THE LIMITS OF LOGICAL EMPIRICISM
SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIE...

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THE LIMITS OF LOGICAL EMPIRICISM

SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE

Editor-in-Chief:

VINCENT F. HENDRICKS, Roskilde University, Roskilde, Denmark JOHN SYMONS, University of Texas at El Paso, U.S.A.

Honorary Editor: JAAKKO HINTIKKA, Boston University, U.S.A.

Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands THEO A.F. KUIPERS, University of Groningen, The Netherlands TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES, Stanford University, California, U.S.A. JAN WOLEN´SKI, Jagiellonian University, Kraków, Poland

VOLUME 334

THE LIMITS OF LOGICAL EMPIRICISM SELECTED PAPERS OF ARTHUR PAP with an Introduction by Sanford Shieh Edited by

ALFONS KEUPINK University of Groningen, The Netherlands

and

SANFORD SHIEH Wesleyan University, Middletown, U.S.A.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-4298-1 (HB) 978-1-4020-4298-0 (HB) 1-4020-4299-X (e-book) 978-1-4020-4299-7 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

Contents

ix xi

Preface Acknowledgments Part I Themes in Pap’s Philosophical Writings Introduction Sanford Shieh 1. Overview of Pap’s Philosophical Work 2. Necessity as Analyticity 3. Necessity as (Implicit) Linguistic Convention 4. The Analytic-Synthetic Distinction: Hypothetical or Functional Necessity 5. The Analytic-Synthetic Distinction: Dispositional and Open Concepts 6. The Limits of Hypothetical Necessity: Formal or Absolute Necessity 7. Logical Consequence and Material Entailment 8. The Method of Conceivability 9. Comparison with Necessity in Contemporary Analytic Metaphysics 10. Logicism 11. Concluding Remarks

3 3 10 12 15 16 23 24 27 30 33 42

Part II Analyticity, A Priority and Necessity 1 On the Meaning of Necessity (1943)

47

2 The Different Kinds of A Priori (1944)

57

3 Logic and the Synthetic A Priori (1949)

77

4 Are all Necessary Propositions Analytic? (1949)

91

v

vi 5 Necessary Propositions and Linguistic Rules (1955) 1. Are there Necessary Propositions? 2. The Confusion of Sentence and Proposition 3. Are Propositions “Logical Constructions”? 4. Necessary Truth and Semantic Systems 5. Implicit Definitions

Contents

109 109 117 122 132 137

Part III Semantic Analysis: Truth, Propositions, and Realism 6 Note on the “Semantic” and the “Absolute” Concepts of Truth (1952) Appendix: Rejoinder to Mrs. Robbins (1953)

147 154

7 Propositions, Sentences, and the Semantic Definition of Truth (1954)

155

8 Belief and Propositions (1957)

165

9 Semantic Examination of Realism (1947) 1. Universals in Re and the Resemblance Theory 2. Platonism and the Existence of Universals

181 181 187

Part IV Philosophy of Logic and Mathematics 10 Logic and the Concept of Entailment (1950)

197

11 Strict Implication, Entailment, and Modal Iteration (1955)

205

12 Mathematics, Abstract Entities, and Modern Semantics (1957) 1. Traditional Problem of Universals 2. Modern Semantics and the Traditional Dispute 3. Classes, Attributes, and the Logical Analysis of Mathematics What Do the Ontological Questions Mean? 4. 13 Extensionality, Attributes, and Classes (1958)

213 213 216 219 226 233

14 A Note on Logic and Existence (1947)

237

15 The Linguistic Hierarchy and the Vicious-Circle Principle (1954)

243

THE LIMITS OF LOGICAL EMPIRICISM Part V

vii

Philosophy of Mind

16 Other Minds and the Principle of Verifiability (1951) 1. The Principle of Verifiability as Generator of Philosophical Theories 2. The Behaviorist’s Confusion about the Notion of Verifiability 3. Are Statements about Other Minds Conclusively Verifiable? 4. Physicalism as an Analytic Thesis

249 254 259 264

17 Semantic Analysis and Psycho-Physical Dualism (1952)

269

249

Part VI Philosophy of Science 18 The Concept of Absolute Emergence (1951)

285

19 Reduction Sentences and Open Concepts (1953) Appendix

316

20 Extensional Logic and Laws of Nature (1955)

317

21 Disposition Concepts and Extensional Logic (1958)

327

22 Are Physical Magnitudes Operationally Definable? (1959) 1. Operational Definition as Contextual Definition of Classificatory Predicates 2. Operational Definition in the Form of Reduction Sentences 3. Physical Magnitudes and the Language of Observables 4. Theoretical Definition and Partial Interpretation The Breakdown of the Analytic-Synthetic Distinction 5. for Partially Interpreted Systems

295

351 351 352 355 358 360

Part VII Arthur Pap’s Life and Writings 23 Arthur Pap (1921-1959) : Intellectual Biography of Arthur Pap

365

Alfons Keupink 24 Arthur Pap: Biographical Notes Pauline Pap

369

viii

Contents

25 A Bibliography of Arthur Pap

375

Alfons Keupink 1. Main Publications 2. Editions 3. Translations 4. Articles, Papers and Reviews

375 375 375 376

References

381

Index

393

Preface

We would like to begin by telling a bit of the somewhat complicated story of how this volume came into being. Several years ago, one of us—Keupink—stumbled across some of Arthur Pap’s major publications1 in a secondhand bookstore in Groningen. As a Ph.D. student in philosophy of science with a special interest in the history of logical positivism, he was taken by the fecundity of Pap’s thought. Here was someone who, to him at least, seemed equally well-versed in all kinds of diﬀerent philosophical traditions (notably ordinary language philosophy and logical empiricism), yet always with something original to say. Keupink quickly made a decision to compile a list of Pap’s writings. He discovered, to his surprise, that Pap had written well over fifty papers during an extremely productive but all too short life. Gradually Keupink formed a plan to edit this material, in order to make it more accessible to a wide philosophical audience. He contacted Kluwer and in the Spring of 2003 submitted a manuscript entitled Arthur Pap: Collected Papers. It contained all of Pap’s papers and Keupink hoped that they could be published in two volumes. Kluwer asked Shieh to review Keupink’s manuscript. Nowadays Pap’s work is relatively unknown in Anglo-American analytic philosophy. Mostly he is read only by philosophers interested in the history of the analytic tradition. Indeed, it is because one of Shieh’s main research interests is in the development of modal logic and the concept of necessity in analytic philosophy that he knew parts of Pap’s magnum opus, Semantics and Necessary Truth, before Kluwer contacted him. As he read through the papers in the proposed collection, he became more and more impressed by the historical and philosophical significance of Pap’s work. The principal criteria that the two of us share for historical-philosophical assessment are: 1 Did the work play an important role in the development of the tradition to which it belongs? 1 Pap

1946b, Pap 1949b, Pap 1955a, Pap 1958c, and Pap 1962.

ix

x

Preface

2 Did it anticipate prominent subsequent developments? 3 Did it provide distinctive solutions or perspectives on problems of contemporary concern, perhaps by pointing out unnoticed problems in contemporary work, perhaps by proposing arguments that are more cogent than contemporary ones, perhaps by allowing us to see the significance of the problems diﬀerently? Shieh’s suggestion was that the best way of making Pap’s work more visible and easily available to the philosophical community is to make a smaller selection of Pap’s papers that best satisfy these criteria, and provide an introduction that clarifies their significance. Shieh was very happy to be asked by Keupink (via a letter written by Prof. Theo Kuipers and Dr. Jeanne Peijnenburg of the University of Groningen), to make the small selection and write the introduction. We hope that the present edition will have the eﬀect of furthering interest in Arthur Pap’s thought and contributing towards a reevaluation of his highly original and stimulating contribution to the development of 20th -century (analytic) philosophy (of science). A K Groningen, the Netherlands, 2005 S S New York, USA, 2005

Acknowledgments

We would like to thank Arthur Pap’s wife, Mrs. Pauline Pap, for her confidence and support; Keupink would like also to thank her for her help with his Intellectual Biography of Pap. We are also very grateful for the help and advice of Prof. Dr. Theo Kuipers, and Dr. Jeanne Peijnenburg. Our other colleagues in Groningen made various constructive remarks on earlier versions of the Intellectual Biography of Pap and the Introduction. In addition, Dr. Peijnenburg carried out the unenviable task of tracking down the copyright holders of the essays herein reprinted, and securing permission for this reprinting; without her help this volume would not have been possible. Acknowledgments are also due to the Netherlands Organization for Scientific Research (NWO), which partly sponsored the research for this project. We would like to thank Andrew Catalano of Wesleyan University and Julia Perkins of History and Theory and Wesleyan University for their invaluable editorial assistance.

Origin of the Essays All permissions granted for the previously published essays by their respective copyright holders are most gratefully acknowledged. There are instances where we have been unable to trace or contact the copyright holder. If notified the publisher will be pleased to rectify any errors or omissions at the earliest opportunity. 1 “On the Meaning of Necessity,” in The Journal of Philosophy 40 (1943), pp. 449-58. Reprinted with permission from The Journal of Philosophy, Columbia University, New York. 2 “The Diﬀerent Kinds of A Priori,” in Philosophical Review 53 (1944), pp. 465-84. Copyright 1944 Cornell University. Reprinted by permission of the publisher. 3 “Logic and the Synthetic A Priori,” in Philosophy and Phenomenological Research 10 (1949), pp. 500-14. Reprinted with permission from

xi

xii

Acknowledgements

the editors of Philosophy and Phenomenological Research, Brown University. 4 “Are all Necessary Propositions Analytic?,” in Philosophical Review 58 (1949), pp. 299-320. Copyright 1949 Cornell University. Reprinted by permission of the publisher. 5 “Necessary Propositions and Linguistic Rules,” in Semantica (Archivio di Filosofia), Roma: Fratelli Bocca (1955), pp. 63-105. Reprinted with permission from the Lodetti family of Libreria Bocca. 6 “The‘Semantic” and the ‘Absolute” Concepts of Truth,” in Philosophical Studies 3, 1952, pp. 1-8. Appendix: Rejoinder to Mrs. Robbins, in Philosophical Studies 4 (1953), pp. 63-4. Reprinted with permission from Springer-Kluwer, Dordrecht. 7 “Propositions, Sentences, and the Semantic Definition of Truth,” in Theoria 20 (1954), pp. 23-35. Reprinted with permission from Theoria, Royal Institute of Technology, Stockholm and Mrs. P. Pap. 8 “Belief and Propositions,” in Philosophy of Science 24 (1957), pp. 12336. Reprinted with permission from The University of Chicago Press, Permissions Department. 9 “Semantic Examination of Realism,” in The Journal of Philosophy 44 (1947), pp. 561-75. Reprinted with permission from The Journal of Philosophy, Columbia University, New York. 10 “Logic and the Concept of Entailment,” in The Journal of Philosophy 47 (1950), pp. 378-87. Reprinted with permission from The Journal of Philosophy, Columbia University, New York. 11 “Strict Implication, Entailment, and Modal Iteration,” in Philosophical Review 64 (1955), pp. 604-13. Copyright 1955 Cornell University. Reprinted by permission of the publisher. 12 “Mathematics, Abstract Entities, and Modern Semantics,” in Scientific Monthly 85, (29 July 1957), pp. 29-40. Reprinted with permission from the American Association for the Advancement of Science, Washington. 13 “Extensionality, Attributes, and Classes,” in Philosophical Studies 9 (1958), pp. 42-6. Reprinted with permission from Springer-Kluwer, Dordrecht. 14 “A Note on Logic and Existence,” in Mind 56 (1947), pp. 72-6. Reprinted with permission from Oxford University Press.

Acknowledgements

xiii

15 “The Linguistic Hierarchy and the Vicious-Circle Principle, in Philosophical Studies 5 (1954), pp. 49-53. Reprinted with permission from Springer-Kluwer, Dordrecht. 16 “Other Minds and the Principle of Verifiability,” in Revue Internationale de Philosophie 5 (1951), pp. 280-306. Reprinted with permission from Revue Internationale de Philosophie, Universit´e Libre de Bruxelles. 17 “Semantic Analysis and Psycho-Physical Dualism,” in Mind 61 (1952), pp. 209-21. Reprinted with permission from Oxford University Press. 18 “The Concept of Absolute Emergence,” in British Journal for the Philosophy of Science 2 (1951), pp. 302-11. Reprinted with permission from Oxford University Press. 19 “Reduction Sentences and Open Concepts,” in Methodos 5, 1953, pp. 3-30. The editors were unable to trace or contact the copyright holder. 20 “Extensional Logic and Laws of Nature,” in Actes du deuxi`eme congr`es international de l’Union International de Philosophie des Sciences, Z¨urich 1954. Neuchˆatel: Editions du Griﬀon, Paris: Dunod (1955), pp. 116-27. Reprinted with permission from Editions du Griﬀon. 21 “Disposition Concepts and Extensional Logic,” in Concepts, Theories and the Mind-Body Problem: Minnesota Studies in the Philosophy of Science, Volume II, University of Minnesota Press (1958), pp. 196-224. Reprinted with permission from The University of Minnesota Press. 22 “Are Physical Magnitudes Operationally Definable?,” in C. West Churchman & P. Ratoosh (eds), Measurement: Definitions and Theories. New York: John Wiley, London: Chapmann & Hall (1959), pp. 177-91. (Publications of contributions to the symposium of the American Association for the Advancement of Science, 1956.) The editors were unable to trace or contact the copyright holder.

A Note on this Edition Minimal editorial changes have been made to the originally published versions of the essays reprinted in this volume. Typographical mistakes are corrected; references, spelling, and punctuation are made uniform throughout the volume. Details of original publication of the essays are given both in chapter 25, a bibliography of Pap’s publications, and in the relevant entries for Pap in the References, starting on page 387 below. All citations of Pap’s works in this volume are exclusively by the entries in the References.

I

THEMES IN PAP’S PHILOSOPHICAL WRITINGS

Introduction Sanford Shieh

1.

Overview of Pap’s Philosophical Work

There are, of course, many styles of philosophizing; but it is sometimes illuminating to think of philosophers as divided into two broad types: the Socratic and the Platonic. The former are the critics, the ones who never cease questioning the grounds of accepted opinions and turning over the details of arguments; the latter are the systematizers, the ones who have a sweeping vision that they take more seriously than the details. On this division Arthur Pap is a Socratic philosopher. As he himself says in describing Semantics and Necessary Truth (Pap 1958c, cited in this Introduction as SNT), “It will hardly escape the reader’s notice that very few definitive conclusions are reached in this book, that perhaps more problems have been formulated than have been solved” (SNT, Preface, xiv). Although Pap was very much an analytic philosopher, a good part of his most philosophically rewarding work are critiques of two prominent strands of analytic philosophy in the 1940s and 1950s. One of these is logical positivism or logical empiricism, the movement stemming from the Vienna Circle of Moritz Schlick and Rudolf Carnap and the Berlin school of Hans Reichenbach, which dominated analytic philosophy, especially in the USA, from the end of World War II through most of the 1960s. The other is ordinary language philosophy, deriving from the work of Gilbert Ryle and the followers of Ludwig Wittgenstein,1 which played a significant and in certain ways oppositional role, in post-war British analytic philosophy until the end of the 1960s. There were many diﬀerences and disputes between positivism and ordinary language philosophy, yet in certain respects their central, as it

1 Interestingly,

J. L. Austin’s work never made much of an impression on Pap. Of course, Austin’s work also failed to make much of an impression on the rest of analytic philosophy until after the period in which Pap was active, when Austin’s ideas about speech acts were taken up in philosophy of language.

3

4

Overview of Pap’s Philosophical Work

were popular, ideas converged. In order to understand Pap’s work and appreciate its relevance to contemporary philosophy, it will be useful to begin with a brief sketch of some major preoccupations of the analytic philosophy of this period that included this convergence. The doctrines I will outline characterize logical empiricism and ordinary language philosophy as philosophical movements, and so should not be confused with the much more nuanced views of, e.g., Carnap and Wittgenstein.

1.1

Intellectual Background: Logical Empiricism and Ordinary Language Philosophy

One of the fundamental motivations of positivism is a perceived contrast between traditional philosophy and the natural sciences. Whereas the sciences seem to display steady if not uninterrupted progress and continued general agreement on results, philosophy seems mired in endless irresolvable disputes. Positivism thus started from the idea that the paradigm of genuine knowledge is empirical scientific knowledge, and from a distrust of metaphysical concepts and theories outside the natural sciences. Positivism pursued two closely related projects. One was an attempt to arrive at criteria for distinguishing genuine procedures for assessing claims to knowledge from spurious ones. Since natural science was the model, these criteria all turned on the existence of impersonal standards, formulated in terms of objective experience, for acceptance and rejection of statements. The other was an attempt to oﬀer a diagnosis of where traditional metaphysics had gone wrong. Perhaps the most well-known thesis of positivism, the verificationist theory of meaning, subserved both of these projects. The principle of verification is a formulation of the shared and objective standards governing the acquisition of knowledge in the sciences: a statement is accepted only on the basis of sensory experiences that establish it as true. This principle goes naturally with an account of meaning: the cognitive meanings of statements are given by the sensory experiences that would verify or falsify them; a statement for which there is no method of verification has thus no objective meaning at all. The language of science was taken to abide by this principle, while that of traditional metaphysics did not. Thus science deals only with cognitively significant statements, while metaphysics traﬃcked in (cognitive) nonsense. The principle of verification came to grief in all sorts of ways,2 of course, but one important problem spurred the formulation of another central tenet of positivism. (Yet another crucial problem will be discussed in section 5 below.) This is the problem of the a priori, as A. J. Ayer called it (Ayer 1952b). Logic and mathematics are indispensable to the practice of modern science; yet nei-

2 See

Hempel 1951 for a detailed account of the problems faced by this principle.

Introduction

5

ther the laws of logic nor the axioms and theorems of mathematics seem to be justified on the basis of experience. The solution adopted by positivism was to hold that logic and mathematics are not based on experience for their truth because they rest, at bottom, on the meanings of our words, the rules governing the use of our language. These meanings or rules are conventional, at least in the sense that their adoption is not constrained to reflect empirical reality, and this explains why it is possible for us to know the truth of whatever is based solely on them without reference to experience. Necessary truth, then, is just truth based on features of language. It should be stressed that, for the most reflective positivists, such as Carnap, meaning rules cannot be constrained by empirical reality, because it is only when the rules are in place that we have a conception of what it is for our acceptance of statements to be responsible to empirical evidence. Thus, not only is necessity of logic and mathematics consistent with the empirical status of science, but they have to have this modal status in order for there to be such a thing as empirical science at all. The solution also represents the principal point of convergence between positivism and ordinary language philosophy. The latter diﬀered from positivism in rejecting science as the paradigm of knowledge and of philosophical method. But, ordinary language philosophers, especially the followers of Wittgenstein, agreed with positivism in taking traditional philosophy to be deficient in (cognitive) meaning. In addition, they went beyond the positivists’ characterization of traditional metaphysics as nonsense. The grand metaphysical theories of traditional philosophy from Plato to Hegel all claimed to provide insight into necessary structures of reality. Armed with its version of the linguistic theory of necessity, ordinary language philosophers took themselves to have exposed these pretensions as mistaking merely conventional features of our language, the rules (of philosophical grammar) that regulate what it is to make sense, for the ultimate truth underlying reality. The linguistic theory of necessity goes naturally with a sharp distinction between the analytic and the synthetic. The meanings of our words, the rules governing their sensical use, are supposed to be unconstrained by empirical reality. As Pap puts it, if these rules are formulated in the guise of statements, “they express stipulations concerning the use of symbols and thus cannot significantly be said to be refuted by facts”3 In contrast, the sentences governed by these rules “are empirically refutable statements ‘about reality’ ” (19, 308). Ordinary language philosophers tend to be suspicious of the notion of analytic truth, since, in their view, a rule of language is strictly speaking neither true nor false, but defines what it is for something to be true or false. But they nevertheless maintain an analytic-synthetic distinction, primarily in the form 3 Chapter

19 below, page 308. In this Introduction, unless otherwise noted, references to Pap’s papers in the present volume will be given in the text, by chapter number and page number.

6

Overview of Pap’s Philosophical Work

of Wittgenstein’s distinction between criteria and symptoms. In the end, their disagreement with the positivists over whether there are analytic truths is a merely verbal one. Both accept a distinction between that which sets the rules for empirical justification and that which, given the rules, are susceptible of being empirically justified. When the shared doctrine is formulated in this way, we can see that what underlies these two major strands of postwar analytic philosophy is a deeplyrooted conception of rationality. The fundamental idea is that participation in rational inquiry presupposes acknowledgment of shared and objective standards for what investigatory results counts in favor of accepting or rejecting what statements. Without this prior agreement on standards, there is no way to separate genuine agreements and disagreements from merely verbal ones; it is unclear whether participants in inquiry even understand one another. Thus a distinction between the rules for conducting empirical inquiry and the results of such inquiry appears inescapable if one is to be rational; this is why the analytic-synthetic distinction appears inescapable.4

1.2

Two Main Themes of Pap’s Work

Although Pap is a Socratic philosopher, it would be false to say that his work consists only of piecemeal and incisive criticisms with no overarching philosophical unities. Two broad themes stand out in his papers, call them anti-reductionism and intuitionism. The first theme is connected to Pap’s critical relation to the analytic philosophy of his time. Pap’s critiques attempt to show, over and over again, the failure of positivism’s and ordinary language philosophy’s attempts to reduce or explain away the categories of traditional metaphysics. Below I will discuss a number of philosophically significant instances of this type of criticism. The second theme emerges from the moral that Pap repeatedly finds in his critiques of reductionism. For example, from his critique of the linguistic theory of necessity he concludes that “the concept of necessity involved in such statements as ‘the conclusion of a valid syllogism follows necessarily from the premises’ and ‘the relation of temporal succession is necessarily . . . asymmetrical’ is not analyzable at all” (SNT, Preface p. xv; emphases mine). Rather, we have to accept that we have some form of irreducible intuition of these concepts and their applications. Another example of Pap’s intuitionism is his argu-

4 In

his “Intellectual Autobiography” Carnap explicitly associates the lack of progress in traditional philosophy with the lack of shared standards therein: “most of the controversies in traditional metaphysics appeared to me sterile and useless.. . . I was depressed by disputations in which the opponents talked at cross purposes; there seemed hardly any chance of mutual understanding, let alone of agreement, because there was not even a common criterion for deciding the controversy” (Carnap 1963, 44-5, emphases mine). My view of the deeper motivations underlying the analytic-synthetic distinction is indebted to Ricketts 1982.

Introduction

7

ment that the relation of logical consequence or entailment cannot be reduced either to syntactic logical form or to model theoretic semantics, so that we have to accept that we have irreducible intuition of material (i.e., non-formal) deductive validity. As we will see, Pap takes this to support the ineliminability of synthetic a priori truths. (As this reference to Kant’s views suggests, the name “intuitionism” refers, not to the mathematical principles of Brouwer, but to the philosophical ideas which he claims to be the basis of his revision of mathematics.) This last point brings me to a general observation. These two broad themes in Pap’s work can be traced to his early training. Pap was a student of Cassirer at Yale, and although his main philosophical works do not fall within the tradition of Cassirer’s neo-Kantianism, they nevertheless manifest an aﬃnity with the kind of pragmatically oriented Kantianism developed by Cassirer. Pap’s most important works applied the best ideas of neo-Kantianism to the logical empiricist tradition, resulting in an internal critique of the latter that revealed its limits.

1.3

The Contemporary Significance of Pap’s Work

In order to indicate the contemporary significance of this work, let me begin by noting that, even to this day, positivism is often taken by many scholars, outside and inside philosophy, to be the very core of analytic philosophy. And, although the identification of analytic philosophy and positivism is a mistake, it is a revealing mistake that contains a substantial grain of truth. Many of the themes of positivism are still with us, in all sorts of surprising ways. For example, perhaps the most prominent contemporary interpreters of Wittgenstein are the British scholars Peter Hacker and Gordon Baker. (See their four volume commentary on the Philosophical Investigations, Baker and Hacker 1980, Baker and Hacker 1988, Hacker 1990, Hacker 2000, as well as Hacker 1986 and Hacker 1996.) Although they strenuously deny that Wittgenstein was a positivist, the views, especially about necessity, that they attribute to Wittgenstein are in the end essentially no diﬀerent from the views of the positivists that Pap criticized. And, to my mind at any rate, Pap’s criticisms apply equally forcefully to the views they attribute to Wittgenstein.5 For another example, the positivists’ distrust of claims outside the natural sciences survive in contemporary attempts to “naturalize” all sorts of things: the mind, ethics, mathematics, truth. Here again, Pap’s criticisms raise serious doubts that fund a plausible skepticism about these naturalization programs. In addition, as we will see in more detail below, Pap’s arguments anticipate a number of contemporary philosophical positions. Pap’s version of Russellian

5I

should say that I take this to cast doubt on Hacker’s and Baker’s interpretations of Wittgenstein.

8

Overview of Pap’s Philosophical Work

logicism, I will argue in section 10, may be understood as a version of the neo-Fregean philosophy of mathematics championed by Crispin Wright and Bob Hale. (See, inter alia, the articles collected in Hale and Wright 2001.) Pap’s critique of the semantic account of logical consequence and his argument for the indispensability of material entailment arrive at positions held by John Etchemendy (Etchemendy 1988a, Etchemendy 1988b, and Etchemendy 1990) and Robert Brandom (Brandom 1994), but by rather diﬀerent routes. Finally, as I will argue in section 9, Pap’s positive views of necessity constitute a defensible alternative to the conceptions of necessity prevalent in contemporary analytic metaphysics.

1.4

Pap in the Analytic Tradition

Let me now say a word about the place of Pap’s work in the history of analytic philosophy. In contemporary Anglo-American analytic philosophy (perhaps especially in America), it is generally thought that two fundamental changes occurred in this philosophical tradition after World War II. First, in the 1950s, W. V. Quine’s critique of the analytic-synthetic distinction led to the waning of logical positivism. Second, in the mid to late 1960s, Saul Kripke made modality philosophically respectable. Pap’s work complicates this picture considerably. As we will see in section 4, Pap had rejected the analytic-synthetic distinction nearly a decade before Quine’s “Two Dogmas of Empiricism” (Quine 1951), for reasons very similar to Quine’s. In addition, Pap’s Semantics and Necessary Truth, published in 1958, was something of a standard reference on necessity in analytic philosophy. This shows how far the analytic tradition had already gone away from the positivist/empiricist reduction or denigration of necessity, some 10-15 years before 1972, when Kripke gave the lectures later published as Naming and Necessity (Kripke 1980). Of course, Pap’s work is not the only significant lacuna in the standard picture of modality in analytic philosophy. Another is Carnap’s Meaning and Necessity, first published in 1947 (Carnap 1947, second edition Carnap 1956a). (It should be noted that Carnap’s interest in modal logic was in many ways a reductionist one, whereas Pap’s views of necessity are just the opposite.) Yet another is the work on the logic, semantics, metaphysics, and epistemology of modality that Ruth Barcan Marcus, Pap’s fellow student at Yale, published from the mid-1940s to the late 1960s (Marcus 1946b; Marcus 1946a; Marcus 1947; and Marcus 1993, chapter 1, esp. appendix 1A, 3, and 4), which advanced many of the theses for which Kripke argued in Naming and Necessity. Finally, there is the work of Jaakko Hintikka, also published in the same period as Marcus’s papers.

Introduction

9

Indeed, the historical picture appears even more perplexing if we ask: what is the basis of Kripke’s view of necessity? Kripke explicitly says that it is intuition. For example, When you ask whether it is necessary or contingent that Nixon won the election, you are asking the intuitive question whether in some counterfactual situation, this man would in fact have lost the election.. . . [S]ome philosophers think that something’s having intuitive content is very inconclusive evidence in favor of it. I think it is very heavy evidence in favor of anything, myself. I really don’t know . . . what more conclusive evidence one can have about anything. (Kripke 1980, 41-2; second and fourth emphases mine)

Indeed, Kripke’s fundamental argument for the coherence of modal notions proceeds by appealing to our intuitive judgment that statements such as “Bush might have won the popular vote in the 2000 US presidential election” and “Weapons of mass destruction might have existed in Iraq in 2003” make sense, even if we’re not sure how we would establish their truth or falsity.6 How, then, is this diﬀerent from Pap’s intuitionism and anti-reductionism? Clearly more work needs to be done if we are to have an accurate history of the analytic tradition.

1.5

Plan of the Introduction

In the remainder of this Introduction, I will discuss the most philosophically interesting and historically significant aspects of Pap’s work represented in the papers of this volume. They are the following: Necessity: sections 2-7 The Analytic-Synthetic Distinction: sections 4 and 5 Dispositional and Open Concepts in Science: section 5 Logical Consequence and Material Entailment: section 7 Logicism: section 10 The Semantic Concept of Truth: section 8.1 The Problem of Other Minds: section 8.2 This is by no means an exhaustive list of the topics of philosophical interest treated by Pap in the papers of this volume. Some other topics which I do not have the space to discuss in this Introduction are listed below in the concluding section 11, at page 42. 6 For

a more detailed account of the intuitive bases of Kripke’s arguments, see Shieh 2001.

Necessity as Analyticity

10

2.

Necessity as Analyticity

In this section I will begin a discussion of Pap’s views of necessity. I will focus on Pap’s critique of one version of the positivists’ linguistic theory of necessity, the thesis that necessity is reducible to analyticity. An extremely familiar formulation of the linguistic theory holds that, e.g., All sisters are female

(1)

is necessarily true because it is analytic, i.e., true by definition. Specifically, since ‘sister’ is defined as, and so synonymous with, ‘female sibling’, and since the mutual substitution of synonymous expressions in a sentence surely doesn’t change its meaning, (1) is synonymous with All female siblings are female

(2)

This sentence is logically true; it is a substitution instance of a valid schema of first-order logic: (∀x)(F x.S x ⊃ S x)

(3)

Thus the obscure notion of a statement’s being more than simply true, but necessarily true, is clarified by reducing it to two notions with no metaphysical baggage: that of meaning, and that of logical truth. The first of Pap’s criticisms of this theory that I will discuss is given in chapters 3 and 4 below. Pap claims that this account of necessity simply fails to cover all truths which we intuitively recognize as necessary. For Pap the basic notion of necessity is independence from empirical facts: “a necessary condition which any adequate analysis of ‘necessary’ must satisfy is that the truth-value of a necessary proposition does not depend on any empirical facts” (4, 92). Pap asks us to consider the following statement: If A precedes B, then B does not precede A

(4)

He argues, I assume that few would regard this statement as factual, i.e., such that it might be conceivably disconfirmed by observations.. . . [But t]o see that [it] is not analytic, in the sense defined, we only need to formalize it, and we obtain “(∀x)(∀y)[xRy ⊃ ∼ yRx],” which is certainly no principle of logic. This statement, then, is not deducible from logic; hence if we want to call it necessary (nonfactual), we have to admit that there are necessary propositions which are not analytic. (4, 96)

It will not do, of course, to respond to this argument by claiming that in such cases we simply don’t know the definitions of the key terms, such as ‘precede’. Such a response presupposes that any necessary truth must be analytic,

Introduction

11

but this presupposition is precisely the linguistic theory of necessity that is being challenged by a putative counter-example. The burden of proof is thus on the defender of the linguistic theory. Note, moreover, a point Pap does not make: if indeed we know that truths such as (4) are necessary without knowing the definition of ‘precede’, then it is plausible that analyticity is not the epistemic basis of necessity; hence the linguistic theory should, at least, oﬀer some explanation of why it is that while ontologically, as it were, necessity is based on analyticity, our epistemic access to necessity does not go through analyticity. Pap’s first argument is clearly not conclusive; it amounts in the end to a challenge to the linguistic theory to meet the burden of proof by coming up with the required definitions. Pap goes further and argues that attempts to meet this burden of proof must overcome a general diﬃculty. One could, of course, give any definition one pleases to any expression. But, if there are no constraints on what definitions may be given for demonstrating analyticity, then it is not clear that there are any true statements that cannot be shown to be analytic and so necessary according to the linguistic theory. It follows that, in meeting the burden of proof, one must abide by criteria of adequacy for definitions. The question then becomes, what are these criteria? What makes a definition a good one? What, in the case of (4), would be an adequate definition of ‘precede’? According to Pap, “most philosophers would agree that no definition of ‘xPy’ (to be used as an abbreviation for ‘x precedes y’) could be adequate unless it entailed the asymmetry of P” (4, 97). There are clearly other commonly agreed constraints: P should, e.g., also be irreflexive and transitive, and it can hold only of events and of whatever other entities there may be with temporal location, etc. Suppose that these et ceteras are suﬃcient to guarantee that a unique relation satisfies them. Let’s write “Q(R)” for the claim that a relation R satisfies these further constraints. Then, one might oﬀer this definition of P: xPy ≡d f (∃R)[(∀x )(∀y )(x Ry ⊃ ∼ y Rx ) . Q(R) . xRy]

(5)

From this definition it is obvious that in second-order logic we can derive (∀x)(∀y)(xPy ⊃ ∼ yPx). But, Pap writes, is it not obvious that acceptance of the definition from which the asymmetry of temporal succession has thus been deduced presupposes acceptance of the very proposition ‘Temporal succession is asymmetrical’ as self-evident? This way of proving that the debated proposition is, in spite of superficial appearances, analytic, is therefore grossly circular. (4, 98)

More generally, the problem for finding definitions to demonstrate analyticity is that if a definition is selected only because it allows for the derivation, from logic alone, of the necessary to be established as analytic, then the analysis of necessity in terms of analyticity is circular. Hence, in order for the

12

Necessity as (Implicit) Linguistic Convention

linguistic theory to work, the criteria of adequacy for definitions must not be based on the necessity of truths expressed using the defined terms. Pap is skeptical of the prospects of avoiding this problem. This is not, it seems to me, because he has an argument which establishes the impossibility of finding adequate definitions independently of recognizing necessary truths. Rather it is because, over and over again, he finds proposed definitions and analyses, especially those of the positivists, turning out, upon examination, to rest on prior recognition of necessary truths. The reader will notice this pattern of argument throughout Pap’s work. When faced with a reductive explanation of X in terms of Y, Pap examines the basis on which we identify Ys, and finds that we do so by recognizing Xs. Pap’s conclusion is then that we have a primitive and irreducible capacity for intuitive knowledge of Xs. It should be clear that a crucial assumption underlying the foregoing considerations is that there are clear instances of truths which we unproblematically recognize as non-factual. One may well ask here: why should an empiricist accept this assumption? Indeed, one way of seeing how Pap diﬀers from Quine is to see that for the latter there are no clear cases of non-factual truths. This thesis is, of course, closely connected with Quine’s rejection of the analyticsynthetic distinction. But the interesting point here is that Pap, as we will see in section 4, also rejects the analytic-synthetic distinction. We will be in a position to understand why Pap’s empiricism goes with the recognition of non-factual truths only by section 6.

3.

Necessity as (Implicit) Linguistic Convention

In this section I turn to Pap’s attack on another version of the linguistic doctrine of necessary truth, one which persists in contemporary philosophy in Hacker and Baker’s version of Wittgenstein. I focus on Pap’s paper “Necessary Propositions and Linguistic Rules” (chapter 5 below, cited in this Introduction as NPLR), which summarizes the major themes of SNT. The problem that Pap raises here follows on the one that we discussed in section 2 above, and applies even if it is possible to overcome the problem of successfully attaining definitions or analyses for certifying the analyticity of intuitively necessary truths. As we saw on page 10 above, part of the linguistic theory’s explanation of the necessity of (1) is the fact that (2) is a logical truth, an instance of the valid schema (3). But, then, even if the notion of analyticity is unproblematic, the linguistic theory doesn’t yet suﬃce as an explanation of necessity, since so far nothing has been said about why a logical truth is necessary or what makes instances of valid schemata necessarily true. So the linguistic theory requires supplementation. Moreover, this supplement had better demonstrate that the

Introduction

13

necessity in question is a matter of linguistic conventions; otherwise necessity will depend on factors other than language. The account of logic that Pap attacks holds that the “primitive” truths of logic are necessary because they implicitly define the meanings of the logical constants. For example, the schemata corresponding to the elimination rules for ‘and’—p . q ⊃ p and p . q ⊃ q—partly define the meanings of ‘and’ and ‘if . . . then’. But now the question is, what is implicit definition? Since these implicit definitions are supposed to be conventional, are they not proposals to use the words ‘and’ and ‘if . . . then’ in certain ways? But are proposals even candidates to be true or false? Do we not think that we can propose any way of using words we like? Surely in making proposals we are not constrained to be faithful to anything. Now, of course this argument by Pap is hardly conclusive. A supporter of the linguistic theory of necessity could reply that Pap has missed the point, because the proposal is precisely that we accept certain statements as true. Moreover, it is precisely because this proposal is unconstrained by empirical evidence that the truth in question is a priori and so necessary. What makes the statement true is just that we have agreed to take it to be true, and nothing we discover about the world need force us to give up this agreement. For the time being, let’s leave this reply and go on to a second version of the linguistic theory. This version holds that the linguistic conventions underlying necessary truths describe, or are based on, the ways in which speakers of a natural language use the words involved in expressing those truths. But then, Pap argues, these conventions are empirical truths about, say, speakers of English. It might be replied that these conventions are not facts about English speakers but rules that constitute what it is to speak English, so failure to follow them results in speaking a language other than English. But, Pap argues, this can’t be right, since natural languages undergo changes. I have heard it said that in Elizabethan England, ‘meat’ was more or less synonymous with ‘groceries’. But if natural languages change, then, intuitively, people in Elizabethan England still spoke English, even though they don’t use the word ‘meat’ in the same way that it is used in contemporary English. It follows then that the way in which words are used at any given time can’t constitute or define what it is to speak a language, and so how a term of the language is used at some given time is an empirical question. The problem that Pap raises for this version of the linguistic theory is that if linguistic conventions are empirical facts about usage then it becomes very unclear how they “can be a reason for the necessity of any statement” (5, 114) As Pap puts the point: “Nobody would say ‘any father must be a parent because there are nine planets’, although undoubtedly ‘there are nine planets’ entails ‘any father is a parent’. Would it be any less absurd to say ‘any father must

14

Necessity as (Implicit) Linguistic Convention

be a parent because, at least according to present linguistic conventions, it is incorrect to apply “father” to an object to which “parent” is inapplicable’?” (5, 114).7 This rhetorical question is unlikely to move adherents of the linguistic theory. Consider again the analytic truth ‘All sisters are female’. Given our present linguistic usage, no experience is relevant to its truth or falsity because the rule is a sister’ to anything in experience is for correctly applying the predicate ‘ is a sibling’ and ‘ is female’ that it is also correct to apply the predicates ‘ to it. There could be no empirical counterexamples to this statement because to recognize anything correctly as a sister requires recognizing it correctly as female. So, to revert to Pap’s example, if it is incorrect to apply ‘father’ to an object to which ‘parent’ is inapplicable, how could anything that is a father fail to be a parent? The only way seems to be that being a father is diﬀerent from being correctly described as a father. But how could that be? The answer to these replies, and the heart of Pap’s argument, is contained in the following passage: Consider [the] necessary proposition: there are no fathers that are not male. If we are a priori certain of it, i.e. in advance of having observed all fathers, past, present, and future, it is surely because we see that fatherhood entails malehood. This is to say that we see a priori the truth of the modal proposition ‘father(x) male(x)’ . . . and hence derive the corollary ‘(∀x)(father(x) ⊃ male(x))’. . . . The point is that since it is our knowledge of the entailment which is the ground of our certainty with regard to the universal proposition, the universal proposition would itself have only the status of an inductive generalization if the entailment were not known a priori. The argument ‘there are at no time fathers who are not male, because fatherhood entails malehood’ has the form ‘p, because N(p)’: for ‘p q’ is definable as ‘N(p ⊃ q).’ Therefore our acceptance of p would be based on empirical evidence if our acceptance of N(p) were based on empirical evidence. Therefore one cannot consistently hold that p is necessary and N(p) contingent. (5, 116)

The crux of this argument is that there is a tension between characterizing a truth as a priori and what the linguistic theory, according to Pap, must take to be its ultimate grounds. Whether the theory takes necessity to be based on proposals or on facts about norms of usages, it implies that one cannot be 7 It’s

worth pointing out that sometimes the terms in which Pap formulates this argument are misleading. For example, he characterizes a linguistic convention by stating that “people who understand English never apply” certain predicates in certain ways. One might take this to be a description of actual language use, and then object to it on the ground that the conventions or rules that make up a language are normative— they state how words ought to be used, not how they invariably are used. But acknowledging the normative character of rules of usage (“grammar” in Baker’s and Hacker’s Wittgensteinian terminology) doesn’t invalidate Pap’s argument. For the standards of correct use in force for a single language can change over time. Moreover, surely it still makes sense to say that what norms are in force in a language at a given time is an empirical fact, or, at the very least, it is a contingent feature of that language which we cannot ascertain merely by reflecting on, e.g., the concept of the English language.

Introduction

15

justified in asserting a statement that is supposed to rest on a priori grounds, unless one is sure that certain facts obtain about the language one is speaking— facts about what proposals for usage have been accepted, what conventions are in force, what norms of usage govern the language. But then whatever knowledge one might express by the statement is not arrived at by reflection on meanings alone. How, then is it a priori? And if it is not a priori, has the linguistic theory explained how it is necessary? This line of argument is the fundamental basis underlying Pap’s objection to the linguistic theory of necessity. None of this criticism, of course, answers the question what makes the laws of logic necessary, or indeed whether they are necessary. Pap does maintain that they are necessary; I discuss his reasons in sections 6 and 7.

4.

The Analytic-Synthetic Distinction: Hypothetical or Functional Necessity

The rejection of the analytic-synthetic distinction is a constant theme in Pap’s philosophical work. It appears as early as 1943 and 1944, in chapters 1 and 2 respectively, and as late as 1963 in the posthumously published “Reduction Sentences and Disposition Concepts” (Pap 1963b). The dates of the early papers points to their significance for the history of analytic philosophy: they are eight years earlier than Quine’s “Two Dogmas of Empiricism” (Quine 1951) and seven years earlier than Morton White’s less well-known “The Analytic and the Synthetic: An Untenable Dualism” (White 1950). In the early papers, which derive from the first part of his dissertation, Pap develops a theory of a type of necessity and apriority which he calls “hypothetical” or “functional” necessity. (I will use these two terms interchangeably.) This is a pragmatic notion of necessity—necessity of means for accomplishing certain purposes—traced ultimately back to Aristotle. The fundamental idea is that, in the course of empirical inquiry, certain statements are taken to be, or treated as, necessary, for the purpose of “systematizing facts, of rendering the body of factual knowledge coherent” (1, 49). Underlying this idea is an abstract picture of scientific inquiry: If a conjunction of traits a-b is found to be repeated without exception, we generalize it into a “universal,” a definitional connection: if A, then B. If, then, experience should one day disclose a contradictory instance, viz., ‘a and not-b,’ we will have the choice between refusing to identify a as an instance of A . . . and considering our law (“if A, then B”) as refuted . . . . (1, 54)

Thus in science we are “free . . . to make empirical truths . . . necessary and thus to deprive them of their intrinsic contingency” (1, 54). But if an empirical law that we have made into a “prescriptive definition” or “rule” comes into conflict with experience, we are equally free to decide it is no longer “good for conducting inquiry,” and start looking for a rule of inquiry “better than it.” In

16

The Analytic-Synthetic Distinction: Dispositional and Open Concepts

Pap’s picturesque language, “experience is free to unmake our makings again” (1, 55). Anyone at all familiar with Quine’s much better-known “Two Dogmas” will have been reminded of its concluding section. Pap’s idea that the a priori is made, in order to perform certain functions, finds an echo in Quine’s claim that “the conceptual scheme of science [i]s a tool . . . for predicting future experience in the light of past experience. Physical objects are conceptually imported into the situation as convenient intermediaries . . . irreducible posits comparable, epistemologically, to the gods of Homer” (Quine 1951, 41). In addition, Quine also sketches an abstract picture of scientific inquiry: [T]otal science is like a field of force whose boundary conditions are experience. A conflict with experience at the periphery occasions readjustments in the interior of the field.. . . [T]he total field is so undetermined by its boundary conditions, experience, that there is much latitude of choice as to what statements to re-evaluate in the light of any single contrary experience. . . . Any statement can be held true come what may . . . . Even a statement very close to the periphery can be held true in the face of recalcitrant experience by pleading hallucination or by amending certain statements of the kind called logical laws. Conversely, by the same token, no statement is immune to revision. Revision even of the logical law of the excluded middle has been proposed as a means of simplifying quantum mechanics; and what diﬀerence is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristotle? (Quine 1951, 39-40)

Similarity is not identity, and there are two important diﬀerences. First, one way in which, in my view, Pap’s critique of the analytic-synthetic distinction is more persuasive than Quine’s is the fact that from his dissertation to his later papers Pap engages much more deeply with actual examples of scientific practices, and so oﬀers more than a highly abstract account of science. These later papers focus on dispositional and what Pap calls “open” concepts in science; I discuss Pap’s views in the next section. The second diﬀerence is that unlike Quine Pap does not hold that the laws of logic are open to revision;8 I discuss this diﬀerence in section 6.

5.

The Analytic-Synthetic Distinction: Dispositional and Open Concepts

As I noted above, one of the fundamental ideas of positivism is that what distinguishes the sciences from metaphysics is acknowledgment of shared empirical standards for the acceptance and rejection of statements; in the verificationist phase of positivism, these standards are given by methods for establishing the truth of statements on the basis of sensory experience. I mentioned 8 Of

course in writings later than “Two Dogmas” Quine was a staunch defender of classical logic. It is, to my mind, a very diﬃcult question whether this represents a genuine change of mind.

Introduction

17

earlier the challenge for the verificationist criterion of meaning posed by the problem of the a priori. Dispositional concepts in science present a less obvious, but more radical, challenge. Indeed, giving a satisfactory account of these concepts is one of the main motivations of Carnap’s “Testability and Meaning” (Carnap 1937; cited in this Introduction as TM), a paper which constituted a pivotal development in positivism.

5.1

Carnap’s Theory of Dispositional Predicates

As Carnap puts it, the verificationist theory of meaning “led to a too narrow restriction of scientific language, excluding not only metaphysical sentences but also certain scientific sentences having factual meaning,” and he argues in TM for “a requirement of confirmability or testability as a criterion of meaning” (TM, 421). Let’s begin by saying what dispositional concepts are. Carnap characterizes them as expressed by “predicates which enunciate the disposition of a point or body for reacting in such and such a way to such and such conditions, e.g. ‘visible’, ‘smellable’, ‘fragile’, ‘tearable’, ‘soluble’, ‘indissoluble’ etc” (TM, 440). The question about these predicates, for a positivist, is: on the basis of what experiences do we correctly ascribe them, i.e., should we accept that something has such a property? Clearly if an object, say a lump of sugar, is observed to be placed in water, and we see that it dissolves, then we would be correct in claiming that it is soluble. This might lead one to think that the meanings of disposition terms can be specified conditionally. In the case of solubility, for instance, we might take the meaning of ‘soluble’ to be given by: is soluble’ is correctly applied to x just in case if x is placed in a liquid, ‘ then x dissolves. But there are two problems with this idea. First, intuitively we think that lumps of sugar which at no time are placed in a liquid are still soluble, and, to use Carnap’s example, a match burnt up completely before it is ever placed in water is “rightly” said to be “not soluble in water” (TM, 440). Second, if the conditional expressions ‘if . . . then’ used to specify the meanings of dispositional predicates are understood truth-functionally, then any forever-untested object would falsify the antecedent of the conditional, rendering the entire conditional true, and the dispositional predicate correctly ascribed to it. This does imply that the forever-untested lump of sugar is soluble; but, at the same time, the forever-untested match would also be soluble. This problem is sometimes called “Carnap’s paradox” in the literature on dispositions. Carnap’s response to these problems is a retreat from the principle of verification. Dispositional predicates are meaningful, but not in virtue of being associated with necessary and suﬃcient observable conditions of application. Rather, they are meaningful in virtue of being associated with a set of suﬃ-

18

The Analytic-Synthetic Distinction: Dispositional and Open Concepts

cient conditions of application, and a set of conditions suﬃcient for applying its negation. For example, associated with solubility in water are If x is placed in water and x dissolves, then x is soluble If x is placed in water and x does not dissolve, then x is not soluble

(6) (7)

Carnap calls statements of these conditions “reduction sentences.” Two reduction sentences which, as in the present example, provide suﬃcient conditions for applying the predicate and its negation is called a “reduction pair.” In general, a dispositional predicate ‘Q3 ’ is associated with the following reduction pair: (8) Q1 ⊃ (Q2 ⊃ Q3 ) Q4 ⊃ (Q5 ⊃∼ Q3 )

(9)

(TM, 441. Note Carnap’s notational convention at TM, 434: “If the sentence ‘(∀x)[− − −]’ is such that ‘− − −’ consists of several [open] sentences which are connected by [truth-functional connectives] and each of which consists of a predicate with ‘x’ as [variable], we allow omission of the [quantifier] and the [variables]. Thus e.g. instead of ‘(∀x)(P1 (x) ⊃ P2 (x))’ we shall write shortly ‘P1 ⊃ P2 ’.”) The present example has the feature that the reduction pair involves the same test condition, i.e., Q1 is the same as Q4 , and the defining reaction for the negative ascription Q5 is just the negation of the defining reaction of the positive ascription, Q2 , i.e., Q5 is ∼ Q2 . In this case the reduction pair is jointly equivalent to (10) Q ⊃ (R ≡ Q3 ) where Q = Q1 = Q4 , R = Q2 and ∼ R = Q5 . This is called a “bilateral reduction sentence.” For our example, the bilateral reduction sentence for solubility is: If x is placed in water, then, x dissolves just in case x is soluble

(11)

The move from verification conditions to reduction sentences does not disturb the basic positivist idea that cognitively significant scientific language is governed by impersonal experiential standards of correctness. The diﬀerence is that reduction sentences do not provide standards of application for objects which don’t fulfill the test conditions. That is, there is an area of indeterminacy where ascriptions of dispositional predicates are not governed by reduction sentences, and so have no truth-values. For this reason, Carnap holds that in these cases reduction sentences do not give the empirical meaning of the dispositional predicate; it follows that reduction sentences are partial meaningspecifications. We will see shortly that this is only one sense of ‘partial’.

Introduction

19

In TM, Carnap takes the specification of the meaning of dispositional predicates by (chains of) reduction sentences to be a model for the way in which scientific predicates in general acquire empirical meaning. Pap’s critique of the analytic-synthetic distinction is based on the attempt to interpret actual scientific practices in terms of partial meaning-specifications by reduction sentences. Before turning to that critique, I pause to discuss Pap’s criticisms of Carnap’s theory of dispositional predicates.

5.2

Critique of Carnap’s Theory of Dispositional Predicates

Pap makes two principal criticisms of Carnap’s theory of dispositional predicates. First, Carnap’s paradox depends on the truth-functional conditional, so one could avoid it by formulating the conditional specification of meaning of dispositional terms using, say, a causal conditional, or Lewis’s strict conditional. Pap’s own preferred account of dispositional concepts uses a non-truth-functional conditional, which he calls “quasi-semantic probability implication” (19, 312). Second, let’s go back to the never tested lump of sugar and the burnt match. They fall in the area of indeterminacy left open by reduction sentences; on the basis of reduction sentences ascriptions of solubility to these objects are neither true nor false. But this means that partial meaning-specifications still fail to account for our judgments that the sugar is soluble, and the match is not. Carnap was quite aware of this point, and he attempts to meet it as follows:

If we establish one reduction pair (or one bilateral reduction sentence) . . . in order to introduce a predicate ‘Q3 ’, the meaning of ‘Q3 ’ is not established completely, but only for the cases in which the test condition is fulfilled. In other cases, e.g. for the match in our previous example, neither the predicate nor its negation can be attributed. We may diminish this region of indeterminateness of the predicate by adding one or several more laws which contain the predicate and connect it with other terms available in our language. . . . In the case of the predicate ‘soluble in water’ we may perhaps add the law stating that two bodies of the same substance are either both soluble or both not soluble. This law would help in the instance of the match; it would, in accordance with common usage, lead to the result ‘the match c is not soluble,’ because other pieces of wood are found to be insoluble on the basis of the first reduction sentence. Nevertheless, a region of indeterminateness remains, though a smaller one. If a body b consists of such a substance that for no body of this substance has the test-condition—in the above example: “being placed into water”—ever been fulfilled, then neither the predicate nor its negation can be attributed to b. This region may then be diminished still further, step by step, by stating new laws. (TM, 445)

20

The Analytic-Synthetic Distinction: Dispositional and Open Concepts

Pap’s criticism of this proposal has become a standard objection in the literature.9 Carnap’s response amounts to claiming that we can extend the conditions given by reduction sentences to never-tested objects, provided that these are of the same kind as some tested objects. Pap asks, in reply, isn’t it conceivable that on some uninhabited planet there exists a species of metal non-existent on this planet which, like the metal species the human race has experimented with, has the disposition called ‘electrical conductivity.’ This question seems to be perfectly meaningful, yet it seems to be condemned to meaninglessness by Carnap’s theory. Carnap might deny this on the ground that our hypothetical bodies are, after all, metals, and that we can avail ourselves of the law ‘all metals are electrical conductors’ in order to make the question whether they also have this disposition significant, just as he availed himself of the law ‘all wood is insoluble in water’ in order to make the question ‘is this match, which has never been placed in water, soluble in water’ significant. But . . . assume a disposition which is unique to a particular individual. For example, a particular human being might have the disposition of feeling nauseated when exposed to the smell of an orange, and it is logically possible . . . that no other organism has that disposition and that the disposition is never actualized for the simple reason that this individual is never exposed to the fatal smell. To be sure, as long as . . . the class of organisms exposed at some time to the smell of an orange . . . is not empty, confirming or disconfirming evidence with respect to the statement ‘if this individual were exposed to the smell of an orange, he would feel nauseated’ is still obtainable. But what if no oranges existed? Then the class . . . would be empty and hence the above statement would definitely be meaningless by Carnap’s theory. And this is strange, since intuitively the truth, and a fortiori the significance, of the non-existential conditional ‘for any x and y, if x were an orange and y smelled x, then y would feel nauseated’ is compatible with ‘there is no x such that x is an orange.’ (Pap 1963b, 561-2)

That is to say, in the end, Carnap’s proposal fails to account for our intuition that objects can have dispositions that are unconnected with observable manifestations, either directly, or via membership in a kind.10

5.3

Open Concepts and the Rejection of the Analytic-Synthetic Distinction

Although he does not state so explicitly, in TM Carnap seems to regard his account of partial meaning-specifications as what he elsewhere calls a “rational reconstruction” of scientific practice. Of particular importance for Pap are two aspects of scientific practice that Carnap interprets in terms of partial meaningspecifications. First, “a property or physical magnitude can be determined by 9 See,

e.g., Mumford 1998, 60 kind-membership revision of Carnap’s original proposal has been worked out formally by Eino Kaila and Thomas Storer (Kaila 1967, Kaila 1941, Storer 1951). These are open to a number of objections, some of which Pap originated. See Pap 1963b for details. 10 The

Introduction

21

diﬀerent methods,” for example, “[t]he intensity of an electric current can be measured . . . by measuring the heat produced in the conductor, or the deviation of a magnetic needle, or the quantity of silver separated out of a solution, or the quantity of hydrogen separated out of water etc.” (TM, 444-5). It follows that no single reduction sentence or reduction pair defines such scientific concepts; this is a way in which they are “partial” distinct from that discussed earlier in which they do not apply to objects not fulfilling the test conditions. Second, Suppose that we introduce a predicate ‘Q’ into the language of science first by a reduction pair and that, later on, step by step, we add more such pairs for ‘Q’ as our knowledge about ‘Q’ increases with further experimental investigations. In the course of this procedure the range of indeterminateness for ‘Q’, i.e. the class of cases for which we have not yet given a meaning to ‘Q’, becomes smaller and smaller. Now at each stage of this development we could lay down a definition for ‘Q’ corresponding to the set of reduction pairs for ‘Q’ established up to that stage. But, in stating the definition, we should have to make an arbitrary decision concerning the cases which are not determined by the set of reduction pairs. A definition determines the meaning of the new term once for all. We could either decide to attribute ‘Q’ in the cases not determined by the set, or to attribute ‘∼ Q’ in these cases. [Suppose Q3 is introduced by a reduction pair, D1 settles the area of indeterminacy negatively, D2 positively.] Although it is possible to lay down either D1 , or D2 , neither procedure is in accordance with the intention of the scientist concerning the use of the predicate ‘Q3 ’. The scientist wishes neither to determine all the cases of the third class positively, nor all of them negatively; he wishes to leave these questions open until the results of further investigations suggest the statement of a new reduction pair; thereby some of the cases so far undetermined become determined positively and some negatively. If we now were to state a definition, we should have to revoke it at such a new stage of the development of science, and to state a new definition, incompatible with the first one. If, on the other hand, we were now to state a reduction pair, we should merely have to add one or more reduction pairs at the new stage; and these pairs will be compatible with the first one. In this latter case we do not correct the determinations laid down in the previous stage but simply supplement them. (TM, 448-9, emphasis mine)

Pap calls scientific concepts with this feature “open concepts.” That it is always possible to add new reduction sentences to the set that defines a particular open concept gives the third sense in which any single reduction sentence is “partial.” Pap’s criticism of Carnap begins, in eﬀect, with a question. When a scientist settles on a set of procedures for determining a physical quantity, and thereby, according to Carnap, implicitly adopts a partial meaning-specification, has she implicitly adopted an analytic truth? In other words, is a partial meaningspecification, as used in science, analytic or synthetic? Pap’s fundamental claim here is that, with respect to open concepts, neither classification is in accord with actual scientific practice. Consider the following two examples.

22

The Analytic-Synthetic Distinction: Dispositional and Open Concepts Suppose a scientist, finding that each metal has a unique melting-point, decides that it is best to define metals by their melting-points. The question before the philosopher of science is how to interpret the meaning of ‘definition’ as used in this context.. . . Suppose, then, that while according to past experience any substance with melting-point M had properties P1 , . . . , Pn (for which reason M was selected as a reliable indicator of these properties), the scientist is suddenly confronted with a specimen having M but lacking all of these properties. Would he really insist that the specimen is an instance of the metal in question and that the generalization ‘all instances of this metal have properties P1 , . . . , Pn ’ had simply been refuted? Should such anomalous specimens turn up frequently, I think it likely that he would be frankly ‘illogical’ and say ‘probably all these specimens ought to be classified as the metal in question, and we’d better give up the definition in terms of melting-point’ . . . . (19, 300) [S]uppose we accepted Mach’s definition of mass in terms of the ratio of the accelerations mutually produced in two interacting particles, which turns [Newton’s] third law of motion, via the second law, into a definitional truth. Let ‘p’ stand for a predication of the defined functor (such as ‘the mass of particle A is m’), and ‘E’ for the evidence which, by the definition, is logically equivalent to p (such as ‘the ratio of the acceleration imparted to A by unit particle B to the acceleration imparted by A to B is equal to m’). Then[, if the definition is not open to empirical refutation,] the outcome of any further tests of p, based on contingent laws connecting mass with other quantities, would be irrelevant to the question of the truth of p if only E is accepted as certain. And if all these other tests yielded a value for the mass of A inconsistent with m, then all the contingent laws would have to be abandoned while p would remain unshaken. Thus, suppose that A and B had equal mass by Mach’s definition (i.e. m = 1), but that we found that A and B produced unequal strains on a spring scale, further that they did not balance on a beam-balance, further that they exerted unequal gravitational attractions on a given mass at a given distance (since the law of gravitation is an independent assumption of mechanics, not deducible from the third law, this is a logical possibility) and so forth. We still would be logically compelled to stand by our hypothesis.. . . I submit that this is highly counterintuitive, and that no scientist would act in [this way]. (19, 299)

The conclusion that Pap draws about the first example is this: If it be asked whether, according to the view here presented, a generalization about a natural kind, like ‘all iron has melting-point M,’ is analytic or synthetic, it must be replied that this dichotomy is inapplicable to propositions involving open concepts. If an explicit definition of ‘iron,’ in the sense of a statement of synonymy, were at hand, then the question would be appropriate; but this is precisely the presupposition which is here denied. (19, 301)

He makes a similar point about the second example: “if a physicist like Mach speaks of a ‘definition’ of ‘mass’ in terms of the third law of motion he does not attach to the word ‘definition’ the meaning underlying the analytic-synthetic distinction” (19, 302). The reader will have recognized, in these examples, concrete illustrations of the abstract description of science underlying Pap’s concept of functional

Introduction

23

necessity. The scientist selects a statement to make into a definition; but this action does not insulate the statement from empirical evidence. In fairness to Carnap, it should be pointed out that in the passage quoted above, he explicitly denies that reduction sentences are definitions. But Carnap’s reason for this is that a definition must cover all cases. So if one wanted a definition for an open concept at any stage of inquiry, one would have to make arbitrary decisions about the area of indeterminacy left by the reduction sentences discovered by that stage. These decisions, however, may conflict with further reduction sentences that might be discovered in the future. What Carnap fails to consider, and Pap’s examples bring out, is the possibility of conflict among reduction sentences, as inquiry progresses. For Carnap open concepts are open only to future narrowings of the area of indeterminacy. Pap shows, however, that they are also open to the possibility that future inquiry could reveal internal inconsistencies among reduction sentences and thereby force us to give up previously adopted procedures for determining physical quantities. Thus, Pap’s point is independent of whether reduction sentences are definitions or not. His point is that if scientific practice can be reconstructed as progressive adoptions of reduction sentences, then these adoptions can be both meaning-fixing and sensitive to empirical evidence. This characteristic of reduction sentences is inconsistent with a sharp analytic-synthetic distinction.

6.

The Limits of Hypothetical Necessity: Formal or Absolute Necessity

For Pap, the freedom that we have in making and unmaking necessities has limits. Quine, as we see from the passage quoted on page 16 above, held that even the laws of logic may be open to revision in response to experience. This is the point at which Pap would part company with Quine. For Pap there is a kind of necessity not reducible to functional necessity; he calls it “formal” or “absolute” necessity: A formally necessary . . . judgment . . . is a judgment whose contradictory is inconsistent. . . . [T]his . . . means that it contradicts the laws of logic, and the latter are themselves functionally necessary in the highest degree, insofar as their rejection would force us to abandon all laws whatever . . . . It is to be noted, though, that formal necessity is not thereby ‘reduced’ to functional necessity. Inconsistency with the laws of logic remains sui generis, although it can be interpreted as, so to speak, the upper limit of material inconsistency. The laws of logic themselves cannot be said to diﬀer merely quantitatively from functionally necessary laws, such as to be defined by the property of having functional necessity in the highest degree. For functional necessity itself is defined in terms of the logical relation of inconsistency, which is itself defined in terms of the laws of logic. (2, 73-74)

I hope that, by now, I have suﬃciently prepared the reader to see the basic line of thinking underlying this distinction in kind between formal and func-

24

Logical Consequence and Material Entailment

tional necessity. Formal necessity is based on the laws of logic, and it is only by reference to these laws that we have a conception of inconsistency; but consistency and inconsistency between experience and statements is the standard by which we judge whether to accept or reject statements. In short, the laws of logic have to be acknowledged in order for us to have any conception of empirical evidence. The intuitive basis of Pap’s view is thus an instance of the conception of rationality wherein there have to be commonly shared rules governing empirical inquiry, and these rules cannot themselves be accepted or rejected on the basis of experience because if they were not in place, there would be no such thing as an objective practice of empirical belief formation. Thus, for Pap, empiricism is itself not possible without the absolutely necessary truths of logic. I note in passing that this line of thinking anticipates significant aspects of Michael Dummett’s criticism of Quine (Dummett 1974). From Dummett’s and Pap’s perspective, Quine’s openness to the possibility of revising the laws of logic bespeaks the abandonment of rationality in science.11 A natural question that arises here is this. We saw that functional necessity is made by us; hence, we know what truths are functionally necessary (at some point in inquiry) by finding out about our decisions. It would seem that to know what truths are formally necessary, we use the laws of logic to determine if their negations are inconsistent. But how do we know what are the laws of logic? And how do know when some statement is inconsistent? There is a clue to the view that Pap eventually develops in this early paper, where he says that “Inconsistency with the laws of logic remains sui generis” (2, 74); this suggests that this inconsistency is unanalyzable and our knowledge of it irreducibly intuitive. We turn now to the topic of logical consequence for an account of Pap’s view of our knowledge of logic, and thus of absolute necessity.

