry obeying all the ﬁve groups of axioms. In other words, the elements of geometry form a system which is incapable of being extended, provided that we regard the ﬁve groups of axioms as valid. (Hilbert 1902, 25) Hilbert commented that the axiom was needed to guarantee that his geometry turn out to be identical to Cartesian geometry. Awodey and Reck (2002) write, “what this last axiom does, against the background of the others, is to make the whole system of axioms categorical. . . . He does not state a theorem that establishes, even implicitly, that his axioms are categorical; he leaves it . . . without proofs” (11). The notion of categoricity was made explicit in the important work of the “postulate theorists,” to which we now turn.

1.4. Completeness and Categoricity in the Work of Veblen and Huntington A few metatheoretical notions that foreshadow later developments emerged during the early years of the twentieth century in the writings of Huntington and Veblen. Huntington and Veblen are part of a group of mathematicians known as the American postulate theorists (Scanlan 1991, 2003). Huntington was concerned with providing “complete” axiomatizations of various mathematical systems, such as the theory of absolute continuous magnitudes (positive real numbers) (1902) and the theory of the algebra of logic (1905). For instance, in 1902 he presented six postulates for the theory of absolute continuous magnitudes, which he claims to form a complete set. A complete set of postulates is characterized by the following properties: 1. The postulates are consistent; 2. They are suﬃcient; 3. They are independent (or irreducible). By consistency he means that there exists an interpretation satisfying the postulates. Condition 2 asserts that there is essentially only one such interpretation possible. Condition 3 says that none of the postulates is a “consequence” of the other ﬁve. A system satisfying the conditions (1) and (2) we would nowadays call “categorical” rather than “complete.” Indeed, the word “categoricity” was introduced in this context by Veblen in a paper on the axiomatization of geometry (1904). Veblen credits Huntington with the idea and Dewey for having suggested the word “categoricity.” The description of the property is interesting: Inasmuch as the terms point and order are undeﬁned one has a right, in thinking of the propositions, to apply the terms in

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connection with any class of objects of which the axioms are valid propositions. It is part of our purpose however to show that there is essentially only one class of which the twelve axioms are valid. In more exact language, any two classes K and K of objects that satisfy the twelve axioms are capable of a one-one correspondence such that if any three elements A, B, C of K are in the order ABC, the corresponding elements of K are also in the order ABC. Consequently any proposition which can be made in terms of points and order either is in contradiction with our axioms or is equally true of all classes that verify our axioms. The validity of any possible statement in these terms is therefore completely determined by the axioms; and so any further axiom would have to be considered redundant. [Note: Even were it not deducible from the axioms by a ﬁnite set of syllogisms] Thus, if our axioms are valid geometrical propositions, they are suﬃcient for the complete determination of Euclidean geometry. A system of axioms such as we have described is called categorical, whereas one to which it is possible to add independent axioms (and which therefore leaves more than one possibility open) is called disjunctive. (Veblen 1904, 346) A number of things are striking about the passage just quoted. First of all, we are used to deﬁne categoricity by appealing directly to the notion of isomorphism.12 What Veblen does is equivalent to specifying the notion of isomorphism for structures satisfying his 12 axioms. However, the fact that he does not make use of the word “isomorphism” is remarkable, as the expression was common currency in group theory already in the nineteenth century. The word “isomorphism” is brought to bear for the ﬁrst time in the deﬁnition of categoricity in Huntington (1906–1907). There he says that “special attention may be called to the discussion of the notion of isomorphism between two systems, and the notion of a suﬃcient, or categorical, set of postulates.” Indeed, on p. 26 (1906–1907), the notion of two systems being isomorphic with respect to addition and multiplication is introduced. We are now very close to the general notion of isomorphism between arbitrary systems satisfying the same set of axioms. The ﬁrst use of the notion of isomorphism between arbitrary systems we have been able to ﬁnd is Bôcher (1904, 128), who claims to have generalized the notion of isomorphism familiar in group theory. Weyl (1910) also gives the deﬁnition of isomorphism between systems in full generality. Second, there is a certain ambiguity between deﬁning categoricity as the property of admitting only one model (up to isomorphism) and conﬂating the notion with a consequence of it, namely, what we would now call semantical completeness.13 Veblen, however, rightly states that in the case of a categorical theory, further axioms would be redundant even if they were not deducible from the axioms by a ﬁnite number of inferences.