7.

Logical Consequence and Material Entailment

The laws of logic, for Pap, are not simply a set of statements of determinate syntactic forms. But neither does he accept the view that there is an explanation of what makes a statement a law of logic, in terms of the meanings of the logical constants. This view, which continues to be widely accepted, holds that deductive implication or entailment is truth preservation in an argument in virtue of the occurrence of the logical constants in that argument. Laws of logic are forms of statements that are logical consequences of any statement whatsoever. The view is closely connected to the one discussed in section 3 (page 13 above), that the “primitive” truths of logic are necessary because

11 For

a more detailed discussion Dummett’s criticisms see Shieh 1997.

Introduction

25

they implicitly define the meanings of the logical constants. Pap’s objections to these ideas are scattered throughout his writings; I will focus on chapter 10, “Logic and the Concept of Entailment” (cited in this Introduction as LCE). The question he sets out to address is: what is a logical constant? Of course we can, and in fact do when teaching elementary logic, identify the logical constants by displaying arguments that are intuitively correct and then showing that intuitive validity is preserved when we vary certain expressions but not when we vary others. We then tell our students that the latter are the logical constants. Now, Pap’s argument about this procedure is very simple. If it constitutes the fundamental ground for classifying expressions as logical, then the notion of logical constant presupposes the notion of (intuitive) deductive validity. But then deductive validity cannot be based, ultimately, on an independent notion of the syntactic forms of statements characterized through the occurrences of logical constants—i.e., cannot be based on the notion of logical form. Pap’s conclusion is this: no adequate epistemological account of the construction of semantic systems of logic can be given without countenancing the concept, held in disrepute by many logicians and philosophers, of material (=non-formal) entailment. It is widely held that entailment is essentially a formal relation, i.e., ‘ “p” entails “q” ’ is held to be equivalent to ‘ “if p, then q” is true by virtue of its logical form’.. . . But . . . judgments of entailment are presupposed by the very process which leads to the definition of the meta-logical concept logical form. For the only way in which this concept can be defined (if it be proper to call such a procedure “definition” at all) is to exhibit logical forms by the use of logical constants. (LCE, 201)

This argument does not, at first, seem to be eﬀective against the dominant account of logical consequence in contemporary philosophy of logic. This semantic account of logical consequence rejects the claim that deductive validity is ultimately based on logical form. It oﬀers, instead, a general explanation of the validity of arguments that also shows why arguments with certain logical forms are valid. The explanation is in terms of the meanings of the logical constants. For example, modus ponens ponendo is a valid form of argument for the following reasons. Given the meaning of the words ‘if . . . then’, a statement of the form p ⊃ q is true just in case p is false or q is true. If the sub-statement p is true, then it is not false. It follows that the truth of p ⊃ q requires the truth of the sub-statement q. That is to say, given the meanings of the logical constants, statements in which they occur have certain truth conditions—in the case of the sentential constants these conditions are given by truth tables. An argument is valid because the occurrence of the logical constants implies that the truth conditions of the premises cannot be fulfilled without those of the conclusion being fulfilled. This just happens to be the case in modus ponens. In this account, intuitive judgments of deductive validity play only the role of identifying certain expressions whose meanings are to be given an analysis.

26

Logical Consequence and Material Entailment

This analysis, rather than any particular set of independently identified syntactic forms, explains why our intuitive judgments hold.12 But an answer to this reply can be gathered from NPLR. Pap writes, the usual way of proving that the ponendo ponens rule has the ‘truth-preserving’ character which any acceptable rule of deduction must have, is to prove that the corresponding calculus formula “if p and (if p, then q), then q” is a tautology. Since a tautology is a truth-function of propositional variables which is true for all values of the variables, such a proof cannot get started until “if, then” is interpreted as a truth-functional connective, specifically in the sense of the symbol of Principia Mathematica ‘⊃’ (material implication). But what is the justification for the truth-functional interpretation of “if, then”? None other than that it preserves that common core of meaning in the various uses of “if, then” . . . which enables us to justify those and only those methods of deduction which we intuitively accept as valid, i.e. as corresponding to entailments. Thus the interpretation of “if, then” in the sense of material implication makes it easy to prove that arguments of the form “p; if p, then q; therefore q” and “if p, then q; not-q; therefore not-p” are always truth-preserving . . . . In that case, however the proof of the truth-preserving character of the ponendo ponens rule is, if not formally circular, based on our intuitive knowledge that any proposition expressed by “p and (if p, then q)” entails the corresponding proposition expressed by “q.”. . . The apprehension of logical [entailment] is thus prior to the adoption of . . . the definition of “if p, then q” as “not both p and not-q,” that render formal proofs possible. Such apprehension alone can explain why we accept just this analysis of the logical constant “if, then” as adequate, and not, say, the analysis “not both q and not-p.” (NPLR, 142-143)

We may thus formulate Pap’s reply to the semantic account of logical consequence thus. The semantic account fails to acknowledge that the analysis of the meanings of the logical constants does not take place in a vacuum, but, rather, is constrained precisely to accord with our intuitive judgments of the validity of arguments. The indispensability of material entailment appears also in considering what is actually accomplished by the logicist “reduction” of arithmetic that I will discuss in section 10. Pap writes, What . . . is the attitude of logicians when they are faced with an evidently nonempirical . . . statement which, though expressing an a priori truth, does not seem to be demonstrable with the sole help of the definitions of specified logical terms? Their endeavor may be described as the analysis of the non-logical concepts that seem to occur essentially by means of logical concepts, so that what appeared as a material entailment reduces to a formal entailment once the proposition is fully analyzed. The obvious illustration of this procedure that comes to one’s mind is the reduction of arithmetic to logic. To take a very simple example, since the relational predicate of arithmetic, ‘being greater than’, does

12 I

give a more detailed account of the contemporary semantic conception of logical consequence in Shieh 1999, sections 2-3, 82-7.

Introduction

27

not belong to the vocabulary of logic, the sentence ‘if G(x, y), then not-G(y, x)’ could not, oﬀhand, be said to express a formal entailment. This postulate of arithmetic, however, can be reduced to pure logic by defining numbers as properties of classes and defining the mentioned arithmetical relation in terms of the concepts ‘similarity’ . . . and ‘proper subclass’. But the definition which enables such a reduction is obviously not an arbitrary stipulation; rather it expresses an analysis of a primitive concept of arithmetic. And I would like to know what else I judge, in judging this analysis to be correct, but that the proposition ‘the number of A is greater than the number of B’ entails and is entailed by the proposition ‘B is similar to a proper subclass of A (where A and B are finite classes)’. But since the non-logical term ‘greater’ occurs essentially in the corresponding conditional statement this is definitely not a formal entailment. We have succeeded in formalizing the entailment from ‘G(x, y)’ to ‘not-G(y, x)’ only by accepting the material entailment just mentioned. (LCE, 201-202)

This leads Pap to a surprising Kantian conclusion: [I]f by a synthetic proposition you mean a proposition not deducible from logic alone, and by an a priori proposition you mean one that is not empirical, and if you define logic by means of an enumeration of a set of concepts called ‘logical constants’—to which there is no alternative in the absence of a satisfactory general definition of ‘logical constant’—then you have to accept the conclusion that synthetic a priori propositions are acknowledged whenever the territory of logic expands. And it appears, then, that in ‘reducing’ the non-geometrical parts of mathematics to logic, the logisticians have not eliminated the synthetic a priori from mathematics; they have merely dislocated it to those regions where mathematical and logical concepts make definitional contact. (LCE, 203)

It should be clear that this conclusion is Kantian not merely because Pap uses the words ‘synthetic a priori’. After all, Russell at one point held that logic was synthetic a priori; but upon examination it turns out that he meant by ‘synthetic’ any proposition that is not an instance of the law that everything is self-identical—clearly not Kant’s view of syntheticity. For a contemporary example, Michael Potter claims that Frege’s logic, in contrast to Kant’s, is ampliative and so synthetic (Potter 2000, 64); Potter means, it turns out, that Frege’s logic is “object-involving,” but gives no explanation of what this means or why this is the case.13 Pap’s view is genuinely Kantian because his point is that we can never eliminate from epistemology propositions that we come to know not on the basis of experience, nor by (discursive) reflection on concepts, but in virtue of intuitive insight into their truth.

8.

The Method of Conceivability

Intuitive insight into entailments and contradictions is for Pap a methodical enterprise. And the method is none other than the one practiced by the great Rationalists such as Descartes and Leibniz: 13 For

more on Potter’s accounts of Kant and Frege see Shieh 2002.

28

The Method of Conceivability In order to be sure that ‘if A, then B’ is a necessary proposition, you must . . . get a negative reply to the question: ‘Can you conceive of an object to which you would apply “A” and would refuse to apply “B’?”’ . . . [W]e . . . discover the necessity of the proposition in an intuitive manner, viz., by trying to conceive of its being false, and failing in the attempt. (2, 106; all emphases in original)

The procedure outlined is continuous with the epistemology of necessity in contemporary analytic philosophy; it is the basis, for instance, of Stephen Yablo’s influential “Is Conceivability a Guide to Necessity?” (Yablo 1993). The reader will see the method of conceivability repeatedly in action in this volume. I sketch here two philosophical interesting applications.

8.1

The Semantic Concept of Truth

In Chapters 6 and 7, Pap argues against the version of the semantic concept of truth adopted by Carnap from Tarski. In Carnap’s formulation, quoted by Pap, it goes: to assert that a sentence is true means the same as to assert the sentence itself ; e.g., the two statements ‘the sentence “The moon is round” is true’ and ‘The moon is round’ are merely two diﬀerent formulations of the same assertion. (Carnap 1942, 26)

Pap’s criticism, it should be noted, is not the standard one that the theory is not capable of handling generalizations using a truth predicate. That criticism, of course, engendered the current cottage industry on deflationary theories of truth which, in one way or the other, argue that something like the use of a truth predicate in generalizations is all there is to the notion of truth. Pap’s criticism is this. The semantic conception has the consequence that predicating truth of a sentence is the same thing as making a statement with that sentence. “Is the same” Pap understands as necessary equivalence. Hence, the semantic concept also implies that, e.g., It is not the case that ‘the moon is round’ is true, and the moon is round

(12)

is self-contradictory. This is where the method of conceivability comes in. If (12) is self-contradictory, then we must fail when we try to conceive of circumstances which are correctly described by it. But, A universe which is just like ours except that it does not contain language, and thus contains no sentences, is surely logically possible. In such a universe it would still be the case that the moon is round, but nothing could be the case in such a universe which logically presupposed the existence of sentences, hence it would not be the case that the sentence “the moon is round” is true. (6, 149)

The conclusion is this. In general, there is no contradiction in conceiving that, by using a sentence, one could make a true description of conditions in

Introduction

29

which there are no sentences or users of them. This means that, intuitively, we cannot avoid the idea of something like a proposition expressed by a sentence which is available to be true, independently of the existence of sentences that express it.

8.2

The Mind-Body Problem

In Chapters 16 and 17, Pap oﬀers a critique of logical behaviorism and physicalism concerning the mental. His principal argument is that the principle of verification does not, contrary to widespread doctrine among positivists, imply either of these doctrines. In order to grasp Pap’s argument, let’s put on the board a formulation of a standard verificationist argument for behaviorism physicalism about the mind: 1 Meaningful statements must be verifiable. 2 If statements about the mental states or processes of another person are made true by non-behavioral or non-physical properties of that person, then they would not be verifiable. 3 Such statements are meaningful. 4 Therefore, such statements are not made true by non-behavioral or nonphysical properties of that person. Pap’s critique begins with a clarification of the principle of verification. On his reading, it must be understood as asserting that the meaning of a statement is the set of conditions whose verification would establish its truth. This has two consequences. First, the circumstances whose verification give the meaning must be entailed by the statement. They can’t be merely good evidence for the statement, which makes its truth more likely, but is compatible with its falsity. Second, the principle by itself has no consequences for what methods of verification are allowable. In particular, the principle does not require that verification is by sensory perception. The principle is consistent with proof, introspection, or ethical intuition as methods of verification. Pap now advances two criticisms. First, statements about mental states do not entail statements about behavior or about physical constitution. Here is where the method of conceivability is used: it is conceivable that I can know that I am angry while being blissfully ignorant of the bodily symptoms by which my feeling is manifested . . . . (16, 257) if it is conceivable that I should know about my anger without knowing about the bodily symptoms of my anger, it is still more easily conceivable that I should

30

Comparison with Necessity in Contemporary Analytic Metaphysics know about my anger without knowing anything about the connection of feelings with brain-events. (16, 258)

It follows that establishing the truth of statements about behavior or physiology does not count as verifying statements about other minds. Second, Pap argues that the possibility of verifying statements of the mental state of another is exactly on a par with the possibility of verifying statements about the past. In both cases the possibility of verification requires a(n indexical) change in the statement being verified: if the speaker had experienced . . . the toothache which he predicated of the other mind, and at that moment had been asked ‘which statement is conclusively verified by your present experience?,’ he would have replied, ‘the statement “I have a toothache now,” ’ not ‘the statement “he has a toothache now”.’ But in just the same way, if I had been standing 50,000 years ago where I am standing now and had observed that it rained, the proposition verified by my observation would then have been formulated by the sentence ‘It is raining here now,’ not by the sentence ‘It rained here 50,000 years ago.’ It could therefore be argued that once a statement about the past were conclusively verified by the speaker it would cease to be a statement about the past. Should we say, then, that public verification, i.e., verification by any observer of any age, including the speaker, of statements about the past is logically impossible? . . . . The main point is that, whichever answer we decide upon, we should give the corresponding answer to the corresponding question concerning the public verifiability of statements about mental events not owned by the speaker. (16, 263)

It follows that if statements about the past are held to be publicly unverifiable, the same will have to be said of statements about other minds; but on this alternative the version of the criterion of verifiability under discussion will simply have to be repudiated, since nobody in his sense would want to lay down a law which leads to the elimination of history along with the elimination of metaphysics . . . . (16, 261)

9.

Comparison with Necessity in Contemporary Analytic Metaphysics

In this section I will be a bit more speculative. I will discuss the relation between Pap’s views of necessity and what has become received wisdom since Kripke. As noted above, Pap’s fundamental conception of necessity is non-factuality. It follows, then, that for Pap there is no radical distinction between necessity and apriority. So, one may well think that Pap’s theory of necessity has been definitively refuted by Kripke’s examples of contingent a priori and necessary a posteriori truths, and so can be of no more than “historical” interest. In this section I will argue for a diﬀerent conclusion; I will show that Pap has the resources to give an interpretation of Kripke’s examples quite diﬀerent from

Introduction

31

those Kripke gives them. Most of my discussion will be on Kripke’s purported contingent a priori truths. Let’s recall the well-known case of the standard meter (Kripke 1980, 54-7). The reference of the expression ‘meter’ (or ‘one meter’) is fixed by stipulating that it is the length of a stick, S , at some time t0 . After making this stipulation, it is possible to know that the statement, S is one meter long at t0

(P)

is true without basing this belief about S on any empirical evidence. So (P) is a priori. But, intuitively, at t0 S could have had a diﬀerent length, if, e.g., it had been heated. In such counterfactual circumstances, the length of S would have been diﬀerent from one meter. Hence S , though true, is only contingently so. Pap can redescribe this case using his concepts of hypothetical and absolute necessity. The apriority of (P) for Kripke is its hypothetical or functional necessity for Pap. The stipulation of the reference of ‘meter’ as the length of S at t0 is equivalent to the introduction of a (possibly new) expression, ‘meter’, by a decision to treat (P) as a rule, and so conventionally true. Obviously, the purpose of making (P) a rule is to provide the basis of a system of measurement. Thus, this decision to treat (P) as true does not occur in a vacuum; it is not, to use Wittgenstein’s words, merely a “ceremony” (Wittgenstein 1968, §258). Rather, in order for the decision to fulfill its function, it has to be accompanied by a (again conventional) practice for using the physical object S as a standard of measurement. Such a decision, together with its accompanying practice, clearly seems consistent with thinking of the object S in ways other than as a standard of measure. In particular, one can apply the method of conceivability and consider whether it is self-contradictory to conceive of S as having a diﬀerent length at t0 than it in fact had. Indeed, Kripke’s argument obviously turns precisely on such conceivability “thought-experiments.” Now, the contingency of (P) rests on the contingency of the length of S . But length is a measurable, i.e., determinable, physical property whose determination requires a standard of measurement. Thus the claim that the length of S is a contingent property is equivalent to the claim that, relative to a standard of measurement, it is contingent what is determined as S ’s (determinable) property of length. But S itself, according to our conventions, is treated as the standard for the metric system of measurement. It follows, then, that the contingency of the length of S can be expressed using S itself. But if we do so, we can use the expression, ‘meter’, that S was used to introduce, in which case we would say The length of S at t0 might not have been one meter

(13)

32

Comparison with Necessity in Contemporary Analytic Metaphysics

i.e. S might not have been one meter long at t0

(14)

And so (P) is (absolutely) contingent. Here, then, we have a description of Kripke’s case in Pap’s terms: it is a case of a statement that is functionally necessary but absolutely contingent. This redescription, prima facie, does not require abandoning the equivalence of necessity with apriority. Moreover, it is easy to see that the apriority of (P) is only pragmatic. Suppose, what is in fact the case, that the length of S varies more than we can tolerate in our metric system of measurement. That would be a pragmatic reason to stop using S as a standard of measurement, i.e., stop taking (P) to be a rule of using the language of the metric system of measurement, and thereby stop taking it to be necessarily true, for this purpose. I should note that anyone familiar with contemporary discussions of twodimensional modal logic would recognize some similarities between my Papian account of the standard meter case and that of Davies and Humberstone in Davies and Humberstone 1980. On their account, there are two notions of necessity: true no matter what world is actual, as against true with respect to all alternatives to a given actual world.14 I will end with just a few brief comments on Kripke’s examples of necessary a posteriori truths. One of Kripke’s examples (Kripke 1980, 102-5) is the statement Hesperus is Phosphorus (15) It is an a posteriori truth because it was established by astronomical investigations. But it is necessary because all identities are instances of a law of logic and so necessary. Now, clearly the claim is that empirical evidence is relevant to establishing the truth of (15). But is this empirical evidence relevant to establishing the truth of an instance of the law of identity? Surely not. Indeed, if we appeal to any evidence at all for laws of logic, we appeal only to intuitive evidence: we try, and fail, to conceive of any object not being identical to itself. Now, do we appeal to such (failed) attempts to conceive of a non-self-identity to establish that (15) is true? Of course not, otherwise astronomy would be a much easier endeavor than it is. Kripke’s argument depends on claiming that the proposition expressed by (15) is the same as that expressed by ‘Hesperus is Hesperus’ or ‘Phosphorus is Phosphorus’, i.e., an identity. As he puts the point,

14I discuss the relation between Kripke’s modal

arguments and two-dimensional modality in Shieh 2001.

Introduction

33

[I]t is only a contingent truth . . . that the star seen over there in the evening is the star seen over there in the morning . . . . But that contingent truth shouldn’t be identified with the statement that Hesperus is Phosphorus . . . . (Kripke 1980, 105)

But, from Pap’s perspective, the important point is that in scientific inquiry, we come to accept or reject (15) on the basis of whether “the star seen over there in the evening is the star seen over there in the morning.” This implies that, in science, we do not take the sort of justification relevant to establishing identities to be relevant to establishing (15). That is, from the perspective of science, (15) and identities do not express propositions with the same modal status. Thus, from this perspective, the burden of proof is on Kripke, to show why his theory of propositions should trump scientific practice.

10.

Logicism

Next I turn to Pap’s work in the philosophy of mathematics, specifically, his papers “Mathematics, Abstract Entities, and Modern Semantics” (chapter 12, cited in this Introduction as MAS) and “Extensionality, Attributes, and Classes” (chapter 13, cited in this Introduction as EAC). Pap begins MAS by recalling the classic philosophical problem posed by mathematics. Philosophers from Plato to Kant have been puzzled by the fact that while geometric proof proceeds by construction on particular, concrete figures, the propositions proved by means of these constructions are taken to hold universally for all geometric figures. Indeed the theorems proved don’t even hold exactly for the concrete figures of the demonstration, since no actual physical figure is, e.g., exactly triangular. Something similar can be said of arithmetic. In order to find the sum of, say, 7 and 5, one might proceed by counting out a set of 7 oranges and a set of 5 apples, and then count all these pieces of fruit. But this establishes more than that 7 oranges and 5 apples make 12 pieces of fruit; it shows that 7+5=12 holds for any classes of entities, not just for fruit. And, of course, should we perform this counting procedure on some other two disjoint sets of 7 and 5 objects, and fail to obtain 12, we see no alternative but to conclude that we miscounted, or that special physical circumstances obtained. As Pap sees it, these features of mathematics suggest that mathematical theorems are not about particular geometrical figures or particular sets of objects, but about what these particulars have in common. And, what these have in common are abstract entities that don’t have spatio-temporal location. For the positivists the idea that mathematics concerns non-spatio-temporal entities is problematic. One problem is that it’s not clear how such entities could be the objects of sense experience, and so of empirical knowledge. But the more serious problem goes beyond this. After all, the positivists could and did accept and account for knowledge of theoretical entities of science that

Logicism

34

are not directly experienceable. The deeper problem is rather that, as we have seen, mathematical theorems seem not to be open to empirical disconfirmation or confirmation. It is this that makes it hard for the positivists to see how mathematical theorems are instances of genuine cognition. The solution adopted by positivism is a logicist account of mathematics,15 an account taken from Frege and Russell. It is the counterpart, for mathematics, of the linguistic theory of necessity. The basic idea of such an account is that, e.g., an arithmetical equation such as ‘3+2=5’ is true because the symbols ‘2’, ‘3’, ‘5’ and ‘+’ are “defined (or tacitly understood) in such a way that [the equation] holds as a consequence of the[ir] meaning[s]” (Hempel 1983, 379). Since mathematical truths hold in virtue of the meanings of the key terms involved, they are not any more open to confirmation or disconfirmation than is the analytic truth ‘All sisters are female’. But now what about the idea that mathematical theorems are about abstract objects? One positivist view is that if mathematical statements are true by virtue of the (conventional) meanings of mathematical vocabulary, then we need not assume that this vocabulary refers to any entities at all. This is for the same reason that, on the linguistic theory of necessity, the truth of the analytic statement ‘all sisters are female’ does not depend on there being entities such as the attributes of sisterhood or femaleness denoted by the words ‘sister’ or ‘female’. Indeed, it does not even depend on there being anything in experience correctly described as sisters or siblings or females. All we need to assume to account for its truth is that there are norms governing the correct use of these words. The main question for Pap is: does logicism really show that mathematics is not about abstract entities? He argues first that, in spite of appearances to the contrary, neither Russell’s or Frege’s logicism aimed to eliminate abstract entities from mathematics. Indeed, Pap proposes that, in order to resolve a central problem of Russell’s logicism, one could take arithmetic to be about nothing other than attributes, entities of traditional metaphysics which the positivists insisted are spurious. Second, Pap argues that the ontological commitments of arithmetic are not settled by this proposal, since statements apparently about attributes can be interpreted nominalistically, by semantic ascent, as meta-linguistic statements about schemata, rather than generalizations quantifying over attributes. I will focus on just the first part of this central argument. It is certainly the case that Frege’s and Russell’s accounts of arithmetic include a set of contextual definitions of number words occurring in statements of applied arithmetic. These definitions allow an interpretation of such statements

15 Or

at least the non-geometric parts of mathematics.

Introduction

35

from which numerals are eliminated, an interpretation avoiding even apparent reference to numbers. These are definitions of numerically definite quantifiers, numerical quantifiers for short. For example, the statement (schema) There are (exactly) two Fs is reinterpreted as the quantified statement (schema) (∃x)(∃y)(x y . (∀z)(Fz ≡ z = x ∨ z = y)) More generally, we define by recursion a sequence of quantifiers, ∃0 , ∃1 , ∃2 , . . . ∃n , . . . :16 ‘There are exactly 0 Fs’ ‘There are exactly n + 1 Fs’ ‘There are exactly n Fs’

(∃0 x)F x (∃n+1 x)F x (∃n x)F x

−(∃x)F x (∃x)(F x . (∃n y)(Fy . y x)), given

But logicism cannot be based only on such definitions. This, Pap points out following Frege and Russell, is because they do not allow us to eliminate numerals from statements such as “2 is an even prime.” These contextual definitions only apply when numerals appear as adjectives, not when they appear as singular terms. They tell us how to interpret phrases of the form ‘There are n . . . ’ as wholes, but do not treat the numeral, n, as an independent part of the phrase. The more serious diﬃculty arising from this problem, which Pap does not make explicit, is that, as Frege in eﬀect argues in Grundlagen der Arithmetik (Frege 1884), such definitions do not allow us to give an interpretation of general statements (apparently) about numbers. For example, it is unclear how we could use these definitions to interpret the Euclidean statement, “For every prime number there is a larger prime.” Such a statement contains a quantifier over numbers; but, as just noted, the subscripts of the numerically definite quantifiers are not independent terms, so we can no more quantify over them then we could replace ‘cat’ in ‘cattle’ with a variable. Since most of the theorems of arithmetic are general statements, the contextual definitions in terms of numerical quantifiers are incapable of giving an account of arithmetic. Russell gives a non-contextual definition of numbers as classes of similar classes, where classes are similar just in case there is a one to one correspondence between the elements of each. More precisely, x and y are similar, x ≈ y,

16 Strictly

speaking no expression is a quantifier unless it occurs together with a variable of quantification. Below I will sometimes be strict and use syntactic variables ‘u’ and ‘v’ to indicate occurrences of a variable of quantification together with a quantifier.

Logicism

36 just in case (∃R){(∀a)(∀b)(∀c)(Rab . Rbc ⊃ b = c) . (∀a)(∀b)(∀c)(Rac . Rbc ⊃ a = b) . (∀a)(a ∈ x ⊃ (∃b)(Rab . b ∈ y)) . (∀a)(a ∈ y ⊃ (∃b)(Rba . b ∈ x))}

For example, the number 2 is the class of all classes similar with two-membered classes. This of course looks circular. But circularity is avoided by use of the numerically definite quantifiers. Thus 2 =d f {x : (∃a)(∃b)(a b.(∀c)(c ∈ x ≡ c = a ∨ c = b))}. Unfortunately Pap does not in these articles make clear that such a definition is not enough for Russell’s logicism. The problem is precisely that which we just mentioned: with these definitions in hand, we can at least begin to interpret statements such as ‘3 is a prime number’ (one needs, of course, a definition of is a prime number’ that yields conditions for it to be applicable to classes ‘ of similar classes), but we still have no device for interpreting arithmetical generalizations. What Russell actually did in order to get around this problem is as follows. He gave definitions of the numeral 0, the relation of (immediate) successor, and the predicate of being a number, all in terms of classes and predicates and relations of classes. And then, using these definitions and the laws of logic, Russell proved the 5 axioms of Peano’s axiomatization of arithmetic. This axiomatization crucially includes the principle of induction, the basis for proving arithmetical generalizations in actual mathematical practice. Here is a formulation of Russell’s definitions in a simple (i.e., not ramified) theory of types. In this theory entities are divided into types; the lowest type, 0, consists of individuals, and the entities of type n + 1 are classes of entities of type n. In order to avoid having to specify types all the time, let’s call type 1 classes sets, type 2 classes classes and type 3 classes Classes. Subscripted variables, xn , yn , zn range over entities of type n. But, again in order to improve readability, we adopt some conventions for variables: a, b, c, . . . x, y, z . . . m, n, . . . X, Y, Z, . . .

range over individuals range over sets range over classes range over Classes

We will be defining the successor relation and the ancestral. Strictly speaking, in Russellian type theory binary relations over type n entities are classes

Introduction

37

of ordered pairs of those entities and so of type n + 3. But we will dispense with this and add binary relation variables over each type greater than 0. In particular, R, S . . .

range over binary relations of classes

There is an extensionality axiom for each type: (∀xn+1 )(∀yn+1 )[(∀zn )(zn ∈ xn+1 ≡ zn ∈ yn+1 ) ⊃ xn+1 = yn+1 ], and a comprehension schema for classes and relations of each type, for any formula φ in which yn+1 does not occur free: (∃yn+1 )(∀xn )(xn ∈ yn+1 ≡ φ) Here are the relevant definitions: 1 Λ = {a : a a},

∅ = {x : x x}

[Λ is the empty set, ∅ is the empty class.]

2 0 = {Λ} 3 x − a = {b : b ∈ x . b a} 4 S (m, n) ≡d f {x : (∃a)(a ∈ x . x − a ∈ m) } ∈ n

[n immediately succeeds m]

5 Her(R)(X) ≡d f (∀n)(n ∈ X ⊃ (∀m)(R(n, m) ⊃ m ∈ X))

[X is hereditary in the series of entities related by R]

6 In(X, R)(n) ≡d f (∀m)(R(n, m) ⊃ m ∈ X) ∗

7 R (m, n) ≡d f (∀X)(Her(R)(X) . In(X, R)(m) ⊃ n ∈ X) 8 R∗= (m, n) ≡d f R∗ (m, n) ∨ m = n 9 n ∈ Z ≡d f S =∗ (0, n).

[n induces X in the R-series] [The strong ancestral of R] [The weak ancestral of R] [n is a number]

Still, even with these definitions Russellian logicism is not free of problems. One of the Peano axioms, in the form adopted by Russell, states that distinct numbers have distinct successors. But, as Pap formulates the problem, given the definition of numbers as classes of similar sets, if only n individuals existed, then the number n + 1, being defined as the class of all classes that have n + 1 members (or that would have exactly n members if exactly one element were withdrawn from them), would be equal to the null class; for in that case no classes with n+1 members would exist. But by parity of reasoning, the successor of n + 1 would also be equal to the null class; therefore n and n + 1, which on the hypothesis made are distinct numbers, would have the same successor. (EAC, 233)

Russell’s way out is to adopt an axiom of infinity, a “law of logic” asserting that there are an infinite number of individuals. This move, of course, seems

38

Logicism

rather ad hoc; to use Russell’s own words, it looks more like theft than honest toil. Moreover, this axiom hardly seems a law of logic. Surely logic consists of those propositions that are true no matter what things we may be talking about and no matter what our (nonlogical) words mean. But then surely laws of logic must be true regardless of what things there are in the world, or, for that matter, of whether there are any things in the world at all. So, while Russell’s axiom of infinity allows him to derive Peano arithmeic, the derivation is only dubiously from the laws of logic. It is thus unclear that Russell has established logicism. Pap’s proposal for saving Russell’s logicism is ingenious. He argues that, even if there are no classes with, say more than 100 individuals, this is quite compatible with there being an attribute or property of having 101 members which just happens not to be instantiated. He writes, Now, we have seen that n+1 = n+2 follows from the assumption that the number of individuals is n, together with the Russellian definition of numbers as classes of similar classes. But if the number n + 1 is, instead, defined as the attribute (of a class) of having n + 1 members, Russell’s conclusion does not follow. For though, on that assumption, both attributes would be empty (inapplicable), they would remain just as distinct as, say, the attributes of being a mermaid and of being a golden mountain, and for just the same reason: they are defined as incompatible attributes. For example, the expressions ‘A has two members’ and ‘A has three members’ are so defined in Principia Mathematica . . . that it would be contradictory to suppose that the same class had both (exactly) two and (exactly) three members. Therefore, even if just one individual existed, the number 2 would retain its conceptual distinctness from the number 3; and this argument, obviously, can be generalized for any finite number n and its successor. (MAS, 223)

As it stands, this proposal is not suﬃcient for accomplishing Pap’s aim of showing how Russell’s logicism is possible without an axiom of infinity. The reason, obviously, is that the proposal does not show how to derive the Peano axioms. What we need is an account of the principles governing attributes, either attributes in general, or those attributes that are the numbers. Of course, if the proposal is to constitute a full defense of Russellian logicism, we will also need an argument to show that the principles governing attributes are logical ones, or, at any rate, are more plausibly logical than is an axiom of infinity.17 I will argue that from Pap’s papers we can derive two conceptions of attributes. One of these receives more textual support, but unfortunately it does not suﬃce for a derivation of the Peano axioms without an axiom of infinity. The other 17 There is an issue here that I’ll come back to. Russell’s own formulation of the axioms of infinity is not the statement that there are infinitely many individuals, or that there exists an inductive class of individuals, as is usual in standard formulations of Zermelo-Fraenkel set theory. Rather, Russell’s formulation is: −∅ ∈ Z, i.e., the empty (type 2) class is not a number. This hardly seems like an assertion that there are infinitely many individuals, although it is equivalent to the latter assertion. Of course, it is equally unclear that Russell’s formulation yields a statement of logic.

Introduction

39

conception is less clearly what Pap had in mind; but it does arguably come closer to the idea of providing a derivation of arithmetic from logical laws alone. What do similar sets have in common? Of course the obvious thing to say is that all such sets have the same number of members. That is, for some finite number n, each such set has the property of having n members. So, we might think of the attribute common to members of a class of similar sets as the property of having n members, for some specific n. This idea, of course, oﬀers no definition of numbers, because the attributes in question are specified using a quantifier over numbers, or a quantifier whose substitution instances are numerals. But, we have seen above that for any finite number n, there is a numerical quantifier, “there are exactly n . . . ” which is definable without using numerals. Moreover, a quantifier can be thought of as expressing a property of the class of entities satisfying the open sentence it binds. E.g., ‘(∀x)F x’ asserts that the extension of the predicate F is identical to the domain of individuals. So, ‘∀’ expresses the property of sets of being identical to the domain of individuals. (Note that, as Pap recognizes very well, for Russell all the talk of classes here should be eliminated in terms of talk about propositional functions; hence, the last claim can be reformulated for Russell as the claim that quantifiers over individuals are properties of propositional functions taking individuals as arguments. From now on I will use a higher order logic that more closely mimics Russell’s logic of propositional functions.) Thus, one way of understanding Pap’s proposal is that the finite cardinal numbers just are those properties expressed by numerical quantifiers. We can formulate a condition for a quantifier (Qv) to be a numerical quantifier, namely, that it holds of a predicate only if it holds of all predicates with similar extensions: (∀F){(Qx)F x ⊃ [(∀G)((Qx)Gx ≡ F ≈ G)]} In the remainder of our discussion of the first interpretation of attribute, I will use the numeral n as the symbol for the numerical quantifier (∃n v) In order to implement this proposal along Russellian lines, we have to define 0, successor, and number in terms of numerical quantifiers. The quantifier that is 0 is, of course, −(∃v). The definition of being a number is also simple if we have succeeded in defining the successor relation; we can take it, as before, as holding of just those quantifiers which stand in the ancestral of the successor relation to 0. The crucial question, however, is, how do we define the successor relation among numerical quantifiers? Pap alludes to Russell’s definition “the number n + 1 [is] defined as the class of all classes that have n + 1 members (or that would have exactly n members if exactly one element were withdrawn from them)” (EAC, 233), and this suggests the following idea. We want the numerical quantifier (nv) to be an immediate successor of a given numerical quantifier (mv) just in case (nv) is a numerical property of all those classes

Logicism

40

with exactly 1 more member than any classes which possess the property expressed by (mv). So, consider any class A. This class would have the property expressed by (nv)—i.e., (nx)(x ∈ A)—just in case the class of y’s distinct from some single fixed element a of A has exactly n members. That is, just in case this class has the property expressed by the numerical quantifier (mv). That is, just in case (mx)(x ∈ A . x a). Thus, a natural Russellian definition of ‘n is the immediate successor of m’, S (m, n), using second-order quantification instead of classes, is this: S (m, n) ≡d f (∀F)[(nx)F x ⊃ (∃x)(F x . (my)(Fy . y x))] But a problem immediately arises. Suppose that there are no classes with n individuals. Then (nx)F x is false, no matter what is assigned to the secondorder variable F. But then the antecedent of (nx)F x ⊃ (∃x)(F x . (my)(Fy . y x)) is true no matter what is assigned to F, and, no matter whether (∃x)(F x . (my)(Fy . y x)) is true. This implies that the second-order generalization defining successor is true no matter what the truth-value of (∃x)(F x . (my)(Fy . y x)). From this it follows that if, say, there are only 100 individuals, then every numerical quantifier (∃n v) for n > 100 is the successor of every other numerical quantifier. So, at least with this definition of successor, we can’t prove Peano’s third axiom without assuming an axiom of infinity.18 All is not lost. The interpretation that I have been pursuing identifies attributes with quantifiers over individuals. But nothing in the intuitive idea that if two classes are similar then they have the same numerical attribute requires this interpretation. One could, instead, take numerical attributes to be entities that are not individuals but are of the same logical type as individuals. And we 18 One might here ask, how is this conclusion related to Pap’s claim, quoted above, that “the expressions ‘A has two members’ and ‘A has three members’ are so defined in Principia Mathematica . . . that it would be contradictory to suppose that the same class had both (exactly) two and (exactly) three members. Therefore, even if just one individual existed, the number 2 would retain its conceptual distinctness from the number 3” (MAS, 223)? The obvious way to understand Pap’s claim is that from (∃2 x)F x and (∃3 x)F x we can derive a contradiction, and this is certainly correct. Indeed, for any distinct numbers m and n, from (∃m x)F x and (∃n x)F x we can derive a contradiction. But this is quite consistent with the conclusion we arrived at above. As we know, the claim that there are exactly n individuals can be expressed quantificationally; for example, (∃x)(∃y)(∀z)(z = x ∨ z = y) expresses the claim that there are exactly two individuals. Moreover, from this claim and (∃3 x)F x we can derive a contradiction. Hence from these two claims we can derive the conjunction (∃2 x)F x . (∃3 x)F x.

41

Introduction

can take the notion of class similarity to be the criterion of identity for these entities. Formally, we introduce a term-forming operator, (Nv), which binds an open sentence with one free individual variable, Φ(v), to form a term denoting the entity that is the number of Φs: “(N x)Φ(x).” Call this the cardinality operator. It is analogous to Russell’s definite description operator ( v); the important point is that the terms that can be formed from this operator are values of a first-order quantifier. Note that I said “a” first-order quantifier, not “the” first-order quantifier. This is because we now use a first-order logic with two sorts of variables, x, y, z . . . for individuals and m, n, . . . for numbers. In addition, we use a second-order logic with predicate and relation variables, each with just one sort. The cardinality operator is governed by axioms based on the intuitive criterion of identity for numbers discussed above: the number of Fs is identical to the number of Gs just in case there is a 1-1 correspondence between the Fs and the Gs. And this holds just in case the classes of Fs and of Gs are similar: (Nu)Fu = (Nv)Gv ≡d f F ≈ G, where u and v are first-order variables of any sort. This specification of identity conditions has been christened Hume’s Principle by George Boolos. It is also, of course, closely related to Cantor’s theory of cardinality. Indeed, some people who don’t like Boolos’s interpretation of Frege call it Cantor’s Principle; most of us now compromise and call it the Cantor-Hume Principle. In terms of the cardinality operator we define 0, immediate predecessor, immediate successor, and the predicate of being a number, in ways that by now will seem very familiar: 1 0 ≡d f (N x)(x x) 2 S (m, n) ≡d f (∃F)(∃x)[F x . (n = (Nz)Fz) . (m = (Nz)(Fz . z x))] (m is the immediate predecessor of n; n is the immediate successor of m) 3 Zn ≡d f S =∗ (0, n) (n is a number just in case 0 is a weak ancestor of n in the immediate successor series)

We also use the following abbreviations: m < n for S ∗ (m, n)

Concluding Remarks

42 m ≤ n for S =∗ (m, n)

I leave out the definitions of the ancestrals; they are straightforward rewritings of the type theoretic definitions. In order to show that an axiom of infinity is not needed, we first prove the principle of induction from these definitions, and then prove, by induction, that every number has an immediate successor. That is, we show that the predicate “ξ has an immediate successor” holds of 0, and is hereditary in the immediate successor-series, restricted to numbers. This predicate is formalized as (∃y)[Zξ . S (ξ, y)] The critical move of the proof is to take the successor of a number to be the number of the members of the immediate successor-series up to that number, i.e., we show that (Nm)[m ≤ k] satisfies the predicate Zξ . S (k, ξ) So the strategy is to prove, by induction on S (ξ, (Nm)(m ≤ ξ)) that (∀n)[Zn ⊃ S (n, (Nm)(m ≤ n))].19 Now, is this logicism? One way to argue that it is is that the only nonlogical axioms express, in part at least, our concept of number. Anyone who understands this concept would have to accept that, no matter what entities one is thinking of, if there is a one to one correspondence between entities satisfying one condition and entities satisfying another, then the same number of entities satisfy each condition. Thus, we can claim that the infinity of the natural numbers follows by the laws of logic from our grasp of the concept of number.20

11.

Concluding Remarks

I hope I have managed in this Introduction to convey something of the interest and importance of Pap’s philosophical work. There are, indeed, quite a few more intriguing discussions in the paper collected here than I could discuss in this space. Here are just some of them: 19 For

full details of the proof see the articles in Demopoulos 1995. it is highly implausible that part of the concept of number is that the empty class is not a number.

20 This is in contrast to Russell’s axiom of infinity:

Introduction

43

Pap’s defense of a Humean theory of causation Pap’s defense of Russell’s ramification in type theory as a solution to the semantic paradoxes, Pap’s semantic account of the realism-nominalism debate, Pap’s critique of the notion of emergent property, and Pap’s formulation of the axioms of a logic of belief. Let me conclude by oﬀering a general characterization of Pap’s philosophical work. Mary Hesse once described Arthur Pap as a “logical empiricist with a bad conscience” (Hesse 1966, 456). To my mind, this is true as far as it goes, but its emphasis is not quite right, nor does it go far enough. Much of Pap’s bad conscience derives, as I have suggested, from allegiance to Cassirer’s neo-Kantianism. Pap wouldn’t give up this allegiance because he saw a deep tension in logical empiricism at its very best, namely, in the work of Carnap. The tension is between Carnap’s adherence to the picture of rational inquiry underlying his continued insistence on an analytic-synthetic distinction, and his attempt to be thoroughgoingly pragmatic, as manifested in his adoption of the Principle of Tolerance (Carnap 1954a, 51ﬀ). What Pap, along with Quine, saw, was that a truly thoroughgoing pragmatism cannot countenance any standards not open to empirical revision, and, equally, a truly thoroughgoing commitment to rationality in inquiry cannot make sense of the possibility that the rules defining inquiry could themselves be changed in response to the empirical evidence they make possible. Quine went with pragmatism, thereby giving up a deeply entrenched conception of rationality. Pap took the other horn of the dilemma, and came to hold that logical empiricism has limits beyond which empiricism cannot go, where there lies nothing other than intuitive knowledge of logic itself.

II

ANALYTICITY, A PRIORITY AND NECESSITY

Chapter 1 ON THE MEANING OF NECESSITY (1943)

In this paper I am mainly concerned with an analysis of the Aristotelian concept of “hypothetical necessity.” It will be defined as a functional synthesis that avoids both the Platonistic reduction of necessity to abstract or mathematical necessity (what the scholastics called “simple” necessity, as contrasted with necessity “secundum quid”) and the empiricistic reduction of necessity (cf. John Stuart Mill) to genetically explicable, yet logically ungrounded, generalization of contingent conjunction. What is characteristic of the Platonistic interpretation of necessity as a formal relation between intensions or essences, is that it involves the banishment of necessity from existence: “Whatever is, might not be,” as Hume said. The empiricist, then, emphasizes that, insofar as a necessary judgment is existential in reference, it represents a generalization of a contingent “conjunction,” which generalization will have a psychological cause, viz., the “generalizing propensity,” in Mill’s phrase, or the “gentle forces” of association, in Hume’s phrase, but no logical ground, and will never represent a necessary connection. The concept of hypothetical necessity helps, as I shall endeavor to show, to avoid the exclusive disjunction, advocated by Hume and his positivistic followers: either existential or necessary, but not both. Something is hypothetically necessary if it is a necessary condition or, functionally speaking, a necessary means for something else. Hypothetical necessity, then, is a matter of consequences rather than a matter of antecedents: antecedents derive their hypothetical necessity from consequences. Metaphorically speaking, hypothetical necessity is prospective. Mathematical or abstract necessity, on the other hand, rather is retrospective; it is a matter of antecedents rather than a matter of consequences: propositions derive their mathematical necessity from the antecedents which they follow from. Hypothetical necessity is predicable of hypotheses or postulates or leading principles; mathematical necessity is predicable of theorems or “reasoned facts.” To explain by hypothetical necessity, in other words, is to explain in terms of “in order to” (“worum”), to explain by abstract necessity is to explain in terms of “because” (“warum”).

47

48

On the Meaning of Necessity (1943)

It might seem, prima facie, that the distinction between hypothetical necessity and mathematical necessity merely reflects opposite ways of reading logical sequences. If “if p, then q” is such a sequence, then we can say “q, because p,” and thus declare q as mathematically necessary; but if we, instead of reading backwards to the antecedent, read forwards to the consequent, then we can say “p, in order that q,” i.e. p is hypothetically necessary with respect to q; it is, in other words, a conceptual means to render q intelligible. One might also point out that, since any proposition is, at least potentially, in different respects both conclusion and premise, it is, in diﬀerent respects, both hypothetically and “simply” necessary. Yet the distinction between these two types of necessity amounts to more than that: if p is merely a suﬃcient condition for q, then, the hypothetically necessary being defined as a necessary condition, it is not hypothetically necessary with respect to q. In order, then, for a condition to be interpreted as a hypothetical necessity, a necessary “conceptual means,” it must be placed within a series of alternative conditions. If hypothetical necessity were an intra-logical concept, a concept, that is, that can be defined without reference to extra-logical notions or operations, then that series of alternative conditions would have to represent an analytical, exhaustive disjunctive set: for p would have to be demonstrated to be the only possible condition for q, and in order to demonstrate that, we must suppose ourselves to know the totality of possible conditions for q, and must, moreover, show the alternative conditions to be impossible or self-contradictory. However, the disjunctive sets of alternative hypotheses or conceptual means which the inquirer has to select from, are in most cases neither exhaustive nor analytic. Hence the choice of one alternative hypothesis rather than another cannot be logically grounded. The hypothesis selected, in other words, cannot be itself logically or mathematically necessary in the sense of representing the conclusion of a disjunctive syllogism whose major premise is an exhaustive and analytic disjunction. In Leibniz’s terms, in such a selection of conceptual means, not the principle of identity but the principle of “what is best” is operative. The selection of one hypothesis rather than another cannot be logically, it can only be teleologically, grounded. A hypothetical necessity is not the only possibility—i.e., a “simple” necessity, a self-evident axiom that stands without alternatives—but it is the best possibility, “best” relatively to the teleological or functional context in which it arises as a hypothetical necessity. If p is suﬃcient to explain q, it is merely good for q; r and s and x may be just as good for q. If p is a hypothetical necessity with respect to q, it is not only good for q, but better than any other hypothesis we know of, and thus, within the context of our limited knowledge, best. Thus, the Ptolemaic Hypothesis was good for explaining the astronomical facts or phenomena; its rejection in favor of the Copernican Hypothesis had no logical ground, but only a functional or pragmatic ground: the latter did the job of explanation better than it, being simpler and hence more convenient.

On the Meaning of Necessity (1943)

49

Such hypothetical necessities, or leading principles, bridge, so to speak, the gap between the contingency of empirical conjunctions and the “simple” necessity of formal connections, for they function essentially as means of systematizing facts, of rendering the body of factual knowledge coherent. It is by their instrumentality that facts acquire representative or signifying capacity, and thus evidential value.1 That is, in any statement of fact there is implicit an analytic or formal statement which determines the evidential context of the former, viz., what the stated fact is evidence for and what is evidence for it. For, as the pragmatic theory of meaning (cf. C. I. Lewis) contends, to understand what f (a) (i.e., this is of such and such a kind) means, one must know what would empirically verify f (a), say g(a). But the criterion which determines that g(a) is evidence for f (a), is the “syntactical” or formal premise (∀x)[ f (x) ⊃ g(x)]: whatever has property f has property g. But, one might object, are not those “syntactical” propositions that serve as criteria of the evidential value of facts independent of experience or formally a priori? Yes and no. Yes, insofar as those formal premises are concerned whose interrelated terms are mathematical concepts, and which, hence, express a priori connections that hold irrespective of whether they are exemplified in experience or not. No, insofar as those formal premises are concerned that do not express formal and necessary, but empirical and contingent, implications. For those syntactical statements represent inductive generalizations, and were in the process of inquiry selected as criteria of evidential value. They are hence, so to speak, only functionally formal, being adopted as criteria instrumental to empirical verification. Logically, such criteria, whose interrelated terms are empirical traits (Dewey’s “generic” propositions) are merely contingent or probable, being dependent on the inductive principle. But functionally, they are necessary; i.e., they are best for such-and-such purposes. The number of empirical properties which a given empirical property is conjoined with, is indefinite; hence there is an indefinite series of properties from which we can select a needed evidential property; there is an indefinite number of competitive potential criteria or hypothetical necessities. The determination of the essential or definitory properties of kinds, therefore, is not a discovery, but a choice. Logically, all the alternative criteria are on the same footing, in that they are all, prior to their adoption as criteria, contingent empirical laws. Adoption of one criterion rather than another, therefore, has pragmatic reasons; it is determined by the “principle of what is best.” Hypothetical necessity, then, we may state in paradoxical language, is necessity qua contingent upon freedom of choice or evaluation. Hypothetical necessity presupposes free choice, for the hypothetically necessary is the best one of alternative means, and nothing is better or best except with respect to 1 The principle of causality can be said to be the leading principle of leading principles, insofar as it abstracts

from any particular content that may be signified, and merely prescribes that facts be treated as signs. Cf. Cassirer 1937.

50

On the Meaning of Necessity (1943)

an act of preference or selection. The tendency to ignore this practical element of choice that enters into theoretical explanation, and hence to convert hypothetical necessities into simple necessities—into possibilities that have no conceivable alternatives, self-evident axioms, that is—is a characteristic trait of rationalism. Thus it is characteristic of the seventeenth-century mathematical determinists—Spinoza, Descartes, Galileo—that they interpreted hypotheses as necessary explanations, i.e., as statements of formal causes or laws discovered by “intellectual intuition” rather than selected. Hypotheses, in other words, were for them rationes essendi, not rationes cognoscendi merely. The heliocentric hypothesis, e.g., is not true in the sense of rendering astronomical phenomena intelligible in terms of simpler formulas than those implied by the geocentric hypothesis, but it is true by correspondence, i.e., the sun, as a matter of fact is at rest, and the earth in motion, not the other way around. The identification of a ratio cognoscendi with a ratio essendi, of a principle of intelligibility with a principle of being, however, is valid only on the assumption that the ratio cognoscendi, the hypothesis which explains the facts, has no alternatives: a possibility can be said to be an actuality only if it is known to be the only possibility; while, as long as our explanation is a logically contingent postulate, we must be content to consider it as a ratio cognoscendi. The hypostatization of rationes cognoscendi into rationes essendi rests therefore on the typically rationalistic assumption that formal causes or hypotheses are not selected as alternative and hence contingent explanations of the facts, but are discovered, by intellectual intuition, as necessary and hence real explanations of the facts—that they are, in other words, not merely logically, but ontologically, prior to the facts. The history of philosophy and science presents us with many instances of this hypostatization of logical priority or hypothetical necessity into ontological priority or simple necessity. Thus, Spinoza, having found a God, i.e., a logical structure, that explains the world, goes on to assert that the world as we find it is necessary simpliciter, i.e., could not be diﬀerent from what it is. Which implies the assumption that God is a ratio essendi, not merely a ratio cognoscendi, which in turn implies the assumption that He is the only possible explanation. In Spinoza’s language: God is substance, and substance is that whose essence (possibility) involves its existence, which definition is satisfied by any possibility that has no alternatives, no “compossibilities.” Leibniz, the mathematician well acquainted with the intrinsic arbitrariness of sets of postulates, was, in that respect, wiser when he asserted that there are many possible worlds, and thus recognized the logical contingency of explanations. But he supplemented this statement of the logical contingency of the actual world by a statement of its teleological necessity: this is the only one among many possible worlds, but it is the best of all possible worlds: it is contingent in terms of the law of identity, but it is necessary in terms of the “principle of what is best.”

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He thus reintroduced into rationalistic philosophy the Aristotelian concept of hypothetical necessity. For, Leibniz’s theological statement that this world is necessary if there is to be a maximum of goodness or harmony has the form which defines the concept of hypothetical necessity: “Such and such is necessary, if such and such end is to be reached, i.e., as a means to such and such end.” To adduce another instance of the hypostazation of logical priority into ontological priority: in the Newtonian scheme, “absolute space” is a hypothesis necessary to make Newton’s first law compatible with his second law. For inertial motion, i.e., uniform motion in a straight-line, has a meaning only in terms of an inertial reference-entity. But, according to the second law, the law of gravitation, all bodies are in accelerated motion relative to each other, hence no reference-body could be found that satisfies the condition of inertia. Therefore, inertial motion must be given meaning in terms of a non-material reference-entity, and as such a non-material reference-entity absolute space is introduced as a hypothesis. This hypothesis then, becomes regarded as a “vera causa,” something which “really exists.” This fallacy of conversion of logical priority into ontological priority, of ascribing ontological status of independent substances to factors that have functional status in the context of inquiry, appears in the empiricistic search for existential archai just as clearly as in the rationalistic search for formal archai. The fallacy I am discussing is due to lack of contextual analysis, to neglect of the intrinsic reference of factors to the total fact—the “situation”—within which they are discriminated, whence flows their hypostatization into atomic antecedents. Factors derive their necessity from the fact within which they are discerned or from which they are abstracted; it is, therefore, self-contradictory to declare them, after having them abstracted, as absolute necessities or ontological priorities, and then to render the total fact from which they were abstracted a problem, a “quaesitum.” The factors a, b, c are (hypothetically) necessary, assuming D is a fact; D, that is, implies a, b, c as its necessary conditions: if D, then a, b, c. But I cannot, then, assuming the absolute necessity of a, b, c go on to ask whether D is a fact or even a possibility: for it is only on the hypothesis that D is a fact that a, b, c were declared as necessities. In other words, we can only ask how D is possible—as Kant asked “how is experience possible,”—we cannot ask whether it is possible. Atomistic empiricism, however, in its search for existential archai, “real archetypes,” “pure” particulars that are “given” to “immediate” perception, commits just that fallacy of mistaking hypothetical necessities for existentially prior data. Thus, Locke starts with the fact of complex ideas and analyzes it into its factors, “simple” ideas. He then ascribes genetic or psychological priority to these logically prior elements of the psychologically prior continuum, thus confusing structural analysis with genetic analysis.

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The fallacious conversion of hypothetical necessities into “simple” necessities results, as we saw, from disregard of the selective activity involved in explanation: a “simple” necessity is, by definition, without alternatives; but a selection has, by definition, alternatives. Since Kant’s philosophy purports to oﬀer a rationale for an absolute separation of theory and practice, one might expect Kant to have succumbed to the very same fallacy. Are Kant’s “synthetic a priori principles” hypothetically necessary, or simply necessary? As C. I. Lewis shows (cf. Lewis 1956), in connection with the development of a functional interpretation of the a priori, it is meaningless or self-contradictory to declare a categorial scheme, i.e., in Kant’s own language “den Inbegriﬀ der Bedingungen der M¨oglichkeit der Erfahrung,” as logically necessary in the sense of standing without alternatives. The a priori, in order to be knowable as such, must have alternatives. For, how could we know that the structural limits of experience are due to the mind rather than to objective reality? They could be validly ascribed to the mind only if a modification of the structure of our mind—i.e., of our system of meanings—would entail a modification of the nature of experience. Hence the assumption of absolute or permanent mindstructure makes the Kantian thesis unverifiable and in that sense meaningless. Now Kant certainly is not guilty of the rationalistic habit of “hypostazation” or of what I called “conversion of logical priorities into ontological priorities.” Indeed, as Cassirer shows, hypostatization or “der allgemeine Hang des Denkens, die reinen Erkenntnismittel in ebensoviele Erkenntnisgegenst¨ande zu verwandeln” (also cf. Dewey’s phrase: “conversion of a function in inquiry into an independent structure” (Dewey 1938, 149)) is the very object of Kant’s critique of “metaphysics.” What was a “metaphysical fact” for the dogmatic rationalist, becomes through Kant’s “transcendental method” a mere conceptual condition of experience or inquiry. However, these “conceptual conditions,” i.e., the categories and the a priori synthetic principles to which they, through the mediation of “schemata,” give rise, have, for Kant, no alternatives, and are thus simply, not hypothetically, necessary. “Die Einheit des Bewußtseins [the highest principle, from which all the particular ‘principles of experience’ are derived] verm¨ogen wir nur dadurch zu erkennen, daß wir sie zur M¨oglichkeit der Erfahrung unentbehrlich brauchen.”2 The emphasis on the functional nature of the categorial scheme, on the fact, that is, that it is meaningless and non-existent apart from its use or application, is clear. But just as clear is the emphasis on the “Unentbehrlichkeit,” the simple necessity, of just that scheme, and this emphasis is fatal from the functionalist point of view. The method by which we prove the indispensability of something to something else, is the method of diﬀerence. But how could this method be applied to prove the in-

2 Cassirer

1922, 587.

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dispensability of Kant’s mind-structure to experience? The mind would have to change its own structure, and then see whether experience would still be as it was. Whether Kant falls within the rationalistic philosophies of science, the tradition of “simple” necessity of self-evident axioms and principles, or whether his “criticism” is closer to the functional-pragmatic interpretation of principles as methodological rules whose choice is pragmatically determined, is a historical question for a Kant scholar to decide. Cassirer tends to assimilate Kant’s doctrine of the a priori to the functional-pragmatic interpretation of the a priori, its interpretation as a methodological rule: “Das a priori muß in rein methodischem Sinne verstanden werden; es ist nicht auf den Inhalt eines bestimmten Axiomensystems festgelegt, sondern es bezieht sich auf den Prozess, in dem, in fortschreitender theoretischer Arbeit, das eine System aus dem andern hervorgeht” (Cassirer 1937, 93). Whether this is a reading into, or a reading out of, Kant, at any rate the typically Kantian conjunction of “a priori” with “synthetic” is essentially an attempt to overcome the dualistic separation of the a priori and the empirical, and is thus opposed to the rationalistic identification of “a priori” with “analytic” or “logically necessary.” For the rationalists the “a priori” or “axiomatic” has intra-logical significance, i.e., its meaning is not defined in terms of empirical application; while the “synthetic a priori” is synthetic just insofar as it is essentially a procedural means, to use Dewey’s term, in existential inquiry. Insofar as it has no alternatives, it is, indeed, axiomatic, “simply” necessary; insofar, however, as it is nothing but a conceptual tool of existential inquiry, a “universal” in Dewey’s sense, it is hypothetically necessary. Kant is still a dualist insofar as those “a priori synthetic” principles are fixed once and for all, and are imposed upon experience ab extra, not being themselves derivable from experience. He is still a rationalist insofar as he ignores the temporal character of inquiry. What is a priori at one time, may have been a posteriori at an earlier time; rules or criteria are themselves derived from, generated by, existence. Something is a priori, in other words, not simpliciter, but secundum quid, i.e., for one phase of the continuum of inquiry; it may be a posteriori for another phase of the same. As Dewey puts it (cf. Dewey 1938, 14): Norms of inquiry are “operationally a priori with respect to further inquiry.” Putting this in Dewey’s language of the “universal” and the “generic”: the universal is a rule operative in the establishment of generic propositions, i.e., empirical laws. But it could not be thus operative if it did not itself represent an empirical law that has been transformed into an a priori or prescriptive law. This is what Dewey means when he emphasizes that inquiry is an immanent or self-contained or—non-viciously—circular process, that it generates its own standards or norms, that the latter are not imposed upon it ab extra: Logical forms “originate out of experiential material, and when constituted introduce

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new ways of operating with prior materials, which ways modify the material out of which they develop” (Dewey 1938, 103). If a conjunction of traits a-b is found to be repeated without exception, we generalize it into a “universal,” a definitional connection: if A, then B. If, then, experience should one day disclose a contradictory instance, viz., “a and not-b,” we will have the choice between refusing to identify a as an instance of A—submitting to the rule set up by ourselves which prescribes the incompatibility of not-b with a—and considering our law (“if A, then B”) as refuted, i.e., considering it as no good any more, considering, that is, that it is not logically necessary but only hypothetically necessary, good for conducting inquiry as long as there is no rule better than it. Suppose, e.g., that we found a man talking, although his heartbeat had stopped. Problem: is he dead or alive? If, as is probable, our conception of death is defined as incompatible with power of speech, while, on the other hand, it is defined as a necessary and suﬃcient condition of absence of heartbeat, this instance would prove our conception of death to be inadequate as a rule for identifying phenomena as cases of death, since it would, on the basis of that rule, have to be identified as a case of both life and death. Thus rules or hypothetical necessities are open to modification by experience. Of course, since, given an incompatibility between a theory and a fact, we are always free to reject the fact as a mere appearance rather than changing our theory, it lies within our power to make our theory final and absolute. Given the law “A is B” and the supposedly contradictory fact “a x is not b,” we are free to say that, if a x is not b, it simply is not a representative case of A, A being defined by B. We are free, in other words, to make empirical truths logically necessary and thus to deprive them of their intrinsic contingency. But thus to conventionalize theory is to cut it oﬀ from its actual interaction with factual knowledge, which interaction is what the progressiveness of science consists in: “La physique progresse parce que, sans cesse, l’exp´erience fait e´ clater de nouveaux d´esaccords entre les lois et les faits” (Duhem 1914, 269). Generic judgments presuppose some a priori knowledge, a posteriori knowledge presupposes some universal judgments, since no object can be identified as being of a kind, unless the kind itself is first defined. But, if the a priori is, as a variable, thus logically necessary, as particular value of that variable it is only hypothetically necessary: it is best for inquiry as long as no rival turns up that proves to do the job better. To quote from C. I. Lewis: “If the criteria of the real are a priori, that is not to say that no conceivable character of experience would lead to alteration of them” (Lewis 1956, 263). To recognize the modifiability by experience of a priori principles is to recognize their empirical origin. In order validly to subsume a percept under a concept, I must be in possession of a hypothetical which defines the concept; but if that hypothetical can serve as a tool of identification and in this sense have existential import, it is because its generating antecedent is itself a categorical all-proposition sta-

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ting an empirical conjunction (and hence being logically an I-proposition).3 In order to buy existential import, or hypothetical necessity, “universals” have to pay the price of contingency; they have to abandon the privilege of logical necessity, the privilege of being eternally analytic and/or irrefutable. If judgments are to be both existential in import and analytic, they must be synthetic in origin. We can make synthetic judgments analytic, convert empirical laws into prescriptive definitions; indeed, as Bradley said, “what is added to-day is implied to-morrow. A synthetic judgment, as soon as it is made, is at once analytic.” But we have, then, to recognize that experience is free to unmake our makings again. Hypothetical necessity is, as it were, the mediating link between empirical contingency and logical necessity. Both the empiricist and the rationalist fail to account for the interaction of empirical and formal knowledge. They both subscribe to the Humean disjunction: either existential or necessary, but not both. For the empiricist (cf. Mill 1851) universal judgments are existential, hence, being problematic generalizations, they are not necessary; for the rationalist, on the other hand, universal judgments are logically or “simply” necessary, such as to have no existential reference. Both of these reductionisms leave out of account the functional enterprise of using universal judgments as conceptual tools for the acquisition of factual knowledge. They ignore, that is, the peculiar logical status of principles that are to be existentially applied, of methodological rules, of “synthetic a priori principles,” viz. hypothetical necessity. The positivistic—allegedly exhaustive—disjunction of judgments into empirical (contingent) and analytic (necessary) judgments takes no account of “synthetic a priori principles,” which are, in Cassirer’s words, “Regeln, gem¨aß denen nach Gesetzen zu suchen und nach denen diese zu finden sind.” The empiricist walks on the plane of particulars and contemns the “high priori roads”; the rationalist walks on the “high priori roads” and contemns the plane of particulars. The functionalist, however, recognizes the functional correlation of plane and high-road: “Die H¨ohenwege sind f¨ur unsere Orientierung in dem Gel¨ande, das wir zu durchschreiten haben, unerl¨aßlich” (Cassirer 1937, 67).