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Third, the distinction hinted at between what is derivable in a ﬁnite number of steps and what follows logically displays a certain awareness of the diﬀerence between a semantical notion of consequence and a syntactical notion of derivability and that the two might come apart. However, Veblen does not elaborate on the issue. Finally, later in the section Veblen claims that the notion of categoricity is also expressed by Hilbert’s axiom of completeness as well as by Huntington’s notion of suﬃciency. In this he reveals an inaccurate understanding of Hilbert’s completeness axiom and of its consequences. Baldus (1928) is devoted to showing the noncategoricity of Hilbert’s axioms for absolute geometry even when the completeness axiom is added. It is, however, true that in the presence of all the other axioms, the system of geometry presented by Hilbert is categorical (see Awodey and Reck 2002).

1.5. Truth in a Structure These developments have relevance also for the discussion of the notion of truth in a structure. In his inﬂuential paper (1986), Hodges raises several historical issues concerning the notion of truth in a structure, which can now be made more precise. Hodges is led to investigate some of the early conceptions of structure and interpretation with the aim of ﬁnding out why Tarski did not deﬁne truth in a structure in his early articles. He rightly points out that algebraists and geometers had been studying “Systeme von Dingen” (systems of objects), that is, what we would call structures or models (on the emergence of the terminology, see itinerary VIII). Thus, for instance, Huntington in (1906–1907) describes the work of the postulate theorist in algebra as being the study of all the systems of objects satisfying certain general laws: “From this point of view our work becomes, in reality, much more general than a study of the system of numbers; it is a study of any system which satisﬁes the conditions laid down in the general laws of §1.”14 Hodges then pays attention to the terminology used by mathematicians of the time to express that a structure A obeys some laws and quotes Skolem (1933) as one of the earliest occurrences where the expression “true in a structure” appears.15 However, here we should point out that the notion of a proposition being true in a system is not unusual during the period. For instance, in Weyl’s (1910) deﬁnition of isomorphism, we read that if there is an isomorphism between two systems, “there is also such a unique correlation between the propositions true with respect to one system and those true with respect to the other, and we can, without falling into error, identify the two systems outright” (Weyl 1910, 301). Moreover, although it is usual in Peano’s school and among the American postulate theorists to talk about a set of postulates being “satisﬁed” or “veriﬁed” in a system (or by an interpretation), without any further comments, sometimes we are also given a clariﬁcation that shows that they were willing to use the notion of truth in a structure. A few examples will suﬃce.

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Let us look at what might be the ﬁrst application of the method for providing proofs of independence. Peano in “Principii di geometria logicamente esposti” (1889) has two signs, 1 (for point) and c ε ab (c is a point internal to the segment ab). Then he considers three categories of entities with a relation deﬁned between them. Finally he adds: Depending on the meaning given to the undeﬁned signs 1 and c ε ab, the axioms might or might not be satisﬁed. If a certain group of axioms is veriﬁed, then all the propositions that are deduced from them will also be true, since the latter propositions are only transformations of those axioms and of those deﬁnitions. (Peano 1889, 77–78) In 1900, Pieri explains that the postulates, just like all conditional propositions are neither true nor false: they only express conditions that can sometimes be veriﬁed and sometimes not. Thus for instance, the equality (x + y)2 = x2 + 2xy + y 2 is true, if x and y are real numbers and false in the case of quaternions (giving for each hypothesis the usual meaning to +, ×, etc.). (Pieri 1901, 388–389) In 1906, Huntington: The only way to avoid this danger [of using more than is stated in the axioms] is to think of our fundamental laws, not as axiomatic propositions about numbers, but as blank forms in which the letters a, b, c, etc. may denote any objects we please and the symbols + and × any rules of combination; such a blank form will become a proposition only when a deﬁnite interpretation is given to the letters and symbols—indeed a true proposition for some interpretations and a false proposition for others. . . From this point of view our work becomes, in reality, much more general than a study of the system of numbers; it is a study of any system which satisﬁes the conditions laid down in the general laws of §1. (Huntington 1906–1907, 2–3)16 In short, it seems that the expression “a system of objects veriﬁes a certain proposition or a set of axioms” is considered to be unproblematic at the time, and it is often read as shorthand for a sentence, or a set of sentences, being true in a system. Of course, this is not to deny that in light of the philosophical discussion emerging from non-Euclidean geometries, a certain care was exercised in talking about “truth” in mathematics, but the issue is resolved exactly by the distinction between axioms and postulates. Whereas the former had been taken to be true tout court, the postulates only make a demand, which might be satisﬁed or not by particular system of objects (see also on the distinction, Huntington 1911, 171–172).