3 If

such a hypothetical represents the definition or analysis of a mathematical concept, then, indeed, it is a priori in the sense of logical necessity. But then its applicability to experience, its hypothetical necessity, is contingent upon Nature’s exemplifying, with a tolerable degree of approximation, its ”interrelated characters.”

Chapter 2 THE DIFFERENT KINDS OF A PRIORI (1944)

I am going to distinguish three kinds of a priori: the formal or analytic a priori, the functional a priori, and the material a priori. With these three kinds of a priori there are associated three types of necessity: formal or logical necessity, as characterizing logical truths, whether the latter be called “tautologies,” as by logical positivists, or “truths of reason,” as by Leibniz; functional necessity (Aristotle’s “hypothetical necessity”), predicable of conceptual means in relation to objectives or ends of inquiry; and the kind of necessity that one might call psychological, if it were not the case that the chief proponents of this kind, the kind of necessity that is traditionally defined by self-evidence or the inconceivability of the opposite, are explicitly opposed to “psychologism” in logic (I am referring to the school of phenomenology, or “Gegenstandstheorie,” as founded by Husserl and Meinong). I Kant defined an analytic a priori judgment as a judgment whose predicate forms part of the meaning of the subject. I am quite aware that this is not what Kant literally said. Kant said, in order for a judgment to be analytic, the concept of the predicate must be contained in the concept of the subject. This formulation is to be avoided, however, because it lends itself to a psychological interpretation, and as a matter of fact in some passages Kant uses explicitly psychological language, as when he says in the concept of body we already think the concept of extension, and therefore the judgment “all bodies are extended” is independent of experience.1 Seizing upon this ambiguity of Kant’s terminology, philosophers have since argued that the distinction between analytic and synthetic, as formulated by Kant, is purely psychological, such that the analytic or synthetic character of a judgment varies with the context of its utterance and cannot be determined apart from that context. Indeed, if the an1 It

should be mentioned, though, that in the paragraph entitled “Of the highest principle of analytic judgments,” in the Critique of Pure Reason, Kant improves on his initial definition of “analytic,” by giving a purely logical definition in terms of the principle of non-contradiction. (Translations from the German are by the author, unless otherwise noted.)

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alytic character of a judgment is defined by the fact that we cannot think of the subject without thinking of the predicate, then the formal a priori would hardly be distinguishable from the material a priori, defined by the inconceivability of the opposite. The formal a priori, then, is not to be defined by the psychological predicate “inconceivability of the opposite,” but by the logical predicate “selfcontradictoriness of the contradictory.” “A is B” is analytic, if B forms part of the definition (and meaning in this logical sense only!) of A, such that “A is not B” reduces to the contradictory judgment “XB is not B,” where X stands for the rest of the defining predicates of A. Such a definition of “analytic” is not exposed to the objection of psychological relativity. For one defines a term— thus converting a floating representation or image into a fixed concept—just in order to render its meaning invariant with respect to the psychological idiosyncrasies of the people who use it. Also, one cannot say that one and the same proposition may be taken as analytic or as synthetic, according to the stage of inquiry and the acquired knowledge of the judging person. For if the predicate B forms part of the meaning of the subject A, the concept denoted by A is diﬀerent from the concept denoted by A when B is not definitory of A; hence the respective propositions are diﬀerent, although the verbal sentences are identical. (A proposition may be said to be a logical entity, which has neither physical existence, like a sentence, nor psychical existence, like an act of judgment; it is the “Sachverhalt” expressed by a sentence.) A “realist” would presumably say that to escape from the psychological relativity of logical necessity by substituting for the thought of the concept the conventional definition of the concept, is to fall from Scylla into Charybdis. If the logical necessity of a proposition depends on the way we define the subjectterm, then logical necessity has an extralogical origin. By merely changing the definitions of our terms, we can destroy and create logical truth; and this kind of relativism is just as detrimental to the dignity of logical truth as the psychological relativism it is intended to amend for. Must we, then, say that “A is B” is logically necessary, only if B forms part of the real definition of A? This, indeed, seems to correspond to Kant’s meaning. Surely, Kant could not have insisted that extension is a defining attribute of bodies while weight is not, if he had recognized the conventional character of definitions. Psychologically, there is no ground for regarding the feeling of pressure or resistance as secondary, and the sense of extension as primary, in the formation of the concept of matter. For the Cartesian school, extension was the essential attribute of matter, in the sense that dynamics was considered as reducible to kinematics and kinematics to analytic geometry. Within Newtonian physics, on the other hand, force is a fundamental concept; masses are idealized to such an extent that they are treated as points, and what constitutes them as physical rather than geometrical points is not that they have extension, but that they are subject to gravitational forces. In this context, therefore, weight would seem

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to be more essential to masses than extension.2 Kant, then, must have distinguished the concept of matter from alternatively definable concepts of matter. In Aristotelian fashion, he must have made an ontological distinction between essential and accidental predicates. As Locke distinguished between “real” and “nominal” essence, so Kant distinguishes, in his lectures on formal logic, between “Realwesen” and “logisches Wesen.” Only the latter is relative to the selective definitions of the inquirer, while the former is absolute, something to be discovered. It is, indeed, true that for Kant not only empirical substances, “material archetypes,” are “Realwesen,” but also the genetically or synthetically defined concepts of mathematics. But such “synthetic” definitions are, for Kant, not altogether arbitrary; they are subject to the laws of “pure intuition.” That the straight line is the shortest distance between two points is, within the domain of pure intuition, just as good a discovery as the solubility of gold in aqua regia is a discovery in the domain of empirical intuition. Every definition is conventional insofar as it involves selectivity. It is no longer possible to ignore the pragmatic element in inquiry, and so far one cannot believe in Aristotle’s “real” definitions. At any rate, one cannot base the formal a priori on necessities of the intuitive kind, like Aristotle’s intuitive discrimination between “essence” and “property,” however considerable a part pure intuition may play in the domain of the material a priori, the a priori of the phenomenologists. As Dewey has shown, “‘substance’ is a logical, not an ontological determination.” That is, “essential” predicates cannot be defined as standing for “inherent” properties of independent, “given” substances, without reference to the objectives of inquiry; they are to be regarded as predicates selected as definitory of a concept. For logic, in other words, substance is always a “logisches Wesen,” not a “Realwesen”; “essential” predicates are definitory predicates, and if we select diﬀerent predicates as definitory, the “substance” will be diﬀerent. The analytic nature of a proposition, therefore, must be recognized as independent of whether the definition of the subject-term be “real” or “nominal.” It is quite true that a definition of a term already in use presupposes (in the genetic order) in most cases—or perhaps always—a synthetic judgment. If the term to be defined had, e.g., already a denotation or extension, the definition will formulate the discovery of a common property of the denotata. As we shall see later, it is highly important to recognize this genetic dependence of analytic truth upon synthetic truth (cf. Kant’s statement: “Where the understanding analyzes, there it must first have synthesized, because it is only as synthesized by it that anything can be given to the faculty of representation” (Kant 1912, 113)). But hence to infer that there is no such thing as a purely 2 Historically

speaking, though, Newton and Kant are in the same boat, in that they both distinguish, in scholastic manner, essential predicates from empirically universal predicates; for Newton, as for Kant, weight is, though empirically universal, not an essential predicate of matter (cf. Cassirer 1922, II, 679-80).

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formal or analytic a priori, would be very confused indeed. Whether a proposition of the form “A is B” is analytic depends on the definition of A, no matter what be the reason for adopting just that definition. And insofar as definitions are conventional, analytic truth or logical necessity is, in part,3 conventional. No state of aﬀairs is logically necessary or logically impossible, a priori the case or a priori not the case, per se, but only relatively to definitions. Insofar as definitions, provided the terms that occur in them have empirical reference, may themselves formulate synthetic, empirical truths, it does not seem to me to be fortunate to say, with the logical positivists, that a formally a priori sentence “says nothing about the world.” In the procedure of science empirical laws are used for the definition of its concepts. For example, mass is defined in terms of Newton’s third law (cf. Mach 1904, 231), or heat-capacity in terms of the principle of the conservation of the quantity of heat (cf. Mach 1919, 1868), or internal energy in terms of the first law of thermodynamics. Of course, once an empirical law has been adopted as an implicit definition of a concept in terms of which it is stated, it ceases to be a contingent truth and becomes, qua definition, irrefutable by experience. All that can happen is that future experience will call for revision or abandonment of the definition. But the reason for such a change of convention is itself a non-conventional state of aﬀairs: it is the fact that the empirical law corresponding to the definition fails to be verified. Provided this concomitance between change of convention and change of empirical truth, between changes in the “meta-language” and changes in the “object-language,” is recognized, there may be no harm in maintaining that what is formally a priori, in the sense of being definitional, says nothing about the empirical world; and we have ourselves insisted that, in order to determine whether a given proposition is formally a priori one need not inquire into the reasons that led to the adoption of just that definition of the subject-term. However, neglect of such genetic considerations easily leads to a futile separation between logical and causal or empirical possibility. If science defines its concepts in terms of causal laws, then causal possibility or impossibility is itself the raison d’ˆetre of logical possibility or impossibility. If mass, e.g., is defined in terms of the dynamic relations expressed by Newton’s third law, or by the law of gravitation, then it will be causally impossible and therefore logically impossible that only one mass should exist. Only if logical possibility should be defined in terms of the psychological concept of conceivability could such a state of aﬀairs be maintained as logically possible. One certainly can perform a “Gedankenexperiment” to the eﬀect of picturing just one mass and nothing

3 The

import of the qualification “in part” will become manifest in the discussion of the material a priori. It is meant to indicate that logical necessity depends on definitions and principles of logic, the latter being themselves not formally a priori, since the formal a priori is defined in terms of them.

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else in the world. But the pictorial or phenomenal meaning of terms must be distinguished from their conceptual, relationally defined meaning. We have so far arrived at a definition of the formal a priori which renders it independent of both psychological context and metaphysical presuppositions like Aristotle’s concept of “real” definitions, based on the distinction between “essence” and “property.” It is furthermore desirable to render it independent of the subject-predicate schema of Aristotelian logic. The standard form of analytic truths is formal implication; as Leibniz pointed out, the “truths of reason” are always conditional in character. In many cases it is, indeed, possible to translate a formal implication ((∀x)[φ(x) ⊃ ψ(x)]) into a categorical A-proposition of subject-predicate form (every φ is a ψ). But such a translation easily leads one to overlook the non-existential character of analytic truths: φ and ψ may define null-classes, while the relation of subsumption, in Aristotelian logic, holds between ontological classes. That the Aristotelian logic of subsumption was not formal at all is evidenced by the fact that subalternation is, in Aristotelian logic, a valid mode of immediate inference. If A-propositions are formulated in categorical form, then, indeed, subalternation seems to be valid; if the classes defined by φ and ψ are interpreted extensionally, then a mere inspection of the meaning of the A-proposition “all φs are ψs” induces us to draw the inference: “some φs are ψs.” But, from (∀x)[φ(x) ⊃ ψ(x)] it by no means follows that (∃x)[φ(x).ψ(x)]; “constant conjunction” cannot be inferred from “necessary connection,” and the above formal implication may, though it need not, express a “necessary connection.” Again, the existential or ontological assumptions implicit in Aristotelian logic are transparent in the so-called square of opposition. As Meinong (cf. Meinong 1907, 43) points out, the negation of the particular negative may be formally equivalent to the universal aﬃrmative; but insofar as the universal aﬃrmative expresses non-existential knowledge (“daseinsfreies Wissen”), while a particular statement is about existents, it cannot be said to be identical in meaning (in a non-formal sense of “meaning”) with a particular statement. According to Meinong’s “Gegenstandstheorie” we must distinguish between the “Objektiv” and the “Objekt” of a judgment. The “Objektiv” of a universal necessary judgment (I disregard, for the moment, the fact that the kind of necessity which Meinong talks about is the necessity of the material a priori, which will be discussed later, not the necessity of the formal a priori) is non-existential; the universal necessary judgment is not concerned with empirical objects at all. The meaning of the judgment “all equilateral triangles are equiangular” is not: “there do not exist any equilateral triangles which are not equiangular”; for we come to deny an existential judgment, whether aﬃrmative or negative, by examining instances and finding that a certain conjunction of traits does not hold in any one instance. But the above universal proposition, using Dewey’s terms, is not a generic proposition: it is not about a conjunction of “characteristics,”

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but about a connection of “characters.” For the purposes of formal logic, indeed, a universal aﬃrmative may be said to be derivable from the negation of the corresponding particular negative by a mere syntactical transformation: ∼ (∃x)[φ(x). ∼ ψ(x)] ⊃ (∀x)[φ(x) ⊃ ψ(x)]. But this transformation is possible only because the symbol ‘⊃’ in the implied statement stands for extensionally defined material implication, the logistic analogue of Hume’s “constant conjunction” (in respect of the absence of logical necessity). If the implied statement, however, is analytic, expressive of “daseinsfreiem Wissen,” the above implication represents an idle syntactical rule—that is of no use in the drawing of inferences: no mathematician would attempt to prove, e.g., that all differentiable functions are continuous by examining instances of diﬀerentiable functions and seeing whether there are any discontinuous ones among them; and if the domain of the quantified variable is infinite, such a procedure would anyway lead only to verification, not to proof. Every universal aﬃrmative is anti-existential, in that it analytically entails negation of an existential statement: (∀x)[φ(x) ⊃ ψ(x)] ⊃ ∼ (∃x)[φ(x) . ∼ ψ(x)]. But to maintain the validity of the converse implication is to confuse material implication and analytic entailment, extensional and intensional universality. It is, e.g., certainly true that no unicorns dislike cake. For, let φ stand for the predicate “being a unicorn” and ψ for the predicate “liking cake.” Then “no unicorns dislike cake” is to be formalized as follows: ∼ (∃x)[φ(x). ∼ ψ(x)]. But in order to prove a conjunction false, all we have to do is to prove the falsity of at least one conjunct. Now, (∃x)[φ(x)] is false, since there are no unicorns. Thus we have proved that no unicorns dislike cake. If we accept the equivalence of the negation of the particular negative to the universal aﬃrmative, we have therefore demonstrated that all unicorns like cake. Surely, it needs the routine of a formal logician not to be puzzled by such startling discoveries! If we define, then, the formal a priori as characteristic of implications rather than of categorical statements, we must explicitly rule out material, extensionally defined implications. It is a necessary condition for the implication (∀x)[φ(x) ⊃ ψ(x)] to be valid, that ∼ (∃x)[φ(x). ∼ ψ(x)]. This condition is at the same time suﬃcient to define material implication. But in order to define analytic implication, we must introduce the modality “self-contradictory.” (∀x)[φ(x) ⊃ ψ(x)] is valid as an analytic implication, and hence formally a priori, if (∃x)[φ(x). ∼ ψ(x)] is not only false, but self-contradictory. The process of converting empirical laws into conventions or implicit definitions (which will in the sequel be illustrated by examples) is just this step from material implication to analytic implication. If analytic implications were based on “real” definitions beyond possibility of revision, then (∃x)[φ(x). ∼ ψ(x)], in the face of (∀x)[φ(x) ⊃ ψ(x)], would not even be a possibility; from the implication and the fact ∼ ψ(a), we could, without further empirical investigation, infer the fact ∼ φ(a). But the above conjunction may occur and induce us to abandon

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the implication as formally a priori or definitional, instead of taking it as a rule prescriptive of what is a fact and what is mere appearance, because the implicatory sentence did formerly stand for a material implication, expressing an empirical law, which we now find to be contradicted by experience and hence unworthy of being any longer used as a definitional rule. One often finds analytic statements defined as statements whose truth follows from the very meaning of the terms; they say nothing about the empirical world in the sense that the recognition of their truth does not presuppose any sort of empirical inquiry. This view is expressed, e.g., by Schlick, in an essay entitled “Gibt es ein materiales Apriori?” (Schlick 1932): “An analytic sentence is a sentence which is true in virtue of its mere form; he who has understood the meaning of a tautology, has at the same time recognized its truth; therefore it is a priori. As regards a synthetic sentence, however, one must first understand its meaning, and thereafter find out whether it is true or false; therefore it is a posteriori” (Schlick 1938, 22). However, it is not quite accurate to say that all we need in order to recognize an analytic statement as true, is to understand the meaning of the terms involved; for, if its truth follows from the meaning of the terms, there are, on the other hand, required logical principles, by which it follows. The foremost of these principles is, of course, the law of non-contradiction itself. “A is B” is true, in virtue of the meaning of A; for, by definition, “A is XB” (where X stands for the defining predicates other than B); then, by the principle of the substitutability of equivalents, “A is B” transforms into “XB is B”; then, by the principle of simplification, this statement is equivalent (in the extensional sense of equivalence, defined by reciprocal implication, or identity of truth-values) to “B is B”; finally, the principle of non-contradiction says “X is X” (where X is a variable, standing for any term); by a rule of substitution, “B is B” is then derivable from the law of non-contradiction, and thus the original statement “A is B” is seen to be true by the law of identity. This pedantic analysis is intended to reveal that the formal a priori can be defined only with reference to principles of logic, and the latter certainly cannot be said to be a priori in the same sense in which statements whose analytic character is determined by these very principles are a priori. The truth of these principles of logic cannot follow from the meaning of their terms, simply because their terms have no meaning at all: they are variables. II Thus the very analysis of what is meant by the “formal a priori” reveals the existence of another kind of a priori without which the formal a priori could not even be defined. I shall call this the material a priori, avoiding the more familiar term “synthetic,” because the latter has been ambiguously applied, by Kant, to both the material and the functional a priori. The principles of logic themselves, which we just saw to be essentially involved in the definition of the formal a priori, are materially a priori. Their truth is a matter neither of de-

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duction nor of induction. We are then left with two alternatives: either they are self-evident, “seen” to be true in pure “Wesensanschauung,” as the phenomenologists would say, or they are conventions. Even though it would certainly be more emancipated to accept the latter alternative, and to dismiss the former alternative as mystical Platonism, I venture to suggest that we are not, here, really confronted with mutually exclusive alternatives. Just as empirical laws of nature are used as conventional definitions of empirical concepts, because they are true in a non-pragmatic sense, so the principles of logic can be used as implicit definitions of logical concepts, like negation, implication etc., because they possess some kind of evidence that is independent of the use that can be made of them. To argue that the law of excluded middle, or the equivalent law of non-contradiction, is analytic because it implicitly defines the meaning of negation, such that its denial would necessarily presuppose it, is no better than to argue that Newton’s third law of the equality of action and reaction is analytic, because it implicitly defines the meaning of mass, or that the first law of thermodynamics is analytic because it implicitly defines internal energy. The principles of logic can, indeed, by Wittgenstein’s truth-table methods, be shown to be “tautologies”; but this method presupposes definitions of the “logical constants” which amount to a recognition of those very principles of logic. Thus, presupposing the following definition of the negation sign ‘∼’: p ∼p T F

F T

one can prove the law of non-contradiction: ∼ (p. ∼ p) to be tautologous, i.e., true no matter what be the truth-values of the components. But what else does the above definition of negation express but the couple of inferences: if p is true, then ∼ p is false, and if p is false, then ∼ p is true? And how do these implications diﬀer from the law of non-contradiction? Just as empirical laws have to be recognized as synthetic truths before they can be used as definitions of empirical concepts in terms of which they are stated, and thus made analytically true, so logical laws have to be recognized as synthetic (“synthetic” in the sense of non-definitional) truths before they can be used as definitions of logical concepts in terms of which they are stated. Once, e.g., we have intuitively recognized the validity of the modus ponens (if ‘p ⊃ q’ is true, and ‘p’ is true, then ‘q’ is true), we can conventionally adopt it as an implicit definition of the symbol denoting implication; thereafter, of course, the validity of the modus ponens will follow from the very meaning of implication, and nobody could deny it unless he used the symbol ‘⊃’ in a diﬀerent sense. Or, to take another example, the principle of mathematical induction can be used as an implicit definition of the concept “finite integer,” because of its intuitive evidence. It would hence be absurd to think one refutes

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the claim of Poincar´e and the intuitionists that the principle of mathematical induction is synthetic a priori by pointing out that it “merely” defines what is meant by a finite integer. This type of consideration applies generally to the axiomatic method of modern mathematics, the method of defining the “primitive notions” by a set of axioms or postulates which they satisfy. The postulates give rise to, are the source of, analytic truths; but they themselves must be regarded as synthetic. As Kant said: “One can indeed recognize a synthetic sentence as true by the law of non-contradiction, but only by presupposing some other synthetic sentence from which it can be inferred, yet never in itself” (Kant 1912, 42). Kant thus admitted that the formal implications that constitute the theorems of pure mathematics are analytic; what is synthetic, according to him, are the basic axioms that give rise to those analytic truths. Of course, as I mentioned already, Kant’s usage of the term “synthetic a priori” is not unambiguous: in insisting on the synthetic nature of the axioms of geometry, he means to point out, first, that their denial is logically possible, and second, that it is intuitively impossible. That Kant was right on the first point—the non-analytic nature of the axioms of Euclidean geometry—has been definitely proven by the development of non-Euclidean geometries. If the parallel-axiom were logically necessary or analytic, its denial could not have led to consistent systems of geometry. But does not an axiom like “the straight line is the shortest distance between two points,” claimed as synthetic by Kant, merely amount to a definition of straight line? Again the answer is: it is a definition of this geometrical concept in the same sense in which physical laws are definitions of physical concepts. Euclid did not arrive at it by an analysis of the concept “straight line” any more than Newton arrived at his second law of motion by an analysis of the concept “force.” The point to notice is that concepts do not “exist” at all prior to the judgments which construct and define them. Axioms do not analyze the meaning of symbols; rather they create meanings, or concepts, and once this creation, this “synthesis,” has occurred, a symbol can be attached to those conceptual creatures; then one can say the axioms “merely” define the meaning of these symbols. In talking that way, one forgets that analysis presupposes synthesis. “La place de la synth`ese a priori n’est pas dans la liaison des termes du jugement, ou dans la d´emonstration de telle ou telle formule num´erique particuli`ere; elle est dans le processus g´en´eral dont d´erive tout nombre particulier, dans la cr´eation des notions elles-mˆemes” (Brunschvicg 1922, 207). If I prefer the term “material a priori” to the Kantian term “synthetic a priori,” for the description of the kind of independence from experience that accrues to axioms, whether logical or mathematical, it is in order to avoid the connotation of intuitive necessity, which at once convicts one of “psychologism.” A materially a priori judgment is such that its contradictory is consistent; in this respect it diﬀers toto caelo from a formally a priori judgment.

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But when Kant described the axioms of geometry as synthetic a priori,4 he furthermore implied the intuitive inconceivability of the opposite; and it is this implication which I would not want to take upon myself, not because I regard it as conceivable that it should be conceivable to anyone that parallels intersect, or, that, in a two-dimensional plane, a line can have more than one parallel, or that two straight lines can enclose a finite space, given the Euclidean intuitive meanings of the terms “straight line” and “parallel,” but simply because “je n’ai pas besoin de cette hypoth`ese” in order to defend the material a priori as a kind of a priori distinct from, and presupposed by, the formal a priori. The only advantage I can see in calling materially a priori judgments conventional is that this term involves recognition of the absence of logical necessity, the logical conceivability (though possibly psychological inconceivability) of alternatives. Indeed, if logical necessity is defined in terms of the principles of logic, it would be absurd to ascribe logical necessity to the principles of logic. Still, being, in their capacity as normative principles, logically independent of experience, they are a priori principles. Shall we, then, call them conventions? Is, in other words, the material a priori perhaps reducible to the functional a priori? No, not reducible to, but compatible with it. Why should utility and a priori truth be incompatible? The principles of logic, it would seem, are useful conventions, not in spite of, but because of their a priori truth. It seems to me to be making a rather cheap use of Occam’s razor, if one dismisses the claim of the apostles of “Wesensanschauung” that there are materially a priori truths, as “psychologism,”5 or, even worse, mysticism. It must, of course, be admitted that some of the allegedly “eidetic truths” resolve, upon analysis, into analytic statements, and are confusedly called materially a priori, just because descriptive terms occur in them. However, the really interesting cases of synthetic a priori truths are aﬀorded by Judgments of Interpretation. 4 It seems to me that the logistic demonstration of the analytic nature of the propositions of pure mathematics by no means conflicts with Kant’s doctrine of the synthetic a priori in mathematics. Metamathematical research has, indeed, revealed that what is asserted, in mathematics, are formal implications between axioms and consequents, and that these implications are valid by the mere rules of logic, being entirely independent of intuition. But, as mentioned already, Kant admitted the analytic nature of these implications. What is synthetic, for him, are the axioms, and hence their consequents, which is quite compatible with the analytic character of the consequences. One might point out that, as Hilbert has shown, the axioms of mathematics are purely formal, and hence not synthetic in the sense of depending on intuition. But the axioms whose synthetic character was defended by Kant are not Hilbert’s formal, uninterpreted ones, but their Euclidean interpretations, and nobody would deny that the interpretation of axioms, the establishment of “Zuordnungsdefinitionen,” requires intuition. If it be said that the axioms qua interpreted do not form part of mathematics at all, why this is a declaration of the way in which the term “mathematics” is used nowadays, which is not the way Kant used the term. Kant, one might say, was not concerned with “pure” mathematics at all, but with “applied” mathematics. His mistake lies only in this, that he regarded Euclidean geometry as the only interpretation of formal geometry that is of any use to physics. And presumably he would not have committed this error if he had lived in the age of Einsteinian physics. 5 Notice that Husserl’s phenomenology started out as an explicit revolt against psychologism! Cf. Husserl 1913, I.

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Consider the judgment: “time is a series,” where a series is defined by the properties of asymmetry and transitivity. Once I have intuited that temporal instants are substitutable as proper values for the variable relata of an asymmetrical and transitive relation, I can, of course, use the formal properties of asymmetry and transitivity to define time; and thereafter, indeed, judgments like “time is irreversible,” or “if Tuesday comes before Wednesday, and Wednesday comes before Thursday, then Tuesday comes before Thursday,” will turn out to be analytic. But the synthetic judgment, presupposed by these analytic judgments, lies in the substitution of temporal instants for the variable relata. This act of interpretation cannot be reduced to the abstraction of a structural property from material instances, and thus to an inductive process. For time is one, it does not have instances in the sense in which a class-concept has instances. Logically speaking, I cannot abstract the structural properties of asymmetry and transitivity from an examination of times, i.e., parts of time. For, in the first place, the very meaning of those properties requires at least 2, or 3, “times,” i.e., elements of time. And, second, if within these minimally extended parts of time those properties do hold, it still does not follow that they would hold within the parts of time that are compounded out of those minimal parts. And to reason from the fact that parts of time have those structural properties to the fact that time as one has those properties, is not at all analogous to inductive extrapolation, the inference from some instances to all instances. For, as Kant pointed out, the relation between time and its parts (like the relation between space and its parts) is essentially diﬀerent from the relation between a class and its members. I can think of a member of a class without thinking of the class, while I cannot think of a part of time otherwise than as a “limitation” of continuous, infinite time. “Idea temporis est singularis, non generalis. Tempus enim quadlibet non cogitatur, nisi tamquam pars unius eiusdem temporis immensi ... Omnia concipis actualia in tempore posita, non sub ipsius notione generali, tamquam nota communi, contenta” (De mundi sensibilis atque intelligibilis forma et principiis, (Kant 2002), paragraph 14). Time, in other words, is a continuum, not a discrete collection, like an extensional class. How does one verify the judgment of interpretation: “time is a continuum?” (“continuum” in the pre-Cantorian sense, i.e., defined solely by “density” or “compactness”). It certainly is not analytic. Then is it an inductive generalization? In that case one would have to admit the possibility that, if the process of dividing time were only continued far enough, we might encounter discrete parts of time, i.e., parts of time not separated from one another by other parts. Is one convicted of “psychologism” if one regards such a state of aﬀairs as inconceivable and hence the judgment “time is a continuum” as “synthetic a priori”? Surely, if one were to insist that “time is a continuum” is no more than an inductive generalization, one would have to admit that it is not the same kind of inductive generalization as those that apply to classes of natural objects. A judgment

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represents an inductive generalization if it presupposes as a necessary assumption the uniformity of Nature. The very concept of “uniformity,” however, is defined in terms of continuous time: the statement of a given functional relation (e.g., the law of gravitation) represents an inductive generalization if it is assumed to hold at all places and at all times; time, however, enters into the equations of physics (whether explicitly or implicitly, through the “time derivatives”) as a continuous variable. Hence, to say “time is a continuum” is an inductive generalization involves an obvious circularity. One can, of course, regard this judgment as “synthetic a priori” in the functional sense of Kant’s attribute: science assumes time to be continuous, just as it assumes space to be continuous, in order to be able to correlate it with the number-continuum, and thus to render a mathematical treatment of motion possible. This is true enough; but it is by no means conflicting to call a judgment both functionally and materially a priori. On the contrary, if the judgment under discussion is functionally a priori, i.e., adopted as a necessary presupposition of science, it is because it is materially a priori, i.e., intuited to be the case. But is not this appeal to “self-evidence,” “inconceivability” of the opposite, outmoded dogmatism? At one time, e.g., energy was assumed to be a continuous function. Nowadays, quantum-physics has produced experimental evidence that it is a discontinuous function. Could not the same happen with respect to time? No, it could not, because, as Kant has shown, time is not an object of experience at all; it is a “constitutive condition” of empirical objects. One can experiment with energy, and other empirical functions; one cannot experiment with time, one experiments in time. Before proceeding to a discussion of the functional a priori, let us emphasize that the material a priori has, by us, been defined negatively rather than positively: a materially a priori judgment is neither analytic nor inductive in the ordinary sense. Phenomenologists define the material a priori positively by self-evidence, disclosed in pure “Wesensanschauung” (eidetic insight): the evidence of the judgment is independent of multiplication of instances; as Goethe characterized Galileo’s discovery of the isochronism of the pendulum: “Ein Fall gilt ihm f¨ur Tausend.” But if one argues for the material a priori in terms of self-evidence, one throws oneself open to the accusation of confusing logic with genetic psychology. No doubt, Galileo did not discover the law of the proportionality of fallen height to t2 by the method of “simple enumeration.” But the question of how a law is discovered must not be confused with the question of the logical validation of that law. Being a statement about empirical objects, Galileo’s law is valid only insofar as it applies to a vast number of instances, unless it be deduced from other empirical laws; and insofar as the premisses from which it might be deduced must themselves be empirical, the reference to many instances will always be logically essential, although it may be genetically, in the order of discovery, inessential.

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The material a priori does not, indeed, belong to the province of formal logic. For formal logic the disjunction: either analytic or inductive, is exhaustive, because the formal logician deals only with “ready-made,” i.e., already defined, concepts, leaving the process of constituting concepts, the process of constructive definition, to the concern of “genetic psychology.” A judgment is analytic if it is true (directly or indirectly) by definition, and the raison d’ˆetre of the definition from which analytic truths derives is of no concern to the formal logician. Being aware of this limited scope of formal logic, Kant felt the need of supplementing it by a more comprehensive “transcendental logic” which investigates the process of noninductive synthesis which precedes and renders possible analysis. Accordingly, “transcendental logic” is, to use a characteristic term of Hermann Cohen’s, occupied with the “Urteil des Ursprungs.” If one dislikes the notion of “Bewußtsein u¨ berhaupt” because of its metaphysical “pathos of obscurity,” and holds that, in any proper sense of the term “Bewußtsein,” Bewußtsein is the subject-matter of psychology, not of logic, then, insofar as the subject-matter of transcendental logic is claimed to be Bewußtsein u¨ berhaupt, one may hold that transcendental logic reduces merely to genetic psychology. However, it is a historical fact that what is commonly called genetic psychology has not concerned itself with the problem of the origin of scientific concepts and their articulations in the form of definitions. Insofar it would be at least as improper usage to call this kind of investigation “genetic psychology” as to call it “logic.” Let me concretely illustrate the diﬀerence between formal and transcendental logic, and the place of the material a priori within the latter. I may refer back to the already discussed judgments “time is irreversible” and “time is a continuum.” The formal logician straightforwardly asks you for the definitions of subject and predicate, and once these definitions have been supplied, he can easily determine whether the judgments under analysis reduce to mere exemplifications of laws of logic and are thus logically true. The origin of the definitions themselves is irrelevant for the purpose of formal logic. Definitions, “primitive notions” and axioms (or rather, with respect to the postulational technique developed by Hilbert, axioms definitory of primitive notions) are the basic elements for formal logic. But there is no reason why these basic elements that are taken for granted, presupposed by formal logic, may not constitute a problem for “transcendental” logic. Let us analyze the following judgment: “If A is higher in pitch than B, and B is higher in pitch than C, then A is higher in pitch than C.” If the relation “being higher in pitch” is defined as a transitive relation, then, of course, this judgment is analytic. Or suppose what is defined as transitive is the more general relation “being higher than.” Then the above judgment may be analyzed into the analytic judgment “If A is higher than B, and B is higher than C, then A is higher than C,” and the synthetic judgment that this analytic judgment is applicable to tones as far as

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their pitches are concerned. Is this synthetic judgment, viz. that tones order themselves according to height of pitch, inductive in any ordinary sense? Is it conceivable that there should be found tones that do not order themselves according to height of pitch, or colors that do not order themselves according to lightness, the way it is conceivable that there should be found white crows, or masses that do not obey the law of gravitation? If it is not, thus a formal logician would presumably argue, it is because tones are defined in terms of that transitive relation, just as the integers are (ordinally) defined in terms of the transitive relation “being greater than.” This kind of reasoning befits a formal logician, because, as mentioned in the discussion of the formal a priori, for formal logic the essential can mean only the definitory. But unless one dogmatically dismisses all phenomenological analysis as “psychologism,” one must admit that it is not na¨ıve scholasticism to say “it is of the nature of tones to have that structural property, and therefore they are defined as having it”; the ground of that definition certainly is not inductive in the same sense in which the ground of the definition of whales as mammals is inductive. Also the definition of “being higher than” in terms of the structural property of transitivity may, from the standpoint of “transcendental logic,” be said to presuppose, genetically, the materially a priori judgment “it is of the nature of the relation ‘being higher than’ to be transitive.” Insight (“Wesensanschauung”) is required whenever a formal definition is applied to a specific case and the nature of the case is such that the validity of the application cannot be inductively verified (unless, indeed, the meaning of the term “induction” be stretched to such an extent that “thought experiment” is regarded as an inductive process). Inasmuch as such insights form an essential part of the cognitive process, it should be conceded to transcendental logic—which “transcends” the scopes of formal logic and inductive logic separately—to register judgments expressing such insights as belonging to a separate category, the category of the material a priori. III whether empirical or “eidetic,” may be made, by Any synthetic a conventional act, into an analytic, formally a priori sentence. But it is usually made formally a priori, in order to be taken as functionally a priori, i.e., a hypothetically necessary presupposition, a “procedural means,” as Dewey would say; or, as Kant called these functionally a priori principles, a “Grundsatz,” in contradistinction to an analytically demonstrable “Lehrsatz.” In the case of empirical systems, these sentences are, prior to their being adopted as functionally a priori, empirically synthetic (in part, at least); in the case of conceptual systems they are “eidetically” synthetic. Thus, the postulates sentence,6

6I

use here the term “sentence” instead of the term “proposition,” in order to obviate the objection that a proposition, interpreted as the “Sachverhalt” expressed by a sentence, ceases to be the same when the sentence that expresses it ceases to be synthetic and becomes analytic.

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which implicitly define the “primitive notions” of Peano’s system of arithmetic (e.g., the principle of mathematical induction: if x is an integer, then (∀P){P(0).(∀y)[P(y) ⊃ P(succ(y))] ⊃ P(x)}, or D. Hilbert’s postulates which implicitly define the primitive notions of pure geometry, are, prior to their use as postulates definitory of the primitive notions of the system, eidetically synthetic (materially a priori). In Carnap’s language of Carnap 1928, such postulates might be characterized as “constitutions” (“constitution” taken as a verbal noun, not as a past participle) of the “basic elements” of the respective system. Once the basic elements, or primitive notions, have thus been constituted, they can be reanalyzed in terms of these constitutive postulates, and the latter will then, of course, appear as analytic. But this analysis presupposes the synthetic process in which the primitive notions were first constituted. Insofar as there is no logical necessity in the choice of a specific set of notions as primitive (or postulationally defined), the basic postulates may be treated as conventions or functional necessities. However, it is especially with respect to the postulates of empirical science, that we must be on our guard against the identification of the conventional with the arbitrary. If the basic postulates of physics have methodological value and can be profitably taken as a priori in the functional sense, it is because they have fundamentum in re. Consider, for example, the principle of the conservation of mechanical energy: its formulation presupposes the definition of potential energy as the negative of the space-integral of force: s1 Fdξi −dV = Fdξi ; hence V = − s2

i

But in this definition the existential assumption is implicit that V exists in the sense of being operationally definable as a function of position only, in contradistinction to kinetic energy which is a function of both position and velocity. Now, that the conservation principle, as a consequence of what I like to call a “definition with existential import,” is not a priori in the sense of being analytic is evidenced by the fact that actually all real systems are nonconservative, since, owing to friction, some energy is always lost during the motion of a system. On the other hand, if the principle were a posteriori, it would have to be rejected as false in the face of this fact of non-conservation. But instead the physicist saves it by introducing a new form of energy, viz. heat energy. The mechanical energy that is dissipated by friction is conserved in the form of heat. This extension of the concept of energy—as expressed by the first law of thermodynamics—is, however, to use Poincar´e’s phrase, though conventional, not arbitrary. For its empirical basis is Mayer’s and Joule’s discovery of the quantitative equivalence of heat and mechanical work. If energy, in this extended form, should still fail to be conservative, the physicist would still search for a further quantitative equivalence between the lost amount of energy and some new form of energy. The conservation principle thus func-

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tions as a leading principle: it expresses the physicists’ faith that something is constant in Nature, and this faith progressively “enacts its own verification,” as James would put it. Again, Newton’s second law functions as an a priori principle; it is, in Dewey’s language, “operationally a priori with respect to further inquiry”; it tells the physicist how to measure force, thus prescribing a method. Methods cannot be directly refuted (in fact, the expression “to refute a method” does not even make syntactical sense); insofar methodological principles are a priori. But they may be indirectly refuted, i.e., they may be proven fruitless by the failure of the empirical laws that gave rise to them to be verified; insofar they have fundamentum in re and are open to revision by experience. The empirical fact that led to the methodological postulate expressed by Newton’s second law is that force manifests itself as change of velocity, while before Galileo force was supposed to be the cause of changes of position. As Mach describes the empirical character of Newton’s second law: “Before Galileo, force was known only as pressure. Now, nobody who has not experienced it can know that pressure does at all produce motion, and still less how pressure passes into motion, that pressure does not determine position nor velocity, but acceleration . . . . It is hence not at all obvious that the factors which determine motion (forces), determine accelerations” (Mach 1904, 142). Theoretically, any postulate, whether it be a formal axiom or an empirical hypothesis, can be adhered to, whatever deduction or induction may disclose. For the so-called experimentum crucis, based on the contrapositive mode of inference, is, if taken to apply to singular postulates, an illusion. One cannot deduce consequences from one singular postulate or hypothesis; there is, for the possibility of deduction, required a set of postulates or hypotheses. Hence, what the falsity (whether empirical or formal) of a deduced consequence entails is not the falsity of one definite postulate, but the inconsistency of a set of postulates. Let us symbolize the set of postulates (no matter whether these be formal postulates or empirical hypotheses) by the conjunction p1 . . . . . pn , and the deduced consequences by q1 . . . . . qn . Then ∼ qi entails ∼ (p1 . . . . . pn ). The negation of a conjunction, however, is equivalent to a disjunction of negations, i.e, ∼ (p1 . . . . . pn ) ≡ ∼ p1 ∨ . . . ∨ ∼ pn . We are thus left with the materially indeterminate conclusion ∼ pi , i.e., we are free to choose among alternative falsities; we are free to decide which one of our postulates we want to abandon and which ones we want to continue to adhere to. If, e.g., the planets should be observed to deviate considerably from the elliptic paths determined by Kepler’s laws, one would have to infer, by contraposition, the falsity of either the law of gravitation or the law of inertia or the law of the parallelogram of forces; provided, indeed, one is assured of a complete knowledge of the initial conditions: it may well be that the deviation from the expected path might be ascribed to the disturbing influence of a force exerted by a hitherto unknown

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planet. Since the law of gravitation is less general than either one of the two latter laws, upon which the very possibility of the geometrical construction of motion depends, the physicist would probably choose to abandon or revise it. It thus appears that whether a hypothesis functions as a priori or not depends on the degree of its generality. It is the most general laws that are under all circumstances adhered to as methodological postulates or leading principles, because the very possibility of science depends on their validity. The very possibility of statistical physics, e.g., or the applicability to physics of the calculus of probability, presupposes the validity of the two synthetic axioms that define a probability-aggregate: the axiom of randomness, which defines the non-causal nature of a series of events (in Schlick’s formulation, this axiom states, that a series of events is non-causal “if, in the case of a very long series of observations, each series to be formed out of the diﬀerent events by permutation (with repetition) has the same average frequency (whereby only the series would have to be small in comparison to the total series of observations)”; (cf. Die Kausalit¨at in der gegenw¨artigen Physik, in Schlick 1938, 71) and the axiom of the existence of limits to series of relative frequencies. These axioms are functionally a priori or, as Kant would say, “synthetic a priori principles of experience,” not because they express “apodiktische Wirklichkeitserkenntnis” (as Schlick wrongly interprets the meaning of Kant’s “synthetic a priori”), a priori insights into the nature of reality, but because they are universal “conditions of possible experience” (where “experience” means “science”). Being undoubtedly synthetic (although they are used to define the subject-matter of statistical physics), they are not formally necessary; their necessity is but functional. Kant always uses the terms “necessity” and “universality” as equivalents; and, indeed, one intuitively feels that there is an inner connection between these categories. We may now give a precise meaning to this equivalence, by showing that functional and formal necessity are, though distinct, continuous with each other in the Leibnizian sense of continuity: formal necessity may be interpreted as a limiting case of functional necessity. A formally necessary, or analytic, judgment, we said, is a judgment whose contradictory is inconsistent. But inconsistency is an irreflexive relation, i.e., no judgment can significantly be said to be inconsistent with itself, but only with other judgments. The diﬀerentia of formal inconsistency, in terms of which formal necessity is defined, as contrasted with material inconsistency, in terms of which functional necessity is defined, is, then, the fact that violation of the most general laws, viz., the laws of logic, is involved: pi , denoting a variable member of the set of postulates p1 , . . . , pn , is functionally necessary if it is so general that its negation would entail the negation of a vast number of empirical laws; ∼ pi would then be said to be to a high degree materially inconsistent. pi , now, is said to be formally necessary, if ∼ pi is formally inconsistent; this, however, means that it

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contradicts the laws of logic, and the latter are themselves functionally necessary in the highest degree, insofar as their rejection would force us to abandon all laws whatever. Formal necessity is thus seen to be equivalent to the highest universality of the laws by inconsistency with which functional necessity is defined. It is to be noted, though, that formal necessity is not thereby “reduced” to functional necessity. Inconsistency with the laws of logic remains sui generis, although it can be interpreted as, so to speak, the upper limit of material inconsistency. The laws of logic themselves cannot be said to diﬀer merely quantitatively from functionally necessary laws, such as to be defined by the property of having functional necessity in the highest degree. For functional necessity itself is defined in terms of the logical relation of inconsistency, which is itself defined in terms of the laws of logic. In summary, it should be noted that, although the formal a priori, the material a priori, and the functional a priori are, as categories or epistemological predicates, distinct, they are quite compatible in the sense of being predicable of one and the same sentence. As a matter of fact, the main intent of this analysis has been to mark out the diﬀerent types of epistemological status that accrue to statements in the process of scientific systematization. Synthetic statements, whether they be empirical or materially a priori, are made into analytic statements in order to be taken as “leading principles” or “conventions.” Hypostatization of the categorial distinction between synthetic truth and conventionalanalytic definition into existential separation, such as to think of the statements which are epistemologically qualified by these categories, as of mutually exclusive classes, gives rise to a radical misconception of science. If definitory or analytic statements are of any use in inquiry, if they have, in other words, existential import, and are not idle nominal definitions, it is because they are synthetic in origin. “Being conventionally definitory” and “being synthetically descriptive” (whether descriptive of “matters of fact” or of “eidetische Sachverhalte”) are no doubt distinct predicates; but this distinctness should not mislead us into throwing “conventions” into one basket, synthetic truths into another; for then we bring the puzzles of applicability upon us. And these puzzles should not be belittled, in the manner of Schlick, who dismisses applicability as a pseudo-problem due to lack of semantical analysis. For example, in his article “Gesetz und Wahrscheinlichkeit” (in Schlick 1938), he “solves” the puzzle that chance should be predictable, that there should be, paradoxically, “laws of chance,” by the simple semantical observation that “the rules of probability apply to chance events for the simple reason that what we call chance events are those events to which they apply.” This is like arguing that there is no problem of induction, since the word “nature” is defined as the totality of events insofar as they obey laws. If there were no uniformities in events, if mathematics, consequently, were not applicable to empirical data, we simply would not possess a concept of nature, hence it would not occur to us to set up

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such a definition. The same considerations apply to the possibility of statistical physics: if irregular or non-causal series of events did not, as a matter of fact, exhibit statistical regularity, in the sense that finite series of relative frequencies approximate to limits, we simply would not come to possess that well-defined concept of “chance-event” of which Schlick is talking; or, at least, it would be a purely formal concept without existential reference. Schlick’s quoted statement typically exemplifies lack of “transcendental logic,” i.e., awareness of the fact that concepts that have, in Kantian phrase, any “reference to objects,” do not exist “ready-made,” but are first constructed by synthetic judgment. Why should the principle of mathematical induction apply to integers? Because integers are defined in terms of that principle; integers that do not obey mathematical induction could not legitimately be called integers. But the question is: quid juris the concept of integer, or, which amounts to the same, quid juris that principle which defines the concept? Again: Why should Nature obey conservation-laws? Because Nature is defined in terms of conservation-laws; but the question is: quid juris such a concept of Nature? Why should there be conservation-laws? Why should that definition have existential import? Why should such a concept of Nature “refer to objects,” rather than being an idle fiction? If one forgets that analysis presupposes synthesis, one is led to perform the “ontological leap”: Why should God exist? Because God is defined as existing. But quid juris the concept of a being among whose definitory properties there is existence itself? Why should such a concept “refer to an object”? Why should such a definition be real?

Chapter 3 LOGIC AND THE SYNTHETIC A PRIORI (1949)

The distinguished American logician, C. H. Langford, recently published a paper (Langford 1949), as brief and alarming as what the title, “A Proof that Synthetic a priori Propositions Exist,” claims for it. Although this publication has, to my knowledge, had no noticeable repercussions in the literature of analytic philosophy, it deserves credit for reopening (for open minds, that is) an issue which according to the logical positivists has been decided once and for all. One of the merits of logical positivism which I would be the last one to deny is to have revealed a typical character of philosophical disagreements, viz., the fact that many (or most, or all?) philosophical controversies are rooted in diﬀerences of verbal usage. I am fairly sure that Langford’s paper constitutes, indeed, further confirmation of this positivistic thesis, for a positivist is not likely to deny the cogency of Langford’s proof of the existence of synthetic a priori propositions in Langford’s sense of “synthetic a priori.” He would rather criticize Langford for having suggested by his terminology an accomplishment which he cannot really claim. I hope, therefore, to shed some light on this issue by scrutinizing the Kantian concepts involved in terms of modern logic. Indeed, it seems to me just as futile to discuss the nature of logic without a clear understanding of the distinctions which Kant strove (though rather unsuccessfully) to clarify as to discuss those distinctions without regard (be it ignorance or oblivion) to modern logic. The line of attack against the Kantian theory of geometry most popular with modern analytic philosophers has been to call attention to the distinction between pure and physical geometry, and to show that synthetic a priori propositions disappear from geometry once this distinction is observed. The “axioms” of pure geometry (more aptly called “postulates”) are propositional functions, so are the derived theorems, and the concepts synthetic-analytic, empirical-a priori significantly apply only to propositions. The propositions of pure geometry really belong to logic (and are hence analytic), since they have the form “if the axioms are true, for a given interpretation of the predicate variables (the so-called primitive terms of the axiom set), then the theorems are true, for

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that interpretation.” On the other hand, once the axioms are interpreted, one obtains either analytic propositions or empirical propositions: if specifically an empirical interpretation is given, the interpreted deductive system refers to physical space (physical geometry) or to some other empirical subject matter. Now, Langford admits, in the cited paper, that if the postulates which have to be added to an “adequate” definition of “cube” in order to derive the theorem “all cubes have twelve edges” are propositional functions, then it cannot be supposed that this geometrical theorem expresses a proposition at all, and that if the postulates are interpreted in terms of physical space, the theorem is not (or, at least, may not be) a priori. Yet, he claims it to be a priori if an interpretation of the postulates in terms of visual space is assumed. But thus he must hold that, however mistaken Kant’s views about physical space may have been (specifically the view, suggested by the apparent finality of Newtonian physics, that physical space must necessarily conform to the axioms and theorems of Euclidian geometry), Kant was right in holding that there is such a thing as “pure intuition” which makes a priori knowledge of synthetic geometrical propositions possible. Langford emphasizes, indeed, that his proof “does not require that all theorems of Euclidean geometry should become true a priori with an appropriate interpretation.” But it seems to me evident that his proof, if valid at all, establishes that all theorems of geometry which require for their demonstration postulates (containing specifically geometrical terms), in addition to explicit definitions, are synthetic a priori propositions, provided only that we suppose them to refer to visual (= idealized?) space. Not that Langford is committed to the Kantian view that we are graced with a power of specifically spatial intuition which puts geometrical knowledge into a category by itself. Consider, for example, a proposition of phenomenological acoustics, like “if x is higher in pitch than y, and y is higher in pitch than z, then x is higher in pitch than z.” This proposition, which we are all inclined to regard as necessary on intuitive grounds (the contradictory is inconceivable) is certainly not derivable from logical principles with the help of definitions: the relational predicate involved admits only of ostensive definition; it defies analysis. And if so, then this proposition would be analytic only if the predicate “higher in pitch than” occurred inessentially in it, i.e. if the proposition “for every R, x, y, z, if xRy and yRz, then xRz” belonged to logic—which, of course, it does not. Some will no doubt say: “Granting, for the sake of the argument, that Langford has established the synthetic character of the proposition ‘all cubes have twelve edges’ (i.e., that it is not demonstrable with the help of explicit definitions, which do not beg the question, alone); has he given any argument at all for the claim that it is a priori (necessary)?” Indeed, Langford just takes it for granted that the proposition is not empirical. And I can hear those who hold all necessary propositions to be definitional truths argue: “What could one mean by saying ‘p is necessary’ if one at the same time admits that p is not demonstrable with the help of logical principles alone? Surely, the concept

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of necessity one has in mind must then be purely psychological, something like the inconceivability of the falsehood of p.” Now, for the sake of those who think, for some such reasons as I just outlined, that Langford’s proof stands and falls (or, rather falls) with his by and large discredited Kantian assumption of a faculty of pure intuition of visual space, I want to show as tersely as I can that it must be possible to know some propositions to be necessary before any can be known to be analytic;1 and that if the concept of synthetic entailment2 be held to be psychological, the concept of analytic truth (as applied to natural languages) will be no better oﬀ. First, what sort of a statement does a philosopher intend to make when he says “all necessary propositions are analytic (and, of course, conversely)?” An empirical statement, like “all dogs have the power to bark?” Obviously not. He intends this statement as an explication (to use Carnap’s term) or analysis (to use Moore’s term) of the concept of logical necessity. It therefore rather resembles such statements as “all cubes are regular solids bounded by square surfaces,” “all fathers are male parents.” I shall now attempt to show (a) that a definition of “logically necessary”3 in terms of “analytic” is untenable since it suﬀers from implicit circularity, and (b) that if a semantic system of logic is taken to include its meta-language (meta-logic), it must be held to contain synthetic propositions (which, of course, are not empirical). Suppose we define an analytic statement as one which is demonstrable with the help of adequate definitions and without the use of extra-logical premises (Langford’s “postulates”). The use of the word “demonstrable” in this definition should make it clear that the concept here defined is a semantic one (corresponding to Carnap’s L-truth), since demonstration, as commonly understood, involves the assertion of the premises from which a deduction has been made as true. An apparently equivalent definition is the one preferred by Quine: a true statement is analytic if either it contains only logical constants, or, provided it is written out in primitive notation, descriptive terms occur vacuously in it, i.e., the statement would remain true if the descriptive terms were arbitrarily replaced by others that are semantically admissible in the context. I want to show that if in these definitions “necessary” were substituted for “analytic,” the definitions would become circular on two accounts. It is important to realize, in the first place, that unless the classification of a statement as belonging to logic or an empirical science respectively is to be wholly arbitrary and devoid of philosophical interest, it will not do to define “analytic” as a predicate which accrues to statements relatively to arbitrary 1 See

also chapter 4. is here called “synthetic entailment” has nothing to do with the notion of causal entailment which is allegedly involved in subjunctive conditionals; for causal statements are at any rate empirical, while a statement expressing a synthetic entailment would be necessary. 3 There exists, to be sure, an established usage for the expression “logically necessary” according to which it is synonymous with “analytic” or “logically true”; and I am obviously departing from this usage here. But we obviously need a qualifying adjective in order to distinguish the sense of “necessary” under discussion from other, irrelevant, senses like “factual necessity,” “practical necessity,” and so forth. 2 What

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definitions of their constituent terms. We obviously want to say, for example, that while one could arbitrarily define “man” in such a way that “all men are mortal” would become a logical truth, the statement as commonly understood simply is empirical. But in that case the definitions from which analytic truth derives will have to be characterized as in some sense adequate. What, then, are the criteria of adequacy of definitions? Extensional equivalence is obviously an insuﬃcient criterion, otherwise it would be adequate to define “equilateral triangle” as meaning “equiangular triangle,” and, worse still, any proposition of physics that has the form of an equivalence (an “if and only if” proposition, in other words) could be made out as analytic and thus belonging to logic. This consideration suggests a further negative criterion of adequacy of definitions, to be added to extensional equivalence of definiendum and definiens: the definition should enable the logical demonstration only of such propositions as are not empirical. But to call a proposition nonempirical is the same as calling it necessary, hence the concept of adequate definition which was used to define analytic truth leads us back to the concept of a necessary proposition, and if “necessary,” here, were synonymous with “analytic,” the definition would be viciously circular.4 To illustrate: in constructing a definition of propositional truth, one will be guided by the criterion “to say of a proposition that it is true is equivalent to asserting that proposition” (if and only if “p” is true, then p), i.e., no definition of truth will be accepted as adequate unless it entails the mentioned proposition. Why not choose as a criterion of adequacy the proposition “if p is true, then, if p is asserted, p is believed by the speaker”? The obvious reason is that this proposition is not necessary, i.e., it is conceivable that people should assert true propositions which they fail to believe (say, because they are liars who, contrary to their knowledge, happen to disbelieve true propositions). Let us, now, illustrate the same point with regard to the definition of a logical constant, say, “not.” Why is the definition embodied in the conventional truth-table (if p is true, then not-p is false, and if p is false, then not-p is true) considered adequate? What does it mean to say, in answer to this question, that it “conforms to ordinary usage”? Suppose we wanted to decide which of the following two definitions of the function “I know that p” conforms to ordinary usage: (a) I believe that p, and p; (b) I believe that p, and p is highly probable on the available evidence. We would, or should, argue somewhat as follows:

4 It

may be noted in passing that C. I. Lewis’s definition of analytic truth, supplemented by his identification of analytic and a priori truth, suﬀers precisely from this circularity. (cf. Lewis 1946, Ch. V). Analytic statements are defined as statements derivable from principles of logic with the help of definitions which are not arbitrary terminological conventions but “explicative statements.” Explicative statements are said to be statements to the eﬀect that the intensions of two terms, “P” and “Q,” are identical. But then we are told that “P” and “Q” have the same intension if they are inter-deducible, i.e., if the formal equivalence “(∀x)[Px ≡ Qx]” is analytic!

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the proposition “if I know that p, then p is true” is clearly necessary, in other words, it would be self-contradictory to claim knowledge of false propositions; but according to definition (b) it would be logically possible that false propositions should be known, since a false proposition may be highly probable on the available evidence; hence definition (a), not (b) is correct.5 In order to conform to the ordinary usage of the defined term, a definition, then, must enable the demonstration of such sentences, and only such sentences, involving the defined term as are ordinarily held to express necessary propositions. Accordingly, the test of adequacy of the definition of “not” is that it enables the logical demonstration of certain fundamental necessary propositions involving the defined constant, such as the law of the excluded middle and the law of noncontradiction. And if we argued that what makes these principles necessary is the fact that they are demonstrable with the help of adequate definitions of the logical constants involved, our argument would evidently be circular. Since the definitions of such logical constants do not form part of the logical system as such but belong to the meta-language, it may be reasonable to demand that the criteria of adequacy themselves be formulated in the meta-language. Thus, one would properly distinguish the tautology of the propositional calculus “for every p, p or not-p” from the meta-linguistic statement “every proposition has either the truth-value ‘true’ or the truth-value ‘false”’ (here “true” and “false” are meta-linguistic terms and hence the “either-or” of this statement is diﬀerent from the “either-or” which is used, but not mentioned, in the calculus). But now we face the following situation: unless this meta-linguistic statement is accepted as necessary, no instruments, as it were, are provided for proving that the law of excluded middle of the object-language is analytic; and if the meta-language is not formalized in terms of a meta-meta-language, the meta-linguistic L.E.M. cannot be analytic; and since we cannot go on building meta-meta...meta-languages forever, some meta-language will have to contain an analogue of the object-linguistic L.E.M. which is at once synthetic and necessary. I anticipate the objection that my argument is completely worthless since it proceeds on the assumption that “analytic” is an absolute concept. The opposition might, indeed, demonstrate the absurdity of my argument by comparing it to the following: a body can be said to move only relatively to a referencebody; but unless we knew that the reference-body is absolutely at rest, we could

5 Incidentally,

it seems to me semantically inaccurate to distinguish, as philosophers frequently do, two kinds of knowledge: certain knowledge and probable knowledge. What could be meant by saying “I know with high probability that the sun will rise”? It could not mean that on the one hand I know, but on the other hand it is (or I am) not certain, for that surely sounds self-contradictory. I think what the intended distinction comes to is merely this: sometimes the proposition “I know that p” (which is always empirical regardless of whether p is empirical or not) is certain and sometimes it is only probable on the evidence. In the latter case one might appropriately say “I am not certain that I know p, but it is probable that I do.”

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not be sure that the first body really moves; hence, if we want to say that some body really moves, we shall have to assume that some other body is absolutely at rest, i.e., at rest regardless of what happens to any other body. The point of the analogy would presumably be that it is just as meaningless to call a statement of a given language synthetic relatively to no meta-language at all, as it is to speak, in old Newtonian fashion, of absolute rest (and the same analogy would, of course, hold for “analytic” and “in absolute motion”). To which I reply: If “analytic” is to be short for “analytic in L,” where L is formalized in terms of some meta-language which includes, among other rules, definitions of the defined terms of L, then “analytic” cannot be regarded as an explicatum of the common notion of logical truth, since by a suitable choice of definitions any statement could then be made out as a logical truth. Thus, to escape from this “conventionalism,” as Lewis calls it, one will have to make reference to a privileged meta-language (just as the earth is the privileged reference-frame tacitly referred to in common-sense statements of the form “x moves” or “x does not move”); and in order to mark out this privileged meta-language one will have to introduce precisely that notion of adequate definitions which leads up to necessary propositions defying formal demonstration. Now, to my second argument for the proposition that a definition of logical necessity in terms of analyticity would be circular. Langford defines “analytic,” as most logicians would, in terms of “logical principle”; and “logical principle” is defined, in the paper already referred to, as “principle involved in the extended function calculus.” But how are we to decide whether a given proposition is involved in the extended function calculus? Either we enumerate all such propositions, so that “logical principle” becomes an abbreviation for a finite disjunction of propositions, or else we shall have to state a common and distinctive property of such propositions by virtue of which they are classifiable as “logical principles.” The former method of definition is impossible since (a) the number of propositions belonging to a system of logic that can be fabricated is unlimited, owing to repeated applicability of the rule of substitution, (b) given such an enumeration of propositions which defines “logical principle,” it would be self-contradictory to suppose that one day a new logical principle should be discovered (or manufactured), since this would mean that an element both belonged and did not belong to the same collection. We are, therefore, faced with the necessity of supplying an explicit definition of “logical principle.”6 6 Some may think that this conclusion can be avoided since the general concept of tautology admits of recur-

sive definition, thus: one first enumerates a set of primitive propositions which one calls “tautologies,” and then extends the term “tautology” to any proposition which is derivable from these primitive propositions with the help of special rules of derivation. I think, however, it is perfectly evident that this amounts to a statement about tautologies and not to an explication of the meaning of “tautology.” The very choice of primitive propositions as well as of rules of derivation must be guided by a prior understanding of what a

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It seems that such a definition will have to make use of the notion of logical constant, e.g., “a logical principle is a true proposition containing only logical constants.” But what do we mean by “logical constant”? While I am unable to give a satisfactory explicit definition, I am sure that such a definition would involve the concept of validity (as predicated of deductive arguments), for the following reason: The rules of formal logic contain no descriptive terms (this is the reason why they are called formal rules), hence before they can be applied to the test of the validity of specific arguments, the latter must be formalized; which means that specific descriptive terms in the argument are replaced with variables until a more or less abstract schema is left over. Not all expressions can be replaced by variables, however, since otherwise it could not be said that the argument has one logical form rather than another: in order to have a specific form of argument, we need some constants. How, then, are we to tell which terms may be replaced by variables (of appropriate type) and which may not? Consider, for example, the syllogism “if Socrates is a man, then Socrates is mortal; Socrates is a man; therefore Socrates is mortal.” We see that if in this argument the proper name “Socrates” is replaced by any other name or description of an individual, the resulting argument would still be valid. Hence, instead of considering this specific argument, we consider any argument of the form “if x is a man, then x is mortal; x is a man; therefore x is mortal.” But then we also notice that the validity of the argument would not be destroyed if any other predicates of the first type were substituted for “man” and “mortal”; which leads us to consider any argument of the form “if Px, then Qx; Px; therefore Qx.” And one further degree of abstraction is seen to be possible: it does not matter what the forms of the constituent propositions are; hence we may introduce propositional variables and consider any argument of the form “if p, then q; p; therefore q.” But what prevents us from pushing this process of formalization one step further by introducing a variable whose values are binary connectives (like “if-then,” “and,” “or”), and studying the schema: pCq (where “C” is a binary-connective-variable); p; therefore q? The answer is obvious: we see at once that not all the argument-forms resulting from substitution of a binary connective for “C” are valid (as, e.g., p or q; p; therefore q). It is no doubt such observations that led logicians to stress the distinction between logical constants, as terms on whose specific meanings the validity of an argument depends, and descriptive constants which occur inessentially (to borrow Quine’s term) with respect to the validity of the argument—although they may occur essentially with respect to the factual truth of propositions en-

tautology is. We might want to say, for example, that not enough rules of derivation, or not enough primitive propositions had been laid down, since there are tautologies which could not be derived in the constructed system; but this would be a self-contradictory statement if “tautology” meant “proposition derivable in the constructed system.”

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tering into the argument. But to say that an argument is valid is to say that its conclusion necessarily follows from its premises; which is to say that the implication from premises to conclusion is a necessary proposition. Which completes the vicious circle I have been endeavoring to demonstrate. If the distinction between analytic and synthetic truths rests, as has been suggested, on the distinction between logical and descriptive constants, and if there should exist no sharp criterion by which the two types of expressions could be distinguished, then the distinction analytic-synthetic is less clear-cut than is commonly supposed. Superficially it looks as though the above description of the function of logical constants in formal logic suggested a perfectly simple explicit definition of “logical constant”: a logical constant is a term which cannot be replaced by a variable in the process of formally testing the validity of arguments which contain it. This definition, however, breaks down under the weight of three objections: (a) an expression which occurs essentially in one argument may occur inessentially in another argument. Take, for example, the identity sign. In the argument “x = y; therefore not-x y)” it occurs inessentially, since any argument of the form “xRy; therefore not-not-xRy” is valid (indeed the word “not” is the only expression in this argument which occurs essentially!). On the other hand, in the context “x = y; Px; therefore Py” the identity sign has an essential occurrence, since “xRy; Px; therefore Py” is not generally valid. (b) On the proposed definition, it will depend on the kind of variables that are available for formalizing arguments, whether an expression belongs to the vocabulary of logic or not. Suppose that we introduced symmetric-relation-variables, i.e., variables taking the names of symmetric relations as values, S , S , S , etc. In that case the argument “x = y; therefore y = x” might be regarded as a substitution-instance of the argument “xS y; therefore yS x,” and since the latter is generally valid, “=” would be classified as a descriptive (inessential) constant. But if the variables at our disposal are less variegated, and we can use only generic relation-variables R, R , etc., then the above argument will have to be considered as an instance of “xRy; therefore yRx,” and since this is not a valid argument-form, we could then with equal plausibility (or implausibility) conclude that “=” is a logical constant. This difficulty cannot be avoided either by stipulating that the formalization should be as abstract or generic as possible. For such a stipulation is presumably equivalent to the demand that the range of the variables used for formalization should correspond to the logical type of the values in question. But what is meant by saying that class C is the logical type to which entity x belongs, if not that “x is a member of C” is true, provided it is significant? Thus a criterion of significance would first be required, and, lest we end up with a circular definition of “logical truth,” we would have to formulate it without using the concepts “analytic” and “self-contradictory.” This, however, is a diﬃcult task which has hardly been tackled yet. In fact, the relation between nonsense and self-

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contradiction has not yet been suﬃciently analyzed. (c) Whether a term has an essential occurrence in an argument all depends on whether the argument is written out in primitive notation or contains definable terms. Thus the term “bounded by squares” occurs essentially in the argument “x is a cube; therefore x is bounded by squares,” but vacuously in the argument “x is a regular solid bounded by squares; therefore x is bounded by squares.” In order to decide with finality, then, whether a term belongs to the essential logical skeleton of arguments, we have to presuppose a completely analyzed language. Nowadays we regard, in the light of the impressive reduction accomplished in Principia, arithmetical terms as part of the logical skeleton of the language of the empirical sciences. But before this meta-mathematical analysis took place, it was natural to regard arithmetical terms as descriptive. In fact, in the sense of the word “descriptive” which most readily comes to one’s mind, such terms as “two,” “three,” clearly are descriptive: they designate observable features of collections; we perceive that a collection has three members just as we might perceive a common physical characteristic of its members. Can we say with finality, then, that, say, geometrical terms, like “straight line,” “circle,” cannot be incorporated into the vocabulary of logic, and that geometrical propositions like “any two points uniquely determine a straight line” could never be written out in primitive notation in such a way that the specifically geometrical terms “point,” “straight line,” “lying on,” drop out as inessential and the proposition becomes derivable from logic? For the reasons just stated, it would be a poor rejoinder to say, unlike arithmetical terms, geometrical terms are descriptive of observable features of the world. It is, moreover, somewhat naive to say flatly “arithmetic is reducible to logic,” or “geometry is not reducible to logic.”7 Apart from the consideration that the confines of the language of logic are largely determined by the intuitive acceptability of certain definitions—somebody might, for example, reject Russell’s contextual definition of descriptive phrases in terms of logical primitives as intuitively inadequate, and on that ground hold that statements about mathematical functions are not really reducible to logic—arguments by which such reducibility is commonly proved involve a subtle element of circularity. To say, for example, that “1 + 1 = 2” is really a truth of logic, is to say that it can be derived from the primitive propositions of Principia with the help of the logistic definitions of the specifically arithmetical terms “1,” “2,” “+.” But on what grounds are those definitions accepted as adequate? Either the grounds are intuitive, in which case one is left without a logical argument 7 What

I mean, here, by a reduction of a system of geometry to logic is a derivation of the postulates of such a system from postulates containing only logical terms. It is frequently said that pure geometry, whose propositions are formal implications in which specifically geometrical terms occur vacuously, is part of logic. This is, however, a rather trivial observation since the word “geometry” here has itself a vacuous occurrence: in the same sense pure mechanics, pure deductive sociology, etc., form part of logic.

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by which the reducibility of arithmetic to logic could be proved in the face of objections from the intuitive inadequacy of, say, the definition of the number one, or else one will have to argue: the definitions are adequate in the sense that they enable the demonstration of the theorems of arithmetic from logical premises alone. Considering, then, the vagueness of the expression “logical term,” I do not find the claim that arithmetic (or any other science which contains, prior to reduction to primitive notation, non-logical terms) is logic any more convincing than the contrary claim. The contrary claim might be supported by two reasons: (a) the whole reduction is circular if the underlying definitions can be defended only by showing that they lead to the desired result, (b) the definitions themselves express synthetic propositions, for a biconditional joining a proposition in purely logical notation with a proposition in partly arithmetical notation cannot be analytic unless arithmetic has already been reduced to logic; but if this reduction could be eﬀected independently of such definitions, there would be no need of setting the latter up as instruments of reduction. By way of clarifying this type of argument (which, it should be kept in mind, is not so much intended as a proof of the existence of synthetic a priori propositions than as a proof that, in default of a clear definition of “logical constant,” it is to a certain measure arbitrary whether a given a priori proposition be called “analytic” or “synthetic”), let us suppose that the confines of a system of logic are drawn, in the manner familiar to formal logicians, with the help of recursive, not explicit, definitions. We arbitrarily mark out some terms, say “not” and “or,” as logical constants, and then specify that any term exclusively definable with the help of “not” and “or” is also a logical constant. This kind of definition leaves us, of course, in the dark as to what “logical constant” means, yet it provides an eﬀective procedure for deciding questions of the form “is x a logical constant in the system L?” If contextual definitions are permitted, then such terms as “all,” “and,” “the,” “there is,” etc., would all be logical constants in this system.8 Yet, how would one decide whether a proposed definition of a derived constant in terms of the selected primitives was adequate? It seems to me that the proof of adequacy would rest on the intuitively accepted validity of certain forms of argument. How could we prove, for example, that “class A has exactly one member” is logically equivalent to “there is an x such that x is a member of A and for any y, if y is a member of A then y is identical with x?” All we can say in the end is that the two statements evidently entail each other. Since what is in question is just a definition of “one” with the help of which the extension of the term “logical constant,” and therewith of the term

8 In

order to reduce “the” to those primitive constants one needs, though, the symbol of identity which can be reduced to the primitive constants only if quantification over predicate-variables is permitted.

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“logical truth,” could be increased, it would be circular to attempt to show that this equivalence is analytic. But what guarantee, now, do we have that any new logical constant which we might discover in the process of analyzing arguments would be definable in terms of our logical primitives? What if what appears to belong to the logical skeleton of our language should defy definitional reduction to the selected primitives? If such a situation should arise, two alternatives would be open to us. We might say that the terms in question are logical constants since they evidently have an essential occurrence in some valid arguments, and that therefore statements containing these constants along with already accepted citizens of the vocabulary of logic are logical truths. Or, we might say that such terms are not logical constants, just because they are not definable in terms of our primitive logical constants, and that therefore statements which contain them essentially are synthetic. Take, for example, the intuitively valid argument: x and y are distinct points in a plane; therefore there exists just one straight line (in that plane) which contains both x and y (where “point” and “straight line” and “plane” are not predicate variables but have the customary geometrical meanings). To be sure, I might, using “point” as an undefined term, define a straight line (in a plane) as a class of points which is uniquely determined by any two members of itself, and relatively to this definition, of course, the argument would be formally valid. But such a definition would obviously be questionbegging in the present context of discussion, just as a definition of “cube” as “regular solid with twelve edges” would be question-begging if it were used as a refutation of Langford’s point. If we assume that “point” and “straight line” and “plane” belong to the undefined vocabulary (which assumption is justified in view of the fact that the meanings of these terms are commonly understood only by virtue of ostensive definition), we see at once that this argument is not formally valid: if “point” and “straight line” and “plane” and “x contains y” are replaced by variables, the resulting argument-form is invalid.9 Should we say,

9 It

might be objected that an explicit definition of “plane” could be constructed which would bring the analytic character of this axiom of Euclidean plane geometry into evidence. Suppose we wanted to verify whether a given surface was plane or curved, would we not endeavor to determine whether one and only one straight line could be drawn through any couple of points in it? And does this not suggest the definition of a plane surface as a surface such that any couple of points in it uniquely determines a straight line? The trouble with this attempt at explicit definition is that it lands us in a vicious circle. For one could similarly ask how one would distinguish a straight line from any other type of line? And with the same plausibility one might say that it is that type of line which, in a plane, is uniquely determined by a couple of points. It is, indeed, sometimes said that the geometrical primitives “reciprocally” define each other, but I have never been able to understand how such “reciprocal” definition diﬀers from viciously circular definition. We are then left, as far as I can see, with two and only two alternatives: either the primitives are predicate variables, in which case the axioms are no propositions at all; or else they have an empirical reference through ostensive definition. It should be noted, however, that the contingent character of such empirically interpreted axioms cannot be inferred from the fact that they refer to qualities of sense experience, no more than it follows that “2 + 2 = 4” is contingent from the fact that it is applicable to classes of empirical objects.

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then, that this is a case of synthetic, or material, entailment since the implication is not derivable from what has been defined as “logic,” but that the essential occurrence of the geometrical terms calls for an extension of our definition of “logical constant”? Or should we say that these geometrical terms could never be called “logical constants” at all, just because they are not definable in terms of the accepted primitive vocabulary of logic? Most logicians would seize the latter alternative, in line with the view that geometrical axioms, unlike the postulates of arithmetic, cannot be reduced to pure logic. But then they might also have refused to admit, say, “all” as a logical constant into a truth-functional logic, since it can be defined in terms of “and” only if infinity (expressed by “. . .”) is admitted as a logical concept—and would it not be arbitrary to do so? And arguments involving “all” essentially would, relatively to such a narrow definition of “logic,” have to be counted as synthetic entailments. The same point may well be illustrated further in terms of Langford’s second candidate to the honorable (or decadent?) title of “synthetic a priori truth,” viz. the proposition “whatever is red is colored.” Langford argues that “x is colored” cannot be formally deduced from “x is red,” since a man who is able to understand the meanings of the extra-logical terms in the premises of a formal argument should also be able to understand the meanings of the extra-logical terms in its conclusion (at least if the argument is written out in primitive notation, and does not have some such trivial form as “p, therefore p or q”); and he argues (convincingly, I think) that a man might well understand the meaning of “red” without understanding the meaning of “colored.” I shall oﬀer an independent, though perhaps similar, argument for the view that this proposition is synthetic. It is natural to suppose that the conditional “if x is red, then x is colored” has the form “if p, then p or q,” since to say “x is colored” is to say “x is blue or green or red or ...” But could such a disjunctive definition, as we may call it, ever be written out? Suppose we wrote it out, eliminating the convenient dots, by putting in as a disjunct each and every color that happens to have received a name. And suppose that thereafter we observed for the first time an object that had a color which, unknown as it was, failed to have a name; would we not want to say that this object is colored? We certainly would, yet if “colored” meant what we defined it to mean, we could not say that. I conclude that “colored,” like “red,” must be regarded as a term whose meaning is grasped only through ostensive definition and which therefore belongs to the primitive vocabulary. But in that case the argument “x is red, therefore x is colored” would be formally valid only if all arguments of the form “x is P, therefore x is Q” were formally valid. Have I shown that the true statement “whatever is red is colored” (and all similar species-genus statements like “whatever is round has shape,” “whatever is hot has temperature”) is synthetic (though presumably necessary)? Only if it can be assumed that “red” and “colored” are not logical constants. This

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assumption seems innocent enough: if these are not descriptive terms, what terms are? Yet, in the first place, the customary manner of distinguishing logical terms from non-logical terms as purely syntactic elements of language from denotative elements of language is, as pointed out already, not quite safe: numbers are, up to a certain point, observable properties of collections, still they are regarded as logical concepts; second, if all we can say by way of defining “logical constant” is that logical constants are those terms which occur essentially in some valid arguments,10 then “red” would have to be admitted as a logical constant. To be sure, this word occurs inessentially in many valid arguments, but the same is true, as was shown, of such full-fledged members of the vocabulary of logic as the identity sign. Perhaps the proper conclusion to be drawn from these observations is that it does not make sense to represent the distinction between logical and nonlogical expressions as absolute. Perhaps all that can be significantly said in answer to the question of what a logical expression is, is that an expression functions logically in the context of an argument in which it occurs essentially (in the already explained sense of Quine’s phrase “essential occurrence”). But in that case the whole problem of whether any necessary propositions could fail to be analytic, i.e., certifiable by reference to logical principles alone, is ill-defined. If, on the other hand, the meaning of “logical principle” should be, more or less arbitrarily, made precise by laying down a number of postulates and rules of derivation and defining “logical principle” as any proposition derivable in this system (including the postulates which are derivable from themselves!), then I think Langford’s proof is irrefutable, since any number of necessary propositions (necessary, that is, in terms of ordinary, pre-analytic usage of the term) could easily be produced which, like “all cubes have twelve edges,” cannot be demonstrated except with the help of extra-logical postulates.11 Of course, what inevitably happens is that such a logical system which, judged in the light of the small number of postulates and primitive terms, ap10 One

might think that much embarrassment could be spared by replacing “valid,” here, with “formally valid.” But such a replacement would make the definition grossly circular, since formal validity is defined in terms of a set of formal rules of deduction which cannot be formulated until after a choice of logical constants has been made for the given language. 11 Whether Langford is right or wrong may well depend on the precise extension assigned to the term “definition” in the characterization of analytic propositions as propositions demonstrable with the sole help of definitions. Specifically, if postulates (or, rather, sets of postulates) should be described as implicit definitions of the primitives they contain, then what Langford calls a “synthetic a priori” proposition is for those who use the term “definition” more liberally simply a species of analytic proposition. Once, however, one operates with the concept of implicit definition, the extension of the concept of logical truth is in danger of becoming paradoxically large, especially if implicit definitions are claimed, as by Schlick in Schlick 1925, to prevail not only in formal mathematical languages but also in the language of science. Thus one might hold that, like the primitives of a system of geometry, the primitives of a system of mechanics (“particle,” “mass,” “force,” etc.), are implicitly defined by the postulates of the science (see e.g., Margenau 1935); in fact, Poincar´e has impressively shown how attempts at explicit definition entangle one in logical circles. But the embarrassing consequence of this position is that the laws which appear as theorems in the

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pears rather meager, expands enormously with the help of definitions: just think of the way in which logic in Russell and Whitehead’s magnum opus swallows up the huge system of classical mathematics! But I see no escape from the conclusion, well worth repeating, that those definitions which syntactically function as rules of translation from one universe of discourse to another and thus enable incorporation of more and more material into logic, express (in a sense of “express” suﬃciently clear to go without analysis in this discussion) necessary or a priori propositions; and since the necessity of those propositions is the ground which makes those definitions cognitively acceptable, it would be circular to prove that they are analytic by reference to the very definitions which they are to support.

postulational development of the science would then have to be regarded either as propositional functions or as logical truths!

Chapter 4 ARE ALL NECESSARY PROPOSITIONS ANALYTIC? (1949)

The title question of this paper admits of two diﬀerent interpretations. It might be a question like “Are all swans white?” or it might be a question like “Are all statements of probability statistical statements?” “Are all causal statements, statements of regular sequence?” etc. If these two types of questions were contrasted with each other by calling the former “empirical” and the latter “philosophical,” little light would be shed on the distinction, since what is to be understood by a “philosophical” question is extremely controversial. Perhaps the following is a clearer way of describing the essential diﬀerence: the concept “swan” is on about the same level of clarity or exactness as the concept “white,” and one can easily decide whether the subject-concept is applicable in a given case independently of knowing whether the predicated concept applies. On the other hand, the second class of questions might be called questions of logical analysis, i.e., the predicated concept is supposed to clarify the subjectconcept. They can thus be interpreted as questions concerning the adequacy of a proposed analysis (frequency theory of probability, regularity theory of causation); and the very form of the question indicates that the suggested analysis will not be accepted as adequate unless it fits all uses of the analyzed concept. Now, when I ask, as several philosophers before me have asked, whether all necessary propositions are analytic, I mean to ask just this sort of a question. I assume that those who, with no hesitation at all, give an aﬃrmative answer to the question, consider their statement as a clarification of a somewhat inexact concept of traditional philosophy, viz., the concept of a necessary truth, by means of a clearer concept. I feel, however, that little will be gained by the substitution of the term “analytic” for the term “necessary,” unless the former is used more clearly and more consistently than it seems to me to be used in many contemporary discussions. And I shall attempt to show in this paper that once the concept “analytic” is used clearly and consistently, it will have to be admitted that there are propositions which no philosopher would hesitate to call “necessary” and which nevertheless we have no good grounds for classi-

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fying as analytic. Moreover, I shall show that even if the concepts “necessary” and “analytic” had the same extension, they would remain diﬀerent concepts. To prove this it will be suﬃcient to show that a proposition may be necessary and synthetic. Probably the most precise analysis of the concept of analytic truth is to be found in the logical writings of Carnap. In Carnap 1954a an analytic sentence is defined as a sentence which is a consequence of any sentence (§10). This definition makes the defined concept, of course, relative to a given language (i.e., “p is analytic” must be regarded as elliptical for “p is analytic in L”), since the syntactic concept “consequence” is defined in terms of the transformation rules for a given object-language. Now, it is clear that this definition is constructed with a view to syntactical investigations into the formal structure of artificial languages such as logical calculi and formalized arithmetic. It is therefore not very useful for philosophers who are interested in the analysis of natural languages which are obviously unprecise in the sense that their formal structure cannot be exhaustively described by stating complete sets of formation rules and transformation rules. Also, no philosopher who proposes “analytic” as the analysans for “necessary” could plausibly mean by “analytic” a syntactic concept, i.e., a concept defined for sentences of an uninterpreted language of which it cannot be said that they are either true or false (uninterpreted as they are) but at best—in case the system is complete, that is—that they are either derivable from the primitive sentences or refutable on the basis of the primitive sentences. For necessity of propositions has always been meant as a semantic concept: a necessary condition which any adequate analysis of “necessary” must satisfy is that the truth-value of a necessary proposition does not depend on any empirical facts. Carnap has since constructed a definition of “analytic” in semantic terms, which yields a concept corresponding to the earlier-defined syntactic concept in the sense that any sentence which is analytic in the syntactic sense (e.g., “p or not-p,” where the logical constants “or” and “not” are not defined by truth-tables but occur as undefined logical symbols in the primitive sentences) becomes analytic in the semantic sense once the language to which it belongs is semantically interpreted. A sentence of a semantic system (i.e., a language interpreted in terms of semantic rules) is said to be analytic or “L-true,” if it is true in every state-description.1 This definition is, of course, reminiscent of the old Leibnizian conception of “truths of reason” as those that hold in any possible world. But this semantics is analogously constructed with a view to the investigation of artificial, completely formalized languages. Specifically, the concept of a state-description is defined for a highly simplified molecular 1A

state-description is a class of atomic sentences of such a kind that the semantic rules of L suﬃce to determine whether any sentence of L is true in the world described by this class of sentences. Thus, if L were a miniature language containing two individual constants “a” and “b,” and two primitive predicates “P” and “Q,” the following would be an example of a state-description: Pa and Qa and Pb and not Qb.

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language containing only predicates of the first level, like “cold,” “blue,” etc. Also, the far-reaching assumption has to be made that the undefined descriptive predicates of the language designate absolutely simple properties and are hence logically independent. Otherwise, further analysis might reveal logical dependence, and what appeared before analysis as a “possible” state-description might turn out to be an inconsistent class of sentences. For such reasons, definitions of “analytic” that are fruitful from the point of view of the semantic analysis of natural languages (including scientific language), which is practiced by both the so-called left-wing positivists and the followers of G. E. Moore, have to be sought elsewhere. This does not mean, however, that we must altogether ignore what formal logicians say about the matter. The following definition by Quine, for example, is illuminating: An analytic statement, as ordinarily conceived, is a definitionally abbreviated substitution instance of a principle of logic. Thus, if the word “father” is introduced into the language as an abbreviation for “male parent,” then “All fathers are male” is synonymous with “All male parents are male,” and, assuming that the type “male” is univocal, this statement reduces to a substitution instance of the logical principle “For every x, P, Q; Px and Qx implies Px .” What are we to understand by a logical principle? Following Quine, a logical principle might be defined as a true statement in which only logical constants occur. This definition raises some problems, to be sure. To begin with, the statement, “something exists,” formalized in the familiar functional calculus by “There is an x and a P such that Px ,”2 would express a logical truth, which some philosophers would find diﬃcult to accept. But the paradox will be mitigated if one considers what would be entailed by the elimination of this statement from logic. According to the customary interpretation of the universal quantifier, “(∀x)Px ” is equivalent to “∼ (∃x) ∼ Px.” It can easily be seen to follow that two statements of the form “(∀x)Px ” and “(∀x) ∼ Px ” are incompatible only if something exists. And would it not be paradoxical if it depended on extralogical facts whether two given propositions are incompatible by their form? Second, it is not easy to give a general definition of “logical constant.” It would obviously be circular to define logical constants as those symbols from definitions of which the truth of logical principles follows. Perhaps we have to be satisfied with a definition by enumeration, just as we cannot define “color” by stating a common property of all colors but only by enumerating all the colors that happen to have names. Such a definition would be theoretically incomplete but practically complete enough. If, for example, we mentioned “or,” “not,” “all,” and any term definable in terms of these, we probably would not omit any logical constant that occurs in the familiar logical, scientific, and 2 The

use of a predicate variable here cannot be circumvented since there are compelling reasons for not admitting the various forms of “to exist” as logical predicates.

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conversational languages (I assume, of course, the reducibility of arithmetic to logic). According to some uses, the statement “All fathers are male” would be called analytic, and the statement “All fathers are fathers” would be called an explicit tautology. But it is clear that those who roughly identify analytic truth with truth certifiable by formal logic alone would include explicit tautologies as a subclass of analytic statements. It appears, therefore, convenient to widen the above definition as follows: A statement is analytic if it is a substitution instance of a logical principle or, in case defined terms occur in it, a definitionally abbreviated substitution instance of a logical principle. This definition commits us to acceptance of an interesting consequence, whether we like it or not: if a statement, like “No part of any surface is both blue and red at the same time,” contains undefined predicates (“blue,” “red”), we cannot know it to be analytic unless replacement of all descriptive terms by appropriate variables leaves us with a principle of logic.3 This point will prove to be important in the subsequent discussion. The question might be raised whether logical principles themselves could be called “analytic” on the basis of the proposed definition. Certainly an adequate definition of analytic truth should allow an aﬃrmative answer to this question. What makes me know that it will rain, if it will rain, is the same as what makes me know the law of identity, “if p, then p,” viz., acquaintance with the meaning of “implies” or “if, then.” It sounds admittedly awkward to say of a statement that it is a substitution instance of itself —but perhaps such language is no more uncommon than, say, the use of implication as a reflexive relation. Thus, stretching language somewhat to suit our purposes, as is quite common in logic and mathematics, logical principles like “if p, then p or q” will be said to be their own substitution instances. And when definitional abbreviations are spoken of, not only definitions of descriptive constants, like “father,” are referred to, but also definitions of logical constants, like “if, then.” This convention enables us to say that “not (p and not p)” is a definitional expansion of “if p, then p,” for example. The fact that a principle of logic is analytic leaves it, of course, an open possibility that it might also be necessary in a sense in which synthetic propositions likewise may be necessary. It will be emphasized, in the sequel, that “p is necessary” does not entail “p is analytic,” although the converse entailment undeniably holds. I pointed out that Carnap’s definitions of “analytic” (or “L-true”) are constructed with reference to (syntactically or semantically) formalized languages 3 There

are certain technical details concerned with a fully satisfactory definition of “logical principle,” such as whether a logical principle may contain free variables or whether all variables must be bound. But these questions are unimportant in this context. Thus I shall call “Px or not-Px ” a logical principle, although customarily variables are used to express indeterminateness rather than universality, and such an expression is, therefore, regarded as a function, not as a statement.

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and have therefore a limited utility. But I should not be misunderstood to imply that reference to a given language ought to be, or can be, avoided in the construction of such a definition, if “analytic” is treated as a predicate of sentences at all.4 The same relativity characterizes the definition proposed above, since “analytic” is defined in terms of the systematically ambiguous term “true.” The so-called semantical antinomies (like the classical antinomy of the liar) are well known to arise from the treatment of truth as an absolute concept, i.e., a property meaningfully predicable of any sentence, no matter on which level of the hierarchy of meta-languages the sentence be formulated. What is to be taken as defined, then, is “analytic in the object-language L,” although a schema is provided by the definition for constructing analogous definitions for each level of language. Another appropriate comment on the proposed definition of “analytic” should be made. It is well known that if our language refers to an infinite domain of individuals, there is no general decision procedure with respect to quantified formulas, i.e., no automatic procedure by which it can be decided, in a finite number of steps, whether such a formula is tautologous, indeterminate, or contradictory. For this reason, it might not be possible to decide in a given case whether the formula which results from a statement suspected as analytic when the descriptive constants are replaced by variables is a logical truth. This is again an admitted theoretical defect of the proposed definition, but not a defect that might really prove fatal to the practice of linguistic analysis. For such undecidable formulas (like Fermat’s theorem, for example) are usually complicated to a degree which the formulas resulting from the formalization of controversial “necessary” statements never are. Thus, the formula corresponding to “No space-time region is both wholly red and wholly blue” would be “∼ (∃x)(∃t)[Pxt.Qxt]” (where we might consider surfaces as constituting the range of “x”), and this formula is certainly not logically true, since we can easily find predicates which, when substituted for “P” and “Q,” would yield a false statement. If “analytic” is thus defined as a semantic predicate of sentences of a language of fixed level, the proposed substitution of “analytic” for “necessary” at once raises the question: Does it make sense to speak of analytic propositions? If it does not, then our new concept cannot replace the old concept of necessity, since obviously necessity is intended as an attribute of propositions. If Leibniz, for example, were asked whether “All fathers are male” and “Alle V¨ater sind M¨anner” are diﬀerent truths of reasons, he would undoubtedly deny it. These two sentences express the same proposition, and it is the proposition which is

4 Carnap, indeed, speaks in his semantical writings at times of analytic or L-true propositions. But he would regard this merely as a convenient mode of speaking: “The proposition that . . . is L-true” is short for “The sentence ‘. . . ’ and any sentence that is L-equivalent to ‘. . . ’ is L-true.”

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said to be necessary. Also, a sentence may obviously be analytic at one time and synthetic at another time, viz., in case the relevant semantic rules undergo a change. But nobody who believes that there are necessary propositions at all would admit that a proposition which is now contingent may become necessary, or vice versa. If we adopt the semantic rule “Nothing is to be called ‘bread’ unless it has nourishing power,” then the proposition expressed by the sentence “Bread has nourishing power” is necessary; and this proposition was necessary also before this semantic rule was adopted, although the sentence by which it is now expressed may at that time have been synthetic. However, the method of logical construction shows a way toward construing reference to analytic propositions as an admissible short cut for talking about classes of sentences that are related in a certain way. To say, “The proposition that all fathers are male is analytic,” might be construed as synonymous with saying “Any sentence which should ever be used, in any language at all, to express what is now meant by saying ‘all fathers are male’ would be analytic.” Those who hold, with C. I. Lewis, that analytic truth is grounded in certain immutable relations of “objective meanings,” not aﬀected by accidental changes of linguistic rules,5 could therefore consistently accept a definition of “analytic” which makes this term primarily predicable of sentences. It will, indeed, be my main point against the linguistic theory of logical necessity, to be discussed shortly, that the necessity of a proposition, whether the proposition be analytic or synthetic, is a fact altogether independent of linguistic conventions. On the other hand, I do find C. I. Lewis and those who share his views concerning the nature of analytic or a priori truth (where “analytic” and “a priori” are regarded as synonyms) guilty of a diﬀerent inconsistency. Analytic truth, they say, is certifiable by logic alone; and I have attempted to clarify what this means by defining “analytic” as above. They also say that what makes a statement analytic is a certain relationship of the meanings of its constituent terms. But it seems to escape their notice that these assertions are by no means equivalent; the first implies, perhaps, the second, but the second, I contend, does not imply the first. Consider a simple statement like “If A precedes B, then B does not precede A.” I assume that few would regard this statement as factual, i.e., such that it might be conceivably disconfirmed by observations.6 And if it is not factual, then it must be true on the basis of its meaning. But it seems that just because all analytic statements are true by the meanings of their terms, it has been somewhat rashly taken for granted that whatever statement is true by what it means is also analytic. To see that the above statement is

5 Cf.

Lewis 1946, chapter 5. hope nobody will make the irrelevant comment that it is meaningless to speak of absolute temporal relations, and that it is empirically possible to disconfirm the statement by a shift of reference-frame. Obviously, I assume that the verb “to precede” is used univocally.

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not analytic, in the sense defined, we only need to formalize it, and we obtain “(∀x)(∀y)[xRy ⊃ ∼yRx],” which is certainly no principle of logic. This statement, then, is not deducible from logic; hence if we want to call it necessary (nonfactual), we have to admit that there are necessary propositions which are not analytic. Some will no doubt reply: “If we knew the analysis of the predicate of this sentence, we would have a definition with the help of which we could demonstrate its analyticity.” But then one would at least have to admit that the statement is not known to be analytic. And since we know it to be true on noninductive evidence, it follows that there is a priori knowledge which is not derived from our (implicit or explicit) knowledge of logic. I anticipate the objection that if the above statement is not factual, then at least I cannot be certain that it is synthetic; after all, I have no ground for asserting that the relation of temporal succession is unanalyzable. But I fully admit that, as Carnap has recently emphasized, we cannot be certain that a given true statement is not analytic unless we assume that our analysis has reached ultimately simple concepts. At least this would seem to be correct as far as nonfactual statements are concerned. What I maintain is only that, if by a necessary proposition we mean a proposition that is true independently of empirical facts (or that is not disconfirmable by observations), then a necessary proposition may be synthetic, and that therefore “analytic” will not do as an analysans for “necessary.” I propose to show, now, that there is a temptation to beg the question at issue in trying to prove that a necessary proposition like the above follows from logic after all. Obviously, to oﬀer such a proof would amount to the construction of a definition of “x precedes y” with the help of which the asymmetry of this temporal relation could be formally deduced. But one could not significantly ask whether a proposed definition is adequate unless one first agreed on certain criteria of adequacy, i.e., propositions which must be deducible from any adequate definition. Thus, most philosophers would agree that no definition of “xPy” (to be used as an abbreviation for “x precedes y”) could be adequate unless it entailed the asymmetry of P. If it should leave this open as a question of fact, it would be discarded as failing to explicate that concept we have in mind. Now, by enumerating all the formal properties which P is to have, one could not construct a definition suﬃciently specific to distinguish P from all formally similar relations with which it might be confused. If I define P as asymmetrical, irreflexive, and transitive, the relation expressed by “x is greater than y,” as holding between real numbers, would also satisfy the definition. But there is a simple device by which uniqueness can be achieved. I only have to add the condition, “The field of P consists of events.” It is easily seen that with the help of this definition our necessary proposition reduces to a substitution instance of the logical truth, “xPy . [(xPy ⊃ ∼ yPx).Q] ⊃ ∼ yPx” (where “Q” represents the remaining defining conditions for the use of “P”). But is

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it not obvious that acceptance of the definition from which the asymmetry of temporal succession has thus been deduced presupposes acceptance of the very proposition “Temporal succession is asymmetrical” as self-evident? This way of proving that the debated proposition is, in spite of superficial appearances, analytic, is therefore grossly circular. I should say, then, that such propositions as “The relation of temporal succession is asymmetrical, transitive, and irreflexive,” “No space-time region is both wholly blue and wholly red,” are necessary, but that nobody has any good ground for saying they are analytic in any formal sense.7 In general, this seems to me to be true of two classes of necessary propositions, of which the first asserts the impossibility for diﬀerent codeterminates (i.e., determinate qualities under a common determinable quality) to characterize the same space-time region, and the second the necessity for certain determinables to accompany each other. A classical representative of each group will be selected for discussion, viz., “Nothing can be simultaneously blue and red all over,” from the first, and “Whatever is colored, is extended,” from the second. I have already insisted that the statement “∼ (∃x)(∃t)(blue xt .red xt )” is not deducible from logic. Indeed, if “blue” and “red” designate unanalyzable qualities, it is diﬃcult to see how analysis could ever reveal that this statement is a substitution instance of a logical truth. Perhaps, however, the above statement (S ) could be formally demonstrated as follows. The sort of entities to which colors may be significantly attributed are surfaces, no matter whether they may be located in physical or perceptual space. But, so the argument runs, if I say “x is blue at t” and also “y is red at t” (where the values of the variables “x” and “y” are names of surfaces), I have already implicitly asserted “x y”; in the same way as simultaneous occupancy of diﬀerent places is tacitly regarded as a criterion of the presence of diﬀerent things at those places. If the proposition here asserted is formalized, we obtain: (∀x)(∀y)(∀t)(if blue xt .red xt , then x y) (T ). Obviously, S follows from T , hence we may say that “if T , then S ” is analytic. But thus we would first have to prove that T is analytic before we could assert that S is. As I follow the rule, “Any proposition is to be held synthetic unless it is derivable from logic alone,” I hold T to be synthetic until such time as conclusive proof of the contrary is produced. And the same applies, of course, to S . Such left-wing positivists as practice an “informal” (or “non-formal”?) method of linguistic analysis will probably disown this kind of discussion as too formal. As I promised, an examination of their linguistic theory of a priori truth is to follow. At the moment I only want to point out that I would not find it enlightening to be told: “S is obviously analytic, since in calling a given 7I

postpone examination of the familiar argument of the “verbalists” that such propositions are analytic in the sense that anybody who denied them would be violating certain linguistic rules.

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part of a surface ‘red’ we already implicitly deny its being blue. ‘Non-blue’, that is, forms part of the meaning of ‘red’.” I do not find this easy argument in the least cogent, since the only meaning I can attach to the statement “ ‘Non-blue’ is part of the meaning of ‘red’ ” is just “ ‘x is red at t’ entails ‘x is not blue at t’,” and the question at issue is just whether such an entailment may be regarded as analytic (or formal). Surely, “non-blue” could not be an element of the concept “red” in the sense in which “male” is an element of the concept “father.” Otherwise it would be diﬃcult to understand why any intelligent philosophers should ever have held it possible that there should be unanalyzable qualities, and specifically that color qualities should be such. Next, let us consider the statement, “If x is colored, then x is extended,” which may be classified together with such necessary propositions as, “if x has a pitch, then x has a degree of loudness,” “if x has size (i.e., length, area, or volume), then x has shape.” Which determinate forms of these determinables are conjoined in a given case is contingent, but that some determinate form of the second should accompany any given determinate form of the first is generally held to be necessary. Here again, the reason for our inability to deduce these propositions from pure logic would seem to be the fact that the involved predicates cannot be analyzed in such a way as to transform the propositions into tautologies; they can only be ostensively defined. I shall examine two counterarguments with which I am familiar. To what else, it is asked, could colors be significantly attributed except surfaces?8 If to nothing else, then only names or descriptions of surfaces are admissible values of “x” in the debated universal statement. But then each substitution instance is analytic, since it has the form, “If this surface is colored, then it is a surface,” which reduces to, “If this is a surface and colored, then it is a surface.” And therefore the universal statement itself, which may be interpreted as the logical product of all its substitution instances, must be analytic. Notice that a similar argument also would prove, if it were valid, the analytic nature of the other necessary propositions of the same category. Pitch can be meaningfully attributed only to tones; it is not false so much as meaningless to say of a smell or feeling that it has a given pitch. In fact, we mean by a tone an event characterized by pitch, loudness, and whatever further determinables be considered “dimensions” of tones. And to say, “If a tone has a certain pitch, then it has a certain loudness,” is, then, surely analytic. But such arguments beg the question. The statement, “Only surfaces can significantly be said to have a color,” diﬀers in an important respect from such statements as “Only animals (including human beings) can significantly be

8 There

may be some who wish to defend the possibility of colored points. Whether such a concept is meaningful is, however, a question of minor importance in this context, since we can easily stretch the usage of “surface” in such a way that points become limiting cases of surfaces.

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called fathers,” or “Only integers can significantly be called odd or even.” For the latter statements involve analyzable predicates and may well be replaced by the statements, “Fathers are defined as a subclass of human beings,” “Oddness and evenness are defined as properties of integers,” while we cannot assume, without begging the very question at issue, that “x is colored” entails by definition “x is a surface.” Similarly, if “x is a tone” is short for “x has a pitch and x has a degree of loudness and . . . ,” then to say that pitch is significantly predicable only of tones is to say that pitch is significantly predicable only of events of which loudness is also predicable. But this semantic statement cannot be replaced by the syntactic statement “‘x has pitch’ entails by definition ‘x has loudness,”’ unless the question at issue is to be begged.9 The second argument in support of the thesis that a necessary proposition like “If x is colored, then x is extended” is analytic can be stated very briefly. If we had the analysis of “x is colored,” we could deduce the consequent from the antecedent and would therefore see that the connection is analytic. Here my reply is twofold. (1) Nothing has been proved that I wish to deny. I contend only that there are necessary propositions, i.e., propositions which are known to be true independently of empirical observations, which are not known to be analytic. One might, indeed, insist that no proposition can be known to be necessary before it is known to be analytic. But I propose to show shortly that this view is untenable. (2) If it is stipulated in advance that an analysis of the antecedent will be correct only if it enables deduction of the consequent, it is not surprising that any correct analysis of the concept in question will reveal the analytic nature of the statement. Consider the following parallel. Everybody would agree that the proposition expressed by the sentence, “if x = y, then y = x,” is necessary; quite independently of our knowledge of logic, one feels that it would be self-contradictory to deny any substitution instance thereof. But as long as the relation of identity of individuals remains unanalyzed, there is no way of deducing it from logic. “If xRy, then yRx” is not true by its form. Now, one will be perfectly safe in claiming that this proposition will turn out to be analytic once the involved relation is correctly analyzed. For formal deducibility of this proposition from logical truths will be one of the criteria of a correct analysis of identity. Indeed, if Leibniz’s definition of identity,

9 It

would be irrelevant to point out that pitch is physically defined in terms of frequency, which is by definition a property of waves with definite amplitude; and that the physical definition of loudness is just amplitude. In the first place, it is causal laws that are here improperly called “definitions”: pitch may be produced by air vibrations of definite frequency, but nobody means to talk about air vibrations when referring to pitch. Let it not be replied that if I am not talking about such inferred physical processes I must be discussing an empirical law of psychology concerning correlations of sensations of pitch with sensations of loudness. I use the word “pitch” as it is used in such sentences as “The pitch of the fire siren periodically rises and falls”: “pitch,” here, refers to a power of producing certain auditory sensations—if the phenomenalist analysis of material object sentences is correct—and such a power would exist even if nobody actually had any auditory sensations.

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(∀P)(Px ≡ Py), is used, the symmetry of identity becomes deducible from the symmetry of equivalence, which is in turn deducible from the commutative law for conjunction. Is it my contention, then, that even a “formal” statement, as it would commonly be called, like, “for any x and y, if x = y, then y = x,” is a synthetic a priori truth? This would, indeed, amount to going more Kantian than Kant himself; for, on the same principle, it could be argued that all logical truths, which Kant at least conceded to be analytic, are synthetic. Take, for example, the commutative law for logical conjunction, just mentioned. Obviously, I cannot prove that “(p and q) ≡ (q and p)” is tautologous, unless I first construct an adequate truth-table defining the use of “and.” But surely one of the criteria of adequacy for such a truth-table definition consists in the possibility of deriving the commutative law as a tautology. If, for example, a “T” were associated with “p and q” when the combination “F T” holds, and an “F” when the combination “T F” holds, the resulting definition would be rejected as inadequate just because it would entail that the commutative law is not a tautology. Indeed, I should belabor the obvious if I were to insist that the laws of logic are not known to be necessary in consequence of the application of the truth-table test, but that the truth-table definitions of the logical connectives are constructed with the purpose of rendering the necessity of the laws of logic (or at least of the simpler ones, like the traditional “laws of thought”) formally demonstrable.10 But my point can be made far more clearly if the term “synthetic a priori” is not used, since it is used neither clearly nor consistently in Kant’s writings. Philosophers who regard “analytic” as the only clear analysans of “necessary” are inclined to hold that we have no good ground for calling a given proposition necessary unless we can formally deduce it from logic. This, however, amounts to putting the cart before the horse. In most cases it is impossible to deduce a proposition from logic unless one or more of the constituent concepts are analyzed—as I have already illustrated more than once. But we accept such an analysis as adequate only if it enables the deduction of all necessary propositions that involve the analyzed concept. We therefore must accept some propositions as necessary before we can even begin a formal deduction.11 One more illustration may be helpful to clarify my thesis. Several logicians are at the present time engaged in the construction of a definition of the central concepts of inductive logic, viz., confirmation and degree of confirmation. But their analytic activities would be altogether aimless if they did not lay

10 I

shall nonetheless belabor this point at some greater length in the sequel. here use “formal deduction” in the sense of “deduction from logical truths alone,” not in the sense of “deduction by (with the help of) logical rules.” In the latter sense, empirical propositions are, of course, likewise capable of formal deduction. 11 I

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down beforehand certain criteria of adequacy, such as the following: the degree of confirmation of a proposition relatively to specified evidence does not vary with the language in which the proposition is formulated; hence, if “degree of confirmation” is treated as a syntactic predicate of sentences, logically equivalent sentences should have the same degree of confirmation relatively to the same evidence. Similarly, if evidence E confirms hypothesis H, and H is logically equivalent to H , then E must also confirm H . Unless these propositions are accepted as intuitively necessary—or, if you prefer, “true by the ordinary meaning of ‘to confirm”’—by all competent inductive logicians, the latter will never agree as to what definition of the concept is adequate. It might be suggested that all we could mean by calling such propositions “necessary” is that if we had suitable definitions we could formally deduce them. But to say of a definition that it is “suitable” is to speak elliptically: suitable for what? Evidently they must be suitable for deducing those very propositions. The proposed analysis of “necessary,” then, reduces to the following: p is necessary if and only if with the help of definitions that enable the deduction of p, it is possible to deduce p. It hardly needs to be explicitly concluded that on the basis of this analysis any proposition would be necessary. Carnap and his followers will undoubtedly protest against this analysis of what they are doing. Those criteria of adequacy which I interpret as preanalytically necessary propositions they would simply call conventions in accordance with which a definition is to be constructed. I should not, of course, deny that a logical analyst may specify such criteria of adequacy without committing himself to any assertion of their necessary truth. Just as a theoretical physicist who is more interested in elegant mathematical deductions than in the discovery of experimental truth may work on the problem of constructing a theory from which some arbitrarily assumed numerical laws would follow, so the logical analyst may formulate his problem merely as the construction of a definition which will satisfy some arbitrarily stipulated conditions. But, to spin this analogy a little further, just as nobody would regard that physicist’s work as being in any way relevant to physics, so the logical analyst’s constructions will have no relevance to the problems of analytic philosophy if the criteria to which they have to conform have no cognitive significance. It might be replied that such conventions are not held to be arbitrary; that, on the contrary, their choice is limited by the dictates of intuitive evidence. But in that case I suggest that what may properly be called a “convention” is the act of selecting some necessary propositions involving the concept to be analyzed as criteria, and not the object selected; the latter is a proposition, and there is no more literal sense in calling a proposition a “convention” than there is in calling a color a sensation or in calling a murdered bird a “good shot.” The results so far obtained may also throw some light on the status of socalled explicative propositions, which occupy a prominent place in analytic

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philosophy. Thinking of the literal meaning of the word “analytic” (dividing, separating), it is, of course, natural to suppose that explicative propositions, like “A father is a male parent,” are analytic. But are they analytic in the sense of being deducible from logic? I want to call attention to the consequences of the triviality that unless certain definitions are supplied, “A father is a male parent” is no more deducible from logic than, say, “A father is a mature person with a keen sense of responsibility.” Relatively to the definition “father =d f male parent,” our explicative statement becomes obviously deducible from the law of identity. But “father” could arbitrarily be defined in such a way that the explicative statement which we regard as necessary would become synthetic, and other statements involving the subject “father,” normally interpreted as empirical statements, would become analytic. Only, such definitions would be rejected as inadequate (in traditional terminology, as merely nominal, not real). We have to admit, then, that by an adequate definition of “father” we understand one with the help of which necessary statements that involve the word “father,” and only such statements, become formally deducible or analytic. It follows that to say of a statement that it is necessary is diﬀerent from saying that, relatively to such and such transformation rules, it is analytic. It is now time to face the objections I expect from the camp of the Wittgensteinian “verbalists.” “Your point is trivial,” they will say. “Nobody has ever maintained that necessary propositions formulated in natural, non-formalized languages are analytic in the sense of being logically demonstrable on the basis of explicitly formulated semantic rules. When we assert that a necessary statement is the same as an analytic statement we use the word ‘analytic’ in the broader sense of ‘true by virtue of explicit or implicit rules of language.”’ The question I now want to raise is: What precisely is meant by an “implicit rule of language”? It is not any sort of insight or intuition, according to the verbalists, which makes a man know the proposition, “If A precedes B, then B cannot precede A” (p), but merely an implicit rule governing the usage of the verb “to precede.” Presumably this means that people familiar with the English language follow the habit of refusing to say “B precedes A” once they have asserted “A precedes B”—provided, of course, that they are serious and mean what they say. Now, it seems to me very obvious (as it may have seemed to such anti-verbalists as Ewing) that this is a grossly incorrect account of what makes a proposition like p necessary. Just suppose that the linguistic habits of English-speaking people changed in such a way that the verb “to precede” came to be used the way the verb “to occur at the same time as” is now used. People would then be disposed to say, on the contrary, “If A precedes B, then B must precede A.” If the verbalist theory were correct, the proposition expressed by the sentence “if xPy , then not-yPx ” would not be necessary, but in fact self-contradictory, in such a changed sociological world. To be sure, the proposition which was formerly expressed by this sentence would remain nec-

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essary, as the verbalist is certain to point out. But if the modality of p is thus invariant with respect to changes in its sentential expression, in what sense can it be said that the modality of a proposition “depends upon” linguistic rules? Notice that I am not misinterpreting the verbalist thesis, as some may have been guilty of doing, to assert that necessary propositions are propositions about linguistic habits and hence a species of empirical propositions. Of course, nobody would maintain (I hope) that in asserting p one makes an assertion about implicit linguistic rules or linguistic habits. What I take the verbalists (like Malcolm, for example) to claim is that the existence of certain linguistic habits relevant to the use of a sentence S is a necessary and suﬃcient condition for the necessity of the proposition meant by S . As I think such a fact is neither a suﬃcient nor a necessary condition for the necessity of a proposition, I reject the verbalist analysis of what a necessary proposition is. It is tempting to regard the existence of a certain linguistic habit relevant to some constituent expressions of S as a suﬃcient condition for the necessary truth of what S means, through some such reasoning as this: If there exists a verbal habit of applying the word “yard” to distances of three feet and only to such distances, then the proposition expressed by “Every yard contains three feet” is identical with the proposition that every yard is a yard; hence, given that linguistic convention and no further facts at all, the truth of the proposition follows. To detect the flaw in this argument we only need to ask, “follows from what?” What is tacitly assumed is that the law of identity, of which the proposition “every yard is a yard” is a substitution instance, is a necessary truth. If it were not, no amount of linguistic conventions would suﬃce to make any proposition necessary. The verbalist may reply that the law of identity (if p, then p) itself derives its necessity from a certain linguistic habit as to the usage of the expression “if, then.” And I would similarly maintain that the existence of such a habit is at best a suﬃcient ground for saying that “the proposition expressed by ‘if p, then p’ is identical with the proposition expressed by ‘not (p and not-p)’, and hence the first proposition is necessary if the second is.” And how could it be maintained that the existence of a certain linguistic habit is a necessary condition for the necessity of a given proposition? If linguistic habits were to change in such a way that, say, a length of two feet came to be called a “yard,” then, of course, the proposition now expressed by the sentence “Every yard contains three feet” is false, and hence not necessary. But surely the proposition which was formerly expressed by that sentence remains necessary? That proposition is eternally necessary, if you wish, in the sense that any sentence which happened to express it would be true independently of empirical facts, including the sociological facts which the verbalists call “implicit rules.” If the rules by which a given sentence now expresses a proposition p were to change in such a way that the same sentence came to express a diﬀerent proposition, p would still be necessary if it ever was.

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Notice that “If not A, then the proposition expressed by S is not necessary” is not synonymous with “If not A, then S does not express a necessary proposition.” If “A” refers to the existence of certain linguistic habits by which the meaning of the sentence S is determined, then the latter statement may be true. But the first statement would be false, since it does not depend on the contingent verbal expression of a proposition whether the latter is contingent or necessary. This would remain the case even if “The proposition p is necessary” were a mere mode of speech, short for “Any sentence which meant p would be necessary.” If linguistic rules change in a certain way, a given sentence may cease to mean p; but it will still be true that it would be necessary if it did mean p. I may as well take the opportunity to call attention to a neat paradox in which the verbalist thesis entangles itself. It asserts the synonymity of the following two statements: (A) it is necessary that every yard contains three feet; (B) “every yard contains three feet” (S ) follows from the rules governing usage of the constituent terms. But rules, especially implicit rules (= linguistic habits), are not propositions from which any proposition could follow. Hence B should be modified as follows: S follows from the proposition asserting the existence of those rules (abbreviate this existential proposition by “S ”). Now, S is empirical, and whatever follows from an empirical proposition, in the ordinary sense of “to follow from,” is itself an empirical proposition;12 which contradicts the original assumption. On the other hand if the statement which A asserts to be necessary is necessary, then it either does not follow from any empirical proposition (viz., if “to follow” is used in the sense in which it is nonsense to make a statement like “‘If it is hot, then it is hot’ follows from ‘It is cold now”’), or else it vacuously follows from any empirical statement. But in the latter case it will be true independently of what linguistic habits happen to exist, if true at all. And therefore B might as well be changed into “‘Every yard contains three feet’ follows from the rules which governed usage of ‘yard’ 50,000 years ago.” In case this paradox should be held to apply only to an unfortunate formulation of verbalism, I proceed to advance a more serious argument against verbalism. To say that it is an implicit semantic rule to apply “B” to anything to which “A” is applicable is presumably equivalent to saying “People who are acquainted with the language never refuse application of ‘B’ to anything to which ‘A’ is applicable.” But how could it be maintained that observation of such a habit is suﬃcient ground for holding that the proposition “If anything 12 It

is, of course, one thing to argue that, on the verbalist theory, necessary propositions are really a species of empirical propositions, and another thing to argue that modal statements of the form “p is necessary” are empirical, if the verbalist theory is correct. I am not sure whether the latter would amount to a pertinent criticism of the verbalist theory, since I am not convinced that “It is necessary that p” entails “It is necessary that it is necessary that p.”

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is A, then it is B” is necessary? Is it not easily conceivable that people use language that way because they firmly believe that whatever has in fact the property A also has in fact the property B? How, then, can observation of such habits be a reliable method for distinguishing necessary propositions from empirical propositions? I notice, for example, that people apply the word “hard” to certain things. Although I have never troubled to carry out the experiment, I am quite sure that if I asked anybody who calls a thing “hard” whether that thing is weightless, he would say “of course not.” But I would not hence infer, and I doubt whether any verbalist would, that the proposition “Nothing that is hard is weightless” is necessary. The verbalist may reply, “You have oversimplified my thesis. To make the sort of observations you describe is not enough. In order to be sure that ‘if A, then B’ is a necessary proposition, you must moreover get a negative reply to the question: “Can you conceive of an object to which you would apply ‘A’ and would refuse to apply ‘B’?”’ This rejoinder, however, amounts to an unconditional surrender of verbalism. If the final test of necessity is the inconceivability of the contradictory of p, then what linguistic rules happen to be followed by people is irrelevant to the question whether p is necessary.13 A knowledge of linguistic rules is necessary only for knowing what proposition it is that a given sentence is used to express. Once this is determined, we all discover the necessity of the proposition in an intuitive manner, viz., by trying to conceive of its being false, and failing in the attempt. Before I embarked on a critique of the linguistic theory of logical necessity, I endeavored to show that “p is necessary” (in the sense of “p is true independently of empirical facts” or, in the Leibnizian language revived in Carnap’s pure semantics, “p is true in any possible world”) cannot be synonymous with “p is analytic,” since the analytic nature of a proposition is presumably knowable by formal demonstration, while an infinite regress would ensue if formal demonstration were the only available method of knowing the necessity of a proposition. In conclusion I shall apply this thesis to the propositions of logic, i.e., true statements containing no descriptive terms (Quine). It is tempting to suppose that with the help of the truth-table method such fundamental proposi13 In

an unpublished paper by a Wittgensteinian friend of mine I have seen the following analysis of the verbalist theory that all necessary propositions are verbal: “Whatever the sentence or combination of signs may be which expresses a given necessary proposition, it is always possible to ascertain the truth of the proposition by ascertaining the syntactic and part of the semantic rules which govern the constituents of the combination.” This statement seems to me to be equivalent to the statement, “If S expresses a necessary proposition, then, in order to know that S is true, it is suﬃcient to know what proposition it expresses.” If this is what the verbalist thesis amounts to, I have no quarrel with it at all; but I should say that “verbalism” is in that case merely a redundant, and moreover misleading, name. Actually, however, I think verbalists want to assert more than this; they want to assert that the necessity of a proposition is somehow produced by linguistic conventions, and this I hold to be a fallacy. That S expresses a necessary proposition is, of course, a consequence of linguistic conventions, simply because it is a consequence of linguistic conventions that it expresses the proposition which it does express. But the verbalists slip in their inference that the necessity of the proposition expressed by S is a result of linguistic conventions.

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tions of logic as the law of the excluded middle or the law of non-contradiction could be shown to be true in any possible world (or, using more formal language, true no matter what the truth-values of their propositional components may be) in purely mechanical fashion, without any appeal to intuitive evidence. Although this view may have the weight of authority behind it, I consider it gravely mistaken. The lights of intuitive evidence can be turned oﬀ only if the T’s and F’s of the truth-tables are handled as arbitrary symbols with no meaning at all. But in that case one obviously does not establish, for example, that “p or not-p” expresses a proposition true in all possible words; one only establishes the far less interesting syntactical theorem that it is a T-formula—which is a result of the same order as, say, that “x2 = 4” is a quadratic equation. In order to establish the semantic theorem first mentioned, I have to interpret “T” and “F” as meaning “true” and “false” respectively. Once this is done, the primitive truth-tables for the primitive connectives “not,” “or” are really semantic rules.14 What, now, is the principle of selection from all the formally possible semantic rules or truth-table definitions? It would seem to be the following: a semantic rule is adequate if it enables the demonstration of the T-character of those basic propositions, like the laws of non-contradiction, excluded middle, etc., which we already know to be necessary in the sense of being true no matter what the empirical facts may be. A keen student of logic should laugh in his teacher’s face if he were told that with the help of the truthtables the “laws of thought” which we always take for granted can be formally demonstrated as necessary propositions. For he should quickly apprehend that in deciding to assign to each elementary proposition at least and at most one of the two truth-values “true” and “false,” one has already assumed the law of the excluded middle and the law of non-contradiction. I do not, of course, deny that the law of the excluded middle, or any of the similarly simple laws into which it is transformable, can be formally demonstrated as a T-formula (or better, tautology, to use the semantic term) without circularity, if only the distinction between object-language and meta-language is observed. What I claim is that its necessity is not known in consequence of such a formal test; that, on the contrary, the semantic rules which render it demonstrable are chosen in such a way that those and only those formulas will turn out as T-formulas which express propositions that are materially known to be necessary truths or follow from such propositions in an axiomatically developed logic. Similar comments would apply to state-description tests of necessity, if such should be proposed. The definition of a necessary proposition as one that holds in any state-description is, of course, formally unobjec-

14 Carnap

interprets them, in Carnap 1942, as truth-rules; but since he accepts Wittgenstein’s principle that to know what a sentence means is to know the truth-conditions of the sentence, he would undoubtedly agree with the above interpretation.

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tionable. But unlike such definitions as “A square is an equilateral rectangle” it does not indicate a method of verifying that the definiendum applies in a given case. The formation rules defining “state-description” are deliberately constructed in such a way that no state-description can be incompatible with such recognized necessary propositions as the law of non-contradiction. One only needs to refer to the “stipulation” that a state-description is to contain any atomic sentence that can be formulated in the given language or its negation, but not both!

Chapter 5 NECESSARY PROPOSITIONS AND LINGUISTIC RULES (1955)

1.

Are there Necessary Propositions?

Logical empiricism, a powerful and in many ways sanitary philosophical movement which was started by the “Vienna Circle,” propagated in England mainly through Wittgenstein, and in the United States mainly through Carnap, has always been committed to some kind of “linguistic” or “conventionalist” theory of necessary propositions, though it would be diﬃcult to pin down a party line as regards the precise form of such a theory. Such jargons as “the laws of logic are rules of the transformation of symbols,” “all a priori knowledge consists in decisions concerning the use of symbols,” are well known to students of logical empiricism. “Logic formulates rules of language—that is why logic is analytic and empty,” writes a famous logical empiricist. In a sense this theory denies that there is such a thing as a priori knowledge, knowledge of necessary propositions. If, as Schlick wrote, “7 + 5 = 12” is just a rule of symbolic transformation, telling us that we may interchange “12” and “7 + 5” in any context, and a proposition is something that is true or false and that may be believed or disbelieved, then this equation does not express a proposition. To say “I know (indeed, I know for certain) that 7 + 5 = 12” would be either to misuse the word “know” or to use it in a diﬀerent sense from what might be called “propositional knowledge”: it would be a case of knowing a rule, like knowing a rule of chess, and therefore the object of knowledge in this sense of the word would not be a specially exalted kind of truth, traditionally called “necessary” or “a priori” truth. In this sense, it would not be incorrect to ascribe to the logical empiricism of the Vienna Circle and its descendants the view that all knowledge in the sense of propositional knowledge—in contrast to knowledge in the sense of acquaintance as well as knowledge in the sense of knowing how to do something (“he knows how to play the piano,” e.g.)— is empirical knowledge. As a matter of fact, this conclusion was in perfect harmony with another basic tenet of logical empiricism, the famous verifiability principle of cognitive meaning. At first glance, the latter may seem to

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be neutral with respect to the question whether a priori knowledge is possible. There are many diﬀerent methods of verification, of establishing the truth of a proposition; why not recognize purely intellectual operations, the sort of operations by which mathematicians and logicians establish their theorems without recourse to empirical data, as one such method? And are not, then, such sentences as “(a + b)2 = a2 + b2 + 2ab” verifiable and therefore, by the verifiability principle, cognitively meaningful, i.e. linguistic expressions of true-or-false propositions? However, “S is verifiable” was meant in the sense of “it is logically possible to verify S .” Clearly, it is not logically possible to verify a self-contradictory sentence1 , like “there are round squares”: if it is self-contradictory to suppose that there is a round square, then it is self-contradictory to suppose that anyone perceives one. Self-contradictory sentences, therefore, must be ruled out as cognitively meaningless. But if a sentence is cognitively meaningless, so is its negation. Therefore analytic sentences must likewise forego cognitive significance: in saying “no squares are round” we express our knowledge of how words are conventionally used, but there is no such thing as knowing that no squares are round—there just is no such proposition waiting to be known. Since the disjunction “either analytic or self-contradictory or empirical” was regarded as exhaustive by the logical empiricists, their verifiability principle of meaning committed them indeed to identification of genuine propositions with empirical propositions, i.e. propositions to be validated by experience and in principle refutable by experience2 . However, most logical empiricists, I think, shy away from the flat doctrine that there are no necessary propositions. After all, they hold that there are no necessary propositions that are not analytic—though some of them have become aware of grave diﬃculties besetting the explication of “analytic”—and they do not mean to assert this as a trivial consequence of there not being any necessary propositions; they do not mean to assert it, that is, in the sense in which one might truly assert that there are no unicorns that do not live in the Bronx zoo. Nevertheless, they hold that the concept of a necessary proposition, which they identify with the concept of an analytic proposition in some suitable sense of “analytic,” is analyzable in terms of concepts referring to language and linguistic rules. Confusedly, but strongly just the same, it is felt that the necessity of a proposition is somehow “rooted” in linguistic rules—the 1 It

is here assumed that “to verify” means “to establish as true.” Schlick must have had this meaning in mind when he drew from his verifiability principle the consequence—criticized later by Carnap, in Carnap 1937—that self-contradictory statements are meaningless. The consequence could easily have been avoided by interpreting “to verify” in the sense of “to establish as true or false,” for if S is self-contradictory it is logically possible to establish the falsehood of S (viz. by analysis and-or formal deduction), and hence it is logically possible to establish the truth of S or the falsehood of S . 2 Both Carnap and Ayer seem to have attempted an escape from this consequence by restricting the domain of application of the verifiability principle (or its liberalized form, the confirmability principle) to synthetic, or “putatively factual” sentences. But such a restriction seems to make the meaning criterion itself redundant: before we could apply it to a sentence, we would have to make sure that the latter is synthetic, but this involves making sure that it expresses a proposition: hence one would have to decide the question of cognitive significance before application, and therefore independently, of the verifiability principle.