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2. Itinerary II. Bertrand Russell’s Mathematical Logic 2.1. From the Paris Congress to the Principles of Mathematics 1900–1903 At the time of the Paris congress, Russell was mainly familiar with the algebra of logic tradition. He certainly knew the works of Boole, Schröder, and Whitehead. Indeed, the earliest drafts of The Principles of Mathematics (1903; POM for short) are based on a logic of part-whole relationship that was closely related to Boole’s logical calculus. He also had already realized the importance of relations and the limitations of a subject-predicate approach to the analysis of sentences. This change was a central one in his abandonment of Hegelianism17 and also led him to the defense of absolute position in space and time against the Leibnizian thesis of the relativity of motion and position, which was the subject of his talk at the International Congress of Philosophy, held in Paris in 1900. However, he had not yet read the works of the Italian school. The encounter with Peano and his school in Paris was of momentous importance for Russell. He had been struggling with the problems of the foundation of mathematics for a number of years and thought that Peano’s system had ﬁnally shown him the way. After returning from the Paris congress, Russell familiarized himself with the publications of Peano and his school, and it became clear to him that “[Peano’s] notation aﬀorded an instrument of logical analysis such as I had been seeking for years” (Russell 1967, 218). In Russell’s autobiography, he claims that “the most important year of my intellectual life was the year 1900 and the most important event in this year was my visit to the International Congress of Philosophy in Paris” (1989, 12). One of the ﬁrst things Russell did was to extend Peano’s calculus with a worked-out theory of relations and this allowed him to develop a large part of Cantor’s work in the new system. This he pursued in his ﬁrst substantial contribution to logic (Russell 1901b, 1902b), which constitutes a bridge between the theory of relations developed by Peirce and Schröder and Peano’s formalization of mathematics. At this stage Russell thinks of relations intensionally, that is, he does not identify them with sets of pairs. The notion of relation is taken as primitive. Then the notion of the domain and co-domain of a relation, among others, are introduced. Finally, the axioms of his theory of relations state, among other things, closure properties with respect to the converse, the complement, the relative product, the union, and the intersection (of relations or classes thereof). He also deﬁnes the notion of function in terms of that of relation (however, in POM they are both taken as primitive). In this work, Russell treats natural numbers as deﬁnable, which stands in stark contrast to his previous view of number as an indeﬁnable primitive. This led him to the famous deﬁnition of “the cardinal number of a class u” as “the class of classes similar to u.” Russell arrived at it independently of Frege, whose deﬁnition was similar, but he was apparently inﬂuenced by Peano, who discussed such a deﬁnition in 1901 without, however, endorsing it. In any case, Peano’s inﬂuence