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problem being how to get beyond such metaphorical suggestions as “rooted.” I wish to prove, now, that though the linguistic theory of necessary propositions claims to do no more than analyze a concept which admittedly has instances, it entails paradoxically that the analyzed concept has no instances. The view that necessary statements owe their truth to linguistic conventions alone, such that they could be deprived of necessity by arbitrarily changing linguistic conventions, has some surface plausibility if one thinks of such statements as “there are no married bachelors” (S ) and of linguistic conventions in the form of explicit definitions, like: “bachelor” is synonymous with “unmarried man.” But the air of plausibility will vanish once we take a look below the surface. The cited definition, or statement of synonymy, entails that S must be true if “there are no married men who are unmarried” (T ) is true. As Quine has put it, in “Truth by Convention” (Quine 1936), explicit definitions transmit truth from one statement to another, they cannot generate truth. Further, if the derivability of S from T is to prove that S is necessary, T itself must be necessary. If the descriptive terms occurring in T , viz. “married” and “men,” are primitives, no further explicit definitions can be invoked as the source of the necessary truth of T ; and if they are not primitive and nevertheless T is necessary, the process of reduction to primitive notation will eventually lead to a statement whose necessary truth is independent of explicit definitions. Now, one might refer to a logical principle of which T is a substitutioninstance as the ground of T ’s necessity: ∼ (∃x)(∃P)(∃Q)(Px.Qx. ∼ Qx). How could definitions in the sense of rules of interchangeability of synonyms be relevant to the assumed necessity of the logical principle? To be sure, if “(p.q) ⊃ p” seems more evident than “∼ (p.q. ∼ p),” we can invoke the definition “p ⊃ q ≡d f ∼ (p. ∼ q)” in order to drive the process of validation still “deeper,” and if we attach the highest degree of self-evidence to logical principles in the disjunctive normal form, we may eventually appeal to “ ∼p ∨∼q ∨ p”—provided “∼” and “∨” are the primitives of our logical system. But clearly we must sooner or later reach necessary statements in primitive notation, whose necessity therefore cannot depend on the kind of linguistic rules we have considered. “But such statements are simply implicit definitions of the primitives occurring in them; the linguistic rules which fix the meanings of the primitives take the form of postulates,” the conventionalist will reply. Although a separate section will be devoted to this notion of “implicit definition,” let us note immediately that the conventionalist who takes this line has already reduced the concept of “necessary truth” to absurdity. For if such allegedly necessary statements as S derive their truth from postulates which are assignments of meanings to expressions, then they derive their truth from sentences to which truth cannot significantly be ascribed—which is to say that they themselves cannot significantly be called “true.” If a sentence like “any two points determine a straight line” is interpreted as a postulate which, though

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incompletely only, specifies what “point” and “straight line” mean, then it says in eﬀect: let “point” mean any element x and “straight line” any class of elements A such that any two x’s determine one and only one A. And such a sentence clearly does not express a proposition—it rather expresses a proposal regarding the use of symbols. Suppose, now, that the conventional definitions which are alleged to be the source of necessary truth are statements about the conventional usage of certain expressions of a natural language (“natural language” is contrasted with “language system”), such as “people who understand English never apply ‘father’ to an object x unless they are willing to apply ‘male’ and ‘parent’ to x and vice versa.” Alternatively, this proposition of descriptive semantics could be expressed as follows: People who understand English use ‘father’ and ‘male parent’ synonymously. It is essential to the argument in process of construction that such statements about linguistic usage be admitted to be empirical. One might deny this on the ground that a language is defined by a set of rules, so that anybody who does not know some of these rules eo ipso does not really understand the language. But this argument confuses natural languages with language-systems. A natural language, unlike a language-system, can without contradiction be said to undergo changes. If any of the rules (rules of sentence formation, semantic rules, rules of deduction) which collectively define a given language-system are changed, a new language-system has been constructed. But the very fact that we are making a contingent historical statement about a given natural language L when we describe, say, changes of meaning undergone by a certain expression of L, indicates that a natural language is not similarly definable by a set of static rules. Accordingly, if “understanding English” were defined as “knowing all the rules (at least those in operation during a certain epoch) of usage of English expressions,” this strong definition itself would be in conflict with the conventional rule governing the usage of “understanding English.” And therefore the claim that a statement like “people who understand English use ‘father’ and ‘male parent’ synonymously” is, if true, simply analytic of what is meant by “understanding English,” is untenable. The point is of suﬃcient importance to deserve a little more elaboration. Suppose the use of a language L were compared to playing a game of chess in the following respect: to say that an expression E is used as an expression of L implies that its use is governed by rules which constitute L. If E were used in accordance with a diﬀerent rule, it would not be an expression of L (example: if the graphic sign “hut” is used to refer to hats, then it belongs to German language; if it is used to refer, not to an article of clothing, but a primitive kind of cottage, then it belongs to the English language). Similarly, moves made with chess figures constitute a game of chess only if they are governed by the rules of chess. Consider now the statement “anybody who plays chess moves the bishop diagonally only, not vertically.” Clearly, this

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is not an empirical generalization about what chess players do; it is a statement analytic of “playing chess.” Those philosophers of language who hope to gain insight into the nature of language by looking at linguistic behavior as the same sort of rule-governed behavior as playing a game—while recognizing, of course, diﬀerences in the motivating purposes—might use the analogy to support the view that statements like “anybody who understands English applies the expression ‘father’ only to males” are analytic. “To understand English” analytically entails “applying ‘father’ to males only,” just as “to know how to play chess” analytically entails “moving the bishop diagonally only,” they might argue. However, this is to overwork an analogy which is helpful up to a certain point. It is in fact to ascribe to natural languages features which only language-systems share with games. The rules of usage of, say, English expressions must be discovered empirically by observing how English speaking people commonly use them; they are not norms which are first stipulated, and then conformed to. Rather linguistic habits develop before the grammarian catalogues them, and the norms are taken notice of ex post facto, if at all, by those who learn the language “naturally.” Of course, in order to determine whether a given person speaks English and understands English, and whether his linguistic behavior, accordingly, is evidence relevant to a generalization about the usage of an English expression, one must test his acquaintance with some rules of usage of English expressions. If, for example, a technique of interrogation is used by the descriptive semanticist engaged in a “field trip,” evidence should first be obtained that the subject who is interviewed understands English to the extent of being able to interpret the interview questions correctly. The point to be emphasized is that once this evidence has been obtained, the question is still logically open how the subject will respond to the interrogation and, therefore, which hypothesis about the rules governing the English expression in question his responses will confirm. To illustrate, suppose we wanted to test the semantic hypothesis that as English speaking people use the verb “to see” in such contexts as “I see a snake,” “A sees an x at time t and at place P” entails “an x exists at t and at P.” Our questionnaire might contain a question like “is it possible (conceivable) that somebody sees a snake at t and at P if there is no snake at t and at P?” Clearly, we shall have to make sure that our subject understands the meaning of “snake,” of “there is,” of “conceivable,” and that he understands the meaning of “to see” to the extent of being able to distinguish between “to see” and “to imagine,” to recognize a sentence like “he saw a sweet smell” as nonsensical, etc. But this test of eligibility as experimental subject need not include a response to the very question through which we want to test alternative hypotheses about the meaning of “to see.” To deny the possibility of such non-circular experimental tests would be as gratuitous as to deny the possibility of a non-circular test of the physical hypothesis that all metals are electrical conductors on the ground that we could not be sure

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whether the material at hand is a metal until it had been tested for electrical conductivity. But once such statements about linguistic usage are admitted to be empirical, the question arises how the fact they state can be a reason for the necessity of any statement, such as “there are no fathers who are not male.” Surely, if by pointing to an empirical fact F we oﬀer a reason for accepting a statement p, then p itself must be an empirical, and so a contingent, statement. It is not that a necessary statement could not be entailed by a contingent statement. Indeed, if “p entails q” is synonymous with “it is impossible that p and not-q”3 , then demonstrably any contingent statement entails any necessary statement. But it is precisely because the entailment from a contingent statement p to a necessary statement q is thus “vacuous,” that p could not be a reason for acceptance of q. Nobody would say “any father must be a parent because there are nine planets,” although undoubtedly “there are nine planets” entails “any father is a parent.” Would it be any less absurd to say “any father must be a parent because, at least according to present linguistic conventions, it is incorrect to apply ‘father’ to an object to which ‘parent’ is inapplicable”? At first sight such a statement may, indeed, seem reasonable. After all, would we still say “any father must be a parent” if, say, “father” were used in the sense in which “brother” is used at present? This superficial impression, however, is due to confusing the admittedly contingent statement of the metalanguage “the sentence ‘any father is a parent’ expresses at the present time a necessary proposition” and the necessary statement whose necessity is to be explained, viz. “any father is a parent.” If the indicated change of linguistic rules occurred, the same sentence would henceforth express a non-necessary proposition, but the proposition previously expressed by it would remain necessary if it ever was. Notice also that, say, a German advocating the same linguistic theory of necessary truth would give a completely diﬀerent reason for holding the same proposition to be necessary, for he would refer to the usage of the German expressions “Vater” and “Elter,” not to the usage of the English expressions “father” and “parent.” This indicates that something has gone wrong somewhere. Some philosophers may be quick to point out that what has gone wrong is that the statement of the object-language “any father is a parent,” whose necessity was to be accounted for, has been confused with the modal statement of the meta-language “‘any father is a parent’ is necessary.” The latter statement, they might say, is not necessary, and therefore it can without absurdity be validated by reference to such empirical facts as linguistic habits. And from the fact that

3 Some

have rejected this definition precisely because it has the “paradoxical” consequences that an impossible proposition entails any proposition and that a necessary proposition is entailed by any proposition. But I think their arguments are untenable. See chapter 11.

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“‘p’ is necessary” is contingent it does not follow, they would say, that “p” is contingent. Therefore the linguistic theory of necessity is compatible with there being necessary propositions. It is important to show carefully the error of this position, for it is held by some analytic philosophers of eminence. Thus G. E. Moore wrote in his essay on “External and Internal Relations” (Moore 1922) that “p entails q” is not itself a necessary proposition, but that “p ⊃ q” is a necessary proposition if “p entails q” is true. Since “p entails q” means “‘p ⊃ q’ is necessary,” Moore’s statement is equivalent to a denial of what I shall call the NN thesis: that if a statement is necessary, then it is necessary that it is necessary (in other words, necessary statements are necessarily, not contingently, necessary). While Moore made this claim en passant, without supporting it by any argument, more recently Strawson has argued in favor of the view that entailmentstatements are contingent statements about the usage of expressions, and that it is therefore a mistake to construct a modal logic with modal operators in the object-language. Also Reichenbach maintains, in Elements of Symbolic Logic (Reichenbach 1947), that the meta-linguistic statement to the eﬀect that a given formula of the object-language is a tautology is, even if true, not itself a tautology. From Quine’s writings on analyticity one gathers that he too regards the NN thesis as at least problematic. For, using “analytic” and “necessary” interchangeably, he holds that the concept of necessity is obscure to the extent that it is wider than the concept of logical truth, because the concept of “synonymy” enters into its definition; and the latter concept he finds obscure because he knows of no operational criterion for deciding questions of synonymy. For example, “all bachelors are unmarried” is not logically true, for the descriptive terms “bachelor” and “unmarried” occur essentially in it, i.e. not all statements of the same form, “all A are B,” are true. If nevertheless it is claimed to be a necessary statement, the reason is that “bachelor” seems to be synonymous with “unmarried man,” and by substitution of “unmarried man” for “bachelor” a logically true statement is obtained. But Quine finds the meaning of such an assertion of synonymy obscure. Now, leaving aside doubts about a sharp criterion of distinction between true statements that are logically true and those that are not, Quine, it seems, would be satisfied with the definition of a necessary statement as one which is either logically true or transformable into such a statement by substitution of synonyms for synonyms, if an operational definition of “synonymy” were at hand. But to ask for an operational definition, in behavioristic terms, of “synonymy” is to presuppose that statements of the form “T is synonymous with T ”4 are empirical. And since the meta-statement

4 Strictly

speaking, such statements are of course incomplete. There should be at least a reference to a group of sign-users and the context of usage. If I omit these variables, it is not because I overlook them, but just for the sake of expository simplicity.

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“ ‘all bachelors are unmarried’ is necessary” would then have to be established empirically, by investigating whether “bachelor” is synonymous for English speaking people with “unmarried man,” it is not itself necessary5 . Now, my argument in support of the NN thesis is that the primary object of a priori knowledge is always a modal proposition. If the latter has the form “N(p),” we pass from our a priori knowledge of the modal proposition to a priori knowledge of the proposition asserted to be necessary via the necessary proposition “if N(p), then p.” Therefore, if “N(p)” means “p can be known a priori, i.e. by just thinking about p, without ‘looking at the world’,” no proposition at all would be necessary if propositions of the form “N(p)” were not necessary. Consider again our simple example of a necessary proposition: there are no fathers that are not male. If we are a priori certain of it, i.e. in advance6 of having observed all fathers, past, present, and future, it is surely because we see that fatherhood entails malehood. This is to say that we see a priori the truth of the modal proposition “father(x) male(x)”—alternatively expressed as an assertion of impossibility “∼ 3(father(x) . ∼ male(x))”—and hence derive the corollary (∀x)(father(x) ⊃ male(x))—alternatively expressed as an assertion of non-existence “(∃x)(father(x).male(x)).” The point is that since it is our knowledge of the entailment which is the ground of our certainty with regard to the universal proposition, the universal proposition would itself have only the status of inductive generalization if the entailment were not known a priori. The argument “there are at no time fathers who are not male, because fatherhood entails malehood” has the form “p, because N(p)”: for “p q” is definable as “N(p ⊃ q).” Therefore our acceptance of p would be based on empirical evidence if our acceptance of N(p) were based on empirical evidence. Therefore one cannot consistently hold that p is necessary and N(p) contingent.

5 Incidentally,

Quine’s definition of “analytic,” as a wider concept than “logically true,” in terms of “synonymy” would be inadequate even if the latter concept could be clarified to Quine’s satisfaction. For concepts of natural kinds which are formed by abstraction usually have a certain amount of intensional vagueness which makes it impossible to give a strict explicit definition of the name of the kind, and yet the intensions are definite enough to permit a number of analytic entailments about the kind to be stated. For example, “all swans are animals” is surely analytic: it is self-contradictory to assert the existence of swans which are not animals. But to complete the definition by stating diﬀerentiating characteristics P1 , . . . , Pn of such a kind that it would be self-contradictory to apply “swan” to an animal which lacked any of them, is very diﬃcult, since we might be strongly inclined to classify an animal lacking Pi as a swan on the ground of its strong resemblance in other respects to animals normally called “swans” (For more details on “intensional vagueness,” see chapter 19 and Hospers 1953, 42-45.) 6 I deliberately exploit the etymological meaning of “a priori” in this context. In spite of Kant’s warning not to confuse the temporal origin of knowledge with its ground of validity, there is an obvious connection between the temporal and the epistemological meaning of “a priori”: to know a universal proposition a priori is to know it in advance of testing its substitution-instances—though observation of some instances of its constituent concepts may be causally necessary for “having” the concepts.

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It is true that no parallel argument can be constructed with respect to modal propositions asserting possibility, specifically compatibility, i.e. possibility of a conjunction of propositions, since no non-modal proposition about the world of particulars is deducible from such a proposition. But once it is granted that propositions of the form “p is necessary” are not contingent, it ought also to be granted that those of the form “p is possible” are not contingent, since “p is necessary” is equivalent to the negation of “not-p is possible.” For from the necessity of “p is necessary” follows the impossibility, and hence noncontingency, of “not-p is possible,” and by contraposition, from the contingency “p is possible” follows the contingency of its negation “not-p is necessary”7 . The case for the NN thesis is strong also if we consider that special class of necessary propositions which are instances of tautologous truth-functions, i.e. functions of propositional variables which are true for all values of the variables. Take, e.g., a proposition of the form: (p ⊃ q) ⊃ (∼ q ⊃ ∼ p). If we knew it to be true merely on the basis of having empirically ascertained the truth-values of the atomic propositions and then computed the truth-value of the molecular proposition with the help of the definitions, given in the form of “truth-tables,” of the connectives, we would not know it to be necessarily true. But to say that we know it to be necessarily true is to say that we know that every proposition of the same form is true, i.e. that it is an instance of a tautologous truth-function. And surely this knowledge is a priori: by reflecting on the meanings of the logical constants “not” and “if, then” we discover that it is an instance of a tautologous truth-function, hence we know a priori that it is necessarily true. Therefore here again our a priori knowledge of the truth of p is based on our a priori knowledge of the necessity of p.

2.

The Confusion of Sentence and Proposition

In my opinion the view just criticized, that modal propositions are contingent propositions about linguistic usage, arises from confusion of sentence and proposition, a confusion which is a special case of a confusion between using and mentioning an expression. If “it is necessary that p” were meant in the sense of “the sentence ‘p’ is used to express a necessary proposition,” then the NN thesis would be false indeed: propositions about what given expressions mean as used in natural language, are surely contingent. But the very appearance of the expression “necessary proposition” in the meta-linguistic interpretation of “it is necessary that p” indicates that the alleged interpretation is no interpretation at all: after all, “S expresses, in L, a necessary proposition”

7 That a contingent proposition has a contingent negation follows by simple commutation from the definition

of contingency: p is contingent ≡d f p is possible and not-p is possible.

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means:(∃p)(Des(S , p, L).N(p))8 , and thus “N(p)” has not been interpreted at all. Indeed, anybody who can grasp the diﬀerence between direct and indirect quotation, ought likewise to see that when a proposition is asserted to be necessary, nothing at all is asserted about the sentence which happens to be used to mention the proposition. In asserting “Aristotle said that the good is that which all men desire” one is not asserting “Aristotle said ‘the good is that which all men desire”’: indeed, the latter proposition is certainly false, since Aristotle did not speak English. In Frege’s terminology, the proposition that p is the sense of the sentence “p”; in Carnap’s terminology, it is the latter’s intension. To be sure, this explanation leaves the diﬃcult question open what exactly is to be understood by “proposition.” To answer this question is to propose a satisfactory explication of the concept of synonymy, i.e. an exact statement of the conditions under which two sentences express the same proposition. But any such explication would have to be guided by what might be called a “pre-analytic” understanding of the sentence-proposition distinction. Such pre-analytic understanding is, for example, manifested by the statement that “Aristotle said that the good is that which all men desire” reports an assertion of a proposition but says nothing about what sentence was used to assert the proposition. Now, the relation between “he said that p” and “he said ‘p”’ is precisely like the relation between “it is necessary that p” and “‘p’ is used to express a necessary proposition” in the respect noted. There is a simple device, recently used by Alonzo Church (Church 1950) to criticize Carnap’s attempt to translate statements of beliefs and assertions (indirect quotations) into a meta-language mentioning sentences but no propositions, for making it plain that statements of the form “it is necessary that p” are not about the sentences replacing the variable “p.” We simply translate both the interpreted sentence of the object-language and its proposed metalinguistic interpretation into another language. The interpretation is correct if and only if these translations are themselves synonymous. For example, “it is necessary that there are no fathers who are not male” is correctly translated into German by the sentence “es ist notwendig, daß es keine V¨ater gibt die nicht m¨annlichen Geslechtes sind” (S 1 ) but “‘there are no fathers who are not male’ expresses a necessary proposition” translates into “ ‘there are no fathers who are not male’ ist Ausdruck eines notwendigen Urteils” (S 2 ), and clearly S 1 and S 2 are not synonymous. Let us return for a moment to the contrast between direct and indirect quotation. The statements “he said that the earth is round” and “he said ‘the earth is round”’ seem to be alike in violating the postulate of extensionality: the extension of a compound expression remains unchanged if a component ex-

8 “Des(S ,

p, L)” reads: sentence S designates proposition p in language L.

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pression is replaced by a diﬀerent expression having the same extension. Thus substitution for the true sentence “the earth is round” of the true sentence “the moon revolves around the earth” would change the truth-value of both statements. Even if we replace the singular term “the earth” by another singular term having the same extension, i.e. denoting the same object, leaving all else unchanged, we might change the truth-value of the indirect quotation just as we—necessarily—change the truth-value of the direct quotation. For example, it might well be the case that the earth is the only planet which is inhabited by philosophers, among other animals. But although on this assumption “the earth is round” is materially equivalent to “the only planet which is inhabited by philosophers, among other animals, is round,” the indirect quotation obtained by making this substitution would be bound to have the opposite truthvalue from “he said that the earth is round,” since the substituted singular term has a diﬀerent sense. But if this apparent similarity leads one to construe indirect quotation as meta-linguistic statements, then one has been misled by a linguistic accident. It is a linguistic accident that we form a name of an expression, in written discourse, by putting the named expression itself within a pair of quotes. If we used instead an entirely diﬀerent expression as name of the expression, it would be obvious that neither the singular term “the earth” nor the sentence “the earth is round” is a component of the direct quotation at all; the latter therefore satisfies the postulate of extensionality trivially9 , and accordingly a careful comparison of the direct and indirect quotations with respect to extensionality reveals a significant dissimilarity rather than similarity. Mutatis mutandis, the same holds for “it is necessary that p” in comparison to “‘p’ expresses a necessary proposition”10 . Logicians who admit the diﬀerences noted but nevertheless adhere, for certain pragmatic reasons, to the postulate of extensionality, may still attempt to interpret modal statements and indirect quotations (as well as other types of apparently non-extensional statements, above all statements about beliefs) in an extensional meta-language. Thus they might be inclined to interpret “it is necessary that p” in a semantic meta-language11 as follows: for any language 9 It

is precisely for this reason that Carnap, at the time firmly committed to the thesis of extensionality, proposed in Carnap 1954a to translate “A believes that p,” respectively “A says that p,” into formal mode of speech simply by “A believes ‘p’,” respectively ‘A says ‘p’.” 10 In Quine 1953b, Quine maintains that “necessity as statement operator,” i.e. statements of the form “it is necessary that p,” is capable of being reconstrued in terms of “necessity as a semantical predicate,” i.e. statements of the form “‘p’ is necessary. But I have not discovered any better reason, in his paper, for this claim than the superficial similarity between “it is necessary that p” and “‘p’ is necessary” consisting in their non-extensionality with respect to “p.” This similarity, to repeat, is superficial because it is entirely accidental that the named sentence “p” is a component of its name “‘p’.” If it were not for considerations of convenience, we might use meta-linguistic names which do not contain their nominata any more than the name “Rome” contains the city of Rome. 11 The semantic meta-language employed by Carnap in Carnap 1942 is not extensional, since it contains the triadic predicate “Des(S , p, L)” which is not extensional with respect to “p”: if “Des(S , p1 , L)” is true, and

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L and sentence S , if S is synonymous with “p” in L, then S is L-true in L, i.e. true by virtue of the semantic rules of L. While postponing a closer examination of this interpretation of modal statements to the next section, let us just note that the argument from translation into another language retains its force even against more complicated meta-linguistic interpretations such as this one. For if a sentence contains a name N of an expression E, then any correct translation of the sentence will contain a synonym of N, not of E; E will be mentioned by it. But no correct translation of an English sentence “it is necessary that p” into another language will mention the English sentence “p”; rather it will mention the sense (intension) of “p.” It was noted above that the contingent meta-linguistic statement “‘p’ expresses a necessary proposition” in no way sheds light on the meaning of “it is necessary that p,” since in fact its intelligibility presupposes an understanding of that meaning; after all, “‘p’ expresses a necessary proposition” asserts “there is a proposition p which ‘p’ expresses and which is necessary.” But many philosophers of language, foremost those influenced by the teachings of Wittgenstein, will object to the analysis of semantic statements in terms of a designation-relation, holding between expressions and designated entities. Thus Carnap, who speaks of propositions as being designated by sentences, and properties as being designated by predicates, just as he speaks of things as being designated by proper names, has been accused (quite unjustly, I think) of confusing meaning with naming.12 Naming, so the argument runs, is a relation of a symbol to an independently existing entity, but to assume that for every meaningful symbol, even those that are not “names” in any usual sense, there exists an entity which is its meaning, is to be guilty of Platonic reification. More formally, the argument may be put as follows: a statement to the eﬀect that an expression which is not a name in any ordinary sense (e.g. a predicate, a sentence) means such and such does not have the logical form “xRy,” though it may share this grammatical form with a statement identifying the designatum of a name, like “‘Peter Shannon’ is the name of the person I had lunch with today.” As Wittgenstein and Russell have emphasized, grammatical similarities between sentences of natural language mislead philosophers to wrong analyses of their meanings: they take grammatical form naively as a clue to “logical form” (where “logical form” may perhaps be defined as the grammatical form

“p1 ” is just materially, not logically, equivalent to “p2 ,” then “Des(S , p2 , L)” is not true. But in (Carnap 1956a) Carnap attempts to extensionalize the semantic meta-language by taking both statements of designation, the one about the proposition p1 and the one about the proposition p2 , as true, yet distinguishing in the meta-meta-language between statements of designation which are conventions or provable on the basis of conventions, and such as are not, or are “F-true.” 12 See e.g. Ryle 1949b.

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of the translation of the sentence into an ideal language)13 . Consider the semantic statements a) “blue” means the color of the sky, b) “die Erde ist rund” means the proposition that the earth is round14 . Since the phrases following the verb are nouns, or noun-clauses, the naive philosophers of language construe them as names of entities constituting the range of “y” in the sentential function “x means y.” But this amounts to no smaller sin, in the eyes of the opponents of what is sometimes called the “representative theory” of language (in contrast to a “functional theory” of language, whatever it may be), than treating all meaningful expressions as names. And to speak of necessary propositions as of entities designated by those sentences which cannot be denied without violating the conventional rules of usage, is to commit just that sin. The detailed reply to this criticism is deferred to the next section where it will be argued 1) that the construal of meaning as a relation whose converse domain consists of such “abstract” entities as properties and propositions does not commit the crime of Platonic “reification,” 2) that the alternative theory of propositions as “logical constructions” out of sentences is untenable. But in the meantime it should be understood that the validity of the NN thesis does not depend on a sentence-proposition distinction which the mentioned philosophers of language find objectionable. For it holds even if for propositions in the sense of (possible or actual) states of aﬀairs designated by meaningful declarative sentences we substitute statements in the sense of sentences as meaning such and such. Thus, instead of saying that the sentences “A is larger than B” and “B is smaller than A” are ordinarily used to designate the same proposition, let us say that they are ordinarily used the make the same statement; for short, we might say that while being diﬀerent sentences they are the same statement. Now, that the sentence “A is larger than B if and only if B is smaller than A” is used to assert a necessary statement, is indeed a contingent fact about the English language. But that the statement which anybody familiar with the rules of contemporary English would make if he uttered this sentence is necessary, is not a contingent fact. If it were, then it would be conceivable that the same statement might not be necessary. But what is conceivable is only that the same sentence might be used to make a diﬀerent statement which is not necessary. Since the NN thesis thus stands independently of any commitment to “propositions,” it follows that a linguistic theory which interprets modal statements as contingent statements about linguistic rules is committed to the denial of the existence of necessary propositions—or necessary statements. 13 For a discussion of problems raised by this definition of “logical form”—a term abounding in the writings

of Wittgenstein and Russell, and crying out for clarification—see Kalish 1952. avoid the formulation “ ‘the earth is round’ means the proposition that the earth is round,” which occurs in Carnap’s semantical writings, because the semantic statement is not informative if the meta-language is the same natural language as the object-language it talks about, unless the semantic statement is a meaninganalysis within the same language—which a statement of the form “‘p’ means that p” is not.

14 I

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Although some proponents of a linguistic theory of necessary truth must be accused of simple carelessness in not observing the distinction between “‘p’ is used to a express a necessary proposition” and “it is necessary that p,” it would be unjust to say generally that logical empiricists who combat “rationalism” overlook the sentence-proposition distinction. Nor can they justly be accused of such crude expressions as “it is nonsense to speak of propositions, for they are by definition unobservable entities.” No, the distinction is usually admitted, but an explication of “proposition” is proposed which somehow conforms to the empiricist criterion of cognitive significance: that statements ostensibly about unobservables be given a cognitive meaning by reduction to statements no matter how complicated, about observables. Just as statements about physical objects are reducible, according to phenomenalism, to statements about sense-data—which is what a phenomenalist means by saying that physical objects are “logical constructions” out of sense-data—so the view is sometimes taken that statements about propositions are shorthand, as it were, for statements about sentences, specifically about classes of synonymous sentences. Thus Ayer writes: “Regarding classes as a species of logical constructions15 , we may define a proposition as a class of sentences which have the same intensional significance for anyone who understands them” (Ayer 1952b, 88). And he clarifies the meaning of this definition in the introduction to the second edition of his “manifesto” of logical positivism as follows): Thus, if I assert, for example, that the proposition p is entailed by the proposition q I am indeed claiming implicitly that the English sentence s which expresses p can be validly derived from the English sentence r which expresses q, but this is not the whole of my claim. For, if I am right, it will also follow that any sentence, whether of the English or any other language, that is equivalent to s can be validly derived, in the language in question, from any sentence that is equivalent to r; and it is this that my use of the word ‘proposition’ indicates. (Ayer 1952b, 6)

This relatively clear statement of the logical-construction-theory of propositions will now be used as a platform for criticizing the theory. I shall ignore the minor defect of the theory in Ayer’s formulation, that logical equivalence instead of a stronger relation of synonymy is used for the logical construction of propositions16 . For whatever the diﬃculties involved in a satisfactory 15 Presumably Ayer has in mind the theory of Principia Mathematica that classes

are “incomplete symbols,” i.e. that statements ostensibly about classes are translatable into statements which do not mention classes (but propositional functions instead). 16 If all logically equivalent sentences express the same proposition, then there is but one analytic proposition and but one self-contradictory proposition. But even a definition of “proposition” from which it only follows that any two synthetic sentences which are logically equivalent express the same proposition has counterintuitive consequences: “Mt. Everest is the highest mountain of Asia” is logically equivalent to “anybody

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explication of “synonymy,” the theory could easily be restated in terms of this stronger relation without thereby becoming immune to the criticisms to be presented. However, one serious confusion should be noted. What exactly does Ayer mean by saying that the assertion “p is entailed by q” implicitly claims that the English sentence s which expresses p can be validly derived from the English sentence r which expresses q? I take him to mean “implicit claim” in the sense in which one might say, e.g., the assertion “I have an older brother” implicitly claims that the speaker’s parents have more than one child, i.e. in the sense of entailment. Ayer, then, claims that “q entails p,” where “p” and “q” are propositional constants (left undetermined), entails “if s expresses p and r expresses q, then s is validly derivable from r.” But here the propositions p and q are mentioned again, in the semantic contexts “s expresses p” and “r expresses q,” which contradicts the aim of the translation. If, on the other hand, the antecedent is omitted, and “the proposition that there are husbands entails the proposition that there are wives,” e.g., is claimed to entail “‘there are wives’ is validly derivable from ‘there are husbands’,” the claim is obviously false: for if the sentences “there are wives” and “there are husbands” happened to express logically independent propositions, which is logically possible, neither would be derivable from the other, and such a contingency, of course, could not alter the necessity of the entailment between the propositions. Now, in the second part of the translation Ayer shifts from “any sentence expressing p” to “any sentence (logically) equivalent to s.” But he is silent about the criterion of logical equivalence. If to say that two sentences are logically equivalent is just to say that they express the same proposition, the translation is implicitly circular again. And it would not help to define “s is logically equivalent to t” syntactically as “the same sentences are derivable from s and t.” Suppose, e.g., that s = “I have two brothers” and t = “I have two male siblings.” From t “I have two siblings” is syntactically derivable, i.e. by applying rules of deduction which refer only to the forms of sentences. Is this sentence syntactically derivable from s? Surely not, unless the definition “brother = male sibling” is used as a rule of substitution. But if with the help of this definition I propose to prove that s and t are logically equivalent, I must first prove that it itself expresses a logical equivalence; otherwise it would be easy to prove, by the debated syntactical criterion of logical equivalence, that “I have two brothers” is, say, logically equivalent to “I have two wives,” by using an arbitrary definition of “brother” as, say, “attractive wife.” And thus we either return to “s and t are logically equivalent because they express the same proposition,” or

who has seen the highest mountain of Asia has seen Mt. Everest,” yet since the proposition expressed by the latter sentence contains the concept of seeing, while the proposition expressed by the former sentence does not, we can hardly say the propositions are the same.

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are faced with an infinite regress in attempting to establish logical equivalence in a purely syntactic way. To maintain that propositions are logical constructions out of sentences is to claim that contextual definitions of statements containing names of propositions or propositional variables can be set up whose definientia mention sentences, either by way of names of sentences or by way of sentential variables (not excluding both), but do not mention propositions by way of names of propositions or by way of propositional variables. Of course, it need not be claimed that this problem of reduction could be solved by a single contextual definition that is applicable to all possible contexts of “propositional talk.” One may have to set up separate contextual definitions for diﬀerent types of statements about propositions, e.g. “it is true that p,” “the proposition that p entails the proposition that q,” “so-and-so believes that p,” “it is logically necessary that p,” “any proposition entailed by a true proposition is true.” Let us examine such a contextual definition for “it is true that p.” Since this is just about the simplest kind of “propositional talk,” one would expect the attempted reduction to succeed in this instance if it is possible at all. But I wish to show that even here it fails. Suppose the objection were raised against the “reduction” of propositions to classes of synonymous sentences that 1) a class is just as abstract an entity as a proposition, so that the reduction, even if it were successful, would not be an application of Occam’s razor at all, 2) this identity could not be meant literally, since there are predicates which are significantly applicable to propositions but not to classes: e.g. it does not make sense to say of a class that it is true, or that it entails another class. The proper reply to this objection would be that what is meant is that statements about propositions are translatable into statements containing universal quantifiers that refer to any sentences satisfying certain conditions, but which do not mention classes at all17 . Let us, then, state formally such a reduction of “it is true that p,” which happens to be identical with the reduction of the “absolute” concept of truth (truth as predicable of propositions) to the “semantic” concept of truth (truth as predicable of sentences of a language-system) in Carnap 1942: for any sentence S and language L, if S designates p in L, then S is true in L. We notice at once that we are still left with a propositional variable, in the semantic context “S designates p in L.” To get rid of the propositional variable—or of constant substituends for the propositional variable, if we consider translation of a specific statement of the form “it is true that p”—in this context is a far more diﬃcult task than it may seem superficially. The obvious suggestion is that we replace “S designates p in L” by “S

17 We

may overlook in this context the fact that a sentence in the sense of a repeatable type of expression, in contrast to tokens exemplifying the type, is itself a class; for the same reply would mutatis mutandis be appropriate to this objection.

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is synonymous with ‘p’ in L.” But apart from the criticisms already advanced in connection with Ayer’s reduction, the argument from translation is applicable again: a translation of “it is true that the earth is round” which mentions the English sentence “the earth is round” cannot be adequate, since if it were, the translated English sentence would not be synonymous with, say, the German sentence “es ist wahr, daß die Erde rund ist.” Further, as G. E. Moore has pointed out (Moore 1946), to say that two expressions have the same meaning is diﬀerent from telling what they mean (just as to say that two classes have the same number is not to say what that number is). However, let us ignore these diﬃculties, and assume for the sake of the argument that “S designates that p” can in some way be translated into a purely extensional meta-language (cf. footnote 11), so that he who cannot countenance propositions as “real entities” may in good conscience employ this mere “fac¸on de parler.” Let us see, then, whether the following equivalence is necessary: It is true that p, if and only if, for every S and every L, if S designates p in L, then S is true in L.

(D)

The answer depends on the sense of “if, then” in the definiens. Is the definiens a generalized material implication (what Russell called “formal implication”)? In that case the definiens will be true of a given proposition, provided there are no sentences designating that proposition in some language. But surely this is a logical possibility. Thus, consider the statement “there was a time when nobody believed or disbelieved or supposed or in any other mode thought of the proposition that there are 9 planets.” Is it not highly probable that it is true? But it entails that there is a proposition p and a time t such that p is not verbally expressed at t. “Yes, but this does not prove that it is logically possible for a proposition to be verbally unexpressed at all times. After all, the very act of saying ‘it is logically possible that nobody should ever have thought of p1 ’ constitutes a verbal expression of p1 , hence the supposition that there is a proposition which is unexpressed at all times cannot be consistently made, just as one cannot consistently suppose that he is never thinking.” This objection, however, rests on the confusion between pragmatic and logical contradiction. A proposition is pragmatically contradictory if its falsehood follows from its assertion; more precisely, “p” is pragmatically contradictory if “p is asserted” entails “p is false.” In this sense the proposition that no proposition is ever asserted is obviously a pragmatic contradiction; but which moderately clear-headed person would seriously doubt its logical possibility? Similarly, the supposition that the proposition that p1 is never at all verbally expressed is pragmatically contradictory, since one cannot assert (or even suppose) that p1 is never verbally expressed without producing, physically or mentally, a sentence which expresses p1 . But this is entirely compat-

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ible with the logical possibility that p1 is never verbally expressed. Whether or not there is such a thing as completely nonverbal thought is an interesting problem of psychology but completely irrelevant to this issue. One cannot think of a proposition without thinking of it: very true. Hence to infer that one cannot consistently think of a proposition as not being thought of (i.e. think of the proposition that a given proposition is not being thought of) is to commit the same howler which Russell attributes to Berkeley: “‘It is impossible for a nephew to exist without an uncle; now Mr. A is a nephew; therefore, it is logically necessary for Mr. A to have an uncle’. It is, of course, logically necessary given that Mr. A is a nephew, but not from anything to be discovered by analysis of Mr. A. So, if something is an object of the senses, some mind is concerned with it; but it does not follow that the same thing could not have existed without being an object of the senses” (Russell 1945, 652). You cannot discover by analysis of a proposition that some mind is thinking of it, whether verbally or not. Now, if p is verbally unexpressed, then by D (whose “if, then” we suppose to be an extensional connective) p is true, but also not-p is true, since there can be no sentence designating not-p if there is no sentence designating p. But that unexpressed propositions are both true and false is an absurd consequence of D which speaks decisively against this rule of translation18 . If the “if-then” in D is interpreted as a strict implication (logical deducibility of consequent from antecedent), the translation does not become more plausible. If a strict implication is true, then, in accordance with the NN thesis, it is necessarily true. But then the translation would entail that only necessary propositions are true, in other words, that there are no true contingent propositions. The same conclusion can be derived from a diﬀerent consideration. We just need to ask ourselves how a conjunction of the form “Des(S , p, L) and T rue(S , L)” could be self-contradictory. Surely, if p is contingent, then it is logically possible that a sentence designating p in some language should be false in that language. To assert the logical impossibility

18 Carnap

might challenge this argument on the ground that the propositional variables of a semantic system range only over propositions for which there are constructible sentences of the system that designate them. But the question whether there are propositions that are not designatable by any sentence of a given semantic system is surely a significant one. Indeed, suppose the language-system to be a “continuous” one, i.e. containing real-number-variables. Consider the proposition, expressible in the system, that a given object has a certain determinate form of a property assumed to be continuously variable, say length. This proposition is one out of a non-denumerably infinite set of similar propositions, yet the set of sentences constructible (“sub specie aeternitatis”) in any language-system would seem to be denumerably infinite. Are we not forced to the conclusion, then, that not all propositions are expressible in language-systems though any given proposition is? Notice that even if the reduction of “absolute truth” to “semantic truth” were satisfactory for statements “it is true that p” within a semantic system, we would be left with the question whether statements outside and about a semantic system, such as “half of the propositions that are not expressible in system L are true,” are thus reducible.

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(self-contradictoriness) of such a conjunction, therefore, is equivalent to asserting the necessity of p. There remains the interpretation of the conditional as a subjunctive conditional which does not hold by logical necessity but is an inductive generalization. At least, one might suggest, the universal conditional of the semantic meta-language has inductive character if the proposition of which truth is predicated is contingent. But, assuming we have found empirically that a sentence S 1 is true if its constituent terms have such and such meanings, are we really drawing an inductive inference in saying “for any S , if S were synonymous with S 1 , S would also be true,” the way it is an inductive inference to infer from the proposition that a given sample of iron melted at temperature t, that any other sample of iron would also melt at temperature t? Clearly not, for it is a contradiction to suppose that S 1 is true, that S 1 is synonymous with S and yet that S is not true. In other words: if a criterion of adequacy to be satisfied by an explication of the diﬀerence between “‘p’ is true” and “it is true that p” is that the latter statement should say more than the former (see the quotation from Ayer, p. 122 above), then the explication in terms of a universal subjunctive conditional referring to any (actual or possible) sentences synonymous with “p” is demonstrably inadequate. If “p” logically entails “q1 . . . . . qn ” then the conjunction “p.q1 . . . . . qn ” has no more logical content than “p” alone. These criticisms apply mutatis mutandis to applications of the logical-construction-theory to other kinds of propositional talk, such as statements of entailments, statements of belief, etc. If one wishes to construe propositions as “pseudo-entities,” “logical constructions,” in the precise sense in which classes are logical constructions in Principia Mathematica, then one will have to show that propositional names, i.e. such expressions as “that Mary is baking pies now” in the context “Clarence believes that Mary is baking pies now”19 are eliminable through contextual definitions. In recent discussions of possible translations of propositional talk into a language acceptable to “nominalists,” attention has been paid primarily to statements of belief. One strong argument against such translatability is Church’s “translation argument,” already discussed above. I would like to add here a few criticisms: In order to translate “A believes that p” into an extensional meta-language, a behavioristic disposition concept must be used, something like Carnap’s “be-

19 It

should be noted that one who countenances propositions as “real entities”—in the formal mode: who employs without compunction a language containing, in the primitive notation, variables ranging over propositions—is not committed to construal of true-or-false statements as names of entities. True-or-false statements intend propositions, but that does not turn them into names any more than predicates are names. In fact the analogy here is fairly close: predicates have properties as intensions, which latter may be named by abstract nouns (“roundness” is a name of the intension of the predicate “round”); similarly names of propositions can be derived from the statements intending them, by using the gerundive (“Mary baking pies now” names the intension of the statement “Mary is baking pies now”).

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ing disposed to an aﬃrmative response” to some kind of sentence. Thus Carnap proposed in Carnap 1956a “A is disposed to an aﬃrmative response to some sentence which is synonymous with ‘p”’ as a tentative analysis, where synonymy is a stronger relation than logical equivalence (explicated by Carnap as “intensional isomorphism”; see Carnap 1956a, para. 13, 14). He was led to the distinction between L-equivalence and synonymy in this context by the following reasoning: if “p” is L-equivalent to “q,” then “A is disposed to an aﬃrmative response to some sentence which is L-equivalent to ‘p”’ entails “A is disposed to an aﬃrmative response to some sentence which is L-equivalent to ‘q”’ (since L-equivalence is transitive). Therefore the analysis of belief in terms of L-equivalence entails that “A believes that p” entails “A believes that q”; yet, since A may not be aware of the L-equivalence of “p” and “q,” it is possible that he should believe that p and yet not believe that q (assume, e.g., that p = (p.(p ⊃ q)), and q = (p.q), and that A does not know the propositional calculus)20 . However, the analysis of belief in terms of synonymy does not overcome this kind of diﬃculty at all; it is open to a perfectly analogous objection. For, suppose that “p” and “q” are not only L-equivalent, but even synonymous. Could not A fail to know that they are synonymous, and hence respond aﬃrmatively to the question “Do you believe that p” but not to the question “Do you believe that q”? But further, “A believes that p” is not a necessary consequence of “A is disposed to an aﬃrmative response to some sentence in some language which is synonymous21 with ‘p’,” since A may interpret the sentences towards which he is thus disposed to express a proposition other than p. To try to obviate this diﬃculty by qualifying A as a subject who understands the questions to which he responds aﬃrmatively or negatively, would be to readmit the banished propositions through the backdoor: for to 20 Since for Carnap L-equivalent sentences express the same proposition, Carnap here asserts that it is possible for a person to believe and yet not to believe the same proposition at the same time! This is not just a question of psychology, nor can such an assertion be defended by pointing to the sad fact that many people hold inconsistent beliefs most of the time; for “A believes at t that p, and not-(A believes at t that p)” is as good an instance of a logical contradiction as any. Carnap, it seems to me, should have drawn the consequence that his weak identity-condition for propositions is not only in conflict with ordinary usage of the word “proposition,” but with the obvious fact that people may fail to be aware of some L-equivalences holding in a language which they understand as well as anybody can be expected to understand any language. 21 An especially serious diﬃculty connected with Carnap’s analysis is that it introduces an interlinguistic synonymy-relation, whereas Carnap’s explications are usually relative to a given language. This is like the diﬀerence between simultaneity defined for events in a given system and simultaneity of events occurring respectively in systems that move relatively to each other. Even if one could plausibly define “A, as expression of L, is L-equivalent to B, as expression of L”’ as “it follows from the combined semantic rules of L and L’ that A and B have the same extension,” the criterion of intensional isomorphism could not be used for interlinguistic synonymy, since two languages may be structurally so diﬀerent that it is impossible to give a syntactic criterion for “correspondence” of constituent signs. Take, e.g., the Latin sentence “ignorabimus” and the English sentence “we shall never know”: since the former is not a compound designator and the latter is, these sentences are not intensionally isomorphic, yet they are obviously synonymous (if Carnap does not agree that this is, preanalytically, an instance of synonymy, then I do not know what his explicandum is).

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say that A understands the question “do you believe that p” is to say that A interprets the sentence p as standing for the same proposition as the interrogator intended by it. Finally, such a behavioristic analysis of belief cannot be adequate for the simple reason that the subject whose beliefs we investigate by interrogation may be a liar. If we take this possibility into account by including the proviso that A is honest (A believes that p = if A were asked, with respect to each and every sentence S which is synonymous with ‘p’, “do you believe that S ?,” or a synonymous question, then, provided A is honest, A would respond aﬃrmatively at least once), the translation becomes grossly circular since “A is honest” means nothing else than “on the evidence that A says that p it is highly probable (or even certain) that A believes that p.” There is a close parallel between the phenomenalist attempt to reduce statements about material objects to statements about sense-data, as well as statements about postulated physical entities, like electrons and force-fields, to statements about homely observables, and the kind of semantic reductionism that is under discussion. Perhaps it will come to be realized more and more widely that semantics needs the postulation of such “abstract entities” as propositions, just as science needs to operate with “constructs” that are not just shorthand devices for formulating highly complex propositions about the phenomenally given. However, while it is unreasonable, and a betrayal of a mere prejudice in favor of a certain kind of language, to dismiss “proposition” as an obscure word unless its meaning can be explained by means of contextual translation into an extensional meta-language, it is a legitimate request that an identity-condition for propositions be formulated. Now, to formulate an identity-condition for a type of entity is to specify a context in which two names of entities of the type in question are interchangeable salva veritate if and only if they designate the same entity of that type. Thus, to mention the least problematic identity-condition, “A” and “B” designate the same class if and only if it does not aﬀect the truth-value of sentences of the form “x ∈ . . .” whether “A” or “B” occupies the dotted place. Similarly, we need a criterion of the form “p and q are the same proposition (or semantically: “p” and “q” designate the same proposition) if and only if “. . . p . . . ≡ . . . q . . .” But what kind of context is symbolized by the dots? To require universal interchangeability would be to lay down a criterion at once ineﬀective and pregnant with paradox. It would be ineﬀective because one of the possible contexts of “p” is the very identity-statement “the proposition that p = the proposition that q” whose truth-value is at issue. And the paradox it is pregnant with is the paradox of analysis: if the synonymy of “p” and “q” entails that nobody can know that p = p unless he knows that p = q, then everybody must know the analysis of every proposition, hence analysis of propositions must be a trivial enterprise

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that does not enlarge anybody’s knowledge22 . Notice that the requirement of universal interchangeability of names of identical entities is equally unreasonable for other kinds of entities that are generally considered less “problematic” than propositions. Thus, if we define identity of individuals x and y, following Leibniz, as “(∀ f )( f x ≡ f y),” where “ f ” ranges over all properties that are meaningfully predicable of individuals, then 1) decision of “x = y” would presuppose itself since “x = y” is one of the values of “ f x,” 2) everybody would have to know every true identity-statement about every individual, since “everybody knows that x = y” would follow from “x = y” and “everybody knows that x = x.”23 Now, I propose as the kind of context in terms of which identity of propositions is to be defined belief-contexts and similar “intentional” contexts—with the exclusion however of the context “A believes (respectively “intends” in some other mode) that p = q,” in order to avoid the paradox of analysis. This definition seems to me to be suggested quite naturally by the reasons for which Carnap’s identity-condition, viz. L-equivalence, is unsatisfactory. “A believes that p but does not believe that q” seems to be perfectly compatible, as Carnap noted, with “‘p’ is L-equivalent to ‘q”’—especially if we consider that any two analytic statements are L-equivalent. But if “‘p’ is L-equivalent to ‘q”’ entailed “the proposition that p = the proposition that q,” there would be incompatibility, since “A believes that p and does not believe that p (at the same time)” is self -incompatible and hence incompatible with any other statement (if “p” is self-contradictory, then “p . q” is self-contradictory). The objection to be faced by his proposal is that such a definition is just as ineﬀective as the one in terms of universal interchangeability, since one cannot decide whether it is logically possible that A believes that p yet not believes that q, independently of deciding whether p = q. But the objection is surely invalid. In fact, the method of proving non-identity of propositions p and q by inviting attention to the evident possibility that someone knows that p yet does not know that q (or mutatis mutandis for other modes of intentionality) is a powerful and perfectly sound method of philosophical criticism24 . One important fruit of

22 For

a detailed discussion of this “paradox of analysis,” see Pap 1955h and Pap 1955a, chapter VI D. be sure, the strong definition of identity would not have this unwelcome consequence if the ramified theory of types were adopted, and the range of “ f ” were accordingly restricted to first-order properties. But the ramified theory of types, according to which it is meaningless to speak of “all the properties” of an entity, turned out to be so restrictive that the axiom of reducibility had to be invoked to mitigate the blow. As it follows from this axiom that if x and y agree in all first-order properties, they also agree in all second-order properties, the above paradox would be reinstated. 24 I deliberately avoid the semantic formulation “it is possible to prove the non-synonymy of ‘p’ and ‘q’ by showing that it is possible for someone to know that ‘p’ is true without knowing that ‘q’ is true.” For, if the latter possibility entailed the non-synonymy of ‘p’ and ‘q’, then no two statements could be synonymous: it is always conceivable that someone fails to understand one of two statements and hence knows one to be true without knowing the truth-value of the other. If, to take account of this objection, the person who allegedly 23 To

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the method is the revelation of the inadequacy of many behavioristic or physicalistic analyses of psychological concepts. Thus, since it is possible for A to know immediately, without induction, that A is in a state of seeing blue, yet not possible for anybody to know without induction that anybody has a certain behavioral disposition, the proposition that A sees blue at a given time cannot be identical with a proposition about a behavioral disposition25 . The choice of intentional contexts for the definition of propositional identity is perfectly natural because propositions are abstracted objects of intentional acts (vide Russell’s suggestive term “propositional attitudes” for what the Brentano school called “intentional acts”). This accords with the traditional definition of a proposition as anything which may be believed or disbelieved or doubted or supposed or asserted, etc. “But, if propositions are thus abstractions from certain kinds of mental states, is not your Platonic realism, according to which propositions may exist without being apprehended by any subject, a gross hypostasis?” Not so. For in asserting the possibility that p1 is not apprehended in any mode by anybody at any time, I merely assert the possibility that nobody apprehend in any mode at any time the proposition that p1 , which is to say that the occurrence of a mental state objectively characterized by p1 is a contingent event. The diﬀerence is merely linguistic: in the one case the propositional noun is grammatical subject and the passive form of “to apprehend” is used, in the other case the propositional noun occurs as the grammatical subject and “to apprehend” in active form. Reification (or “hypostasis”) is the crime committed by those who suﬀer from the compulsion to think of every significant noun as referring to a spatio-temporal particular, not by those “realists” who recognize diﬀerent types of entities. If a philosopher who smells an infantile belief in ghosts and fairies, which is incompatible with what Russell has called a “robust sense of reality,” whenever he hears a fellow philosopher mention entities that are not spatio-temporal, asks me, with an aﬀected air of puzzlement “but where are those propositions? Where is Plato’s heaven?,” I have nothing to reply except to invite him to familiarize himself with type distinctions, and so with the diﬀerence between meaningful and meaningless questions. And if a sophisticated philosopher of language, reared at Oxford, who professes to be well aware of the “category mistake,” claims that propositional nouns are merely incomplete symbols, that they do

might know one to be true without knowing the other to be true is qualified as one who understands both statements, then Goodman’s criticism (see Goodman 1952a) seems inescapable: one who understands both statements must already know whether or not they are synonymous, hence the test in terms of knowledgepossibilities becomes redundant. However, my formulation of the method is invulnerable to Goodman’s criticism, since knowledge of the truth of the proposition that p does not presuppose understanding of the sentence “p” (cf. Pap 1955h). 25 An excellent critique, along this line, of dispositional analysis of psychological concepts may be found in A. C. Ewing’s article (Ewing 1949).

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not refer to anything, he should be invited to make good his claim by answering the presented critique of the logical-construction-theory of propositions.

4.

Necessary Truth and Semantic Systems

It will be recalled that my argument in support of the claim that there are no necessary propositions at all if the linguistic theory of necessary propositions is correct, rested on two premises: 1) the NN thesis, 2) modal statements are, according to the linguistic theory, contingent statements about expressions—if they have a truth-value at all, and are not prescriptive rules of usage misleadingly formulated as declarative sentences. The second premise, however, is meant to refer to natural languages. If modal statements are meant as relative to a language-system, specifically those interpreted language-systems which Carnap calls “semantic systems,” then we get a diﬀerent picture. Thus it can easily be shown that the NN thesis holds if “N(p)” is interpreted as “‘p’ is L-true in L” (where L is a semantic system). Hence one might argue that acceptance of the NN thesis does not compel the conception of necessity as an intrinsic property of extra-linguistic propositions, but is perfectly compatible with the view that “necessary” is to be construed as a relational predicate whose first argument is a sentence and whose second argument is a system of linguistic rules that determines the meaning of the sentence. Before evaluating this species of linguistic theory, whose chief exponent is Carnap, let us clarify the issue. What does it mean to say that necessity is an “intrinsic” property of a proposition? The explanation “it means that a necessary proposition would not be the same proposition if it were not necessary” is not suﬃciently clear. It might give rise to the complaint that it does not at all diﬀerentiate intrinsic from non-intrinsic properties, that indeed any property of any entity is “intrinsic” to the entity in that sense. For is not, quite generally, “x has P and y does not have P” incompatible with “x = y”? And this objection is perfectly valid. It seems to make sense to say “but couldn’t the same proposition which happens to be believed by A not be believed by A, whereas it is a contradiction to suppose that the same proposition which is necessary might not be necessary?,” but the appearance of sense is due to an incomplete formulation. To say of p that it is believed by A is not to ascribe a property, relational or not, to p at all. Since nearly every non-angelic rational being believes some proposition at one time and not at another time, an enormous number of propositions would have incompatible properties if “being believed by A” expressed a property, and the law of non-contradiction would, to the satisfaction of the “dialectical” philosophers, break down not only for the dynamic world of changing particulars but even for the static world of abstractions. Of course, all this excitement stems just from a simple semantic mistake; only “being believed by A at time t” expresses a property, and then our proposition turns out to have the perfectly

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compatible properties “being believed by A at time t1 ” and “not being believed by A at time t2 .” However, that the same proposition should be believed by A at t1 and not be believed by A at time t1 , is just as impossible as that the same proposition should be both necessary and not necessary. On the other hand, the distinction which we vaguely apprehend in feeling that there is, after all, something “in” the intrinsic-extrinsic distinction is simply the distinction between properties expressed by one-place predicates and properties expressed by relational predicates. “p is believed” is an incomplete statement, neither true nor false; it requires expansion into “p is believed by A at time t”; and to say that necessity is an intrinsic property of a proposition is simply to say that statements of the form “p is necessary” (where the values of “p” are propositions, not sentences) are complete. In fact, the expansion “p is necessary in L” yields a meaningless statement, just as would the expansion “p is necessary in New York,” if p is an extra-linguistic proposition. It should be noted that the criterion of whether a predicate expresses a property (can occur as predicate of complete statements) is not syntactic, but rather whether or not predications of the predicate satisfy the law of noncontradiction. “A is to the left of B” is syntactically a complete sentence, unlike “A is between B.” But “being to the left of B” does not express a property any more than “being between B,” since if it did “A is to the left of B” would be both true and false: if A is to the left of B relatively to an observer north of A and B, then A is to the right of B relatively to an observer south of A and B. Now, if “necessary” is construed as a semantic predicate, i.e. a predicate applicable to sentences of an interpreted language (in contrast to sentences of a purely syntactic calculus, which can be characterized as provable, refutable, or undecidable, but not as true, and hence not as necessary), then statements of the form “S is necessary” (here “S ” is a sentential variable) are indeed incomplete: S 1 may be necessary in L1 , and non-necessary in L2 . The basic question at issue, then, can be formulated as follows: is “p is necessary” irreducibly complete, or can it be translated into a complete statement containing a corresponding, semantic and dyadic, predicate “L-true in L”? Carnap defines “S is L-true in L” as “S holds in all the state-descriptions of L,” where “state-description of L” is defined as a conjunction containing for every atomic sentence of L either it or its negation but not both. Thus “Pa. ∼ Qa.Pb.Qb” is a state-description of a semantic system containing just the two individual constants “a” and “b” and the two primitive (one-place) predicates “P” and “Q.” That an S holds in all state-descriptions of L can be discovered without running through all the state-descriptions of L individually. Indeed, if this were not the case, then L-truth relatively to an infinite language, i.e. a language containing quantifiers of unlimited range, would be undecidable. Whether L be finite or not, we can give a finite proof for the Ltrue character of an L-true sentence of L by looking at the semantic rules of

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L, especially those fixing the meanings of the logical constants (the semantic rules for the logical connectives in particular are usually given in the form of truth-tables). To illustrate by a simple example: “Pa ∨ ∼ Pa” holds in all statedescriptions of any L containing “a,” “P” and the logical constants “∨” and “∼ ,” because “Pa” holds in all those state-descriptions of which it is a conjunctive component and “∼ Pa” holds in all those state-descriptions of which it is a conjunctive component, and by the above definition of “state-description,” every state-description of the specified L contains either “Pa” or “∼ Pa” as a conjunctive component. It is clear, then, that “S is L-true in L” is decidable by inspecting the semantic rules of L, just as “S is true in L” is thus decidable if S is L-true in L26 . That the NN thesis holds if “N” is interpreted the way Carnap interprets it is particularly obvious in the case of those necessary statements, like “there are no married bachelors,” whose necessity depends on synonymy-relations. For whether the statement “‘bachelor’ is synonymous-in-L with ‘unmarried man”’ is true can be decided by just looking up the semantic rules of L, hence it is Ltrue (in the meta-language of L) if it is true at all. If one should object that the “looking up” of semantic rules involves after all sense-perception of signs, and that therefore the statement of synonymy rests on empirical evidence after all, he would consistently have to dismiss the very notion of necessary statement, whether explicated in Carnap’s way or in some other way, as absurd. Of course one cannot determine the logical status of a statement without perceiving signs, or imagining signs of a kind some instances of which one has perceived at some time. But the NN thesis, being a conditional proposition, is perfectly compatible with the proposition, be it absurd or not, that there are no necessary statements. Now, whether “‘p’ is L-true in L” can serve as an explication for “p is necessary” surely depends on what L is picked as reference-frame, so to speak. It is necessary that there are no married bachelors, it will surely be agreed. But how can we tell whether the sentence “there are no married bachelors” is Ltrue in L? It all depends on what L is meant. If L contains the usual semantic rules for the logical constants and besides the semantic rule “‘bachelor’ means the property being an unmarried man,” then I am confident that the quoted sentence is L-true in L. My point is that it is nonsense to propose “‘p’ is Ltrue in L” as an interpretation of the complete statement “it is necessary that p” if “L” is a variable. Carnap proposes “to take as the explicatum for logical necessity that property of propositions which corresponds to the L-truth of sentences,” and accordingly lays down “the following convention for ‘N’: For any sentence ‘. . .’, ‘N(. . .)’ is true if and only if ‘. . .’ is L-true” (Carnap 1956a,

26 Cf.

my arguments in support of the NN thesis on pp. 116-117, also Carnap 1956a, 174.

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174). But “ ‘p’ is L-true” is elliptical for “ ‘p’ is L-true in L,” and similarly “ ‘N(. . .)’ is true” is elliptical for “ ‘N(. . .)’ is true in L.” Carnap’s convention, therefore, must have been intended by him in the following sense: For any semantic system L and for any sentence ‘. . .’, ‘N(. . .)’ is true in L if and only if ‘. . .’ is L-true in L. But then I don’t see how it is relevant to the explication of, not “ ‘N(. . .)’ is true in L,” but “N(. . .).” If an analogous convention were laid down for “N(. . .),” it would read either: (a) for any sentence ‘. . .’, and for any L, N(. . .) if and only if ‘. . .’ is L-true in L, or: (b) for any sentence ‘. . .’, N(. . .) if and only if ‘. . .’ is L-true in L. Now, that (a) is false can easily be seen as follows. Consider the substitution-instance with “there are no married bachelors” substituted for the dots in the context “N(. . .),” “ ‘there are no married bachelors’ ” substituted for “ ‘. . .’,” and a language containing the semantic rule “‘bachelor’ means the property being a married man” as value of “L,” which we shall denote by “L1 ”: it is necessary that there are no married bachelors if and only if “there are no married bachelors” is L-true in L1 . But the quoted sentence is not L-true in L1 ; and if the meta-meta-language in which convention (a) is formulated contains the usual semantic rule for “bachelor,” then “it is necessary that there are no married bachelors” is true in it; hence the biconditional is false27 . On the other hand, convention (b) does not even allow us to derive a substitution-instance which is a true-or-false statement, since it contains “L” as a free variable. One cannot answer the question whether it is necessary that there are no married bachelors if and only if “there are no married bachelors” is L-true in L, until L is specified. At best, “it is necessary that p” could be interpreted as “ ‘p’ is L-true in every L which is logically adequate,” where a logically adequate L is a semantic system whose state-descriptions are, according to the customary meanings of the terms, descriptions of possible worlds. Since a possible world is any world that conforms to all necessary propositions, this interpretation slips, none too subtly, the concept of “necessary propositions,” which was to be replaced by a language-relative concept of L-truth, in again through the backdoor. Indeed, that Carnap is guided by an apprehension of entailments and incompatibilities which are “absolute” in the sense of not being relative to a language, in his very choice of the linguistic reference-frame for the construction of his semantic “explicatum” L-truth, is obvious. Thus he noticed that no more than one member of a family of codeterminate predicates, like the family of color predicates, can be admitted as primitive into a descriptive semantic system. For, suppose that both “blue” and “red” were primitives of L. Then conjunctions like “blue(a).red(a)” would occur in some state-descriptions of L, and conse-

27 Perhaps the reader will follow the argument more easily if its form is made explicit: “(∀x)(p ≡ f x)” entails “p ≡ f a”; but “p” is true and “ f a” false; hence the universal premise is false.

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quently (∀x)(blue(x) ⊃∼ red(x))” would not be L-true in L28 . But the explication of “N(p)” as “‘p’ is L-true in L” would then be inadequate, since some sentences which according to their customary interpretation express necessary propositions would fail to be L-true in L. Again, it was noticed that relational descriptive predicates would cause the same sort of trouble if they were admitted as primitives. For, it is surely necessary that the relation Warmer, e.g., is asymmetrical. But some state-descriptions would contain conjunctions like “warmer(a, b).warmer(b, a),” hence “(∀x)(∀y)(warmer(x, y) ⊃∼ warmer(y, x))” would not be L-true in L. After considering various possible solutions of these diﬃculties, Carnap has indicated his preference for the method of “meaning-postulates” and a corresponding redefinition of L-truth. The primitive predicates are introduced into the language-system in the context of postulates which might be regarded as partial explications of their meanings. Thus, if various codeterminate predicates (determinate predicates falling under the same determinable) P1 , . . . , Pn are introduced as primitives by incompatibility postulates of the form “(∀x) (Pi (x) ⊃ ∼ Pr (x))” (where i is not r), we see right away that they are codeterminate, though the question is left open which specific qualities they designate. And if a relational predicate “R” is introduced by postulates expressing asymmetry and transitivity, then we know that it designates an asymmetric and transitive relation, though we don’t know whether it is the relation Warmer, the relation Earlier, or the relation Louder etc. The original definition of “S is L-true in L” as “S holds in all state-descriptions of L” is now replaced by “the conditional with the conjunction of the meaning-postulates of L as antecedent and S as consequent holds in all state-descriptions of L” (Carnap 1952). But what determines the choice of meaning-postulates? Is it not one’s insight into the necessary character of the propositions ordinarily expressed by such and such sentences? Is not Carnap here again constructing the languagesystem in such a way that the extension of the systematic concept (the “explicatum”) “sentence which is L-true in L” should coincide as closely as possible with the extension of the presystematic concept (the “explicandum”) “sentence which, according to the customary meanings of its terms, expresses a necessary proposition”? Carnap knows that the proposition, say, that the relation Warmer is asymmetrical (here “Warmer” is used as a name of the relation which is usually meant by English speaking people when they use the relational predicate “warmer”), is necessary, and this a priori knowledge which is not knowledge of, or about, linguistic rules, but an apprehension of incompatibility (of the propositional functions that x is warmer than y at time t and that y is warmer than x at time t), motivates his adoption of the corresponding

28 Cf.

Carnap 1950b, §18c.