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is noticeable in Russell’s abandonment of the Boolean leanings of his previous logic in favor of Peano’s mathematical logic. Russell now accepted, except for a few changes, Peano’s symbolism. One of Peano’s advances had been a clear distinction between sentences such as “Socrates is mortal” and “All men are mortal,” which were previously conﬂated as being of the same structure. Despite the similar surface structure, the ﬁrst one indicates a membership relation between Socrates and the class of mortals, whereas the second indicates an inclusion between classes. In Peano’s symbolism we have s ε φ(x) for the ﬁrst and φ(x) ⊃x ψ(x) for the second. With this distinction Peano was able to deﬁne the relation of subsumption between two classes by means of implication. In a letter to Jourdain in 1910, Russell writes: Until I got hold of Peano, it had never struck me that Symbolic Logic would be any use for the Principles of mathematics, because I knew the Boolean stuﬀ and found it useless. It was Peano’s ε, together with the discovery that relations could be ﬁtted into his system, that led me to adopt Symbolic Logic. (Grattan-Guinness 1977, 133) What Peano had opened for Russell was the possibility of considering the mathematical concepts as deﬁnable in terms of logical concepts. In particular, an analysis in terms of membership and implication is instrumental in accounting for the generality of mathematical propositions. Russell’s logicism ﬁnds its ﬁrst formulation in a popular article written in 1901 where he claims that all the indeﬁnables and indemonstrables in pure mathematics stem from general logic: “All pure mathematics—Arithmetic, Analysis, and Geometry—is built up of the primitive ideas of logic, and its propositions are deduced from the general axioms of logic” (1901a, 367). This is the project that informed the Principles of Mathematics (1903). The construction of mathematics out of logic is carried out by ﬁrst developing arithmetic through the deﬁnition of the cardinal number of a class as the class of classes similar to it. Then the development of analysis is carried out by deﬁning real numbers as sets of rationals satisfying appropriate conditions. (For a detailed reconstruction see, among others, Vuillemin 1968, RodriguezConsuegra 1991, Landini 1998, Grattan-Guinness 2000.) The main diﬃculty in reconstructing Russell’s logic at this stage consists in the presence of logical notions mixed with linguistic and ontological categories (denotation, deﬁnition). Moreover, Russell does not present his logic by means of a formal language. After Russell ﬁnished preparing POM, he also began studying Frege with care (around June 1902). Under his inﬂuence, Russell began to notice the limitations in Peano’s treatment of symbolic logic, such as the lack of diﬀerent symbols for class union and the disjunction of propositions, or material implication and class inclusion. Moreover, he changed his symbolism for universal and existential quantiﬁcation to (x)f (x) and (Ex)f (x). He adopted from Frege the symbol for the assertion of a proposition. His letter to Frege of June 16, 1902, contained the famous paradox, which had devastating consequences for Frege’s system:

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Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. Therefore we must conclude that w is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a deﬁnable collection does not form a totality. (Russell 1902a, 125) The ﬁrst paradox does not involve the notion of class but only that of predicate. Let Imp(w) stand for “w cannot be predicated of itself,” that is, ∼w(w). Now we ask: Is Imp(Imp) true or ∼Imp(Imp)? From either one of the possibilities the opposite follows. However, what is known as Russell’s paradox is the second one oﬀered in the letter to Frege. In his work Grundgesetzte der Arithmetik (Frege 1893, 1903), Frege had developed a logicist project that aimed at reconstructing arithmetic and analysis out of general logical laws. One of the basic assumptions made by Frege (Basic Law V) implies that every propositional function has an extension, where extensions are a kind of object. In modern terms we could say that Frege’s Basic Law V implies that for any property F (x) there exists a set y = { x : F (x) }. Russell’s paradox consists in noticing that for the speciﬁc F (x) given by x ∈ / x, Frege’s principle leads to asserting the existence of the set y = {x : x ∈ / x }. Now if one asks whether y ∈ y or y ∈ / y from either one of the assumptions one derives the opposite conclusion. The consequences of Russell’s paradox for Frege’s logicism and Frege’s attempts to cope with it are well known, and we will not recount them here (see Garciadiego 1992). Frege’s proposed emendation to his Basic Law V, while consistent, turns out to be inconsistent as soon as one postulates that there are at least two objects (Quine 1955a).18 Extensive research on the development that led to Russell’s paradox has shown that Russell already obtained the essentials of his paradox in the ﬁrst half of 1901 (Garciadiego 1992; Moore 1994) while working on Cantor’s set theory. Indeed, Cantor himself already noticed that treating the cardinal numbers (resp., ordinal numbers) as a completed totality would lead to contradictions. This led him to distinguish, in letters to Dedekind, between “consistent multiplicities,” that is, classes that can be considered as completed totalities, from “inconsistent multiplicities,” that is, classes that cannot, on pain of contradiction, be considered as completed totalities. Unaware of Cantor’s distinction between consistent and inconsistent multiplicities Russell in 1901 convinced himself that Cantor had “been guilty of a very subtle fallacy” (1901a, 375). His reasoning was that the number of all things is the greatest of all cardinal numbers. However, Cantor proved that for every cardinal number there is a cardinal number strictly bigger than it. Within a few months this conundrum led to Russell’s paradox. In POM we ﬁnd, in addition to the two paradoxes we have discus