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meaning-postulate. Carnap, however, does not admit that such “conventions” are motivated by any sort of a priori knowledge, and it is pretty obvious to me that he is dodging the issue by committing just the confusion between the empirical proposition that a given sentence is ordinarily used to express a necessary proposition, and the non-empirical proposition that the expressed proposition is necessary, which I have already called attention to. This seems to me clearly brought out by the following passage: Suppose that the author of a system wishes the predicates ‘B’ and ‘M’ to designate the properties Bachelor and Married, respectively. How does he know that the properties are incompatible and that therefore he has to lay down postulate P1 ?29 This is not a matter of knowledge but of decision. His knowledge or belief that English words ‘bachelor’ and ‘married’ are always or usually understood in such a way that they are incompatible may influence his decision if he has the intention to reflect in his system some of the meaning relations of English words. (Carnap 1952)

Carnap seems to say that there is no such thing as knowledge that the properties B and M are incompatible, but only the empirical knowledge that the words “B” and “M” are “always or usually understood in such a way that they are incompatible.” But he overlooks that the knowledge that two predicates are commonly so meant as to be incompatible is compounded of an empirical and an a priori component. For, to say that “P” and “Q” are so meant as to be incompatible is to say: (∃P)(∃Q)(Des(‘P’, P).Des(‘Q’, Q).Inc(P, Q))—where “Des” symbolizes the designation-relation of descriptive semantics which is really at least a triadic relation, since reference must be made to a specified group of sign-users or sign-interpreters, but which I here simplify as a dyadic relation on the assumption that the “pragmatic” variable is kept constant. Suppose that P1 and Q1 are the properties for which this existential statement holds. Then “Des(‘P’, P1 )” and “Des(‘Q’, Q1 )” express empirical propositions, but the proposition expressed by “Inc(P1 , Q1 )” is, if true, necessary.

5.

Implicit Definitions

The reply which Carnap and other “conventionalists” with respect to a priori knowledge may be expected to make is that my argument is based on the mistaken presupposition that the designata of predicates are somehow given independently of the meaning-postulates that express their logical relations or properties. But if, to illustrate, observations of linguistic behavior led me to the semantic hypotheses that Frenchmen mean by the words “bleu” and “rouge” the properties Blue and Red, and I later noticed that they frequently applied these predicates to the same object (“cet objet est bleu et dur et chaud et rouge”), 29 P 1

= (∀x)(Bx ⊃∼ Mx).

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would I not revise my semantic hypotheses? Does not the sentence asserting the incompatibility of these properties function as an implicit definition of the predicates, then? Similarly, if a visiting anthropologist who wants to discover synonymies between words of the tribe language and English words with the assistance of an interpreter who is able to question the natives, is in particular looking for a synonym of the English word “warmer,” and arrives, after some questioning done by the interpreter, at the hypothesis that, say, “krut” is the synonym, he would surely abandon that hypothesis if the natives subsequently gave an aﬃrmative answer to the question whether krutness is a symmetrical relation30 . Is not the sentence “for any x and y, if x is warmer than y, then y is not warmer than x” an implicit definition of “warmer,” then? But let us ask what exactly is meant by saying of a sentence that it is an implicit definition. “Being an implicit definition” surely refers to a functional, not a structural property of a sentence, unlike such predicates as “containing three words,” “disjunctive,” “conditional,” etc. That is, a sentence can significantly be said to be used as implicit definition in certain contexts, not to be an implicit definition intrinsically. Now, I propose as analysis of “S is used as implicit definition in context C”: S does not have the form of an explicit definition, but is used in C as a means for explaining the meaning of one or more constituent terms. Thus, if somebody with a mediocre knowledge of English were to say “I met a happily married bachelor yesterday,” I might say to him “look here, no bachelors are married” in order to get him to see that he was misusing the word “bachelor.” Again, it is conceivable that I might say “if a is earlier than b, then b is later than a, but b can’t be earlier than a if a is earlier than b” to a foreigner who frequently used the word “earlier” where he should have used the word “later,” or vice versa, just in order to teach him the diﬀerence in meaning between “earlier” and “later.” Undoubtedly, the philosophers who, like Wittgenstein certainly and Carnap probably, feel that to speak of necessity as an intrinsic property of propositions is to speak a kind of metaphysics which arises from a fundamental misunderstanding of language, hold that “the proposition that p is necessary,” as a sentence of a natural language, has either no cognitive meaning at all or else means just “the sentence ‘p’, as well as its synonyms, is used, not to assert a fact, but as an implicit definition,” where “being used as an implicit definition” means more or less what it has been analyzed to mean. Yet, is it not obvious that sentences which are ordinarily held to express contingent propositions can exercise just the same function? Thus I might complete my explanation of the diﬀerence in meaning between “earlier” and “later” by asserting the contingent propositions a) event a happened earlier than event b, b) event b happened later than event a, using the words “earlier” 30 Naturally, in the absence of evidence that the natives have studied symbolic logic, the interpreter will have

to ask this question in a roundabout way.

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and “later” to express them. The method of teaching the meanings of words by implicit definition instead of by explicit definition is comparable to challenging the instructed person to solve a crossword puzzle. It amounts to saying that the word “X” means something of which such and such statements containing “X” are true, and it does not matter in principle whether these statements express contingent or necessary propositions. If the use, for the purpose of semantic instruction, of the latter kind of statements is preferable it is because the probability that the instructed person shares the instructor’s belief that p is greater if p is a necessary proposition than if p is a contingent proposition. Suppose, e.g., that B has seen crows and by abstraction formed a concept of crows, but does not understand the meaning of the English word “crow” and furthermore does not believe that all crows are black. If A then tells him “crows are a species of birds all of whom are black,” this assertion will not lead to the goal of communicating to B at least part of the meaning of “crow.” It may be replied that, indeed, sentences expressing contingent propositions may occasionally be used as implicit definitions, but that they are at any rate sometimes, or even most of the time, used to assert facts, whereas the distinctive characteristic of so-called “necessary” statements is that they are always used as implicit definitions and are never used to assert facts. Now, if “fact” here is simply a synonym for “empirical” fact, it is of course undeniable that necessary statements are never used to assert facts, since if they were in given contexts so used, they would, by definition of “necessary,” not be necessary statements in those contexts. But suppose that “fact” is used in the sense in which “it is a fact that p” is cognitively synonymous with “p.” We use the expression in this sense, as an “assertion sign,” when we make statements like “the fact that p, entails that q.” But it is used in just the same sense, whether the premised proposition p be contingent or necessary. If, e.g., it is proper to say “the fact that p, entails that q,” where “p” and “q” stand for contingent propositions, why should it not be proper to say “the fact that p entails r, entails that not-r entails not-p”? It may be countered that it is naive to infer from the grammatical propriety of the expression “the fact that p” that “p” is used to assert a proposition; the utterance “it is an undeniable fact that no bachelors are married” may just be an emphatic way of urging the person addressed to observe a conventional way of speaking. But remembering that propositions are abstractions from intentional states of mind (or dispositions manifesting themselves in such states), the relevant question to ask here is simply whether it is significant to make a statement of the form “I believe (resp. disbelieve, doubt, etc.) that p” where p is a necessary proposition. This will have to be granted if it is granted that “I believe that (p entails q)” is significant, since “p entails q” expresses, if true, a necessary proposition. And of course this must be granted. I remember a time when I did not believe that “∼ q” is entailed by

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“p ≡ q. ∼ p”; finding that an authoritative logician made this claim, I applied the methods of the propositional calculus, and now I believe (and probably know) that the entailment holds. The answer given by the linguistic theory to the perennial question why some universal statements, like the laws of logic and of arithmetic, but also statements like “the relation Warmer is asymmetrical” which cannot plausibly be classified as logical truths since a descriptive term occurs essentially in them, are absolutely undeniable although it is impossible to observe all the cases to which they apply, is that to deny them is equivalent to changing their meanings, to violate conventional linguistic rules. But this answer, popular as it is with empiricists who want to get rid of the “inner eye of Reason,” cries out for clarification. Suppose the assertion made by these philosophers is the following: to say that S expresses, according to the customary interpretation of its terms, a necessary proposition is equivalent to saying that if anyone were to deny S then he would take S to express a diﬀerent proposition, so that his disagreement with those who would stake their lives on the truth of S is merely verbal. Let us waive the objection that one might, after all, correctly interpret a sentence S to express a proposition p which is objectively necessary but disbelieve it just the same, just the way I personally once disbelieved the necessary proposition expressed by the sentence “(∀p)(∀q)[(p ≡ q . ∼ p) ⊃ ∼q].” For the “conventionalist” might with good reason say that this could not happen if the necessary propositions in question are of the “self-evident” variety, like the laws of non-contradiction and of the excluded middle. But let us ask whether “if, then” in the conditional “if anyone were to deny S then he would take S to express a diﬀerent proposition” stands for necessary implication or for factual implication. If the former, then the claim that is made by the conventionalist is that it is self-contradictory to suppose that a “self-evident” necessary proposition be disbelieved. Now, in the first place, it is not self-evident that such a supposition is self-contradictory. At any rate, if anyone maintains that he can derive a contradiction from the statement “A believes that there are two particular events (not kinds of events!) x and y, such that x happens before y and y happens before x,” and from similar statements, the burden of proof rests on him. But second, reference to linguistic rules would be conspicuously absent from such an analysis, hence it would be obscure in just what sense our theory has given a “linguistic” explanation of necessity. The other horn of the dilemma is that the conditional is meant to express a factual implication only: from the fact that A says, for example, “there are (or may be) events x and y such that x is earlier than y and y also earlier than x” it can be inferred with high probability that A uses some constituent term, especially the term “earlier,” in an unconventional sense. But since this condition would obviously be satisfied if the sentence in question expressed conventionally a contingent proposition

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which is probably believed by A, our theory gives no criterion of necessity at all on this alternative. Suppose we found a man who, having convinced himself antecedently of the truth of the empirical proposition “all metals conduct electricity” and “this is a metallic object,” discovered that the object in question does not conduct electricity, and thereupon concluded that the principle of deductive inference “if all A are B and x is an A, then x is a B” is invalid. Undoubtedly we would say that, if he is not just perpetrating a joke, he must be misinterpreting at least one of the logical constants, perhaps “if, then,” which enters into the formulation of the principle. But to say that our conclusion that the man must have misunderstood this abstract sentence is inevitable because the latter implicitly defines the logical constants, is not illuminating since it amounts to a mere repetition, in obscurer language, of the inference to be justified. How, after all, is one to prove that it is impossible for a man to believe the conjunctive proposition that all A are B and x is an A and x is not a B (where A and B are classes and x is an individual)? If C, who utters sentence S in order to assert proposition p, is cocksure that D, who verbally contradicts him, has misunderstood him, i.e. is not really contradicting p, this is because C is cocksure that D believed the proposition that p just as firmly as he himself. The view that those necessary statements in particular which are called “logical truths”—the exact boundary between logical truths and necessary truths that do not belong to logic being a controversial subject which I cannot go into here—are implicit definitions of logical constants, and hence terminological proposals or announcements and hence do not express “truths” at all, seems to be inspired by a dangerously misleading analogy between the axiomatization of bodies of descriptive statements containing extra-logical terms, like geometry or mechanics, and the axiomatization of logic itself. The axioms of a system of pure geometry, for example, implicitly define the geometrical primitives in the sense of specifying the structural meanings they have within the deductive system—not to be confused with their material meanings in empirical applications of the system. The truths of, say, pure Euclidean geometry, are, not the axioms themselves, nor the theorems themselves, but logical truths of the following form: for all values of x, y, z if F(x, y, z), then G(x, y, z), where “x,” “y,” “z” represent the primitives of the system that admit of various interpretations, “F(x, y, z)” the conjunction of the axioms, and “G(x, y, z)” a theorem. The axiomatic system is, as it were, embedded into a logical calculus whose axioms and theorems are used as rules for deducing the theorems of the axiomatic system from its axioms. Now it is tempting to construe the axioms of the logical calculus in turn as implicit definitions of the primitives of the logical calculus, like “or,” “not,” “all,” “if, then.” Suppose, for example, that a propositional calculus contained as sole primitives “if, then,” “not” and “and,” occurring in the following axioms: 1) if (if p, then q) and (if q, then r), then (if p, then r); 2)

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if (if p, then q) and not-q, then not-p; 3) if p and q, then p. We might take the conventionalist line, now, that these axioms are not truths at all, but simply explanations of the meanings of “if, then,” “and” and “not” in the calculus. If we replace these connectives by corresponding operation variables, two of them binary and one of them singularly, writing: 1) R(S (R(p, q), R(q, r)), R(p, r)); 2) R(S (R(p, q).Nq), N p); 3) R(S (p, q), p), we would indeed be left with formulae which are syntactically meaningful insofar as, given certain formal rules of derivation, certain other formulae are derivable from them, but their assertion as true would at once be recognized as nonsensical. Yet, the logical truths would simply reappear in the meta-language in which the rules of derivation are formulated in the non-symbolic, “natural” idiom. Thus, no formal proof at all can be conducted without the ponendo ponens rule: if “p” is an axiom or theorem of the calculus, and “R(p, q)” is an axiom or theorem of the calculus, then “q” is a theorem of the calculus. To say that this rule again is an “implicit definition” of “if, then” and the other connectives that occur in it, in the same sense in which this was asserted of 1), 2) and 3), is to start on an infinite regress of formalization. Some twin of our principle of inference will always remain alive on the highest level of the meta-linguistic hierarchy, which cannot be said to be an implicit definition in the same sense. Now, what is the justification for adoption of the above rule? Is this like asking for the justification of the adoption of a particular rule of a game? On the risk of acquiring the reputation of an impenitent reactionary, I would side with “logical realists” in firmly denying that deductive logic is analogous to a game of chess in this respect. Is it not obvious that this rule is laid down with a view to interpretation of the calculus? Specifically, the formalizing logician thinks of the “if p, then q” which was formalized as “R(p, q)” in the first place. And he knows that the stated rule will invariably lead from true axioms to true theorems if the calculus interpretation includes the interpretation of “R(p, q)” in the sense of “if p, then q.” But how can he know this without proving it as a “meta-theorem,” which presupposes formalization of the meta-language and application of an analogous rule of inference formulated in the meta-meta-language? Now, the usual way of proving that the ponendo ponens rule has the “truthpreserving” character which any acceptable rule of deduction must have, is to prove that the corresponding calculus formula “if p and (if p, then q), then q” is a tautology. Since a tautology is a truth-function of propositional variables which is true for all values of the variables, such a proof cannot get started until “if, then” is interpreted as a truth-functional connective, specifically in the sense of the symbol of Principia Mathematica “⊃” (material implication). Once this has been done, the proof is elementary: by definition, “p ⊃ q” is false if and only if “p” is true and “q” is false; hence “(p.(p ⊃ q)) ⊃ q” could be false only if “q” were false and “p.(p ⊃ q)” true. But the latter is impossible since in order for “p.(p ⊃ q)” to be true, “p” must be true, and the joint truth of

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“p” and falsehood of “q” is incompatible with the truth of “p ⊃ q” and hence with the truth of “p.(p ⊃ q).” But what is the justification for the truth-functional interpretation of “if, then”? None other than that it preserves that common core of meaning in the various uses of “if, then” (to assert causal connections, logical deducibility, resolution to act in a certain way if such and such conditions are realized, etc.) which enables us to justify those and only those methods of deduction which we intuitively accept as valid, i.e. as corresponding to entailments31 . Thus the interpretation of “if, then” in the sense of material implication makes it easy to prove that arguments of the forms “p; if p, then q; therefore q” and “if p, then q; not-q; therefore not-p” are always truth-preserving (i.e. leading from true premises to true conclusions) whereas arguments of the forms “q; if p, then q; therefore p” and “if p, then q; not-p; therefore not-q” are not. In that case, however the proof of the truth-preserving character of the ponendo ponens rule is, if not formally circular, based on our intuitive knowledge that any proposition expressed by “p and (if p, then q)” entails the corresponding proposition expressed by “q.” Such intuitive apprehension of entailments may then motivate our “adoption” of the criterion of adequacy: no analysis of “if, then” is adequate unless it enables a formal demonstration of the ponendo ponens entailment. The apprehension of logical necessity is thus prior to the adoption of linguistic conventions, such as the definition of “if p, then q” as “not both p and not-q,” that render formal proofs possible. Such apprehension alone can explain why we accept just this analysis of the logical constant “if, then” as adequate, and not, say, the analysis “not both q and not-p.” To explain, therefore, the necessity of the customary principles of deductive inference in terms of “linguistic conventions” is to put the cart before the horse.

31 Cf.

the following statement by I. Copi: “although most conditional statements assert more than a merely material implication between antecedent and consequent, we now propose to symbolize any occurrence of ‘if-then’ by the truth-functional connective ‘⊃’. It must be admitted that such symbolization abstracts from or ignores part of the meaning of most conditional statements. But the proposal can be justified on the ground that the validity of all valid arguments involving conditionals is preserved when the conditionals are regarded as asserting material implications only . . . ” (Copi 1954, 18).

III

SEMANTIC ANALYSIS: TRUTH, PROPOSITIONS, AND REALISM

Chapter 6 NOTE ON THE “SEMANTIC” AND THE “ABSOLUTE” CONCEPTS OF TRUTH (1952)

In Carnap 1942, Carnap tells us, following Tarski, that a criterion of adequacy to be satisfied by an acceptable definition of truth is that to assert that a sentence is true means the same as to assert the sentence itself ; e.g., the two statements “the sentence ‘The moon is round’ is true” and “The moon is round” are merely two diﬀerent formulations of the same assertion. (Carnap 1942, 26)

If we are permitted an interlinguistic use of “if and only if” (a usage involved in Carnap’s own formulation of truth-rules for atomic sentences, such as “‘P(a)’ is true if and only if a has property P”), we may formalize this informally stated criterion of adequacy as the following necessary truth:1 (∀S )(∀p)[if S designates p, then (S is true if and only if p)], where the variable “S ” takes names of sentences as substituends (or the values of S are sentences) and the variable “p” takes sentences as substituends (or the values of p are propositions, designata of sentences). Obviously, in committing himself to the above criterion of adequacy, Carnap (or, for that matter, Tarski) is committed to the view that the following proposition is self-contradictory: not-(‘the moon is round’ is true) and (the moon is round). (For reference purposes, this conjunction will be abbreviated by “not-S 1 and p1 ,” where it is understood, of course, that the proposition p1 is the designatum of the sentence S 1 .) Later on in the book, however, Carnap discusses a “corresponding” absolute concept of truth which is not semantical at all since it is predicable of 1 Since

by a “criterion of adequacy” is meant a proposition which must be logically demonstrable on the basis of an adequate definition of the concept in question, it follows indeed that in laying down “p” as a criterion of adequacy one asserts its necessity, though preanalytically.

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propositions, not sentences, such that no reference to a definite language is required in predications of this absolute concept. Thus sentences of the form “it is true that p” are complete as they stand; being themselves object-linguistic they do not refer to any language at all. On the other hand, sentences of the form “S is true” are elliptical; in order to become unambiguous they must be expanded into “S is true in language L.” Or, as it might be put, it is logically possible that the sentences “S 1 is true” and “S 1 is not true” should both be true, since the former sentence may be short for “S 1 is true in L1 ” and the latter sentence short for “S 1 is not true in L2 .” But a statement of the form “it is true that p and it is not true that p,” being logically equivalent to “p and not-p,” would necessarily be self-contradictory. It would appear, then, that Carnap asserts the self-contradictoriness of both of the following conjunctions: not-(S 1 is true in L1 ) and p1

(1)

(where it is understood, and signalized by the use of identical subscripts, that p1 is the proposition designated by S 1 in L1 ), and it is not true that p1 , and p1 .

(2)

What I wish to prove in this note is that (1) is not self-contradictory, though (2) is self-contradictory, and that for this reason predications of the absolute concept of truth cannot be regarded as merely diﬀerent formulations of the corresponding semantic statements. I shall further argue that the theory of propositions as possible states of aﬀairs, which was subsequently sketched by Carnap in (Carnap 1956a), brings him close to a wholly nonsemantical conception of truth which is defensible, unlike the semantic conception, by reference to ordinary usage, and moreover involves no commitment to Platonism. My argument is exceedingly simple: “S 1 is true” could not be true unless the sentence named by “S 1 ” existed, while “p1 ,” the object-linguistic sentence, does not entail the existence of any sentence at all; hence “S 1 is true” and “p1 ” cannot be logically equivalent. That “S 1 is true” does entail the existence of at least one sentence follows from the following general logical principle: if “F(x)” is any sentential function at all (here “F” is not meant as a predicate variable but as an undetermined constant predicate), and xi an admissible value of the variable x, then “F(xi ),” the singular statement, entails “(∃x)F(x).” If the sentential function belongs to the object-language, then a familiar illustration of the principle would be as follows: if John is tall, then there are tall individuals and, a fortiori, there are individuals. If, now, “F” happens to be the semantic predicate “true,” and the variable x ranges over object-linguistic sentences as its values, it follows that “S 1 is true,” which is an instance of the form “F(xi ),” entails that there are true sentences and, a fortiori, that there are sentences.

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The same result can be obtained via the theory of descriptions in case the grammatical subject of a semantic predication of truth should be, not a name of a sentence such as the usual quotes, but a definite description of a sentence, as in “The paradoxical sentence on p. 200 is true.” For this is analyzable into “there is one and only one paradoxical sentence on p. 200, and that sentence is true.” It was with a view to the theory of descriptions that the negation in (1) was given the inelegant form “not-(S 1 is true in L1 )” instead of the more colloquial form “S 1 is not true in L1 .” For, suppose that “S 1 ” is an abbreviation of a description of a sentence. In that case the latter negation would naturally be interpreted to mean that the described sentence exists and is not true, while the former negation, being equivalent to the disjunction “either S 1 does not exist, or S 1 exists and is not true,” would be true in case S 1 did not exist. The conjunction “S 1 is not true in L1 , and p1 ” is, indeed, self-contradictory; but “S 1 is not true” is not the contradictory of “S 1 is true,” just as “the king of Switzerland is not jolly” is not the contradictory of “the king of Switzerland is jolly.” To illustrate the point in terms of Carnap’s sample sentence “the moon is round” (which I prefer to Tarski’s “snow is white” only because it is less worn in the vast literature on the semantic concept of truth): A universe which is just like ours except that it does not contain language, and thus contains no sentences, is surely logically possible. In such a universe it would still be the case that the moon is round, but nothing could be the case in such a universe which logically presupposed the existence of sentences, hence it would not be the case that the sentence “the moon is round” is true. I have already indicated one possible source of the erroneous belief that (1) is self-contradictory, viz. confusion of “S 1 is not true” (the contrary of “S 1 is true”) with “not-(S 1 is true)” (the proper contradictory). Another possible source is the confusion between formal and pragmatic contradiction. A sentence is pragmatically self-contradictory if it is necessarily falsified by the occurrence of a token of itself but not by what it asserts; in other words, if S is pragmatically self-contradictory, then a contradiction is deducible, not from S alone, but only from the conjunction “S , and there exists a token of S .” In this sense “no proposition is ever asserted” is pragmatically contradictory but not formally so (unlike “no proposition is true”). Now, if anyone should feel that from “the moon is round” it follows that the sentence “the moon is round” exists, this feeling would evidently be due to an unnoticed shift from the object-linguistic premise (which is a statement of the physical language) to the statement that the premise has been asserted (which is a statement belonging to the pragmatic meta-language). Now, I propose to show that Carnap, in addition to being wrong in maintaining the self-contradictoriness of (1), is moreover inconsistent in holding both

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(1) and (2) to be self-contradictory. As has been shown, in holding (1) to be self-contradictory one is committed to the view that “p1 ” (e.g., “the moon is round”) entails the existence of at least one sentence; for if it does not entail this consequence (and it surely does not), then it might be true even though “S 1 is true” is false on account of the nonexistence of S 1 . Now, in Carnap 1942 Carnap implicitly asserts the logical equivalence of “it is true that p1 ” and “p1 ” by laying down definition 17-1: p is true ≡d f p.2 He also holds, as we have already seen, “p1 ” to be logically equivalent to “S 1 is true.” If he were consistent, he would accordingly have to hold that “S 1 is true” is logically equivalent to “it is true that p1 .” But as a criterion of adequacy to be satisfied by an acceptable definition of the absolute concept of truth, i.e., of such sentences as “it is true that p1 ,” he lays down the following logical equivalence: (a proposition) p is true if and only if for every language L and for every sentence S , if S designates p in L, then S is true in L. (slightly altered version of T 17-A in Carnap 1942)

Hence consistency would require him to hold that “S 1 is true” is logically equivalent to: (∀S )(∀L)(if S designates p1 in L, then S is true in L).

(3)

Now, if “p1 ” entails the existence of at least one sentence, and “p1 ” is logically equivalent to “it is true that p1 ,” then “it is true that p1 ” cannot be logically equivalent to (3), for (3) is a nonexistential universal implication: you cannot deduce from (3) that there exist any values of S and L which satisfy the antecedent! After what has been said already, it hardly needs to be added that the proper way to resolve the inconsistency is to recognize that “p1 ” does not entail the existence of sentences, not to reject T 17-A. As a matter of fact, if the latter criterion of adequacy were employed as the definition of “it is true that p” (Carnap leaves it undecided whether any other definitions, likewise satisfying T 17-A, would be preferable), we would have a precise formulation of the logical-construction theory of propositions according to which propositions are classes of sentences. It has not been suﬃciently realized perhaps that, even if one holds statements about propositions to be reducible to statements about sentences in the sense of T 17-A, one can consistently hold that propositions may exist without being verbally expressed. But T 17-A clearly shows this to be the case, since by this theorem (or definition) “(∃p)(p is true)” is logically equivalent to 2 I am replacing Carnap’s “p is true” by “it is true that p,” because one cannot derive from “p is true” by substitution grammatical sentences if, as is customary, statements are substituted for “p.”

The “Semantic” and the “Absolute” Concepts of Truth (1952)

(∃p)(∀S )(∀L)(if S designates p in L, then S is true in L),

151 (4)

which still does not assert the existence of sentences. It may be objected, justifiably, that T 17-A can hardly be regarded as a formulation of the logical-construction theory of propositions, for this reason: it shows, indeed, how predications of truth upon propositions may be dispensed with in favor of predications of truth upon classes of sentences, but it does not show how any reference to propositions as entities is eliminable since the propositional variable remains with us in the context of the function “S designates p.” But the sketched logical construction of propositions out of sentences can be purified by replacing the relation of designation, which calls for a propositional variable, with the relation of synonymy. The expression “for any sentence S , if S designates that the moon is round, . . . ” may simply be replaced with the expression “for any sentence S , if S is synonymous with ‘the moon is round’, . . . ´’ (and if it is objected, in keeping with current fashion, that the relation of synonymy is not clear, it is easy to reply that it is at least as clear as the relation of designation). My purpose is not to argue in favor of the logical-construction theory of propositions, with its implication that the absolute use of “true” (predicating it upon propositions) is merely a fa¸con de parler which is reducible to the semantic use of “true” (predicating it upon sentences). Rather I wish to show that even if one endorses the thesis of the reducibility of the language of abstract entities to the nominalistic language, it still makes good sense to speak of the existence of propositions which are not verbally expressed. In accordance with the above proposal of replacing “designation” with “synonymy,”3 we obtain the following translation of “p1 is true”: (∀S )(∀L)(if S is synonymous with ‘p1 ’ in L, then S is true in L).

(5)

But this universal implication does not entail the existence of sentences any more than the corresponding implication containing “S designates p.” If the existence of the sentence “p1 ” were deducible from it, as a superficial glance might make one think, then “(∃S )(S is synonymous with ‘p1 ’)” would be deducible from it (for “p1 ” itself is such an S ). But the latter existential sentence 3I

would like to remind the reader that I am here only concerned with the implications of the logicalconstruction theory of propositions, not with the question of its tenability. I have, as a matter of fact, serious doubts whether propositional variables are eliminable in the indicated fashion, since the synonymy of “S 1 designates that p1 ” and “S 1 is synonymous with S 1 ” is questionable for this reason: given the assumption that the language containing S 1 is univocal, the latter statement is a tautology; but the former statement is clearly contingent since its translation into another language leaves us with an informative statement of descriptive semantics. For example, “‘the moon is round’ designates, in English, that the moon is round” translates into “‘the moon is round’ heißt, auf Englisch, daß der Mond rund ist.”

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is not deducible, by the principle that no universal implication entails the existence of values satisfying its antecedent. Now to the second thesis I promised to argue, viz. that the absolute use of “true,” involved in statements of the form “it is true that p,” is far more common than the semantic use, and can be logically reconstructed in terms of the theory of propositions as possibilities which Carnap sketches (perhaps all too briefly) in Carnap 1956a. For one thing, if a sentence like “it is true that X voted against the motion” were synonymous with the semantic statement “‘X voted against the motion’ is true,” then its correct translation into, say, German, would be: “‘X voted against the motion’ ist wahr”; for the sentence thus construed would contain a name of the sentence asserted as true which belongs equally to all languages containing the quoting device. But while a translation of “he is the author of a book entitled ‘Meaning and Truth”’ into “er is Verfasser eines Buches betitelt ‘Meaning and Truth”’ is undoubtedly correct, I doubt whether anybody would accept the above as a translation of the English statement of the form “it is true that p.”4 A more important point against the semantic conception of truth, however, is that the interpretation of “true” as a systematically ambiguous predicate goes against ordinary usage. I think ordinary usage of “true” would justify a criterion of adequacy, to be satisfied by an acceptable explication of the concept of truth, according to which “true” in the meta-linguistic statement “it is true that language L contains some atomic statements” means the same as “true” in the object-linguistic statement “it is true that some things are blue.” One might reply that one is surely justified in departing, in the logical reconstruction of a term T , from the ordinary usage of T in case the latter is inconsistent in the sense of giving rise to contradictions; and that the reconstruction of “true” as a systematically ambiguous term is demanded precisely by the well-known antinomy of the liar which can be constructed in natural languages. But while splitting “true” into “true relatively to object-linguistic sentences” and “true relatively to meta-linguistic sentences” is one way of solving the antinomy, it is not the only way. It can, of course, equally be solved by forbidding self-referential statements, i.e., splitting “statement” into “object-linguistic statement” and “meta-linguistic statement.”5 Whether the 4 The

same point is made by P. F. Strawson, in his subtle article “Truth” (Strawson 1949), at 84. does not seem to be any inconsistency between the use of “true” as an absolute predicate, i.e., a predicate primarily applicable to propositions, not sentences, and Tarski’s method of solving the antinomy by the prohibition of “semantically closed” languages. The stratification of sentences yields indirectly a stratification of propositions, where “object-linguistic proposition” would mean “proposition expressed by an object-linguistic sentence,” and similarly for the higher levels. Even though a given meta-linguistic proposition might entail that some propositions are verbally expressed (e.g., the meta-linguistic proposition that “the moon is round” designates that the moon is round), it could never entail that it itself is verbally expressed, and thus the concept of verbally unexpressed propositions is equally applicable to meta-linguistic propositions.

5 There

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153

price which this method of solving the antinomy costs in terms of violence done to ordinary usage is just as high as the price to be paid for adoption of the former method is at least an open question. I am not denying that “S 1 is true” is ambiguous until it is expanded into “S 1 is true in L1 ,” just as “x is in motion” is ambiguous until it is expanded into “x is in motion relatively to y.” But this fact lends no support to the view that “true” is ambiguous in the sense that it might mean “true in L1 ” or “true in L2 ”— just as, say, “sentence” is ambiguous in that it might mean “sentence relatively to formation-rules F1 ” or “sentence relatively to formation-rules F2 .” For the ambiguity of “S 1 is true” is easily accounted for in terms of the ambiguity of the sentence of which “true” is predicated. The expansion “. . . in L1 ” (or, more specifically, “relatively to semantic rules S R1 ”) simply amounts to an indication of what proposition, expressed by S 1 , it is that truth is predicated of. To argue that “true” is systematically ambiguous because the meaning of “it is true that p1 ” varies with the semantic rules determining the meaning of “p1 ,” is no more plausible than to argue that, say, “believed” is systematically ambiguous because the meaning of “it is believed that p1 ” varies with semantic rules in just the same way. In Carnap 1956a, specifically in the section on “Extensions and Intensions of Sentences,” Carnap moves pretty close to an absolute theory of truth, i.e., a theory according to which “true” is primarily predicable of propositions and only derivatively of sentences: a true sentence is a sentence designating a true proposition. He speaks of true propositions as of possibilities exemplified by a fact and of false propositions as of unexemplified possibilities. An absolute theory of truth as such a relation of exemplification holding between propositions and facts, by analogy to exemplification as a relation between properties (universals) and individuals, has been worked out by Baylis, in Baylis 1948, and I cannot go into an examination of this theory in this brief note. I wish to confine myself here to a reply to the frequent objection against absolute theories of truth that they necessarily involve a foggy Platonic metaphysics of abstract entities. What, it is asked, could be meant by the “existence of unrealized possibilities”? Is not Carnap’s talk of “intensions” or “designata” a revival, in technical jargon, of Meinong’s “Gegenstandstheorie” which Russell put in its place long ago? It seems to me that Carnap could easily silence these accusations by applying once again his fruitful maxim (if I may put it in Russellian paraphrase): wherever possible, translate from the material to the formal mode of speech. Consider the sentence “the possibility that Mary should be baking pies now exists, and exists no matter whether it is exemplified or not.” If this is deemed obscure, let us translate this piece of “ontology” into the semantic meta-language, as follows: Let “ S 1” represent the sentence “it is possible that Mary should be baking pies now,” and “S 2 ” the sentence “Mary is baking pies now.” Then

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the truth of S 1 is compatible with the falsehood of S 2 ; in other words, S 1 does not entail S 2 . Thus the assertion of the existence of possibilities reduces to the assertion of the truth of certain modal propositions. Surely, we cannot accuse a man of talking obscure metaphysics because he freely uses “possible” as a primitive term?

Appendix: Rejoinder to Mrs. Robbins (1953) [Editors’ note: Pap here replies to B. L. Robbins’ criticisms of chapter 6 in Robbins 1953.] There is, indeed, one serious misrepresentation of my argument in Mrs. Robbins’ “Some Remarks on Semantic Systems” (Robbins 1953). It may be described as a confusion of material and logical equivalence. She claims that I reject: (1) S 1 is true in L1 if and only if p1 . But I don’t. What I reject is the claim that this biconditional is necessarily true, not that it is true. After all, in order to show that the biconditional is false I would have to show that either “p1 ” is true and “S 1 is true” is false or that the latter statement is true and the former statement false! What I claimed was that it is logically possible that “p1 ” is true while “S 1 is true” is false. It follows that Mrs. Robbins’ reductio ad absurdum of my argument on Robbins 1953, 26, is fired at a strawman. For while the negation of “p ≡ q” is indeed equivalent to “p ≡∼ q,” the negation of “‘p ≡ q’ is necessary” is obviously not equivalent to “p ≡∼ q”! The same oversight leads her to impute to me the claim that (1) is inconsistent with (2): it is true that p1 if and only if p1 . What I actually claimed, on page 4 of my article (January 1952), is that it is inconsistent to hold both (1) and (2) to be necessary equivalences. I did not deny that (1) is provable in the semantic systems of Tarski and Carnap on the basis of their definitions of truth. My point is that insofar as their definitions turn (1) into a necessary proposition they do not accord with the ordinary meaning of “true”; for (1), taken as a statement of natural language, does not express a necessary truth. This non-accordance with the ordinary meaning of “true” is the “error in the construction of the definition of truth” which Mrs. Robbins challenges me to detect. To take an analogy, suppose I constructed a definition of “significant” which makes the following statement provable in my system: There is a statement S such that S is significant but not-S is not significant. She would presumably reject my definition just because it entails a statement which she finds, on the basis of the ordinary meaning of the defined term, unacceptable. To ask for an independent argument against the definition would be unfair; the only way you can refute a theory, be it scientific or philosophical, is by showing that it entails unacceptable consequences. As for the “gratuitousness” of my eﬀort “to prove that predications of the absolute concept of truth cannot be regarded as merely diﬀerent formulations of the corresponding semantic statements,” I would like to say this: “it is true that p” diﬀers from ordinary truth-functional compounds (“if p, then q,” “p or q,” etc.) in that it contains the name of an intension, “that p1 ” names the intension of the sentence “S 1 .” Since Carnap still maintains the thesis of extensionality, one would expect him to hold that names of intensions are eliminable. I wanted to show that “that p” is not eliminable from the context “it is true that p” via translation into “S is true (in L).” Mrs. Robbins may say that it is easily eliminable since “it is true that p” is synonymous with “p.” But when Carnap says in Carnap 1956a that a (factual) proposition is something which may or may not be exemplified, one gathers that truth is conceived as a genuine property of a proposition (its being exemplified) just as nonemptiness is a genuine property of a property (something is predicated of a property when it is said to be nonempty, though this can be said without using the second-level predicate).

Chapter 7 PROPOSITIONS, SENTENCES, AND THE SEMANTIC DEFINITION OF TRUTH (1954)

Those philosophers who conduct analyses of concepts in a formal way generally lay down formal and material conditions of adequacy for the definition which is to be constructed. The formal conditions of adequacy concern the formal features of the language in which the definition is to be constructed; an example would be the rule that any variable which occurs free (unbound by quantifiers) in the definiendum must likewise occur free in the definiens. The material conditions of adequacy, however, are philosophically more interesting: they are sentences which must be provable on the basis of an adequate definition, and which are intended to guarantee that the defined term A designates on the basis of the constructed definition the same concept it designates in its ordinary usage in the “natural” language (whether conversational or scientific)—or at least a closely similar concept.1 The material conditions of adequacy, then, are sentences containing A which must themselves satisfy the following requirements: R1 they are necessary statements (for they are supposed to be provable on the basis of a definition, without the help of empirical premises) R2 they contain A in its usual sense (otherwise one would not be justified in calling them conditions of adequacy, for an “adequate” definition of A is a definition formulating the usual sense of A).2 The separation of these requirements is, indeed, somewhat artificial; one might simply state the requirement that the material conditions of adequacy 1 That

the “explicandum” and the “explicatum” are diﬀerent concepts is shown by the fact that some necessary statements involving the explicandum cease to be necessary when the explicatum is substituted for the explicandum. Thus material implication is diﬀerent from the concept usually meant by “implication,” because in the usual sense of “implication” it is necessary that “for any p there is a q such that p does not imply q nor not-q,” while this is false for material implication. 2 Frequently, what is called the condition of adequacy is, not the statement which is to be proved on the basis of the definition, but that the statement be provable on the basis of the definition. This, however, is an inconsequential diﬀerence of terminology.

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be sentences which, in the usual sense of the definiendum, express necessary propositions. But for expository purposes this artificial separation may be salutary. By way of illustration, suppose the concept of truth were so defined that the statement “whatever is believed by everybody is true” is a logical consequence of the definition. This statement is, however, ineligible as a material condition of adequacy for the definition of truth, for, since it is not self-contradictory to suppose that at a given time a given false proposition is believed by everybody, R1 is violated. Should one object that the statement is necessary for the simple reason that it is a logical consequence of a definition, then R2 would be violated, since the statement would be necessary only because the definition gives “true” an unusual sense. As is well known, Tarski selects as material conditions of adequacy for the definition of truth sentences of the form: p is true if and only if p

(T)

where “p” is a functor that turns into the usual kind of sentence-name, viz. the sentence itself put within quotes, when a sentence is substituted for its argument “p.” (Notice, however, that if by the “values” of a variable we understand the entities designated by the substitutable constants, then the values of “p” are propositions, and the values of “p” are sentences.) What I wish to show in the following is that statements of the form (T) satisfy neither R1 nor R2. I begin by showing that R1 is not fulfilled, i.e. that statements of the form (T) are contingent, not necessary. Admittedly, Tarski has not explicitly claimed that such biconditionals are, according to the ordinary meaning of “true,” necessarily true, but only that they are true. Yet the former claim is implicit in his selecting them as sentences that should be provable by means of the semantic definition of truth. The thesis to be established, that biconditionals of the form (T) are contingent, may be made intuitively plausible by the following reasoning. Let us consider the example “‘the moon is round’ is true if and only if the moon is round.” Now, it is consistently thinkable that, while the moon is indeed round, the sentence “the moon is round” is not true for the simple reason that it does not express the proposition that the moon is round, but instead some false proposition. From the proposition that the moon is round we can logically deduce such propositions as that the earth’s satellite is round (assuming the identity “the moon = the earth’s satellite” to be analytic), or that there exists at least one round celestial body, not however the proposition that the sentence “the moon is round” expresses the proposition that the moon is round. In other words, the truth-value of the semantic proposition that “the moon is round” is true depends on what proposition is expressed by this sentence, while the truthvalue of a proposition of astronomy hardly depends upon semantic facts. To be sure, nobody would ever say “the moon is round, but ‘the moon is round’ is not true.” Similarly, nobody would ever say “John is healthy, but the individual I am talking about is not named ‘John”’: the speaker’s belief that the subject of

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the proposition he asserts (viz. the proposition that John is healthy) is named “John” causes him to express the proposition by the words “John is healthy,” hence the belief indicated by the statement “but ... is not named ‘John”’ cancels, as it were, the belief which led him to say “John is healthy.” It would be disastrous, however, to confuse this kind of “contradiction,” which might be called pragmatic, with a logical contradiction. It certainly is no logical contradiction to suppose that John is healthy but is not named “John.” In order to put this argument more formally, let us introduce a descriptive function: ( p)(Des(“p”, p)), where “p” is a propositional variable, “ ‘p’ ” a sentential variable, and “Des” stands for the semantic relation of designation. If the domain of this relation consists of sentences of a semantic system, then a third term-variable is required, ranging over semantic systems, since the propositions designated by the sentences of a semantic system are determined by the semantic rules of the semantic system. Thus, if in such a system L we have the semantic rules “‘John’ designates John,” and “‘intelligent’ designates the property intelligent,” we can write:3 That John is intelligent = ( p)(Des(“John is intelligent”, p, L)). However, since we are at present concerned with sentences of a natural language (attempting to show that sentences of the form (T) do not, as sentences of a natural language, express necessary propositions), we need not complete the above descriptive function in this way. Instead, we have to complete it by adding a time-argument, since the same sentence may at one time be used to express one proposition and at another time to express a diﬀerent proposition.4 We thus get the three-termed descriptive function: ( p)(Des(“p”, p, t)). It appears, now, that semantic sentences of the form “‘p’ is true” are to a considerable degree incomplete, requiring completion to ( p)Des(“p”, p, t) is true. 3 Propositions are related to sentences the way properties are related to predicates, i.e. they are meanings, something which synonymous sentences have in common. But no expressions, except perhaps the highly problematical “logically proper names,” name their own meanings. A predicate is not a name of a property, and analogously sentences are not names of propositions. However, names are sometimes constructed in order to talk about the meanings of expressions. Thus “blueness” is a name of the meaning of “blue,” and analogously “that p,” in such contexts as “X believes that p,” “it is true that p,” is a name of the meaning of “p.” 4 Strictly speaking, several more variables would have to be taken into account, e.g. the language-user, the situation of utterance of a token of the sentence etc. But since the time variable is of special relevance to a major question that will be discussed in the sequel, viz. whether “true” is a time-independent predicate, it is here made explicit in preference to other suppressed variables.

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If the constant “p1 ” abbreviates the sentence “the moon is round” (or designates the proposition that the moon is round), and the meta-linguistic constant “ ‘p1 ’ ” abbreviates the name of the sentence “the moon is round,” and t0 is a particular time during which we may suppose the relevant linguistic conventions to be invariant, then the proposition whose necessity is here disputed may be formulated thus: ( p)Des(“p1 ”, p, t0 ) is true if and only if p1 . It is easy to see that this proposition can be negated without self-contradiction. For, consider the conjunction, incompatible with it: p1 and ∼ (( p)Des(“p1 ”, p, t0 ) is true). If the contingent statement: Des(“p1 ”, p, t0 ), were true, this conjunction would indeed be contradictory, since “p1 and ∼ (p1 is true)” is, on account of the logical equivalence “p1 is true if and only if p1 ”—here truth, being attributed to a proposition, is not a semantic concept at all!—equivalent to “p1 and ∼ p1 .” But just because the above statement of designation is contingent, the conjunction could be true; and it would be true if “p1 ” designated at t0 a false proposition while the proposition p1 is true. It must be concluded that the conjunction “p1 and ∼(‘p1 ’ is true)” expresses a possible state of aﬀairs. It would be a grave error to suppose that this possibility is ruled out by the convention that instances of (T) are to be constructed by substituting for “ ‘p’ ” the names of sentences substituted for “p,” such that “‘p1 ’ is true if and only if p1 ” could not fail for the reason that “ ‘p1 ’ ” is not the name of the sentence “p1 .” For the relation of designation which is involved in the conjunction that was argued to be logically possible goes from sentences to propositions, not from names of sentences to sentences. A defender of (T) as a material condition of adequacy for the systematic definition of truth may reply that my argument is irrelevant since it is based on the absolute concept of truth (in the terminology employed by Carnap in Carnap 1942, henceforth to be referred to by “ta ”), i.e. truth as predicable of propositions, whereas Tarski explicitly uses “true” as a predicate applicable to sentences (henceforth to be referred to by “t s ”), and repudiates propositions as obscure entities. However, if “ ‘p’ is true” is not interpreted in the sense of “the proposition designated by ‘p’ is true,” then it is, as a sentence of a natural language, simply incomplete; in which case we have no means of assessing the truth-values of such sentences, and therefore no means of evaluating the claim that the instances of (T) are according to ordinary usage of “true” true—let alone necessarily true. In the first place, natural languages allow the construction of ambiguous sentences. If we regard sentences of the form “‘p’ is true” as complete and at the same time take sentences of a natural language as the values of “ ‘p’,” we get into conflict with the law of non-contradiction, since the occurrence of an ambiguous sentence “p” could lead to both “‘p’ is true” and “ ‘∼ p’ is true.” And the natural way of saving the law of non-contradiction is then to expand “‘p’ is true” into “the proposition p1 , designated by ‘p’ in

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some usages, is true,” and “ ‘∼ p’ is true” into “the proposition p2 , designated by ‘∼ p’ in some usages, is true,” where p1 and p2 are compatible propositions. That predications of “true s ,” as a monadic predicate, upon sentences are incomplete statements is further evident from the time-dependence of “true s .” When Carnap argues, in Carnap 1949, on the contrary that “true,” unlike “confirmed,” is a time-independent predicate, in the sense that it is a complete statement to say of a sentence that it is true and the question “at what time is it true?” is meaningless (just as it would be meaningless to ask at what time an entailment holds between two given propositions), he must be taken to refer to “truea ,” otherwise his claim is indefensible. For, it clearly makes sense to suppose that a sentence is true at one time and false at another time because at diﬀerent times it designates diﬀerent propositions. This is particularly evident if we consider sentences containing the indicator term “now,” which express a diﬀerent proposition each time they are uttered or written. Further, it is not unnatural to say that if an object has a given property P at one time and does not have it at another time, then the sentence “x has P” is true at one time and false at another time. The reply which probably would be made by the defenders of the theory of the time-independence of truth is that the incompleteness of “‘p’ is true” is, in the alleged cases, of course due to the incompleteness of “p.” For example, to say “John is shaved” is to make an incomplete, ambiguous statement; if truth, however, is predicated of the complete statement “John is shaved at 9 a.m. April 4 1954,” then it is nonsense to go on asking at what time the statement is true. But thus the theory is saved by restricting it to truth-predications upon complete sentences, specifically sentences complete with respect to time. Yet, since “S designates the proposition that p” is always incomplete with respect to time if S belongs to a natural language, it is always significant to ask at what time S is true, provided that “S is true” means “S designates a true proposition.” In order to vindicate Carnap’s claim, therefore, the assumption that truth is predicated of sentences unambiguous as to time is not suﬃcient. We must moreover interpret “‘p’ is true” in the sense of “the proposition that p is truea .”5 For example, it is perfectly meaningful to say “the sentence ‘John is shaved at 9 a.m. April 4 1954’ is true at 9 a.m. April 4 1954, but will be false at any time thereafter,” because it is conceivable that after the mentioned date any of the constituent expressions of that sentence change their meanings in such a way that the same sentence henceforth expresses a false proposition. However, what would be meaningless is the statement (about a proposition) “it is true at 9 a.m. April 4 1954 that John

5 I deliberately write “the proposition that p ...” instead of “the proposition designated by ‘p’ ...” because those who accept the theory of descriptions may say that a statement about a proposition still makes an assertion about contingent usage if the proposition is referred to by a semantic description, such as “the proposition designated by ‘p’.”

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is shaved at 9 a.m. April 4 1954.” For, “it is truea that p,” unlike “‘p’ is true s ,” expresses a proposition whose truth-value is unaﬀected by contingencies of linguistic conventions (unless, of course, p is about linguistic conventions), and therefore may be regarded as cognitively synonymous with “p.”6 But it would be even ungrammatical to say “John is shaved at 9 a.m. April 4 1954, at 9 a.m. April 4 1954.” Thus a close analysis reveals that, even if a completely unambiguous language is presupposed, the claim of the time-independent character of truth holds only for the absolute concept of truth, i.e. truth as predicable of propositions. Now, the argument to the eﬀect that “true s ” is time-dependent even if the sentences of which it is predicated are unambiguous as to time could be countered by construing this predicate as relational, as in fact it is construed in Carnap 1942: S is true s in language L, where in fixing L we fix the semantic rules that determine the meaning of S . For it could be significant to suppose that S , as meaning p (where the coordination of the proposition p to S is just what the semantic rules of L eﬀect), changed its truth-value, only if it were significant to suppose that p changed its truth-value; but, as we have just seen, “truea ” is time-independent. Notice that if predications of “true s ” are completed in this way, then the argument, presented above, to the contingency of the instances of (T) from the contingency of linguistic conventions ceases to be valid. For while the sentence “‘the moon is round’ is true” could change its truth-value owing to a change of meaning of the sentence “the moon is round” without the moon itself changing its shape, this fate could not befall the sentence “‘the moon is round’, as meaning that the moon is round, is true.” My argument was based on the assumption that “truea ” is needed for the completion of “S is true s ” into “the proposition designated by S at t is truea .” And since S could at diﬀerent times designate propositions of diﬀerent truth-values, it followed that S could change its truth-value. But if S designates diﬀerent propositions at diﬀerent times then L has changed; and the possibility of S being true s in L1 and false s in L2 is, of course, consistent with the time-independence of “true s (respectively false s ) in L.” However, there remain two arguments against the claim that (T) is a condition of adequacy satisfying R1 and R2. In the first place, (T) violates R2 because the semantic use of “true” just is not its ordinary use at all. If in re-

6 Carnap

obliterates this important distinction when he says, “We use the term (‘true’) here in such a sense that to assert that a sentence is true means the same as to assert the sentence itself ; e.g. the two statements “The moon is round” and “The sentence ‘The moon is round’ is true” are merely two diﬀerent formulations of the same assertion” (Carnap 1942, 26). Notice that “to assert” is inconsistently used by Carnap in that in “to assert that a sentence is true” the object of the assertion is a proposition (albeit a proposition about a sentence), whereas in “to assert the sentence itself” a sentence is the object of assertion. I should say it is a proper usage of “assert” to say that a sentence is used to assert something, but not to say that a sentence is asserted.

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sponse to a statement one says “that’s true,” “that” refers to the proposition he takes the speaker to have asserted by means of the uttered sentence, not to the uttered sentence.7 If the speaker repeated his assertion by means of a slightly diﬀerent, but clearly synonymous, sentence, and then asked “is that true?,” it would be perfectly proper to reply “I have already admitted it.” Further, “it is true that the moon is round,” or “the proposition that the moon is round is true”—which expressions are surely more usual in ordinary language than “‘the moon is round’ is true”—cannot be adequately translated into a metalanguage, whether the proposed translation by which the propositional name “that the moon is round” (cf. note 3, p. 157) is to be removed be “‘the moon is round’ is true” or “all sentences synonymous with ‘the moon is round’ is true” or “some sentence synonymous with ‘the moon is round’ is true.” Such a translation must be inadequate because of the principle that if S 2 is a correct translation of S 1 , then any sentence which is a correct translation of S 2 must be a correct translation of S 1 . For, any of those meta-linguistic translations contain a name of the English sentence “the moon is round,” hence any correct translation of them into another meta-language (e.g. “ ‘the moon is round’ ist wahr,” if the meta-language be German) would have to contain a name of an English sentence; yet no correct translation of “it is true that the moon is round” into another language could mention the English sentence “the moon is round.” A still more important argument against such translations, however, which at the same time proves that (T) violates R1 as well, can be constructed on the premise that no statement of the form “ f (a),” where “a” is a proper name or definite description, can be true unless the entity denoted by “a” exists.8 For, a contingent consequence which follows from “‘the moon is round’ is true” by virtue of this premise is the existence of a sentence, whereas “it is true that the moon is round” has no such consequence (which is a further reason why it may be considered cognitively synonymous with “the moon is round”). This argument against the logical necessity of the instances of (T), unlike the argument from the contingency of linguistic conventions, survives the expansion of “S is true s ” to “S is true s in L,” for the latter sort of statement entails the existence of a sentence no less than the incomplete statement which it completes. That predications of truth upon propositions do not entail the existence of sentences is implicitly acknowledged by Carnap, when he proposes as a translation of “p is true” into the semantic meta-language “for any sentence S and semantic system L, if S designates p in L, then S is true in L”: for the truth of

7 Cf.

D. R. Cousin’s illuminating article (Cousin 1950). significantly diﬀerent variant of this premise is obtainable by substituting “can be significant” for “can be true.” But while this diﬀerence is, indeed, significant (cf. Pap 1953c), a similar argument leading to the same conclusion could nonetheless be constructed.

8A

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this universal conditional is compatible with there not existing any sentences that designate the proposition of which truth is predicated. However, for this very reason the attempted reduction of “truea ” to “true s ” is demonstrably inadequate. For, let us ask how the “if-then” of the semantic meta-language is to be interpreted. Suppose that material implication is meant, such that the translation may be formalized: (∀S )(∀L)(Des(S , p, L) ⊃ true s (S , L)). Then it would follow that any verbally unexpressed proposition must be true.9 If, on the other hand, the translation is meant as an entailment, a no less paradoxical consequence follows. For, let us ask what kind of p is such that from the fact that a given S designates p in a given L it follows that S is true in L? The obvious answer is: a necessary proposition. With regard to a contingent proposition it cannot be logically impossible that a given sentence expresses it in a given language and yet that sentence is false; in fact, if p is false, then any sentence designating it (in L) will be false (in L). In order, then, for the falsehood (in L) of S to be logically incompatible with the fact that S designates (in L) p, it must be logically impossible for p to be false. The translation thus entails that only necessary propositions are true! In order to escape from this dilemma one may be tempted to try as translation a contingent, but existential conditional: (∀S )(∀L)(Des(S , p, L) ⊃ true s (S , L)).(∃S )(∃L)Des(S , p, L). It would then follow that only expressed propositions can be true. And since it would be an absurd asymmetry to hold that propositions cannot be true unless they are expressed yet can be false without being expressed, the view to be examined is really that it is logically impossible that there should exist unexpressed propositions. I propose to prove, however, that if we make the sentence-proposition distinction at all (and I don’t see how one can think clearly, above all about semantic questions, without making this distinction), then we must grant that it is possible that there should be unexpressed propositions, true or false. Let t be the time which was the first time when a given proposition p was, graphically or orally, expressed, such that it is true to say with respect to p “before t nobody said or wrote that p.” In fact, if only t is some time before the origin of language, this will be true with respect to any proposition. Obviously, such a statement, involving what is commonly 9 This

counterintuitive consequence is reminiscent of Carnap’s reductio ad absurdum of explicit definitions (in an extensional language) of disposition predicates, in “Testability and Meaning” (Carnap 1937). Indeed, consistency would require him to say that “truea ” is reducible to “true s ,” yet not explicitly definable in terms of it—unless the ideal of an extensional language be abandoned and “if-then” be construed as the subjunctive connective which so far has resisted all attempts to define it in an extensional language.

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called indirect quotation, is not about the sentence “p,” but about its meaning, the proposition that p. For example, it may be true to say “before t nobody said or wrote ‘the earth is round’,” yet false to say “before t nobody said or wrote that the earth is round” because before t some Greek uttered a Greek sentence expressing the proposition that the earth is round. Now, by the rule of existential generalization, “before t nobody said or wrote that p” entails “(∃t)(∃p)(Unexpressed(p, t)).” And this entails, by a simple shift of quantifiers, “(∃p)(∃t)(Unexpressed(p, t))]”: there is a proposition which is at-sometime-unexpressed. But that there should come a time when such a, at-sometime-unexpressed, proposition comes to be expressed, is clearly a contingent event. Therefore the statement “there is a proposition which is at-all-timesunexpressed” ((∃p)[(∀t)(Unexpressed(p, t))]) must be conceded to describe a logical possibility. Conceivably it may even be provable in some way analogous to the way in which mathematicians prove that there are unnamed numbers (though it would, of course, be contradictory to mention a particular member of the class thus proved to have members). If all this is correct, then “truea ” is not reducible to “true s ,” at least not by the method shown by Carnap in Carnap 1942. Further, since predications of “truea ,” i.e. statements of the form “it is true that p,” which, as I have argued, cannot be translated into meta-statements containing “true s ,” are perfectly clear provided that “p” is clear, I do not see why the “Platonistic” commitment, which according to some critics of Carnap’s semantics is involved in such uses of absolute concepts, should worry us. Existential generalization is a mode of deductive inference which makes explicit a part of what is asserted by its singular premise; thus “(∃x) f x” expresses part of the information expressed by “ f (a),” regardless of what the type of “x” may be, and accordingly it is impossible that one who understands “ f (a)” and who understands the logical constant “there is,” should fail to understand “(∃x) f x.” But then anybody who understands, for example, “that the earth is round, is true, yet was not believed by anybody 3000 years ago,” ought to understand “there is something which is true yet was not believed by anybody 3000 years ago.” And if it is further explained that the word “proposition” refers to the sort of things (entities) that can significantly be said to be true, to be believed, to be entailed, to be meant by declarative sentences etc., no obscurity should surround statements beginning with “there are propositions which ....” To be sure, in the spirit of Occam’s razor logicians and semanticists may hail the reduction to the indispensable minimum of the types of entities to which scientific discourse is committed. In this spirit, one may hope that quantification over propositional variables is dispensable in favor of quantification over sentential variables. It will be impossible, however, to carry this reduction through, unless singular statements involving names of propositions, like “it is true that p”—here “that p” names a proposition, not “p,” which is not a name at all (cf. footnote 3)—“it is be-

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lieved by x that p,” can be translated into statements devoid of such names. Once such a translation has been achieved, and not before, one may justifiably look upon propositional names as “incomplete” symbols in the sense in which class-names are incomplete symbols in Principia Mathematica, i.e. existential generalization from “ f (p1 )” to “(∃p) f (p)” will then be illegitimate. To suppose that propositional names are eliminable without diﬃculty at least from truth-predications, since “it is true that p” is cognitively synonymous with “p,” would be to overlook that the subjects of truth-predications are more frequently descriptions, than names, of propositions. Thus, consider “the first proposition he asserted in his speech is true.” Even if the proposition satisfying the description could be named, say, “the proposition that he would attempt to prove two propositions,” the statement containing the description is not logically equivalent to the statement containing the name, since it is a contingent fact that the named proposition satisfies the description. Therefore we cannot move by mere substitution of synonyms from “ p which satisfies φ is true,” via “it is true that p1 ,” to “p1 ”: for, the identity “p1 = ( p)φp” is usually factual. And if the description be eliminated from the truth-predication by means of the contextual definition of descriptions known as the “theory of descriptions,” we are left with a statement in which “truea ” occurs in the context of propositional variables. But he who dreads propositional names will dread propositional variables at least as much,10 and therefore an elimination of names or descriptions of propositions serves no philosophical purpose if it necessitates the employment of propositional variables instead.

10 Thus

Quine says repeatedly that the ontological commitments of a theory or a piece of discourse are more reliably learnt from the latter’s variables of quantification than from its names, since names may be eliminable shorthand devices, like e.g. class-names in Principia Mathematica.

Chapter 8 BELIEF AND PROPOSITIONS (1957)

The repudiation of propositions as “obscure entities,” which is prevalent among logicians and philosophers of “nominalistic” persuasion, is frequently justified by pointing out that no agreement seems ever to have been reached about the identity-condition of propositions. And if we cannot specify, so they argue, under what conditions two sentences express the same proposition, then we use the word “proposition” without any clear meaning. Quine, for example, feels far less uneasy about quantification over class-variables than about quantification over attribute-variables and propositional variables, because there is a clear criterion for deciding whether we are dealing with two classes or with only one class referred to by diﬀerent predicates: two classes are identical if they have the same membership. And such a criterion of identity is alleged to be lacking for intensions. The problem is a serious one and cannot be disposed of by applying a generalized Leibnizian principle of identity of indiscernibles: two propositions are identical if they agree in all properties. For this would mean that two sentences express the same proposition if one can be substituted for the other in any context without change of truth-value. But either this criterion is applied to extensional contexts only, or it is also applied to modal and intentional1 contexts. If the former, it leads to the untenable conclusion that all sentences of the same truth-value express the same proposition (“proposition,” then, becomes a redundant term; to ask what proposition is expressed by a given sentence would simply be asking whether the sentence is true or false). If the latter, then the criterion seems to be ineﬀective for several reasons. First, a non-extensional language permits the construction of such non-extensional sentences as “the proposition that p = the proposition that q” (such sentences are non-extensional with respect to “p” and “q” because obviously diﬀerent propositions may have identical truth-values). And how is one to decide whether “p” is substitutable for “q” on the right-hand side of this identity without changing the truth-value 1 By

an intentional context of “p” I mean a statement like “A believes that p,” “A doubts that p” etc.

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of the identity-statement, unless one already knows whether or not “p” and “q” express the same proposition? Second, consider a modal context like “it is necessary that (A is a father if and only if A is a male parent).” This modal statement is true if and only if “A is a father” is L-equivalent to “A is a male parent,” but although L-equivalence is not, as I shall explain presently, in general a suﬃcient condition for identity of propositions, I doubt whether the Lequivalence of these two sentences could be established independently of the assumption of their synonymy2 —which is the question to be decided by the substitution test. It may be replied that Leibniz’s principle leads to similar diﬃculties when applied to individuals, and that the intensionalist, therefore, is really no worse oﬀ than the nominalist who uses the concept of “individual” without philosophical qualms. Indeed, there is something in this objection. If the predicatevariable in “(∀P)(Px ≡ Py),” the Leibnizian definiens for “x = y,” is completely unrestricted except as to type, then peculiar consequences follow: in the first place, “x = y” would be a value of “Px,” hence the attempt to decide an identity-statement on the basis of such a definition would be circular. Second, if an intensional function “A knows that x = y” is admitted as value of “Px,” it would even follow that no identity-statement about individuals can be both informative and true. For, since undoubtedly everyone knows that x = x, whatever individual x may be, “x = y” could not then be true unless everyone knew that x = y. This is a generalization of the paradox of analysis and of what Carnap has called the “antinomy of the name-relation,”3 anticipated by Russell’s observation that King George wished to know whether Scott was the author of Waverley, but not whether Scott was Scott. Quine has shown that unrestricted substitutivity of identity also leads to paradox if modal functions are admitted: it used to be thought that the morning star is identical with the evening star, but this was to overlook that the morning star is necessarily identical with itself whereas it is not necessarily identical with the evening star.4 Nevertheless, the nominalists are right in feeling that identity of individuals is less problematic than identity of intensions, such as propositions. For, whether or not a consistent calculus of modal functions—to be distinguished from modal operators, prefixed to names of propositions—be possible, none of the mentioned paradoxes would arise if the Leibnizian definition were restricted to first-order functions in the sense of the ramified theory of types. This does not mean that the ramified theory of types, which is widely held to 2 Given

the synonymy of “father” and “male parent,” the above modal statement is of course derivable from the truth of modal logic “it is necessary that A is a father if and only if A is a father.” 3 See Carnap 1956a, §31. 4 Incidentally, some logicians would solve Quine’s paradox of necessary identity by pointing out that it cannot arise in the primitive notation in which the identity sign does not occur between descriptions but between logically proper names: there the identity-statements have either the form “a = a” or the form “a = b,” and hence are L-determinate since diﬀerent individual constants are defined as names of diﬀerent individuals. But as the informative identity-statements of natural language always involve descriptions, I doubt whether this solution would satisfy anyone who doubted on philosophical grounds whether an identity-statement like “the morning star is identical with the evening star” could be true.

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be dead, must be resuscitated in order to rescue the concept of identity of individuals. Expressions like “all the properties (of a given type, but of any order) of x” may be admitted as perfectly meaningful, yet a definition of “x = y” as “x and y share all extensional first-order properties, i.e. extensional properties not defined in terms of a totality of properties” is perfectly adequate. Nobody, for example, would seriously doubt the identity of Scott and the author of Waverley just because a king some time ago was doubtful of this identity while being perfectly certain of Scott’s self-identity. It is perfectly satisfactory, then, to define identity of individuals as agreement with respect to all extensional first-order properties, whereas a similar restriction of substitutivity of identity does not, as we have seen, work in the case of intensions. According to Carnap two designators have the same intension if and only if they are L-equivalent. A special case of this criterion is that two declarative sentences express the same proposition if and only if they are L-equivalent. But this will not do either. For, an obvious criterion of adequacy which an explication of “proposition” (via the explication of synonymy of declarative sentences) should satisfy is that “A believes that p, and p ≡ q” should entail “A believes that q.” Yet, any two logically necessary statements are L-equivalent, but it could hardly be maintained that, where “p” and “q” are logically necessary, “A believes that p” entails “A believes that q.” For example, anybody with a rudimentary knowledge of the propositional calculus will believe a simple tautology like “((p ⊃ q). ∼ q) ⊃∼ p,” yet there are tautologies with respect to which he could profess neither belief nor disbelief because he does not recognize them as tautologies. In general, it is surely possible that, being familiar with the semantic and syntactic rules for the symbols of a logical system, one understands the sentences of the system, i.e. knows what propositions they express, yet does not know whether the propositions expressed are logically necessary. We all understand the sentence “for n greater than 2, there are no solutions for the equation: xn + yn = zn ,” i.e. know what proposition it expresses, but according to my information it is not yet known whether the proposition is logically necessary. But according to the L-equivalence criterion of propositional identity, we already believe this proposition if it is logically necessary! In order to bring the L-equivalence criterion into accord with the concept of (propositional) belief, one would in fact have to stipulate, as a postulate partially defining “belief”: ([A believes that p] and p entails q) entails (A believes that q). But this postulate is obviously not satisfied by the concept ordinarily meant by “belief.” It is only “A believes that p” together with “A believes that p entails q” that could be said to entail “A believes that q.” The L-equivalence criterion does not seem to be satisfied by contingent propositions either. This can again be shown in terms of the evident requirement that “A believes that p, and p ≡ q” should entail “A believes that q.” It will surely be granted that it is impossible to have a propositional attitude,

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whether belief or disbelief or doubt or any other, towards a proposition some of whose constituent concepts one does not “have.” A man who does not understand the meanings of color predicates, e.g., cannot have a propositional attitude towards a proposition containing a color concept. Now, take any true contingent proposition p; it is L-equivalent to the proposition “if anyone asserted that p, he would make a true assertion.” It seems to me to be logically possible that a man who “has” the descriptive and logical concepts that are constituent of p should fail to have the concept of asserting, and therefore believe p without having a propositional attitude towards this conditional proposition which is L-equivalent to it. I am not, however, convinced by this argument for the thesis that even in the case of contingent propositions L-equivalence is not a suﬃcient condition for identity. For its validity depends on the notion of “concept constituent of a proposition,” which requires to be clarified. Is the concept expressed by “red,” for example, a constituent of the proposition expressed by “a is round and red, or round and not red”? Now, it is not, according to a plausible definition of “constituent concept”: the concept expressed by a constituent expression of a sentence is a constituent of the proposition expressed by the sentence, if and only if the expression occurs essentially in the sentence. Clearly, “red” does not occur essentially in the above disjunction of conjunctions. But it might be argued that “assert” likewise fails to occur essentially in the sentence “if anyone asserted that p, he would make a true assertion.” For, the latter can be transformed into “if anyone asserted that p, he would assert a true proposition.” Now, a verb occurs inessentially in a sentence if any consistent and grammatically admissible substitution of a diﬀerent verb for it leaves the truth-value of the sentence unchanged. But the only verbs that can be inserted for the dots in “so-and-so . . . that p” without producing nonsense are intensional verbs, i.e. verbs like “assert,” “believe,” “disbelieve,” “presuppose” etc., and it is clear that any such substitution into “if anyone . . . that p, he would . . . a true proposition” is truth-preserving. Nevertheless, I do not think that by reference to inessential occurrence of terms the failure of logically equivalent contingent propositions to be identical can always be shown to be merely apparent. “x is orange” is logically equivalent to “x is intermediate-in-color between red and yellow,” but all the descriptive terms occur essentially, hence we can argue that a man having a propositional attitude towards the proposition expressed by the first sentence might fail to have a propositional attitude towards the proposition expressed by the second sentence, since he might, say, have a concept of the color orange without having a concept of the color red: for the latter concept is a genuine constituent of the proposition expressed by the second sentence. Let us return to our adequacy criterion: A believes that p, and p ≡ q, entails that A believes that q. Would it be satisfied if we strengthened the identitycondition by requiring, instead of merely L-equivalence, intensional isomor-

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phism of “p” and “q”? It has been argued recently that even this extremely strong identity-condition—I shall argue presently that it is much too strong— does not necessarily satisfy it. Thus Benson Mates (Mates 1952) seems to think that, though whoever believes that all Greeks are Greeks believes that all Greeks are Greeks, it is not necessarily the case that whoever believes that all Greeks are Greeks believes that all Greeks are Hellenes; yet, on the assumption that “Greek” and “Hellene” are L-equivalent, the sentence “whoever believes that all Greeks are Greeks, believes that all Greeks are Greeks” (D) is intensionally isomorphic with the sentence “whoever believes that all Greeks are Greeks, believes that all Greeks are Hellenes” (D ). If we substitute D and D for “p” and “q,” and agree with Mates that it is possible to doubt that D but not possible to doubt that D, then I suppose we must agree with him that no explication at all of synonymy which allows diﬀerent sentences to be synonymous could satisfy our adequacy criterion. However, while it is of course conceivable that a person respond aﬃrmatively to the question “do you believe that everybody believes that all Greeks are Greeks” yet negatively to the question “do you believe that everybody believes that all Greeks are Hellenes,” this does not establish Mates’ contention. For if the subject were asked to support his doubt whether everybody believes that all Greeks are Hellenes, he could only do so by pointing out that somebody may have an imperfect knowledge of the English language so as to fail to know that “Greek” and “Hellene” are synonyms, and therefore fail to know that the proposition expressed by “all Greeks are Greeks” is the same as the proposition expressed by “all Greeks are Hellenes.”5 But then Mates’ argument is simply based on the confusion between “A believes that p” and “A believes that ‘p’ expresses a true proposition” (as I have argued in Pap 1955b). Mates’ argument does not prove that no explication of synonymy that is compatible with applicability of “synonymous” to pairs of distinct sentences must fail to satisfy our adequacy criterion (substitutivity in belief sentences). If it proves anything, it proves the inadequacy of a superficial behavioristic analysis of “belief” according to which a subject’s response to questions about his beliefs is conclusive evidence for or against hypotheses about his beliefs.6 It would indeed be strange if an experimental linguist reported that some people do not accept the law of identity, his evidence being that some people who professed belief when they were asked whether all Greeks are Greeks, expressed doubt when they were asked whether all Greeks are Hellenes although the two sentences express (to him, the interrogating linguist!) the same proposition. Such a response would normally be 5 It

is but for the sake of the argument that I assume here that a trivial tautology of the form “all A are A” expresses a proposition at all. 6 It is interesting to notice, though, that on Carnap’s behavioristic analysis of “belief” viz. “A believes that p = A is disposed to an aﬃrmative response to some sentence that is intensionally isomorphic to ‘p’,” it is logically impossible that A should believe that D, yet fail to believe that D .

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taken as evidence, rather, that “Greek” and “Hellene” do not mean the same to the subject. On the other hand, intensional isomorphism in Carnap’s sense7 is too strong an explicatum for “synonymy,” because a simple designator cannot be intensionally isomorphic with a compound designator. I believe that “father” is synonymous with “male parent,” “ignoramus” with “we do not know,” “x3 ” with “x · x · x”; further, “p and q” with “q and p,” “p or q” with “q or p,” “some A are not B” with “some A are non-B.” And since none of these pairs is intensionally isomorphic, I conclude that either Carnap’s explication is incorrect or else his explicandum is not a concept of synonymy that is of interest for philosophical analysis. Carnap’s reply to a similar objection by Linsky (Linsky 1949) was that there is a family of stronger and weaker synonymy-concepts and that it is unfair to criticize an explication of a stronger synonymy-concept on the ground that it does not fit a weaker synonymy-concept. But this reply is insuﬃcient, since the explicandum which analytic philosophers are interested in is a semantic relation of which the pair “father, male parent,” e.g., is an instance; and second, since in a language in which “x3 ” is explicitly introduced as abbreviation for “x · x · x,” “33 ” is surely synonymous with “3 · 3 · 3” in precisely the same sense in which, say, “3 · 3 · 3” is synonymous with “3 × 3 × 3.” The time is ripe for suggestion of a new approach to the problem of propositional identity. To begin with, it is utopian to look for an absolute criterion. Instead, propositional identity should be relativized to specified kinds of substitution-contexts. The weakest concept of propositional identity is propositional identity relative to extensional contexts: relative to an extensional language, “p ≡ q” is a suﬃcient condition for “ f (p) ≡ f (q).” Hence there is here no need to distinguish propositions from truth-values: the range of the sentential variables in the propositional calculus comprises only two values, the true and the false. The idea that a law of the propositional calculus is a truthfunctional tautology could then be expressed as follows: if “ f (p)” is a tautology, then “ f (T). f (F)” holds, if “ f (p, q)” is a tautology, then “ f (T, T). f (F, T). f (T, F). f (F, F)” holds, etc. The next stronger concept of propositional identity is propositional identity relative to modal contexts: relative to modal logic, strict equivalence, not material equivalence, is the suﬃcient condition for truthpreserving substitutivity of statements. “If p is L-equivalent to q, then f (p) ≡ f (q)” expresses this stronger identity-condition. Accordingly, there are distinct propositions of the same truth-value in modal logic. However, relative to modal logic there is but one necessary proposition and but one impossible proposition, since all necessary statements are strictly equivalent and all impossible statements are strictly equivalent. Now, we have seen that it is the

7 See

Carnap 1956a, §§14, 15.

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adequacy condition concerning belief which forces us to distinguish one necessary proposition from another. This is to say that L-equivalent statements are not necessarily interchangeable in intensional contexts. The strongest identitycondition for propositions accordingly reads: if “x believes that p”8 is strictly equivalent to “x believes that q”—which entails that “p” is strictly equivalent to “q”—then f (p) ≡ f (q). Or alternatively, f (p) ≡ f (q) if p is strictly equivalent to q and moreover B(x, p) ≡ B(x, q). Relative to an extensional language, propositional identity means material equivalence; relative to a modal language it means strict equivalence; relative to an intensional language it means strict equivalence of the corresponding statements of belief. The objection is likely to be raised that no operational (“eﬀective”) criterion of synonymy has been provided at all. For the only possible strict proof of the strict equivalence of “x believes that p” and “x believes that q” would rest on the proof of “p = q,” and hence the whole procedure would be circular. Now, in the first place, the same circularity arises already in connection with the weaker criterion of propositional identity which is in eﬀect Carnap’s criterion: in order to prove, e.g., that “father” is synonymous with “male parent” one would have to prove that the biconditional joining them is just as necessary as the trivial biconditional “all and only fathers are fathers,” which proof could be conducted only by substitution of “male parent” for one occurrence of “father” in the trivial biconditional, which substitution could be justified only by the assertion of synonymy which is in question. The lesson to be learnt from this, it seems to me, is that the clarification achieved by analyzing “proposition” (via “same proposition”) in terms of modal or intentional concepts, in a sense of “analysis” which requires that the analysans be clearer than the analysandum, is illusory. In the following, I shall propose an alternative method of, as it were, simultaneous clarification of the category-term “proposition” and the non-extensional operators, both modal and intentional, through axiomatic definition. In the meantime, it might be pointed out that at least negative conclusions about synonymy can be arrived at without apparent circularity through the test of substitutivity in intensional contexts. I think the following type of argument is perfectly respectable: “p” and “q” do not express the same proposition, because it is possible to know that p without knowing that q. It is by this type of argument that I try to convince students who are accustomed to a loose use of the word “definition,” that so-called definitions of color predicates in terms of wavelengths do not express the ordinary meanings of the color predicates, and that physicalistic definitions of mentalistic terms likewise do not express the meanings of the latter: it is possible to know that a thing is blue, or that one is feeling sad, without knowing which is the wave-

8 This

statement-form will henceforth be abbreviated: B(x, p).

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length of the light the thing is disposed to reflect, and without knowing anything about one’s physiological condition or behavior at a time one is feeling sad. And just to forestall an irrelevant debate about the criterion of “knowledge,” I hastily add that the argument remains unaﬀected by a substitution of “belief” for “knowledge”: if “p” and “q” are synonymous, then the statement “it is possible to believe that p without believing that q” is self-contradictory. Hence, if the latter statement is not self-contradictory, the sentences are not synonymous. Nelson Goodman in Goodman 1952a has decried this negative test of synonymy as a pseudo-test, on the following ground: with respect to any two statements at all, it is possible that one should know one to be true without knowing the truth-value of the other, since one might not understand the other. And if the test is applied only to persons who understand both statements, then it becomes redundant since one could not be said to understand both of two statements unless one knows whether or not they are synonymous. However, Goodman’s objection does not apply to my formulation of the test, for the latter involves statements of the form “A knows that p,” which are object-linguistic and non-semantic, not meta-linguistic statements of the form “A knows that ‘p’ is true.” The first part of Goodman’s argument presupposes that “A knows that ‘p’ is true” entails “A knows what proposition is expressed by ‘p’.” But such semantic knowledge is obviously not presupposed by A’s knowledge of the proposition that p. Now, it must be admitted that to say two sentences express the same proposition if they are interchangeable, not only in extensional contexts, but also in modal contexts, is not illuminating unless the meanings of the modal operators, “necessary,” “possible”9 etc. are understood. But the relevant meaning of “possible,” e.g., can, apart from illustrations, be indicated only by giving a set of axioms which are satisfied by the intended meaning: if p, then possibly p; if necessarily p, then possibly p; if possibly (p and q), then possibly p and possibly q, but not conversely (for “not-p” is substitutable for “q”) etc. Such an axiomatic definition10 at the same time defines “proposition”: the propositions to which modal logic is “committed” are the values of the sentential

9 Notice

that only one primitive modality is required, e.g. “possible.” is sometimes objected to axiomatic (or “implicit”) definitions that they are not unique, since there is more than one model for any consistent (and nontrivial) set of axioms. But neither is there any guarantee that an explicit definition is unique in a sense in which an implicit definition is not. For either the primitives occurring in the definiens are implicitly defined by the axioms of the deductive theory in which the defined term occurs, and then their ambiguity is communicated to the defined term (e.g., since “straight line” as a primitive of a geometrical system admits of several interpretations, so does “triangle” which is explicitly defined in terms of “straight line”). Or else they are interpreted by means of ostensive definition. But there is no guarantee that an intended meaning is really communicated by means of ostensive definition, since the latter limits it only to a property shared by all the instances pointed to (and absent from all the negative instances pointed to, if such are used). At any rate, ostensive definition is applicable only to descriptive terms, not to logical constants, hence only the first alternative is relevant in the present context.

10 It

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variables in the axioms of a system of modal logic. If, indeed, the variables are given completely unrestricted ranges, then the terms which would otherwise be used in the meta-language to characterize the subject-matter referred to by the axioms simply appear as additional primitives in the axioms. For example, one could diminish the number of primitives in the axioms of formal Euclidean geometry, by using special variables ranging over points, special variables ranging over straight lines, and special variables ranging over planes. This would be analogous to using the term “proposition” in the meta-language to define the ranges of the bound variables of modal propositional logic. But we could alternatively put it as a primitive into the axioms, just the way one normally handles “point,” “straight line,” “plane” as primitives along with the relational predicates that are used to make assertions about these entities; and then it would be obvious that “proposition” is defined simultaneously with the modal operators. Following this approach, let us formulate the stricter requirements of propositional identity that are imposed by belief-contexts by laying down axioms of a logic of belief, and saying that propositions are the values of the sentential variables in those axioms. The latter may be conceived as implicitly defining at once “belief” and “proposition.” I proceed to formulate five axioms of such a logic of belief which in terms of the intended meaning of “belief” are selfevident—indeed this is to assert a tautology since the axioms serve the purpose of explicating the meaning of this new primitive.11 A1 A2 A3 A4 A5

(B(x, p) B(x, q)) (p q) (B(x, p).B(x, (p q)) B(x, q) B(x, ∼ p) ∼ (B(x, p)) B(x, (p.q)) B(x, p).B(x, q) ∼ (B(x, p) p)

The sense of A1 is that we cannot analytically infer from the fact that a person believes p that he believes q, if p does not entail q—though “p entails q” is compatible with “somebody believes p but does not believe q.” Of course, if a person believes p without believing q although the latter proposition is entailed by the proposition he believes, this must be because he is not aware of the entailment. Hence A2 says, not that “p entails q” warrants the inference of “A believes q” from “A believes p,” but that “A believes that (p entails q)” does. A3 is likely to be disputed by those who insist that people frequently 11 Strictly

speaking, “believe” requires a time variable, so that a triadic, not dyadic, predicate should be used to express the concept of belief. The axioms are of course meant as universally quantified with respect to the (omitted) time variable. A5 is added in order to diﬀerentiate belief from knowledge, for A1 -A4 are satisfied by the relation of knowledge as well as by the relation of belief. It should be noticed that the possibilities of interpretation of the axioms are severely limited by the meta-linguistic explanation that “x” ranges over persons and “p” and “q” take declarative sentences as substituends.

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believe contradictory propositions; they may say that what is here implicitly defined is rational belief, not actual belief. I would justify the axiom, however, by the consideration that if a man believes incompatible propositions, this is because he is not aware of the incompatibility. As a matter of fact, from A3 together with A2 we can deduce: B(x, p).B(x, p ∼ q) ∼ (B(x, q)

(T 1 )

which means that it is impossible to believe propositions which one believes to be incompatible. This theorem may seem to be much weaker than ∼ 3(B(x, p. ∼ p))

(T 2 )

deducible from A4 and A3 ,12 since substitution of “∼ p” for “q” in T 1 yields the apparently more cautious entailment: B(x, p).B(x, p ∼ ∼ p) ∼ (B(x, ∼ p)). But the caution is unnecessary since a man who grasps the ordinary meaning of “not” cannot fail to see that the propositions expressed by “p” and “not-p” are contradictory. It would be futile to try to prove empirically that a man may (at the same time) believe explicitly contradictory propositions by obtaining aﬃrmative responses from the same person to two sentences which, as interpreted by the interrogator, are contradictory. I should think one would inevitably infer from such responses that the subject has misinterpreted at least one sentence. If I found a man insisting with great earnestness that the same part of a surface can at the same time be both green and red, I would conclude that he is either an excellent actor or else does not mean by “green” or “red” what one usually means. However, it is possible to believe propositions which are incompatible if their incompatibility is not self-evident. Therefore the inference from an affirmative response to several sentences which in their intended interpretation are contradictory to the subject’s misinterpretation of at least one sentence, is not always warranted. This is the reason why the second conjunct of the antecedent of A2 is “B(x, (p q))” instead of “p q.” A definition of “understanding ‘p”’ from which it follows that one does not understand “p” unless one knows all that is entailed by “p,” is both arbitrary and ineﬀective—though it cannot be denied that the test of whether one has understood “p” includes a test for awareness of some consequences of “p.” The specified axioms serve as adequacy criteria for behavioristic interpretations of “belief,” in much the same way as interpretations of “probability” will 12 Indirect proof: Suppose B(A, (p. ∼ p)). By A , this entails B(A, p).B(A, ∼ p). By simplification, B(A, ∼ p). 4 By A2 this entails ∼ (B(A, p)). Since the hypothesis thus entails the contradiction B(A, p). ∼ (B(A, p)), it is self-contradictory.

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usually be accepted as adequate only if they satisfy the axioms of the calculus of probability. By a behavioristic interpretation is meant an interpretation of “belief” as a disposition manifested in responses to specified stimuli. For example, “A believes that p = A is disposed to respond aﬃrmatively to some sentence synonymous with ‘p’,” as suggested by Carnap in Carnap 1956a. When combined with Carnap’s L-equivalence criterion of propositional identity, this interpretation leads to the contradictory consequence that a person may believe and also not believe (at the same time) one and the same proposition, since he may respond aﬃrmatively to some sentence synonymous in L with “p” yet fail to respond aﬃrmatively to some sentence synonymous in L with “q” though “p” and “q” are L-equivalent in L. This would of course be due to his not recognizing the L-equivalence in question, but as already pointed out it would be gratuitous hence to infer that he misinterprets at least one of the two sentences. This contradiction can be avoided by strengthening the requirements for propositional identity, so as to make “p ≡ q” entail “A believes that p if and only if A believes that q.” For if p ≡ q in the sense that “p” and “q” are interchangeable even in belief-contexts, then obviously it must be the case that a person believes that p if and only if he believes that q. And furthermore, on this assumption of strong synonymy of “p” and “q,” it is logically necessary that A is disposed to respond aﬃrmatively to some sentence synonymous with “p” if and only if he is so disposed towards some sentence synonymous with “q.” But having defined synonymy in terms of belief and belief implicitly in terms of a set of axioms, we face the question whether the behavioristic definition of belief satisfies those axioms. Now, it obviously does not satisfy A3 , for example, since it is quite possible that A respond aﬃrmatively to some sentence synonymous with “∼ p” and also to some sentence synonymous with “p,” because he does not interpret these sentences as contradictory. This indicates that the behavioristic definition is inadequate unless an important “mentalistic” condition is added to the linguistic stimulus: at best we may infer “A believes that p” from “A was asked to respond aﬃrmatively or negatively to the sentence S , and A interpreted S to mean the proposition p, and A responded aﬃrmatively to S .” I say “at best” because even this better grounded inference is not necessary: A may assert the proposition that p without believing it; he may be lying, in other words. This consideration shows that an interpretation of “belief” in behavioristic terms can hardly be adequate if it takes the form of an explicit definition in terms of a causal implication, unless the antecedent of the latter contains a “ceteris paribus” clause which covers our ignorance of relevant “intervening variables” (such as correct interpretation). The same objection would apply to introduction of “belief” by reduction sentences conceived as analytic, like

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Carnap’s “bilateral” reduction sentences.13 Carnap, therefore, is surely on the right track in advocating, more recently,14 that “belief” be treated as a theoretical construct which is implicitly and incompletely defined by the postulates of a psychological theory—though whatever theoretical postulates Carnap may have in mind must be supplemented by reduction sentences which tie the term to the observation-language in the loose way of indefinite probability implications. The important point in the present context is that “proposition,” being defined correlatively with “belief,” will inevitably share the latter’s openness of meaning.15 Notice that the terms “belief” and “intension”—a proposition being a special kind of intension—are almost inseparable in reduction sentences connecting them with behavioristic terms. As we saw, no reliable inference can be drawn either from a belief-hypothesis to a verbal response or conversely unless an assumption about an act of interpretation is warranted. But likewise, a hypothesis about a habit of interpretation—e.g., A is in the habit of interpreting “dog” to refer to a quadruped of kind K—is not highly confirmable by bare observation of linguistic behavior, for it is highly relevant to the question what a person means by an expression to know what beliefs motivate him to apply the expression to such and such objects. This interconnection of the constructs “belief” and “intension” might be expressed by the following postulate: if predicate “P” has property P as its intension for A, then, ceteris paribus, A applies “P” to an object x if and only if A believes that x has P. The “ceteris paribus” proviso covers such, not exhaustively known, conditions as suitable stimuli to a verbal utterance, a desire to express the belief and not to mislead, absence of relevant inhibitions, etc. The postulates for other kinds of intensions, like propositions conceived as intensions of declarative sentences, would be similar. Such postulates are perhaps less “analytic” than the axioms stated above, in the sense that observations of human responses would be more likely to lead to their revision than to a revision of the more “logical” axioms. But a strict analytic-synthetic distinction must be abandoned if a postulational method of meaning-specification is adopted.16 It remains to contrast the outlined approach to the definition of “intensional” concepts—which in its recognition of semantical “constructs” that are not definable in the nominalistic vocabulary is akin to the liberalization of strictly positivistic meaning criteria with respect to physics and psychology—with the approach of the logical-constructionists. According to the theory of Principia Mathematica, classes are logical constructions in the sense that names

13 See

Carnap 1937, §8. Carnap 1954b. 15 For a detailed discussion of “openness of meaning” see chapter 19, Pap 1963b, and Hempel 1952, section II. 16 This is lucidly argued by Hempel, in Hempel 1954. 14 See

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of classes as well as class-variables are contextually eliminable. It is in this sense that according to old-fashioned phenomenalism material objects are logical constructions out of sense-data: expressions referring to material objects were supposed to be in principle eliminable by translation of material-objectstatements into sense-data-language. In just the same sense Ayer, for example, regards propositions as logical constructions out of sentences—a convenient fa¸con de parler that can in principle be dispensed with. Since I have advanced detailed arguments against this kind of “reductionism” elsewhere,17 I shall confine myself here to one fundamental criticism. The logical-constructionists hold that statements ostensively referring to such pseudoentities as propositions are shorthand for meta-linguistic statements about classes of synonymous sentences. And this implies that they are translatable into a meta-language containing sentential variables and names of sentences (formed by means of the familiar quotes), but no propositional variables nor names of propositions. But surely the statement “there are propositions which are not expressed” is not so analyzable. Therefore the logical-constructionists could make good their claim only if they could show that, unlike statements about expressed propositions, this statement is meaningless. Now, the latter is incomplete in two respects: it does not specify a time, nor a specific language. Let us take it in the strongest form: there are propositions which are not expressed in any language at any time. It is not my purpose to argue for the truth, but only for the meaningfulness, of this statement. For this purpose we may begin with the innocent and undoubtedly true statement: there was a time when nobody believed, or even thought of, the proposition that the earth is round, and when no language at all existed. The proposition that the earth is round, therefore, was not expressed at that time. Therefore there is a proposition which was not expressed at that time. But that a proposition comes to be expressed in a language, is a contingency. It is therefore possible that this proposition should have remained unexpressed forever—though we could not have mentioned this possibility had it been actualized. And if “it is possible that p is at all times unexpressed” is true, then “p is at all times unexpressed” must be a meaningful statement. That it is a pragmatically self-refuting statement, in the sense that its falsehood follows from its assertion, is irrelevant. In the same sense “I am not asserting anything now” is pragmatically self-refuting, yet it is logically possible that the person denoted by “I” should not be asserting anything at the time denoted by “now.” The enemies of propositions will be quick to point to the fallacy in the foregoing argument: the step from “the proposition that p1 is not expressed at t” to “there is a p such that p is not expressed at t” begs the question. It presupposes that “that p1 ” is a genuine name, which it can be only if there are

17 See

chapter 5, especially section 3.

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propositions. It is like the inference from “roundness is a property shared by all nickels” to “there is a property which is shared by all nickels.” But since the apparent name “roundness” is contextually eliminable by translation of the above sentence into “all nickels are round,” the apparent basis for existential generalization disappears.18 Indeed, if the claim of the contextual eliminability of propositional names and variables can be made good, then the nominalist will be justified in saying that there are no propositions—or in a more tolerant vein, that we need not assume that there are. Since I have expressed serious doubts about the possibility of such contextual elimination, some may label (or libel?) me as one holding a metaphysical belief in propositions as “real entities.” I would like to conclude, therefore, by inquiring what it means to believe that “there are propositions.” Quine has oﬀered a much debated criterion of a man’s “commitment” to a special kind of ontology: Look at the variables of quantification in his language; if they belong to the primitive notation, then the user of that language is committed to the belief that the entities over which they range exist. But he has not, to my knowledge, discussed the question what it means to say that such entities exist. Now, within a realistic language we encounter only qualified existence assertions, not what might be called categorial existence assertions: “there are attributes (or propositions) satisfying the function f (φ) (or f (p))”—e.g., “there are propositions which nobody believes,” “there are attributes which are possessed by only one individual”—not “there are attributes (or propositions).” In Carnap’s terminology, categorial existence assertions are external to a given language.19 We may not like Carnap’s claim that the corresponding questions like “are there propositions” are devoid of cognitive meaning (unlike the questions concerning qualified existence which are formulated within a given language), but whosoever claims the contrary should oﬀer an interpretation of such questions. I suggest that we take Quine’s criterion of ontological commitment as the very definition of ontological commitment and thereby assign a cognitive meaning to categorial existence assertions. That is, to say “there are propositions” is to say that the propositional names and variables which we employ in order to say what we want to say are not contextually eliminable, that they belong to the “ultimate furniture” of our language. Thus nebulous questions about the ultimate furniture of the (extra-linguistic) universe are reduced to less nebulous questions about the ultimate furniture of cognitive language. The question whether propositions and other abstract entities exist is not, indeed, decidable empirically, not even in the indirect empirical way in which scientists decide whether atoms and electrons exist, but it is nonetheless a cognitive question: it is decidable the way questions of 18 See 19 See

Quine 1949. Carnap 1950a.

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semantic analysis are decidable, by examining whether proposed translations into a language with a specified primitive vocabulary preserve the meanings of the translated statements. Of course, in what precise sense of “meaning” such philosophical translations into ideal languages are required to preserve meanings, is another question—fortunately beyond the scope of this paper.

Chapter 9 SEMANTIC EXAMINATION OF REALISM (1947)

1.

Universals in Re and the Resemblance Theory

It is by no means beyond dispute what precisely the terms “realism” and “nominalism,” in the age-long controversy about the status of universals, have stood for. Without any regard to historical complexities and shifts of meaning, I shall, in this paper, define “realism” and “nominalism” as follows: According to realism, universals exist, to employ the scholastic phrase, in re, i.e., one and the same property (in the wide sense in which both qualities and relations are properties) is often simultaneously exemplified by several particulars. “Property” and “universal” are here used as synonyms. A property, in this usage, is the intension (or logical connotation) of any predicate, of whichever degree (relations are thus the intensions of predicates of degree 2 or any higher degree). According to the nominalists, on the other hand, there are no “ontological” universals. In the impressive language of metaphysicians, “only particulars have ontological status,” according to nominalism. There are, indeed, general words; but it is a mistake to suppose that, like proper names and definite descriptions, general words stand for, or refer to, an entity. Predicates (which are general words) are, indeed, applicable to several particulars that resemble each other in certain respects. But if the word has a unique referent, the latter is not a universal whose identical presence constitutes the resemblance, but at best a class of similar particulars. It is not always clear whether nominalists deny the existence of universals, substituting instead the existence of classes that are constituted by resemblance, or whether they maintain that universals are classes. The latter statement means presumably that the terms “universal” and “class” are synonyms. However, such a view cannot be taken seriously. Any material equivalence of the form “for every x, if and only if x has the property F, then x has the property G” expresses the fact that the class of particulars having the property F is identical with the class of particulars having the property G, where F and G are distinct properties since they do not mutually entail their presence. Hence, if “universal” is used in the sense of “property,” “universal” and “class” cannot

181

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be synonyms. A universal is an entity capable of having instances; a class has members, but no instances. To speak of an instance of a class makes no more sense than to speak of a member of a property or universal. No doubt many of the controversies between nominalists and realists are due to the fact that the term “universal” is used in diﬀerent senses by the disputants. Besides the confusion mentioned above, the tendency to speak of universals as though they were mental entities may be cited. Berkeley’s attack on abstract ideas, for example, has nothing to do with the issue whether there are universals in re or whether there are only classes constituted by resemblance and predicates with multiple applicability. Ideas, whether they be abstract thoughts or concrete memory images, are themselves particulars, since they are dated events (even though it might be questioned whether, like physical particulars, they can be spatially located). Hence, in identifying universals with ideas, one could not be using the word “universal” in the sense in which universals are contrastable with particulars. Nominalism, in the sense in which I have defined this traditional doctrine, may assume a stronger or a weaker form. By the stronger variety of nominalism I mean the flat assertion that it is never the case that the same property is instanced in more than one particular at the same time.1 In its weaker variety, nominalism merely asserts that it may always be doubted whether one and the same property is instanced in more than one particular. Now, it seems to me that the only way in which the nominalist could attempt to defend the stronger of the two assertions is by refuting any apparent contradictory instance of his universal negative proposition. But how else could this be achieved than by showing that in each proposed counter-instance it may be doubted whether the same property is instanced more than once? Hence it will suﬃce, for the refutation of nominalism,2 to show that such multiple exemplification of the same universal cannot always be doubted. Nominalism has historically gone hand in hand with empiricism, while realism has been fought as a metaphysical school. Thus the nominalist is prone to point out that classes of similar particulars are all that is empirically given, whereas universals identically present in diﬀerent particulars are purely metaphysical, non-verifiable entities. It is my endeavor to show that realism, in the sense defined, can be established on purely semantic grounds and is hence in no way opposed to empiricism. 1 Particulars

constitute a genus, of which things (Broad’s “continuants”), events (Broad’s “occurrents”), and processes are species. Traditionally, in speaking of particulars most attention seems to have been paid to things with qualities lasting over a finite duration. Strictly speaking, of course, the strong variety of nominalists would have to contend not only that spatially separate things sharing an identical property do not exist, but even that a thing is not ever in a definite state for more than an instant of time. For if the latter were not contended, it would be admitted that several successive events (which, like things, are particulars) may be instances of the same universal. 2 The reader should keep in mind that when I speak of the “refutation of nominalism, ” I mean by “nominalism” only what I said I mean. There may be senses of the word in which nominalism is quite unobjectionable. For example, if by “nominalism” be meant a semantically-founded criticism of Platonic reifications, I have no quarrels with it at all.

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First, let us see what precisely is involved in the claim that only similarities, not identical properties, are empirically given. The favorite examples used by the defenders of the resemblance theory are color shades. Can I be sure that two juxtaposed color patches are instances of precisely the same specific shade of red? Suppose that this may, indeed, be doubted. Can it be doubted, then, that we have here, at least, two instances of the determinable “redness”? Suppose this is also considered doubtful, since, after all, color terms are vague, and we might be confronted with a borderline case where one of the compared patches still fell within the class “red” while the other just fell outside it, even though extraordinary discrimination might be required to detect this fact. Surely, all doubt must vanish if we push our abstractive procedure one step further and consider the determinable “being colored.” When we compare patches or surfaces with respect to color and we make comparative judgments (a resembles b more than c in hue, but it resembles c more than b in color saturation, e.g.), we certainly imply that the compared surfaces all have the self-same property of being colored. In fact, the respect in which they are said to be similar or dissimilar is precisely a property (usually a determinable) identically present in all of the compared particulars; otherwise there would be no “ground of comparison.” To take another example: we may be in doubt as to whether there are two hats in the universe of precisely the same size; but how could we even raise the question whether they have the same determinate size, unless the hats shared the determinable property of having size (i.e., some size or other)? It could not be replied that determinables are universals manufactured by the mind and not really perceived in things. For if determinates are perceived, determinables must be perceived also. When I perceive a red patch I cannot fail to perceive a colored patch; and in fact I may perceive a patch as having some color or other without being able to tell which determinate color it is. My first point, then, is that resemblance is always resemblance in a certain respect, and if we only choose as our “respect” a suﬃciently abstract property, we can eliminate all doubt as to the presence of an identical universal in the compared instances. And since the more abstract properties that constitute a ground of resemblance are usually related to the more concrete (specific) grounds of resemblance as determinable to determinate, it cannot be argued that while the latter are perceivable, the former are mental constructions only. At least I strongly doubt whether a philosopher could convince an ordinary mortal that he never saw any human being even though he saw plenty of women and plenty of men.3

3 It

might be added that it is hard to be sure in any given case whether a given determinate is really an “ultimate” determinate in the sense of not being itself a determinable with respect to still more specific determinates; hence the boundary line between perceivable and merely conceivable determinables would be fairly arbitrary. Thus, if coloredness is held to be not strictly perceivable since whatever we perceive must

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Now to my second point against the resemblance theory. Resemblance is itself a property or universal, even though a relational one. Let “R1 ” stand for the relation of resemblance between particulars; “R1 ” is thus a polyadic predicate of type 1. Suppose we have two perceptual judgments: aR1 b and bR1 c. To be consistent with their resemblance theory, the nominalists could not assert “R1 = R1 ” (where the first occurrence of the predicate refers to the context a, b, and the second occurrence to the context b, c). Instead they have to introduce a relation of resemblance holding between first-order resemblances: R1 R2 R1 , where “R2 ” is of type 2. The infinite regress is obvious. Not that the infinite regress by itself proves the falsity of the resemblance theory. However, let us apply our abstractive procedure again. In which respects does relation R1 in the context a, b resemble relation R1 in the context b, c? One obvious respect in which this second-order resemblance holds is the respect of being a relation. If a and b are related in a certain way, and b and c are related in a certain way, it may, indeed, be doubted whether they are related in quite the same way. But how could it be doubted that they are both related? Now, being related is a universal, even though an extremely generic or abstract one; and being related by an n-adic relation (where n is a constant greater than 1) is a universal,4 a little more specific. I confess to being absolutely convinced that this universal is instanced in the pair a, b as well as in the pair b, c. I proceed now to the specifically semantic argument I promised. We saw that with regard to the atomic sentences “aR1 b” and “bR1 c” the nominalist, to be consistent, would have to deny that R1 = R1 (where “=” is the identity sign). But what could such a denial amount to, if not the assertion that the symbol “R1 ” in the first application (“aR1 b”) has a diﬀerent meaning from the symbol “R1 ” in its second application (“bR1 c”)? To take a concrete example: suppose I pointed to an American Indian and said “he is red,” and then pointed to a communist, saying likewise “he is red.” These sentences have the syntactic form of the atomic sentences, in an artificial symbolic language, “Pa” and “Pb” (where “P” is the symbolic substitute for “red”). Suppose I added to my symbolic object-language the sentence “P P.” How should we interpret it? I think it could only be regarded as a misleading formulation of what is properly expressed by the following sentence from the semantic part of the meta-language: the predicate “P” has diﬀerent meanings, in the sense of conhave a specific color, why not say that redness likewise cannot be perceived, since whatever is red must have a specific shade of red? 4 “Being a relation” and “being related,” as well as “being an n-adic relation” and “being related by an n-adic relation,” must, of course, be distinguished as predicates of diﬀerent types and diﬀerent degrees. A couple of related particulars constitute an instance of the relation (of type 1) “being related by a dyadic relation,” but not of the property (of type 2) “being a dyadic relation.” In this connection it might also be noticed that the relation “being an instance of” is diﬀerent from the relation of determinate to determinables. Determinates are always themselves universals, while instances may be particulars; and the former relation is analogous to class-membership while the latter relation, being transitive, is analogous to class-inclusion.

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noting diﬀerent properties, in diﬀerent applications (such as the application of “red” to both the American Indian and the communist). Thus the doubt as to whether an identical property can ever be instanced more than once is in no way distinguishable from the doubt as to whether any predicate may be univocally applied to more than one instance. When we say of a given predicate that it is non-univocal or ambiguous, we mean that in diﬀerent classes of usages it refers to diﬀerent properties. The predicate “red” is ambiguous in this sense, as shown by the above illustration. But the mere fact that a predicate is applicable to more than one instance does not make it ambiguous. Yet if we can never be sure that the same property is present in diﬀerent instances, then we cannot be sure, from an examination of actual usage of predicates, that there are any univocal predicates at all. For how else can we establish the univocality of a predicate than by observing the instances to which it is applied by competent users of the language, and noting the recurrence of a common property in those instances?5 This lack of certainty with regard to univocality of predicates, which seems to be entailed by the resemblance theory, does not mean merely that natural languages are inevitably beset with ambiguities and that an ideal language in which all predicates are univocal is—just an ideal. For, if multiple applicability constitutes ambiguity, then any predicate is ambiguous in virtue of being a predicate, and the only univocal names would be proper names which are literally “proper”—unlike proper names in the grammarian’s sense of “proper name”—to one and only one individual. The question “Are a and b instances of the same property P?” is translatable from the “material mode of speech” into the “formal mode of speech”: “Is the predicate ‘P’ applicable to both a and b?” In whichever mode, material or formal, the question be stated, it is undeniably a factual one (assuming, of course, that “P” is a non-logical, descriptive predicate). But is the question “Is ‘P’ applicable to more than one instance?” likewise factual? In some cases it certainly is. Thus it is a fact that the property “being dictator of Germany between 1935 and 1940” has only one instance. However, one will find that as a rule such unit classes are specified in terms of predicates which are compounded out of simpler predicates, and that the latter are applicable to more than one instance. In our example, this is true of “being dictator between 1935 and 1940” (Mussolini is another instance) and still more of the still simpler predicate “being dictator.” Now, the meaning of complex descriptive predicates is a function of the meanings of the constituent predicates. But since no descriptive predicate in actual use is infinitely complex, we must begin with simple predicates

5 Naturally, the instances to which a univocal predicate is applied will always have more than one property in

common and the instances to which an ambiguous predicate is applied will also share common properties, however glaring the ambiguity may be. Hence, in any such semantic investigation a prior judgment as to which properties could possibly be meant by the predicate is indispensable.

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whose meaning can only be denotatively shown. With regard to such simple predicates it is not a factual question at all whether they apply to more than one instance. For they can acquire a meaning (an intension) for the language user only by being applied to several instances. Hence it follows from the very fact that they are meaningful that they specify classes of several members. It is an elementary truth about the process of learning one’s native language that it is impossible to give an ostensive definition of a predicate by pointing to one and no more than one particular. For the particular that is denoted has a variety of properties; and how is the instructed person to tell which of these properties is being pointed out to him? Thus, if I point to a white billiard ball and tell a child “that is white,” the child might say “white” the next time he sees a red billiard ball. To prevent such misunderstandings, I have to point to other white things which share, besides their color, as small a number of properties as possible with the first thing and with each other. Suppose that no particular except that memorable billiard ball possessed the property of whiteness. In that case, “white” would not function as a predicate at all, but it would be indistinguishable in function from a proper name. For “a is white” (where “a” refers to the particular which we suppose to have, besides whiteness, several properties which are exhibited in other instances as well) would be a conventional baptismal, as it were, involving no judgment of similarity. Of course, an act of recognition would be involved if the same term “white” were applied to a on several occasions. But the same sort of recognition is involved in saying “this is John Smith,” if one has met John Smith before, and we would not say that “John Smith” is for that reason a predicate and not a proper name.6 The alleged diﬃculties besetting the realistic theory of multiple exemplification of an identical universal are all of them purely linguistic—or rather there are no genuine diﬃculties but only apparent diﬃculties arising through the tricks which language plays upon some of us. One speaks of universals as of entities named by predicates just as particulars are entities named by proper names. This reification of universals is facilitated by the grammatical accident that not only adjectives but also nouns may function as predicates. Adjectives like “cubical” or “human” explicitly indicate properties; nouns like “cube” or “man” make us think of abstract substances. Substances may resemble each

6 Theoretically

it is, indeed, possible to render an ostensive definition unambiguous by a purely eliminative process, involving no multiplication of instances of the ostensively defined quality. I might point to a second billiard ball which resembles the first in all respects except the color, and utter the words “not white.” This eliminative method of removing ambiguities as to which property of the particular pointed at is being pointed out is indispensable when it is predicates designating determinates that are ostensively defined. If I pointed only to white things and not to non-white things, the child might come to think that “white” means what “color” means. However, it remains true that a simple predicate must refer to a universal that has more than one instance. For, suppose, indeed, the white surface pointed at for purposes of definition had no duplicates at other places, and moreover disappeared as soon as it was perceived. At least it is divisible into spatial parts, and each of its parts is an instance of whiteness just as much as the whole surface.

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other; hence such relations must hold between universals and particulars likewise, especially between a particular and a universal of which the former is an instance. Thus we get Aristotle’s “third man,” as the respect in which a particular man and the universal man resemble each other. The confusion, of course, is one of logical types: the domain and the converse domain of the relation of resemblance must contain entities of the same logical type; the difference of logical type between the property chosen as ground of resemblance and each of the compared entities must be the same, viz., one. Again, there is the old paradox, discussed in Plato’s Parmenides, of multiple location. How can one and the same entity be simultaneously located at diﬀerent places?7 But this, again, is paradoxical only because “same” is surreptitiously given the meaning of numerical identity, which is a property that can be significantly attributed only to particulars, not to universals. If I say, for example, “sphericity has the dispositional property of being simultaneously exemplifiable at several places,” my ontological language is likely to raise puzzles; if universals have dispositions, are they not analogous to particulars, like pieces of sugar, which have the disposition to dissolve in water? But if they have dispositions, then these dispositions are likely to become actualized some time. Like particulars, then, universals have a history; in Whitehead’s idiom, they “ingress into actual entities.” Which is their locus before ingression takes place? The divine mind, a Platonic heaven of subsistence, human minds? There is no end to pseudo-questions of this sort. But the sentence which generates them is of the kind aptly termed by Carnap “pseudo-object-sentences.” Once we replace it by the corresponding meta-linguistic sentence “‘spherical’ is a predicate which is in principle applicable to more than one instance,” the metaphysical mystery vanishes. For what do I add to the information conveyed by “‘spherical’ is a predicate,” if I continue “which is in principle applicable to more than one instance”? I merely make explicit what characterizes a predicate as such, i.e., in contradistinction to a proper name.

2.

Platonism and the Existence of Universals

So far, in discussing “realism,” I have neglected the historical distinction between Platonic and Aristotelian realism. I confess that it is far from clear to me what the phrases by which the distinction is commonly expressed mean. According to the Platonists, so it is said, universals constitute an autonomous 7 It

is noteworthy that while many philosophers found simultaneous multiple location of universals in space paradoxical, multiple location of a single universal in time did not give rise to any puzzles. But just as multiple location in space is paradoxical if it is associated with continuants (common sense “things”), so multiple location in time is paradoxical if it is associated with another kind of particulars, viz., events; while the latter kind of multiple location is perfectly consistent with the nature of continuants. This suggests that those philosophers thought (or think) of universals not so much as another class of particulars, but more specifically as another class of continuants.

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realm; they are “independent of particulars.” The Aristotelian, on the other hand, is supposedly more earth-bound; and while he admits that universals as well as particulars exist (presumably in opposition to the nominalists), he insists that universals have no “independent being,” but exist only in particulars as attributes.8 Now the question is what literal meaning we can attach to the phrase “universals have a being independent of particulars,” as well as to the phrase “the being of universals depends upon the being of particulars.” If the former, the Platonic, phrase means “no universals are exemplified by particulars,” and the latter, Aristotelian phrase means “all universals are exemplified by particulars,” then both Platonism and Aristotelianism are plainly false views. The simple truth (or truism) is that some universals are exemplified and some are not. Suppose, then, we analyze the Platonic phrase as follows: “universals would exist, even if they had no instances.” And suppose that this is what the Aristotelian realist means to deny. The controversy, couched in these terms, looks seductively like a controversy concerning the existence or non-existence of a certain relation of causal dependence. Would noises exist if no auditory nerves existed? Of course not, since a noise is a sensation which arises when, under otherwise normal conditions, certain physical stimuli hit the auditory nerves. Waiving all that is inessential in this analogy, it may be presumed that the Aristotelian conceives universals to be as intimately dependent upon particulars as noises are dependent upon auditory nerves. The Platonist thinks otherwise. But unless it is specified in which sense one uses the word “existence” when one speaks of the existence of universals, it is absurd to take sides in the controversy. What could the Platonist mean in contending that whiteness, unadulterated, pure, and absolute, would still exist even if no white particulars existed? And what would the Aristotelian be denying if he denied this? Now, if we examine ordinary uses of the verb “to exist,” we find that existence assertions have the form “the so-and-so exists” or “x’s of such and such a kind exist.” In the former case we assert that a certain description applies to an individual, in the latter case we assert that a certain class has members or, in intensional language, that a certain property has instances. In this ordinary usage, the verb “to exist” has, like all verbs, a tense. We can significantly ask “do bears still exist in Switzerland?” or “will there ever exist a dictator in the United States?” or “how long did the Athens of Pericles exist” etc. A moment’s reflection shows that no such questions could be significantly raised about pure universals. It makes sense to ask “when did the human race (i.e., human beings) come into existence”; but it would be nonsense to ask “when

8 The

apprehensive reader will naturally wonder what the Aristotelian realist would have to say about (a) unexemplified universals, such as “golden mountain” or “mermaid”; (b) universals that cannot possibly characterize particulars, since they are properties of properties, e.g., “universal” or “color.”

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did manhood come into existence?” Indeed, the lovers of subsistence will probably say “You have simply put your finger upon the decisive diﬀerence between universals and particulars. Particulars are in time, but universals are timeless and unchanging.” This reply, however, is misleading for the following reason. “Unchanging” and “timeless” are predicates which designate, as they are commonly used, a logically possible, though rarely actualized, state of particulars (specifically, of continuants or processes). Thus the Sphinx may be said to be unchanging and to be timeless in the sense of persisting unaltered through time; the same may be said of the planetary revolutions and, in general, of any periodic process. To say of a thing that it changes is to say that at diﬀerent successive instants it has diﬀerent properties, i.e., exemplifies diﬀerent universals. This being the way the terms “changing” and “unchanging” are ordinarily used, the only sensible interpretation of which the statement “universals are unchanging” appears to be susceptible is: the predicate “changing” cannot be significantly applied to universals (indeed, such application would violate the theory of types). Suppose a man said “smells have a certain brightness of their own, colors are not unique in that respect.” You then point out to him that such brightness is very peculiar, since the brightness of colors varies with conditions of illumination, while presumably such optical changes could not aﬀect the brightness of smells. And he replies “of course, that’s because smells are smells and colors are colors; this is just the decisive diﬀerence.” I think this reply is on par with the Platonist’s statement “the reason why existence, as predicated of universals, is tenseless, is just that it is of the nature of universals to be timeless.” In the above analogy, brightness corresponds to existence, the smells correspond to universals, and the changes in optical conditions correspond to the flow of time which cannot aﬀect the existence of universals. The Platonists have failed to give a meaning to the verb “to exist” in its application to universals, just as, in the above analogy, no meaning is provided for the word “bright” as applied to smells. To be sure, most Platonists would prefer to say universals subsist; they more or less dimly recognize the misuse of language involved in the statement “universals exist.” But how does this new word help? It is as though, to refer again to the analogy, one exchanged the word “brightness” for the word “grightness,” saying “colors alone have brightness, but smells have grightness; that’s what makes them smells rather than colors.” That statements like “whiteness exists” have no sense even though they have the same grammatical form as statements which do make sense seems to be furthermore evident from the following consideration. “Whiteness” is undoubtedly synonymous with the corresponding adjective “white” in the sense that both of these designative expressions have the same intension. But “white exists” is unquestionably meaningless, since the quoted expression is not even a sentence conforming to grammatical syntax. Hence, if “whiteness exists”

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were admitted as a meaningful sentence, one would have to accept the extraordinary consequence that a sentence may be synonymous with an expression which is not a sentence.9 Indeed, it would be a worthwhile objective to see whether among people whose language does not admit the construction of abstract nouns corresponding to adjectives or verbs anything similar to the Western Platonic belief in the existence of universals may be found. Perhaps, however, Platonism cannot be disposed of so easily. Historically, Platonism has grown up concomitantly with mathematics, and maybe Platonists can be found as frequently among mathematicians as among metaphysicians. Something ought to be said, therefore, about the use of the verb “to exist” in connection with mathematical entities, specifically numbers. Let us note at the outset that numbers are universals in precisely the same sense as qualities, relations, and processes (i.e., kinds of processes, designated by verbs) are universals: they are properties, and as such belong to the converse domain of the relation “being an instance of.” As meta-mathematical analysis by Frege and Russell has revealed, the natural numbers are properties of properties or, in Russell’s extensional language, classes of classes.10 For example, fiveness is a property of the property “being a finger on my right hand” as well as of the property “being a toe on my left foot.” To say “the fingers on my right hand are five” is to say “the property ‘finger on my right hand’ has five instances.” Again, to say “God is one” is to say (assuming “one” is used as a numerical predicate, and not as a metaphysical predicate whose meaning is ineﬀable) “there is one and only one God,” i.e., the property “divine” has one and only one instance. It is in the sense here illustrated that numbers are said to be properties of properties.11 What, now, could be meant by the assertion that the natural number n exists? According to Russell’s analysis of number it means that there are properties which have just n instances. The latter existen-

9 One

might object that there are cases where a single word is synonymous with a whole sentence, as with “alas” and “I feel terrible.” However, “synonymous,” here, could not be used in the sense in which to say of two expressions that they are synonymous entails that they necessarily have the same truth-value; for there is no sense in attributing truth or falsehood to a mere exclamation. If “alas” is held to be synonymous with “I feel terrible” or some such introspective report, this could only mean that both kinds of expressions literally “express” the same kind of mental state of the speaker, and would hence be interpreted as causal signs of the same kind of state. 10 Frege and Russell’s analyses may conveniently be combined by saying that the intension of a numeral (which latter is a symbol of type 2) is a property of a property, while its extension is a class of similar classes. 11 The main consideration that led Russell to his extensional analysis of numbers as classes of similar classes is this: suppose one defined the number 5, e.g., as the common property of all properties that have 5 instances (the definition is only apparently circular, since with the help of symbolic logic one can express the fact that a property has n instances without using the concept of the number n). How does one know that there is only one such common property? To insure the fulfilment of this uniqueness condition, Russell substitutes for the common property of all properties having n instances the class of all classes similar (in the sense of one-one correspondence) to a given class of n members. This definition of number evidently does not militate against the characterization of a number as a property of a property.

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tial statement is itself translatable into “there are classes (of individuals) which have just n members.” Thus, ultimately, if n individuals (of a certain kind) exist, the number n exists. And in order to insure the existence of all the natural numbers,12 Russell had to assume the existence of infinitely many individuals (axiom of infinity). Many philosophers feel uncomfortable about this consequence of Russell’s analysis, viz., that the existence of numbers should depend upon the existence of individuals, and in fact that only a finite number of numbers could be assumed to exist if the universe happened to contain only a finite number of individuals. But one thing clearly speaks in favor of Russell’s approach: it involves a clear analysis of the notion of existence. The existence of numbers “depends upon” the existence of individuals simply in the sense that numbers are properties, and to say of a property that it exists means that it has instances. Since numbers are properties of type 2, their existence definitionally reduces to the fact that they are properties of properties which have instances. In Russell’s logic, the statement “fiveness exists” is analyzable in one and only one way, “there is an x, such that x is five”13 (where the values which x, consistently with the theory of types, may assume, are properties of individuals), or, in the class calculus “the class of quintets has members.” Any idiomatic statement which contains “exist” or “exists” as grammatical predicate can be formalized in such a way that it exhibits the structure of an assertion of classmembership. Examples: “zebras exist” = “the class of zebras has members”; the author of Tom Jones exists = “the class of authors of Tom Jones has one and only one member”;14 “the even prime exists” = “the class of even primes has one and only one member.” Russell himself occasionally reduced existential statements to statements about sentential functions, thus: “zebras exist” = “the sentential function ‘x is a zebra’ is true for some values of ‘x’.” However, this is not an analysis in the ordinary sense, since here the analysans is expressed in the meta-language, while usually both analysandum and analysans are expressed in the same object-language (this is, of course, consistent with the fact that the statement expressing the analysis is meta-linguistic).

12 Once

the existence of the natural numbers is certain, the existence of all the other kinds of numbers (rational, irrational, real, etc.) follows, since the latter are all “constructable” out of the former. 13 For the sake of illustration, I assume a symbolic language which contains number concepts as primitives. In the system of Whitehead and Russell, of course, there are no such primitives, since numbers are defined in terms of purely logical concepts. 14 Sentences having a proper name as grammatical subject and existence as grammatical predicate are either meaningless or disguised assertions of class-membership. Thus “Hitler exists” might mean “Hitler is now alive.” But if “exists” is used in a tenseless sense, the statement is meaningless, since there cannot be proper names of non-existent individuals. In the grammarian’s sense of “proper name,” “Apollo” or “Jehovah” or “Saint Nicholas” are, indeed, proper names. But they are not proper names in the sense in which the referent of such a name must be, directly or indirectly, given through denotation; for, obviously, these mythological names are merely abbreviations for definite descriptions, which is not the case with strictly proper names.

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Platonism and the Existence of Universals

We have already seen that those Platonists who maintain that universals exist could hardly be using “exist” in the sense here analyzed. For then they would either mean that all universals have instances or that some have; but they could not mean the former, since that is so patently false that it could not escape their notice that it is false, and since it is moreover typical of Platonists to assert the existence of unexemplified essences (such as perfect beauty and perfect sphericity); and they could not mean the latter, since that is too trivially true to deserve emphasis. But as long as no alternative analysis is forthcoming (and to my knowledge none has yet come forth), the much debated statement that universals exist is meaningless. The analysis in terms of the notion of classmembership certainly fits the mathematician’s usage of the verb “to exist.” To ask, for example, whether the square root of 2 exists is precisely synonymous with the question “is there an x such that x2 = 2” or “does the class of integers whose square is equal to 2 have a member?” The psychological root of the puzzle about the existence of universals such as numbers may well be this: most ordinary uses of “exist” are—prima facie in contradiction to the class-membership analysis—not tenseless.15 Since existence is usually predicated of particulars that come into being and perish, a sentence of the form “x’s exist” or “the so-and-so exists” automatically evokes the question “where, and when?” Such questions can be meaningfully addressed to properties of type 1: when and where does this property have instances? (or, in the equivalent language of classes, “. . . this class have members?”). Therefore, when we speak of the existence of properties of higher type, the same quest for spatio-temporal specification obtrudes itself, and our inability to satisfy it leaves us in a mystery. But all that is rational in our feeling that numbers and other properties of non-elementary type exist and still do not exist in the way particulars exist (i.e., in space and time) is that these properties are not properties of individuals with spatio-temporal position. Of course, it may be presumed that hardly any Platonist would be satisfied with “P has instances, or the class determined by P has members” as the analysis of “P, the universal, exists.” For he feels that the universal would still exist even if it were not exemplified. After all, before being exemplified, the universal must already be, must it not? But this question arises from a confusion between logical intensions to which temporal predicates (such as “existing before exemplification”) cannot be significantly applied and ideas which are mental events or mental dispositions.16 If “universals exist, even if they 15 Actually,

however, there is no contradiction. For a statement involving a temporal use of the verb “to exist” may be transformed into an assertion of class-membership by introducing temporal predicates. Thus the statement “there will be an atomic war” may be transformed into “the class of future atomic wars has a member.” 16 In a statement like “I have no idea of complex numbers,” “idea” obviously refers to a disposition, not to an event, since I do not intend to assert merely that I am not apprehending the nature of complex numbers at the

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have no instances” means “we may have ideas of kinds of things that do not exist,” then, of course, the statement is both meaningful and true. We can all form ideas of golden mountains and dragons. Who would deny it? Conceptual apprehension of universals is possible, whether the apprehended universal have instances or not. “But, surely, unless unexemplified universals had some sort of being, they could not be apprehended?” Well, if “x has being” means “x is a possible object of apprehension,” it must be admitted that any universal which we happen to apprehend has “being,” no matter whether it be exemplified in the actual universe or not. For, as the scholastics said, “de esse ad posse valet consequentia”; in plain English, whatever is the case, must be possible. This, however, is entirely too trivial to be insisted upon. It may well be that the Platonists, with their talk of universals as independent “essences,” are led by the nose by grammatical forms. The statement “I think of a universal” has the same grammatical form as the statement, to pick one at random, “I hear of the death of Roosevelt”: the death of Roosevelt would have occurred even if I had never heard of it, and unless it had occurred in the first place it is unlikely that I would have heard of it. In this case, the grammatical object of the verb corresponds to an object that exists independently of, and in a sense prior to, the activity expressed by the verb. This leads one to suppose that it is the same with the object of the verb “to think of” or “to think about.” But perhaps thinking of universals is like dancing dances or singing songs or smelling smells.

present moment. Even though, however, the word may more often refer to a disposition than to an actual mental event or process, ideas in this latter sense are logically prior to ideas in the former sense; for, in psychology as well as in physics, dispositions are defined in terms of events or states evoked by appropriate stimuli.

IV

PHILOSOPHY OF LOGIC AND MATHEMATICS

Chapter 10 LOGIC AND THE CONCEPT OF ENTAILMENT (1950)

The main thesis of this paper will be best approached by raising a question of the philosophy of logic (or “meta-logic”) which most practicing logicians neglect to raise, presumably for the same reason that most mathematicians neglect to raise philosophical questions about mathematics: what is a logical constant? The problem of defining what is meant by a “logical constant” (logical term, logical sign) is crucial for a satisfactory theory of logical truth, since it seems impossible to analyze the latter concept without using the concept of a logical constant. Definitions of logical truth which do not use this concept are easily shown to be unsatisfactory. If, for example, we define a logical truth as a statement which is true by the very meanings of its terms, we are either defining a concept of psychology, not of logic, or else the definition is implicitly circular. The former is the case if we interpret the definition to say that anybody who understands what the constituent terms of the statement mean (who understands, in other words, what proposition the sentence is used to expressed) will assent to it; and the definition is circular if it tells us that the statement will turn out to be derivable from logic alone once the definitions of its terms are supplied. Again, it is implicitly circular to define a logically true statement as one that cannot be denied without self-contradiction. For, surely, we want to say that p is logically true if a contradiction is derivable from not-p with the help of logic alone, without the use of factual premises. A definition which, prima facie, is free from the vice of circularity is the following one: a logical truth is a true statement which either contains only logical constants (besides variables) or is derivable from such a statement by substitution (this is essentially the definition preferred by Quine).1 It remains to be seen, however, whether this appearance will stand the test of analysis. The crucial question is obviously whether we could construct an independent definition of “logical constant.” 1 See

Quine 1947b.

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The customary procedure of logicians who define their meta-logical concepts with respect to a specified postulational system is to define the logical constants simply by enumeration. But while such definitions serve the function of criteria of application, they clearly cannot be regarded as analyses of intended meanings. To give an analogy, suppose we defined “colored” by enumerating n known colors, i.e., colored = C1 or C2 ... or Cn . And suppose we subsequently became acquainted with a new color which we name “Cn+1 .” On the basis of our definition it would be self-contradictory to say that Cn+1 is a color, or at any rate we could not say that it is a color in the same sense as the initially enumerated ones. Thus, so-called definitions by enumeration do not tell us anything about the meaning of the defined predicate, and the same is true of many recursive definitions. In fact, recursive definitions of “logical constant” given by logicians usually reduce themselves to an enumeration of logical signs with the addition that any sign definable in terms of these alone is also a logical sign. The problem of defining this basic meta-logical concept explicitly, however, cannot be said to have been solved.2 Frequently the explanation is given that logical signs are purely formal (or syntactic) constituents of sentences, as though it were perfectly clear what was meant by that. But in classifying a sentence as having such and such a form we presumably point out what it has in common with other sentences. Why, then, could not, for example, the sentences “the sky is blue” and “all exam-booklets are blue” be said to be formally similar on account of sharing the constituent “blue,” if it is all right to call the sentences “the sky is blue” and “the weather is nasty” formally similar because they contain the “is” of predication as a common element? If all we can reply is that the latter is a formal sign while “blue” is a descriptive sign, the above explanation leaves us no wiser than we were before. Logical terms are, as usually understood, contrasted with descriptive terms, and if, therefore, an independent definition of “descriptive term” were at hand, a logical term could simply be defined as a non-descriptive term. It will appear, however, that such an independent definition presents grave diﬃculties. To begin with, it would not be clarifying to define a descriptive term as one that refers to an observable feature of the world as long as we have no clear criterion of observability. Are numbers, for example, observable features of the world? This could not plausibly be denied since numbers are observable properties of collections (counting is surely a mode of observation), although of a higher type than the properties which could be ascribed to the elements of the collection singly; and yet number-predicates would by most logicians be classified as logical terms, in view of the logistic reduction of arithmetic. Again, it would not be illuminating to define descriptive terms as those terms that may function as values of variables, where variables are divided, say, into individual, predicate, and propositional variables. For, if the logician were asked why, 2A

critical comment on the proposed solution by Professor Reichenbach, in Reichenbach 1947, will be found later in this chapter.

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for example, no connective-variables, i.e., variables taking connectives as values, occur in his system, he would presumably reply that connectives are not descriptive terms. Perhaps we shall fare better if, instead of looking for an explicit definition, we try a contextual definition, like the following: T occurs as a descriptive term in argument A, if A would remain valid, respectively invalid, when any other syntactically admissible term is substituted for T in all its occurrences. But this definition is open to objections from two angles: (1) if it is our aim to define “logical term” negatively, as “non-descriptive term,” then this definition makes it impossible to define a valid argument as one such that the implication from premises to conclusion is true by virtue of the meanings of the logical constants involved. At least this is an objection from the point of view of those who are not satisfied with accepting “valid” as a meta-logical primitive. (2) While the mentioned condition is no doubt necessary, in line with the idea that the logical validity of an argument does not depend upon the subject-matter which the argument is about (briefly, the idea that logic is a formal science), it is not suﬃcient. For example, the argument “x = y, therefore not-not-(x = y)” would remain valid no matter what relation were substituted for identity, and still one would not call “identical” a descriptive term. The point is that a term may occur inessentially in an argument without being descriptive. It should be mentioned, though, that this objection would lose its force if the distinction between logical and descriptive terms were altogether functional, i.e., if “T is descriptive” should be regarded as elliptical for “T is descriptive in A.” But while we may in the end have to accept this possibility—which would lead to the, perhaps surprising, consequence that no general definition of logical truth could be given, let us first see whether we have better luck in beginning with a positive definition of “logical constant.” The process by which terms used in deductive arguments are actually identified as logical and as determining the logical form of the argument, is to replace constants with variables until only those constants are left over on whose meanings the validity of the argument depends. But the definition, thus suggested, in terms of essential occurrence in deductive arguments, has already been seen to be unsatisfactory, since one and the same term may occur essentially in one argument and inessentially in another. This is particularly obvious if our arguments contain defined terms: in “x is a triangle, therefore x has three sides,” “triangle” occurs essentially, but in “x is a triangle, all triangles have property P, therefore x has P,” “triangle” occurs vacuously. The explicit definition of “logical term” recently proposed by Reichenbach (Reichenbach 1947, §55) seems to me, indeed, to break down because of this circumstance, viz., that the concepts “logical term” and “term occurring essentially in every necessary implication in which it occurs” do not have the same extension. Reichenbach attempts to clarify the distinction between logical and descriptive terms by means of the distinction between expressive and denotative terms.

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Denotative terms are values of individual, predicate, or propositional variables, and expressive terms are those which do not denote. A logical term is then defined as an indispensable expressive term, or one definable in terms of such. This definition leads, however, to such embarrassing consequences as that the connectives “or,” “and,” etc., are not logical terms. For while “or,” for example, may be used, and mostly is used, as an expressive term, it could be used as value of a two-term predicate variable. Instead of writing “p or not-p” one would then write “or (p, not-p).” To be sure, as such a denotative term it is definable by means of the corresponding expressive term, but ipso facto the latter is not indispensable: definitional eliminability works both ways. If I understand Reichenbach correctly, he hopes to overcome this diﬃculty by defining “logical term” with respect to a language in which the following notational convention is observed: in a tautologous formula only such denotative terms may occur as have a vacuous occurrence. Thus we should not express universality by a second-level predicate “Un” and write, for example, Un (p or not-p), since “Un” here would occur essentially, i.e., the function obtainable from this proposition by substituting an appropriate variable for the secondlevel predicate would not be universally assertible. But this convention will not do the job of saving the initial definition since it presupposes that the concepts “logical term” and “term having an essential occurrence in tautologies” have the same extension, which they do not. To add an illustration to the one already oﬀered, in the tautology “p ∨ q ⊃ p ∨ q,” the logical term “∨” occurs inessentially. And any tautology that contains defined predicates illustrates the possibility of essential occurrence, in tautologies, of non-logical terms.3 But even if these anomalous cases of vacuous occurrence of logical constants could be discounted for some reason, the general definition of “logical constant” in terms of “essential occurrence in deductive arguments” would be open to further objections. In the first place, since terms that would not normally be classified as “logical” may occur essentially in arguments containing defined terms, one would have to specify either that the arguments referred to are to be formally valid or invalid, or that they are to contain no definable pred3 In personal correspondence, Professor Reichenbach has answered my objection to his definition of “logical

term” by pointing out that while logical terms may, indeed, occur vacuously in some tautologies they do not occur vacuously in all tautologies and that this is the reason why they cannot be treated as values of variables. The suggested definition of “logical term” in terms of “term occurring essentially in some tautologies” leads, however, into the following dilemma. Either “tautology” is so used that only sentences in primitive notation could be tautologies, or more broadly (and more in accordance with ordinary usage) so that sentences containing defined terms could also be tautologies. In the former case, there could be no defined logical terms at all, since obviously a defined logical term cannot occur essentially in sentences which contain no defined terms. In the second case, however, terms that are ordinarily regarded as descriptive, like “bachelor,” would turn out to be logical, since they occur essentially in such tautologies as “all bachelors are unmarried men.” If, to avoid the latter consequence, the definition of “logical term” be restricted to non-descriptive languages, it becomes circular again, since a non-descriptive language is presumably a language containing only logical constants.

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icates. This, however, is a real dilemma. Relatively to the first specification, the definition is circular, since an argument is said to be formally valid if it is valid by virtue of the meanings of the logical terms alone; and relatively to the second specification we get a concept which is applicable only to fictitious completely analyzed languages. Instead of hunting any longer after a satisfactory general definition of “logical constant,” let us now focus attention upon an important consequence of our mainly negative results. It has been seen that the discrimination of logical terms from descriptive terms amounts to identification of those terms in a given deductive argument on whose meanings the validity of the arguments depends.4 But since any formal test of validity presupposes identification of logical constants (how could I know to what degree a given argument under scrutiny is to be formalized, unless I knew which terms may not be replaced by variables?), validity could not here, on pain of circularity, be established by a formal test. This suggests that no adequate epistemological account of the construction of semantic systems of logic can be given without countenancing the concept, held in disrepute by many logicians and philosophers, of material (= non-formal) entailment. It is widely held that entailment is essentially a formal relation, i.e., “ ‘p’ entails ‘q’ ” is held to be equivalent to “ ‘if p, then q’ is true by virtue of its logical form.” But the latter statement is presumably equivalent to the statement “‘if p, then q’ is true by virtue of the meanings of the logical terms involved, all other ingredients occurring vacuously.” But I have shown that judgments of entailment are presupposed by the very process which leads to the definition of the meta-logical concept logical form. For the only way in which this concept can be defined (if it be proper to call such a procedure “definition” at all) is to exhibit logical forms by the use of logical constants. What, indeed, is the attitude of logicians when they are faced with an evidently non-empirical conditional statement which, though expressing an a priori truth, does not seem to be demonstrable with the sole help of the definitions of specified logical terms? Their endeavor may be described as the analysis of the non-logical concepts that seem to occur essentially by means of logical

4 The

frequently debated thesis that logic is a purely formal science, in the sense that questions of logical validity can be answered without any appeal to meanings, is ambiguous. If the contention is that, provided no defined descriptive terms occur, the meanings of the descriptive terms are irrelevant to questions of logical validity, that is true enough—although, as I have suggested, possibly truistic. But if the claim is that no semantic rules at all need be consulted in such investigations, then “logic” is defined as a science that is competent to answer only questions of purely syntactic derivability, which therefore does not concern itself with relations of truth-values, and is therefore inapplicable to problems of logical validity as they arise in natural languages.

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concepts, so that what appeared as a material entailment5 reduces to a formal entailment once the proposition is fully analyzed. The obvious illustration of this procedure that comes to one’s mind is the reduction of arithmetic to logic. To take a very simple example, since the relational predicate of arithmetic, “being greater than,” does not belong to the vocabulary of logic, the sentence “if G(x, y), then not-G(y, x)” could not, oﬀhand, be said to express a formal entailment. This postulate of arithmetic, however, can be reduced to pure logic by defining numbers as properties of classes and defining the mentioned arithmetical relation in terms of the logical concepts “similarity” (= one-one correspondence) and “proper subclass.” But the definition which enables such a reduction is obviously not an arbitrary stipulation; rather it expresses an analysis of a primitive concept of arithmetic. And I would like to know what else I judge, in judging this analysis to be correct, but that the proposition “the number of A is greater than the number of B” entails and is entailed by the proposition “B is similar to a proper subclass of A (where A and B are finite classes).” But since the non-logical term “greater” occurs essentially in the corresponding conditional statement6 this is definitely not a formal entailment. We have succeeded in formalizing the entailment from “G(x, y)” to “not-G(y, x)” only by accepting the material entailment just mentioned. I anticipate the objection that analysis of a concept belonging to an interpreted language must not be confused with interpretation of the primitives of a postulational system; that there is no sense in speaking here of an entailment holding between a proposition of arithmetic and a proposition of logic, since a postulate of uninterpreted arithmetic is no proposition. I submit, however, that if the logistic thesis is to be redeemed from the charge of triviality, it must be taken to assert that the terms of interpreted arithmetic are reducible to logic. For the thesis that uninterpreted arithmetic, as erected upon Peano’s postulates, is logic could only mean one or the other of equally trivial propositions: (a) assertions of the form “if the postulates are true, for a given interpretation, then the theorems are true, for that interpretation” belong to logic; (b) Peano’s postulates are satisfied by a logical interpretation. (a) is trivial since in this sense any uninterpreted postulational system is logic, (b) is trivial, since the logical interpretation is by no means the only one which satisfies Peano’s postulates. Any class of objects which is the field of an asymmetrical one-one relation, which contains an object which does not stand in the converse relation to any member of the field, and whose members are characterized by properties hereditary with respect to

5 The

only reason I use the qualifying adjective “material” rather than “synthetic” is that “material” is the natural term to use in contrast to “formal.” My use of the word “material” in this context has, of course, nothing to do with its use in the phrase “material implication.” 6 By the conditional statement corresponding to the entailment-statement “‘p’ entails ‘q”’ I mean the statement “if p, then q,” which occurs in the object-language and does not contain names of propositions.

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the ordering relation, would be a model for Peano’s postulates. If the model is sociological, for example, one could then with equal plausibility say that arithmetic is reducible to sociology. If I were to permit myself the use of inexact metaphorical language in the midst of an austerely exact discussion, I would say that as logic expands by naturalizing more and more alien concepts in its economic household, one material entailment is used to kill oﬀ another, and ipso facto material entailments are there to stay. Furthermore, that we can reduce an apparently non-formal entailment to a formal entailment by supplying a correct analysis for the troublesome non-logical terms is no surprise. For is not such reducibility a tacit criterion of a correct analysis? Would Russell and Whitehead, for example, have proposed those definitions of the primitives of arithmetic in terms of logical concepts which they did propose, if they had not enabled them to reduce, say, Peano’s postulates to pure logic? Or, to take a simpler example, how could one controvert the claim that the admittedly necessary connection between the attributes “colored” and “extended” is at bottom a formal one, which would no doubt become evident if we knew the correct analysis of the attribute “colored”? Why, no such analysis would, of course, be judged correct unless it enabled the purely formal deduction of the attribute “extended.” I have deliberately postponed the dropping of the bomb to the concluding section of my paper, lest I prejudice from the outset my chances of getting an impartial hearing. Here is the bomb: if by a synthetic proposition you mean a proposition not deducible from logic alone, and by an a priori proposition you mean one that is not empirical, and if you define logic by means of an enumeration of a set of concepts called “logical constants”—to which there is no alternative in the absence of a satisfactory general definition of “logical constant”—then you have to accept the conclusion that synthetic a priori propositions are acknowledged whenever the territory of logic expands. And it appears, then, that in “reducing” the non-geometrical parts of mathematics to logic, the logisticians have not eliminated the synthetic a priori7 from mathematics; they have merely dislocated it to those regions where mathematical and logical concepts make definitional contact. It may be clarifying to refer to an analogous logical situation in the reduction of one empirical science to another, say thermodynamics to mechanics. Once the temperature of a gas is defined as

7I

do not intend to resuscitate the ghost of Kantian epistemology by using this Kantian expression, and would be sorry if my terminology had this eﬀect. I would justify my usage by pointing out that the term “synthetic” as well as the term “a priori” is in good standing with philosophical analysts who are fully emancipated from metaphysics; so why should the term “synthetic a priori” be disreputable? If, indeed, the term “analytic” is used so broadly, as, e.g., in C. I. Lewis, that any statement which can be established by reflecting upon meanings is analytic, then “analytic” becomes synonymous with “a priori” or “nonempirical” and the thesis of the analytic character of all a priori truth becomes irrefutable on account of triviality (see, on this point, chapters 3 and 4).

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average molecular kinetic energy, thermodynamic laws containing the variable “temperature” are translatable into mechanical language. But in terms of the meaning attached to “temperature” in the language of thermodynamics before such a reduction took place, the proposition “the temperature of a gas is proportional to the average kinetic energy of its molecules” was obviously synthetic. As far as I can see, the only ground on which the suggested analogy between reduction in the empirical sciences and reduction in the formal sciences could be questioned would be the following: one might hold that if a term of formal science S is analyzed with the help of terms of formal science S , the term of S has no independent meaning; that it only receives a meaning by virtue of its definitional reduction to the terms of S . In order to refute this view it will suffice to point out that it is equivalent to the view that such analyses are arbitrary stipulations. For to see the untenability of such a position—which is motivated by what I regard as an irrational dread of intuitionism—we only need to observe that according to it the question whether S is reducible to S would be nonsensical; all one could sensibly ask would be whether it is convenient to reduce S to S . In conclusion, I shall briefly reply to the comment, which I may well anticipate, that I have here argued in terms of an absolute concept of entailment which is an intuitionistic superstition as far behind the times as, say, the Newtonian superstition of absolute space, time, and motion. The thesis that “‘p’ entails ‘q”’ is, if meaningful at all, to be construed as elliptical for “‘q’ is derivable from ‘p’ in terms of the transformation-rules (including definitions) of a specified language-system” seems to me to involve a fatal paradox. The paradox is simply that through the construction of suitable definitions statements which would normally be held as logically independent could be made to entail one another. And if it be replied that those definitions, of course, must be adequate, the paradox is merely dodged, not solved. For, in order to determine whether a definition of A in terms of B is adequate, philosophers normally do not resort to statistical investigations of linguistic habits— if they did, how could the results of philosophical analysis obtained in one language-community be of any philosophical interest to philosophers in other language-communities?—but instead ask themselves whether it would be selfcontradictory to predicate at once A and non-B. But this is to ask whether a predication of A entails a predication of B. And it should be obvious that a consistent adherence to the criticized theory of entailment makes this procedure either circular or infinitely regressive.

Chapter 11 STRICT IMPLICATION, ENTAILMENT, AND MODAL ITERATION (1955)

Ever since C. I. Lewis oﬀered the concept of “strict implication,” defined explicitly in terms of logical possibility (p q ≡d f ∼ 3(p . ∼ q)) and implicitly by the axioms of his system of strict implication, as corresponding to what is ordinarily meant by “deducibility” or “entailment,” there have been analytic philosophers who denied this correspondence. They denied it specifically because of the paradoxes of strict implication: that a necessary proposition is strictly implied by any proposition and an impossible proposition strictly implies any proposition. These theorems, it is maintained, do not hold for the logical relation ordinarily associated, both in science and in conversational language, with the word “entailment.” It is my aim in this paper to show that it is extremely diﬃcult, if not downright hopeless, to maintain this distinction. I shall refer specifically to a subtle paper by C. Lewy (Lewy 1950), which deals with the intriguing problem of modal iteration, and which emphatically endorses the distinction here to be scrutinized. Let me begin by presenting a brief argument against the distinction which seems to me conclusive, though I do not intend to rest my case on it. It is simply that anybody who wishes to maintain the distinction must abandon one or the other of two propositions which seem equally unquestionable: (a) p is necessary if and only if ∼ p is impossible, (b) p entails q if and only if it is necessary that (if p, then q)—where “if p, then q” is a material implication. For, from the conjunction of (a) and (b) we can deduce: p entails q if and only if (p and ∼ q) is impossible. Since the impossibility of the latter conjunction follows from the impossibility of p, we have already the conclusion that an impossible proposition entails any proposition—which is what those who insist on the diﬀerence between entailment and strict implication deny. Notice that this argument does not presuppose that either (a) or (b) are definitions of “necessity” and “entailment” respectively. The conclusion follows even if the equivalences asserted by (a) and (b) are just material. In the above mentioned essay, Lewy confesses that he cannot completely de-

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fine “entailment,” but thinks that he is nevertheless justified in distinguishing entailment from strict implication because he can state two conditions which are necessary for p to entail q, over and above the condition that p strictly imply q, and which are not necessary for p to strictly imply q: p entails q only if (1) “R counts in favor of p” strictly implies “R counts in favor of q” and (2) “R counts against q” strictly implies “R counts against p.” Lewy uses “counting in favor of” as a primitive concept; the illustrations he gives show that it is meant as a very broad concept which covers both “confirming evidence” in the sense of inductive logic and deductive entailment as special cases. Thus he would say presumably that a sample of black ravens counts in favor of “all ravens are black,” but also that the proposition “all birds are black” would, if it were true, count in favor of “all ravens are black.” He gives the following examples of strict implications which are not entailments because they either do not satisfy (1) or do not satisfy (2): “the proposition that there is nobody who is a brother and is not male is necessary” strictly implies “there is nobody who is a sister and is not female,” because the implied proposition is necessary and a necessary proposition is strictly implied by any proposition. But it is no entailment, he says, because (1) is not satisfied. (1) is not satisfied because it is logically possible that there should be an R1 which counts in favor of the first proposition but is irrelevant to the truth of the second proposition. Lewy does not produce an example of such an R, but he might have produced the following: the concept being a brother is identical with the concept being a male sibling. Perhaps he would say that this proposition—the classical example of a “correct analysis” in Moore’s sense—entails, and therefore counts in favor of, the modal proposition “it is necessary that there are no brothers that are not male”; and surely we could agree that this proposition is irrelevant to the strictly implied proposition, since the latter does not contain the concept being a brother at all. His example of a strict implication which fails to satisfy condition (2) is, however, more convincing, since it involves nothing more problematic than that empirical evidence is, favorably or unfavorably, relevant only to contingent propositions: the false contingent proposition “Cambridge is larger than London” is strictly implied by the impossible proposition “there is somebody who is a brother and is not male,” but while there is empirical evidence counting against the implicate, there can be no empirical evidence that counts against the logically impossible implicans. If it were significant to say “since so far no brother has been found anywhere that was not male, it is unlikely that there is somebody who is a brother and is not male,” then the sentence “there is somebody who is a brother and is not male” would presumably express a contingent proposition. Lewy, then, assumes the following principle: if p entails q, then, for any R, “R counts in favor of p” strictly implies “R counts in favor of q”; and he be1I

take Lewy’s “R” to be a propositional variable, such that “counting in favor of” designates, like “strict implication” and “entailment” in his usage, a relation between propositions.

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lieves that this principle (together with the analogous principle corresponding to condition (2) above) serves to diﬀerentiate entailment from strict implication, because it is false if, for “entails,” “strictly implies” is substituted. I wish to show, however, that Lewy’s principle is false, its falsehood being a consequence of another principle nicely established by Lewy himself in the very same essay, viz., that a contingent proposition may entail, not just strictly imply, a necessary proposition. For, let R1 be empirical evidence which counts in favor of a contingent proposition p, and let q be a necessary proposition entailed by p. Since by the very meaning of “necessary proposition,” no empirical evidence can be relevant to a necessary proposition (and, a fortiori, cannot count in favor of it), R1 cannot count in favor of q; but this, by Lewy’s principle, contradicts the assumption that p entails q.2 In order to gain a clear insight into this subtle matter, it is necessary to consider the kind of entailment from a contingent proposition to a necessary proposition adduced by Lewy. The contingent proposition (J) “There is somebody who is French and is not under fifty years of age,” he says, entails the necessary modal proposition (H) “The proposition that not-J is not necessary.” While Lewy just takes it to be intuitively evident that J entails H, we can postpone, if not dispense with, the appeal to intuitive evidence by deriving this entailment from the formal entailments: (1) 2p entails p, (2) p entails ∼ ∼p.3 It seems, therefore, that Lewy is right in holding that J entails H. But consider, now, the singular proposition (A) “Pierre is French and is not under fifty years of age.” Inasmuch as A entails J, it counts in favor of J. Yet, if a necessary proposition is, by definition, such that no empirical evidence can be relevant to it, and the modal proposition H is, as maintained by Lewy, necessary, must we not conclude that A does not count in favor of H? Suppose, however, that Lewy replied: A does count in favor of H, because, entailment being transitive, “A entails H” follows from “A entails J” and “J entails H”; and if A entails H, then A counts in favor of H. To this I would make a threefold rejoinder: (1) If one holds that empirical evidence may count in favor of a necessary proposition, it may prove very diﬃcult, if not impossible, to explain the distinction between necessary and contingent propositions. (2) Lewy’s principle is presumably oﬀered as a partial analysis 2 One

might think that Lewy’s principle could be saved by considering strict implication itself as a case of “counting in favor of,” such that “p counts in favor of q” would be equivalent to “p is confirming evidence for q, or p entails q, or p strictly implies q.” But if “counting in favor of” were, in gratuitous violation of the ordinary usage of the expression, construed in this way, then Lewy’s attempt to diﬀerentiate entailment from strict implication would come to naught anyway: since strict implication satisfies the syllogism principle “if (if p, then q), then, if (if r, then p) then (if r, then q),” the proposition “for any p and q, if p strictly implies q, then ‘R counts in favor of p’ strictly implies ‘R counts in favor of q’ ” would not fail for the “paradoxical” case of a necessary q being strictly implied by any p. For, for any R, it will be impossible that R counts in favor of p but does not count in favor of q, simply because it will be impossible that R does not strictly imply q. 3 From (1): ∼ p entails ∼ 2p (3); from (3): ∼ ∼ p entails ∼2(∼ p) (4); from (2) and (4): p entails ∼2(∼ p) (5); from (5): J entails ∼ 2(∼ J).

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of the relation of entailment, which describes a method for deciding whether a given strict implication is also an entailment. Thus we ought to be able to decide whether the strict implication from J to H is also an entailment, by deciding whether “R counts in favor of J” strictly implies “R counts in favor of H.” But then the question is begged if we infer “‘A counts in favor of J’ entails ‘A counts in favor of H,”’ and hence “‘A counts in favor of J’ strictly implies ‘A counts in favor of H,”’ from “J entails H.” Of course, this strict implication follows, by virtue of Lewy’s principle, from “J entails H,” but he ought to show that it holds, independently of the assumption of the entailment. (3) The appeal to the transitivity of entailment becomes irrelevant to the question at issue, viz., whether the entailment from J to H satisfies Lewy’s principle, if instead of A we consider some empirical proposition which confers a high probability upon A, and therewith upon J, e.g., “Pierre is French and his birth certificate—which probably has not been falsified—indicates that he is not under fifty.” Such a proposition will not entail H, as Lewy surely would admit. If, then, it is held to “count in favor of” H, this must be in the sense of “making H highly probable.” Yet, is it not nonsense to say, with respect to a necessary proposition p, “The available empirical evidence makes it highly probable that p is true”? Now, Lewy’s attempt to diﬀerentiate entailment from strict implication might still be successful if a flaw could be found in Lewy’s demonstration of an entailment from a contingent to a necessary proposition. Perhaps modal propositions, like H, are contingent? If only we were permitted to use Lewy’s principle, we could easily prove that propositions to the eﬀect that some other proposition is necessary (propositions of the form “It is necessary that p”) are not contingent—and one would expect that the same holds for any kind of modal proposition, and thus for H in particular. For suppose that 2p is contingent while p is necessary. If 2p is contingent, then there is an empirical R which, if it were true, would count in favor of 2p. But 2p entails p. Hence, by Lewy’s principle, R would count in favor of p. But this contradicts the hypothesis that p is necessary (compare the first rejoinder above). Therefore 2p cannot be contingent. However, it would be poor strategy to use this argument in the present context. For, we set out to prove the noncontingency of modal propositions in order to defend the claim, made and substantiated by Lewy, that a contingent proposition may entail a necessary proposition (i.e., that there are pairs of propositions (p, q) such that p is contingent and q is necessary and p entails q), and this claim we wanted to defend in order to be able to refute Lewy’s principle. Yet, the above argument presupposed Lewy’s principle, hence we would assume the validity of the latter in proving its invalidity. But actually nothing more elaborate is required to see that modal propositions are noncontingent than reflection on the meaning of the words “necessary” and “contingent.” When we call a proposition “necessary” we are saying

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that it can be known to be true without empirical investigation by reflecting upon meanings and, in some cases, applying logical principles. Now, it is clear that we do not make any sort of empirical investigation in order to answer a question like “Is it necessary that anything which is a brother is also male?” Should one object that it can be answered only by discovering the rules governing the use of the words “brother” and “male,” then one would be guilty of confusing the proposition about verbal usage “The sentence ‘anything which is a brother is also male’ expresses, in ordinary usage, a necessary proposition”—which is admittedly empirical—with the proposition in question, which is not about a sentence at all, but rather about a proposition. The distinction is analogous to the distinction between “The property being a round square is L-empty” and “The expression ‘being a round square’ designates an L-empty property”; it is of course a contingent fact that a given expression is used to designate the property it designates, but this has no tendency to prove that the designated property has whatever properties it has contingently, not necessarily. Some philosophers no doubt will say that these distinctions are intelligible only if one conceives of propositions and properties as of entities named by expressions (what has been satirized as the “‘Fido’ means Fido” pattern of semantic analysis). Meaning, they would say, has been fallaciously assimilated to naming, as though “meaning” were a transitive verb that can occur in sentences of the form xRy, where the values of x are linguistic expressions and the values of y—meant meanings! But I think the question at issue can be settled without any prior commitment for or against Platonistic semantics. For while a philosopher may have misgivings about propositions as extra-linguistic entities designatable by sentences, he will surely accept the distinction between sentences and statements, where a sentence is a special kind of sequence of physical marks or noises, and a statement is a sentence as meaning such and such, “meaning” here being used as an intransitive verb, a verb without object. With this terminology the above distinction can be reproduced as follows: that the sentence “If anything is a brother, then it is male” is used to make a necessary statement, is indeed a contingent fact if it is a fact at all; but that the statement which this sentence is ordinarily used to make is necessary, is not a contingent fact. There is no possible world in which this sentence, as meaning what it usually means, is false; a world in which it would be false is a world in which it would be used with a diﬀerent meaning. But this we find out by reflecting on the meaning the sentence is commonly used to convey, not by ascertaining empirically what meaning it is that the sentence is commonly used to convey. That statements to the eﬀect that a given statement is necessary are themselves necessary has been denied by Strawson (Strawson 1948) on the ground that modal statements are contingent meta-statements about the uses of expressions. Thus, “It is necessary that there are no brothers that are not male” turns

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on this theory presumably into an empirical statement about the usage of the expressions “brother” and “male” (the linguistic theory of logical necessity advocated by Strawson and others who shun the “inner eye of reason” would also lay stress on the prescriptive function of such modal statements, their “recording our determination” to adhere to a certain usage, as Ayer put it, but this aspect of the theory is not under discussion now). Strawson faces the obvious objection to this theory, viz., that the modal statement talks about the concept of being a brother, or, if you wish, the meaning of the expression “brother,” not about the expression “brother,” in the usual manner: a concept is a class of synonymous expressions, hence it can be conceded that a statement is made about the concept of being a brother without surrendering the claim that a statement is made about the expression “brother.” The modal statement, if correctly interpreted, is a meta-statement which mentions the expression “brother” in order to define a class of expressions, viz., the class of expressions synonymous with “brother,” and it makes an assertion at one stroke about each member of this class. Thus it says not only that “brother” is inapplicable to x unless “male” is also applicable to x, but further that if “Bruder” is synonymous with “brother” and “m¨annlich” synonymous with “male,” then “Bruder” is inapplicable to x unless “m¨annlich” is also applicable to x, etc. According to this theory, then, the statement about the concept being a brother, viz., “being a brother entails being male,” is a statement of the same sort as the statement about the expression “brother,” only it says much more of the same sort. This additional content can be expressed as a conjunction of conditionals of the form “If E is synonymous with E1 and D is synonymous with E2 , then E is inapplicable to x unless D is applicable to x.” Yet, conditionals of this form are surely analytic: it is surely self-contradictory to suppose that there existed two expressions E and D which are respectively synonymous with “brother” and “male,” but are such that E is correctly applicable to something to which D is not correctly applicable. But a conjunction of a contingent statement and an analytic statement asserts no more than the contingent statement alone.4 Therefore Strawson has utterly failed to analyze the diﬀerence between the statement about “brother” and the statement about the concept being a brother. He might reply that on his theory the concept of synonymy is not used in the meta-linguistic translation of “being a brother entails being male”; that the latter is to be translated into a conjunction of categorical statements about the uses of synonyms of “brother” and “male,” such statements as “ ‘Bruder’ is not correctly applicable to x unless ‘m¨annlich’ is correctly applicable to x.” But this analysis is even

4 Perhaps

it would be more accurate to say that the conditionals in question are, not analytic in themselves, but analytically entailed by the statement about the English expressions “E1 is not correctly applicable to x unless E2 is applicable to x.” But the same conclusion would follow: if p analytically entails q, then (p and q) has no more factual content than p alone.

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less plausible, since it entails that only a person who knows something about all languages could know an entailment-statement to be true. That necessary propositions are necessarily necessary is particularly evident in the case of tautologies of propositional logic. Consider, e.g., “[(A ⊃ B). ∼ B] ⊃ ∼ A” (F), where “A” and “B” abbreviate definite statements substitutable for the variables “p” and “q” of the propositional calculus. The same method of analysis of logical ranges, the “truth-table” method, which assures us of the truth of this statement though we may be ignorant of the truth-values of the component statements, also assures us of its necessary truth, its truth in all “possible worlds.” For the unbroken column of T’s under the major operator signifies not only that this statement is true but also that any statement of the same form is true. Hence, if the possibility of establishing the truth of a statement without appeal to empirical evidence marks it as necessary, then not only F but likewise “It is necessary that F” must be held to be necessary. If this attempt to show that modal statements like H are, as maintained by Lewy, not contingent, has been successful, then we must conclude that Lewy’s principle is invalid and therefore does not enable us to distinguish the relations of entailment and strict implication. Perhaps, however, the distinction can be drawn in some other way. Let us try the following suggestion: an entailment between p and q is decidable, if it is decidable at all, without knowledge of the modalities of p and q. Notice that this condition may be satisfied even though one or both of the propositions standing in the entailment-relation are modal propositions: just as (p.q) entails p regardless of whether (p.q) is a contingent, necessary, or impossible proposition, so 2(p) entails p regardless of whether 2(p) is a contingent, necessary, or impossible proposition; similarly, we can say with certainty that p entails that p is possible (where “possible,” of course, is so used that “p is possible” is compatible with “p is true,” “p is false,” and “p is necessary”) prior to having settled the question whether or not “p is possible” is contingent. Now, “p strictly implies q” is defined as “It is impossible that p and ∼ q.” From this definition it follows that (p and ∼ p) strictly implies q, whatever propositions p and q may be, because (p and ∼ p) is impossible, and the conjunction of an impossible proposition with any other proposition is of course likewise impossible. It seems, therefore, that the condition above stipulated for entailment is not satisfied by the “paradoxical” strict implications: we know that q is entailed by (p and ∼ p) because we know that (p and ∼ p) is impossible; and we know that [∼ (p and ∼ p)] is entailed by q, because we know that [∼ (p and ∼ p)] is necessary. Yet, this attempt to diﬀerentiate entailment from strict implication overlooks that the “paradoxes” of strict implication do not depend on the stated explicit

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definition of strict implication5 at all, but can be derived within a system of strict implication in which this concept is but implicitly defined, by a set of axiomatic strict implications, just as well. As was shown by Lewis himself, the following strict implications (supplemented by the rule of substitution and the ponendo ponens rule) suﬃce for the derivation: (p.q) strictly implies p; p strictly implies (p∨q); [(p∨q). ∼ p] strictly implies q. These strict implications, however, satisfy the condition imposed upon entailment: that no assumptions about the modalities of the terms of the entailment are required to see that the entailment holds. I conclude that it is obscure what the alleged distinction between strict implication and entailment is. Perhaps, however, the feeling that there is a distinction may be traced to the following origin: those strict implications which are “paradoxical” are inferentially useless. If the antecedent of a strict implication is impossible, we cannot use the implication as a basis for proving the consequence, since the antecedent is not assertible; and if the consequent is necessary, the implication is useless as a rule of inference simply because no premise is required for the assertion of the consequent (it is “unconditionally” assertible). But if it is only by reference to inferential utility that strict implication is distinguishable from entailment, then there is no basis for saying that these are distinct logical relations; for the concept of inference, and any concept defined in terms of it, is of course no logical concept at all.

5 Alternatively, “p strictly implies q” could be defined as “2(p ⊃ q)”; if the definition of “2(p)” as “∼ 3 ∼ p” is added, the above definition turns into a theorem about strict implication.

Chapter 12 MATHEMATICS, ABSTRACT ENTITIES, AND MODERN SEMANTICS (1957)

A science can be in a highly advanced state, even though its logical foundations are far from being clarified. Mechanics, for example, reached a stage of astounding perfection by the end of the 18th century, mainly through the genius of Galileo and Newton, although much room was left for controversy about the meanings of its fundamental concepts: length, simultaneity, mass, and force. Even the meaning of the simple law of inertia remained controversial right up to Einstein’s “unification” of inertia and gravitation. Similarly, mathematics was not prevented from reaching breathtaking heights of perfection by the Grundlagenstreit (dispute about foundations) which began in the 19th century and still continues. The fundamental question of the philosophy of mathematics concerns the very nature of its subject matter. Traditionally, mathematics was defined as the science of quantity, but this definition was decisively criticized by Bertrand Russell in terms of a conception which reduces pure, abstract mathematics to pure, abstract logic, thereby lifting any restriction to a special subject matter. But though pure mathematics in this conception is, in contrast to the special empirical sciences, unrestricted in subject matter, its propositions are analyzed as referring to classes and attributes. In probing into the foundations of mathematics, therefore, one cannot avoid facing, sooner or later, the old metaphysical problem of the status of abstract entities. The latter has been tied up with the problems of linguistic meaning and reference ever since Plato but never before as closely as at the present time. Modern semantics, as cultivated by analytic philosophers in the United States and England, grew up in close contact with symbolic logic. Accordingly, the problem to be discussed in this article should be of equal concern to mathematicians who reflect on the foundations of their science, to semanticists, and to symbolic logicians.

1.

Traditional Problem of Universals

When a high-school teacher demonstrates that the sum of the interior angles of a triangle equals 180◦ , he usually draws a triangle on the blackboard, then

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draws a straight line through the top vertex parallel to the base, in order to be able to reduce the proof of the Euclidean theorem to the proposition that “corresponding” angles are equal. How would he satisfy a “philosophic” pupil who protested that he had not established that all triangles whatsoever have the property in question but only that the particular triangle on the blackboard had it? Obviously, the proper reply would be this: We can be perfectly sure that every Euclidean triangle has the property, without illustrating the proof over and over again by diﬀerent triangles, because any properties of the particular triangle that distinguish it from other triangles—such as its being equilateral, or its comparatively small size, or its being bounded by white lines—were disregarded in the proof. In the language of Plato: What the mathematician is thinking about when he discovers the geometric proof is not a particular triangle but triangularity as such—in other words, that which all particular triangles have in common and by virtue of which they are triangles. It is this sort of thing that various philosophers, past and present, under Plato’s influence call a “universal,” or an “intelligible form,” or an “essence.” These entities were supposed by both Plato and Aristotle to be the objects of scientific thought, as contrasted with “particulars” that are given in senseexperience with a unique location in space and time. Of course, these universals did not have to be geometric forms. In the case of every predicate whatsoever, the things to which the predicate is applicable can be distinguished from the property the possession of which by a particular thing is the criterion of the predicate’s applicability: redness is to be distinguished from red things, hardness from hard things, humanity from particular human beings, and so on. Again, numbers seemed to be a kind of universals, for, to ask a question which unphilosophic people never ask at all: What is the number 2? It surely is not the symbol “2,” for we can use diﬀerent symbols to talk about the same number (for example, “II,” “two,” “zwei”). It must, then, be something that the symbol stands for. But it cannot be identified with a particular pair of objects, say a pair of gloves, or a pair of apples, or a pair consisting of wife and husband. It seems, rather, to be something that all particular pairs have in common (“twoness”), an object of abstract thought, not a sensible object. Once universals are conceived as objects of thought, the realm of universals will be found to be populated not only with those that are somehow “exemplified” in the particulars we sense but also with unexemplified universals: no physical sphere corresponds exactly to the mathematical concept of a sphere; hence, the universal sphericity which the mathematician thinks about is not, strictly speaking, exemplified in the physical world. And even if one never physically constructed a regular polygon with 1000 sides, this mathematical object could be reasoned about, and in order that it may be reasoned about it must in some sense “exist,” say the Platonists. In the heaven of Platonic forms, then, we find not only perfect mathematical objects but also unexemplified forms that move the imagination of less intellectual people: mermaidhood, centaurhood, unicornhood, and so on. The Platonists’ argument seems to be

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that, in order that something may be thought of, it must in some sense be; we can obviously think of mermaids, though there are no physical mermaids in space and time, but we could not have such thoughts unless there existed the universal mermaidhood. Since these universals do not exist in space and time, some Platonists try to avoid confusion by calling the sort of “being” that is to be ascribed to them “subsistence.”1 It is important to understand that universals as traditionally conceived, especially by Platonists, are not mental in the sense in which a thought or an act of imagination or a feeling is mental. They are, rather, postulated objects of thought whose subsistence is alleged to be independent of their being mentally apprehended in any way. It is for this reason that Platonism is usually called a kind of “realism.” Universals are held to be real in the sense that there would be such entities even if no consciousness existed, just as a physical realist holds that physical objects are there, whether or not any conscious organism or disembodied mind is aware of them. The most important traditional arguments, then, for there being universals over and above the world of sensible particulars in space and time are the following two. (i) With the exception of proper names, all meaningful words are general. To say that they are meaningful is to say that they stand for something. But, unlike proper names, words like horse, round, and red do not stand for particular things. Therefore they stand for universals. And these universals would be “there” somehow, even if they were never referred to by means of words. (ii) Every thought has an object; to think is essentially to think of something, and the object of one’s thought is clearly distinct from the thinking of it. When two mathematicians, for example, converse with each other about the properties of a certain number, they assume without hesitation that they are thinking about the same object and that they are merely trying to discover the properties of that object, in the same sense that a chemist is trying to discover the properties of a compound he is experimenting with, though by essentially diﬀerent methods. The square root of two would have been irrational even if no mind had ever thought about it, let alone discovered its irrationality. This second argument is much more tempting than the first. As we shall see presently, the first argument is rather easily disposed of by exposing the underlying confusion between meaningful words and names. But the second argument is a natural outgrowth of a psychological situation with which even non-mathematicians are familiar. Imagine a child who has just learned the meaning of “square number” by means of the examples 4 = 2×2, 9 = 3×3, and curiously investigates how many square numbers there are between 1 and 100 and whether their frequency increases or decreases as the numbers increase.

1 See,

for example, Montague 1958, chapter 4; Russell 1912, chapter 9.

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He seems to explore a domain of objects that have invariable properties in no way dependent on human thoughts, and yet these objects are not physical objects with spatial location. The child may at first suppose himself to be talking about marks on paper when he says “9 is a square number but 10 is not,” but you can easily convince him that these are merely symbols we use for referring to invisible numbers: the sentences “9 is a square number” and “nine is a square number” refer to the same number, but the symbols “9” and “nine” are obviously diﬀerent.

2.

Modern Semantics and the Traditional Dispute

Many traditional philosophers, especially Occam and his followers in medieval philosophy and the British empiricists since Hobbes, have tried to discredit Platonism as a belief that has no better rational foundation than a belief in ghosts and fairies. Whatever exists, said the nominalists, is a particular in space and time, and there is no need to postulate any “subsistent” abstract entities. A general word does not stand for a universal but is just an economic device for referring to several particular things that are in certain respects similar. According to the realists, a meaningful general word, like man, stands for a universal, and when we judge that a particular object is correctly called “man,” we compare it, as it were, to our idea of the universal. But the nominalists denied that we can form ideas of universals—“abstract ideas”—at all. To quote the subtlest of them, George Berkeley: it is thought that every name has, or ought to have, one only precise and settled signification, which inclines men to think there are certain abstract, determinate ideas that constitute the true and only immediate signification of each general name; and that it is by the mediation of these abstract ideas that a general name comes to signify any particular thing. Whereas, in truth, there is no such thing as one precise and definite signification annexed to any general name, they all signifying indiﬀerently a great number of particular ideas. (Berkeley 1710, sec. 18)

This is a striking anticipation of modern semantic analysis. Berkeley is criticizing the scholastic belief, ultimately derived from Plato, that the generality of a word consists in its representing a determinate entity which is not a particular and of which we have an “abstract” idea. Rather, he held, the generality of a word is its capacity to evoke any one of a set of similar particular ideas. If I apply the general word triangle to a particular figure, I do not do so as a result of finding the particular figure to correspond to an abstract idea of triangularity—there is no such thing—but simply because I recognize it as similar in a certain respect to objects with which I have been conditioned to associate the word triangle. Berkeley might have said, as some contemporary analytic philosophers have said explicitly, that the meaningfulness of a word does not presuppose the existence of an entity which is what the word means.

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That the word triangle is unambiguous in the context of geometric discourse does not entail that it functions in that context as a name of a unique entity, call it a “universal.” One chief source of Platonism, according to this point of view, is the naive and mainly unconscious assumption that every meaningful word is a name of something. But the verb “to mean” is, in the relevant respect, more like the verbs “to dream” and “to wish” than like the verbs “to name” and “to eat”; there must be something that can be named in order that an act of naming may take place, and there must be something that can be eaten in order that a process of eating may occur. But would it not be grotesque if I argued to the existence of green dogs that can sing Verdi arias from the fact that I dreamed of such an animal, or the existence of a woman who is my wife and combines more virtues than any other woman from the fact that I wish for such a wife? In grammatical terminology, this criticism of Platonism accuses it of construing “to mean” as a transitive verb, whereas this verb is obviously intransitive. The chief philosophic motive behind Bertrand Russell’s theory of descriptions.2 was to expose this confusion and thereby to inhibit the tendency to postulate entities that do not exist in the physical universe. To borrow Russell’s own example, consider the sentence “the present king of France is bald.” On a crude level of analysis one might say that, since the sentence is meaningful (all constituent words are meaningful and are arranged in a syntactically correct way), it must be about something. And what is it about if not the present king of France? But there is no present king of France! Is it, then, about a Platonic essence or Idea that happens to be unexemplified in the physical world? Russell’s answer was that no such metaphysical postulations are required to account for the significance of the sentence. We have only to distinguish between contextual meaning and denotation: A definite description, like “the present king of France,” has meaning in context but does not denote anything at all. It has meaning in the context of sentences that contain it, and this meaning can be explained by means of a synonymous sentence in which no definite descriptions occur: “there is one and only one individual which is present king of France, and that individual is bald.”3 Once meaning and denotation are thus distinguished—a symbol may be meaningful (in context) without denoting anything—the argument for the existence of universals from the meaningfulness of predicates collapses, for a predicate can be meaningful in the context of entire sentences without being a name of anything. This, in fact, is a fundamental tenet of contemporary semantic nominalism (I call this philosophic school “semantic” nominalism to distinguish it from metaphysical nominalism, because its members tend to avoid such metaphys-

2 It

was first formulated in Russell 1905. See also Russell 1950, chapter 16. course, “France” is, in turn, an abbreviation for some definite description, but Russell believed that by repeated application of his rule of translation, any definite description could eventually be eliminated.

3 Of

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ical locutions as, “only particular things or events in space and time are real, universals are abstractions that have no reality”): to this school, predicates are contextually meaningful, and to explain the meaning of a predicate is to formulate a rule for applying the predicate to particular objects, but predicates are not names of anything. The predicate “blue” is not related to its meaning in the way that the name “John” is related to John. Indeed, it is already misleading, the semantic nominalists urge, to speak of the meaning of “blue,” since this expression suggests an elusive entity. It is preferable to speak of the rule governing the use of “blue,” and to give this rule is to formulate the conditions under which it is correct to apply “blue” to a thing. In the case of “blue,” the rule of usage cannot be formulated without pointing at particulars. In this respect the rule of usage for, say, “mermaid” is, of course, of a diﬀerent sort, since we can be said to understand what “mermaid” means, although we have never encountered any mermaids. But, the semantic nominalist insists, to grasp this meaning is not to be face to face, as it were, with a universal, call it mermaidhood, but just to know under what conditions it would be correct to apply the predicate “is a mermaid” to an object.4 An important application of the theory of contextual meaning to mathematics arises if one reflects on the meaning of infinitesimals, expressed by “dx,” “dy,” and so on. Mathematicians used infinitesimals successfully long before they were clear about their meaning. At first glance there is an air of hocus-pocus about infinitesimals: how can a genuine quantity be neither zero—insofar as an infinite sum of infinitesimals equals a finite quantity—nor finite—insofar as a finite sum of infinitesimals does not equal a finite quantity? And how can the ratio of dy (where y is a function of x) to dx equal a finite number if neither dx nor dy are finite numbers? The diﬀerential quotients themselves seemed perfectly legitimate, since they admitted of geometric interpretation, namely, as measures of the slopes of tangents. But what troubled both mathematicians, who used the device of infinitesimals without being able to justify it “philosophically” (as contrasted with “pragmatic” justification: it works!), and philosophers, who were critical of the calculus (Berkeley, for example), was that the components of the diﬀerential quotient had no intelligible meaning. The theory of contextual meaning, however, justifies the use of infinitesimals as “incomplete symbols,” in Russell’s phrase—that is, as symbols which do not denote any funny kinds of numbers but have meaning in context. dy = c” means that the limit approached by a seAn equation of the form “ dx

4 A programmatic formulation of this semantic theory, which derives from L. Wittgenstein and M. Schlick, is

as follows: the meaning of an expression consists in the rules for its use. Wittgenstein, a dominant member of the “Vienna Circle” (the founders of logical positivism see Kraft 1953), exerted a powerful influence on English analytic philosophy. For a lucid application of Wittgensteinian semantics to the problem of universals, see Lazerowitz 1946.

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∆y quence of diﬀerence quotients ∆x , as ∆x becomes smaller and smaller, equals c, and the meaning of “limit” can be explained without postulating any numbers other than finite ones. When one has explained the meanings of equations that involve diﬀerential quotients, both such as equate them to a constant and such as express them as a function of a variable, one has explained the contextual meaning of diﬀerential quotients and of their components.

3.

Classes, Attributes, and the Logical Analysis of Mathematics

It was stated in the beginning of this article that it is diﬃcult to avoid Platonism if one becomes philosophic at all and asks what a number, as distinct from a numeral, is. It seems that a number is either a universal—that is, a common property of classes of discrete objects that can be set into one-one correspondence—or else a class of such “similar” classes. The number two in particular, then, is revealed as either the universal “twoness” or the class of all pairs. Although classes are, for good or for bad reasons, somehow more acceptable to nominalists than are universals (or attributes, as we shall henceforth call them), they are likewise abstract entities. By “the class of all men,” for example, a logician does not mean a group of men that could possibly be seen but the conceptually apprehended totality of men past, present, and future.5 Can these Platonic temptations be overcome by means of the theory of contextual meaning? That is, can it be maintained that numerals have only contextual meaning and do not denote any entities at all? Now, contextual definitions of the natural numbers in terms of logical constants and non-numerical variables can be constructed, as was shown by Frege and Russell6 independently (although Frege has historical priority and influenced Russell’s thinking on the foundations of arithmetic); and these contextual definitions clarify, in a very important way, the language of applied arithmetic. Let us begin with the first natural number, zero. What is meant by a statement like “the number of French judges in the Supreme Court is zero”? Obviously, it means that there are no such persons, or that the class of such persons

5 There

is, incidentally, a neat proof that classes are diﬀerent from wholes, whether the parts of the whole be spatially or temporally contiguous or discrete. If x is part of a whole y and y is part of a larger whole z, then x is part of z. But if x is a member of class y and y is a member of class z, then x is not a member of z. For example, John is a member of the class of men, the latter is a member of the class of classes of living organisms, but John is not a member of the latter class, since he is not a class. Furthermore, the same whole may be conceptualized as diﬀerent classes: we can consider an organic body, for example, as a composite of cells but also as a composite of molecules. We then have a composite of cells that is identical with a composite of molecules; but a class of molecules cannot be identical with a class of cells, for, since cells are diﬀerent from molecules, nothing can be a member of both classes. 6 For an elementary exposition of “logicism” (as the reduction of mathematics to logic, attempted by Frege and Russell, is called), see Hempel 1983.

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is empty. Here, “no,” “there are,” and whatever symbol is used to denote membership in a class are logical constants. Note that zero, just like the larger numbers, is predicable of classes, not of individuals; that is, statements of the form “there is nothing of the kind K” are meaningful, but statements of the form “this thing is not” or “this thing is nothing” are not. Next, consider the number one. Again, we cannot predicate unity in the arithmetical sense of an individual: “the sun is one” makes no sense; “there is one (and only one) sun” makes sense. That is, we might say that the class of solar objects has exactly one member, and this means, according to the FregeRussell analysis, that there is an x such that x is a member of the class of solar objects, and for any y, if y is a member of the class of solar objects, then y is identical with x. Similarly, “there are two things of kind K” means that there is an x and a y, y distinct from x, such that x and y are of kind K, and for any z, if z is of kind K, then z is identical with x or with y, and so on for the larger numbers. It should be noted that the contextual definition of the arithmetical predicate of classes “one” is not circular, because “there is an x” does not have to be read as “there is at least one x” but may be read as “it is not the case that there is no x.” Are these definitions correct? Do they formulate what we mean by arithmetical symbols in the context of applied arithmetic? This question cannot be answered unless criteria are specified which must be satisfied by correct definitions. Now, an important criterion of correctness which these contextual definitions can be shown to satisfy is that the equations of arithmetic which are actually used in deducing numerical statements about empirical subject matters from other numerical statements should be logically demonstrable. For example, we would expect from adequate contextual definitions of “2,” “3,” and “5” that they make possible a formal proof (that is, a proof based on nothing but laws of logic) of the proposition “if a person has 2 daughters and 3 sons, then he (or she) has 5 children.” It may be thought that this proposition of applied arithmetic is just a special case of the general equation “2+3=5” and that there is no further problem, since the equation itself is simply “true by definition.” But insofar as the equation is just a provable formula in an uninterpreted system of arithmetic, its constituent arithmetical symbols do not have the sort of meaning that is presupposed by empirical applicability. In such a system (a “formal calculus” in the terminology of the logicians) 1 is defined as the successor of 0, 2, as the successor of 1, 3, as the successor of 2, and so on, where 0 and successor are primitive terms. These primitives also enter into the recursive definition of +: x + y = (x + y) ; x + 0 = x (here, “. . . ” is short for “the successor of . . . ”). In terms of these definitions, and of the rule that equals are interchangeable, it is indeed possible to transform 2+3, step by step, into 5, but as long as the primitives remain uninterpreted, the defined symbols likewise are without interpretation. The definition 2=1 , which occurs within the

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formal calculus, does not enable one to decide whether a given class has two or more or fewer members; indeed, there is nothing within the formal calculus itself to suggest even that numerals designate properties of classes that can be ascertained by counting. We require, then, an interpretation of the arithmetical primitives in terms of an understood vocabulary before we can even significantly raise the question of whether the equations are true. And if we want to justify the use of 2+3=5 as a rule for deducing “x has 5 children” from “x has 2 sons and 3 daughters” (with the additional help of the definition or theorem “x is a child of y = x is a son or daughter of y”), we have to assign to the numerals just those meanings which they have in such empirical contexts as “x has two sons.” This is precisely what the Russellian contextual definitions in terms of logical constants accomplish; they at once clarify the meanings of empirical statements in which numbers are applied and justify the belief that “x has 5 children” logically follows from “x has 2 sons and 3 daughters.” However, as both Frege and Russell clearly saw, such contextual definitions are insuﬃcient for the elucidation of numerical symbols. In eﬀect, what has been defined so far is only the statement form “class A has n members.” But we still do not know what 2, for example, means in the context “2 is a prime number.” Here, 2 does not occur as predicate but as subject. This is the kind of consideration that led Frege and Russell to ask what the number 2 itself is. Formally speaking, in the context “2 is a prime number,” the symbol “2” is a name, not an incomplete symbol. What is it the name of? Two alternative answers may be considered: (i) the class of all classes with two members, (ii) the attribute of being a class with two members. Answer (i) is a special case of Russell’s general definition of numbers as classes of similar classes. Two classes are said to be similar if there is a one-one relation which relates every member of one class to a member of the other class, and a relation R is one-one if at most one entity has R to a given entity and a given entity has R to at most one entity (for example, being the immediate successor of; being the positive square root of; being the wife of, in a monogamous society). It is natural to think of a number as the common property of all mutually similar classes. But this definition is open to the objection that there is no guarantee that a set of similar classes have just one property in common.7 For example, if all individuals have a certain property f in common, then all possible pairs that could be formed out of them would have the common property “containing as members individuals with property f .” By the foregoing definition, therefore, the number 2 would fail to be unique. On the other hand, the

7 See

Russell 1938, §110.

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class of all pair-classes is necessarily unique, and this is why Russell preferred to define numbers as classes of similar classes. Nevertheless, there is at least one reason why it may be better to regard numbers as special kinds of attributes, in the sense of definition (ii). The formal diﬀerence between classes and attributes is that attributes which apply to exactly the same entities may still be distinct, whereas classes with the same membership are identical. Thus, we can conceive of a universe in which all round things are blue and all blue things are round. If such were the world, the class of blue things would be identical with the class of round things, yet the attributes blueness and roundness, the one a color the other a shape, would remain perfectly distinct. Or, to illustrate by a mathematical example, the class of natural numbers between 1 and 3 is the same as the class of even primes, but the corresponding attributes are clearly diﬀerent; we describe the number two diﬀerently if we describe it as the natural number that is less than 3 and greater than 1, or as the only even number that is a prime number. Again, many distinct attributes are not possessed by anything at all: being a unicorn, being a blue dog that can speak French, being a golden mountain, and so forth. But they all correspond to the same class, the null class. There cannot be more than one null class, for if there were two, one would have to have a member which the other does not have, and this contradicts the hypothesis that neither has any members. Now, to come closer to the defect of definition (i), which is of interest here, the arithmetical meaning of the term successor (as used, for example, in the definition of 1 as the successor of 0) is such that necessarily distinct numbers have distinct successors. This, in fact, is one of the five postulates upon which the Italian mathematician and logician Peano erected the arithmetic of natural numbers. And one of Russell’s avowed aims in “logicizing” the concepts of arithmetic was to so define Peano’s primitives (“zero,” “successor,” “natural number”) that all the postulates (and therewith the theorems) of uninterpreted arithmetic turn into logically necessary propositions—propositions that are formally deducible from the purely logical axioms of Principia Mathematica8 (chapter 1-3).9 Russell noticed, however, with considerable intellectual discomfort, that his definition of natural numbers as classes of similar classes of individuals does not make Peano’s axiom that distinct numbers have distinct successors logically necessary. For suppose (what is conceivable without self-contradiction) that the number of individuals10 in the universe were some finite number n. Then the number n+1, defined as the class of all classes with 8 See

Russell 1950. Hempel 1983. 10 The concept “individual” is but negatively defined in Whitehead and Russell 1925: anything that is neither a class nor an attribute nor a proposition. The question is thus left open whether the individuals are observable things, or postulated particles, or physical events, or whatnot. 9 See

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n+1 members, would be the null class, since there would be no classes with n+1 members. For the same reason, n+2 would be the null class. Hence, we would get n+1=n+2, although n, being a nonempty class, is distinct from n+1. For reasons connected with the theory of types (a theory devised as a solution of logical paradoxes, which we cannot and need not explain here), Russell saw no other escape from this diﬃculty than to postulate that the number of individuals is not finite (“axiom of infinity”; see Russell 1950, chapter 13). Now, we have seen that n+1=n+2 follows from the assumption that the number of individuals is n, together with the Russellian definition of numbers as classes of similar classes. But if the number n+1 is, instead, defined as the attribute (of a class) of having n+1 members, Russell’s conclusion does not follow. For though, on that assumption, both attributes would be empty (inapplicable), they would remain just as distinct as, say, the attributes of being a mermaid and of being a golden mountain, and for just the same reason: they are defined as incompatible attributes. For example, the expressions “A has two members” and “A has three members” are so defined in Principia Mathematica (see the foregoing sample contextual definitions) that it would be contradictory to suppose that the same class had both (exactly) two and (exactly) three members. Therefore, even if just one individual existed, the number 2 would retain its conceptual distinctness from the number 3; and this argument, obviously, can be generalized for any finite number n and its successor. Apart from the question of the axiom of infinity, it is, within the framework of the logical analysis of mathematics contained in Whitehead and Russell’s monumental Principia Mathematica, unimportant whether numbers be conceived as classes (of classes) or as attributes (of classes), for names of classes are, in either case, contextually eliminable, according to Russell’s theory of classes as “incomplete symbols.” According to this theory, a statement that ascribes an attribute F to a class that is defined as the totality of entities that have attribute G is analyzed as follows: there is a predicative function θ such that θ is formally equivalent to G and such that θ has F.11 In this context, function means, not numerical function, but propositional function, and for the present purposes propositional functions may be identified with attributes. Two attributes are “formally equivalent” if they apply to the same entities, and a predicative function is an attribute that does not presuppose, in its definition, a totality of attributes. For a full understanding of this theory of classes, familiarity with the theory of types, especially the “ramified” theory with its distinction between the type and the order of a function, would be required.12 But for the present discussion,

11 See

Whitehead and Russell 1925, 71f. Whitehead and Russell 1925, Introduction and chapter 2. More easily intelligible expositions of the ramified theory of types are Lewis and Langford 1951, chapter 13, and Copi 1954, appendix B.

12 See

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all that matters is that in the “primitive” notation of Principia Mathematica, no names of classes but, instead, variables ranging over attributes (propositional functions) occur. For a semantic nominalist holds that the language of mathematics involves an obscure Platonism as long as there are in its primitive notation either names of, or variables ranging over, abstract entities; and he regards attributes as abstract entities.13 Take, for example, the simple statement that two individuals, x and y, diﬀer in at least two respects. This means that there is an attribute f and an attribute g such that f is not equal to g and such that x has f and y does not have f and x has g and y does not have g, or y has f and x does not have f and x has g and y does not have g, and so on. But here f and g are variables ranging over attributes; therefore such an analysis is unacceptable to a semantic nominalist. Now consider the more complicated statement: 2 + 2 = 4. The logical analysis performed by Frege and Russell shows that we do not have to regard the symbols “2” and “4” as names of mysterious entities. Even the names “the class of all classes with two members” and “the class of all classes with four members” are, in principle, eliminable. But what cannot be got rid of, apparently, are variables ranging over attributes. For if, in conformity with Russell’s view that classes are logical fictions, not real entities, we define numbers as certain kinds of attributes of attributes (in the sense in which “being a color,” for example, is an attribute of attributes, namely, of blueness, redness, and so on), we obtain the following analysis of 2 + 2 = 4: for any attributes f , g, h, if just two individuals have f and just two individuals have g, and nothing has both f and g, and if an individual has h if and only if it has either f or g, then just four individuals have h (illustration: f = being a coin in my left pocket at time t0 , g = being a coin in my right pocket at time t0 , h = being a coin in one or the other of my pockets at t0 ). Is mathematics, then, irremediably Platonistic? Or is there some way of eliminating attribute variables so as to satisfy the demands of semantic nominalism? The issue is a highly technical one, but the following suggestion should be intelligible to those without any knowledge of symbolic logic. As was explained in earlier paragraphs, the nominalist insists that a predicate can be meaningful without being the name of anything. For this reason he refuses to transcribe (i) “x is red” into (ii) “x has the attribute redness.” From (ii) we may deduce (iii) “there is an attribute f such that x has f ,” but since “is red” has meaning only in context, it is not a name of a value of a variable. The deduction of (iii) from (i) is as legitimate as the deduction of (iv) “there is something which I am now thinking of” from (v) “I am now thinking of a unicorn.” The

13 What

I call “semantic nominalism” has been formulated by W. V. Quine, in Quine 1949 and in Quine 1947a. See especially Quine 1953a, chapters 1 and 6. The program of nominalistic reconstruction of mathematics is sketched by W. V. Quine and N. Goodman in Goodman and Quine 1947.

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transition from (v) to (iv) is tempting because it is similar to the perfectly valid deduction of (vi) “there is something which I am now eating” from (vii) “I am now eating an apple.” But, whereas we could not eat apples if there were no apples, we can think of unicorns in spite of there not being any. If we allow the deduction of (iv) from (v), we get entangled in a verbal contradiction, since the something which (iv) says I am thinking of is, of course, a unicorn, and then there must be unicorns after all! To resolve the contradiction, the Platonist may say that the object of my thought is the attribute of being a unicorn, not a concrete unicorn. The nominalist, on the other hand, criticizes the translation from (v) to (iv) on the ground that “unicorn” is not a name but a predicate (or part of the predicate “is a unicorn”). Correctly analyzed, (v), unlike (vii), does not assert a relationship between two independently existing entities but simply characterizes my thought as what we might call a “unicornish” thought. Similarly, “this is a picture of a unicorn” characterizes the picture as unicornish; it does not mean that this has the relation being a picture of to an independently existing entity—in the way that this is the meaning of “this is a picture of my son.” Bearing in mind the distinction between predicates (which may not be replaced by variables ranging over some kind of entities) and names, let us return to the logical analysis of 2 + 2 = 4. To satisfy the nominalist, we must replace “have f ,” where f represents a name of an attribute, with “are f .” But then f , g, and h become letters that represent predicates, not names of attributes; hence, they are not variables over which we may “quantify,” as the logicians say. The prefix (quantifier) “for any attributes f , g, h” must, accordingly, be cut oﬀ, for the letters f , g, and h do not represent names of attributes at all. What, then, is the law of arithmetic 2 + 2 = 4 in logical and, at the same time, nominalistic interpretation? The surprising answer is that there is not one such law, there is not a sentence containing, besides variables, only logical constants which we can point to and say “this sentence expresses the arithmetical law that 2 and 2 make 4,” for “if just two individuals are f and just two individuals are g ...” is a schema, not a statement; it does not express a proposition at all. In order to obtain propositions out of it, we must substitute specific predicates for the schematic letters in it, and there are as many such propositions as there are triplets of predicates applicable to individuals. If the language of arithmetic is reconstructed nominalistically, then, no laws of pure arithmetic can be formulated in the object-language—that is, the language which, whatever it may refer to, does not refer to symbols. Instead, one has to formulate in the meta-language (the language that is used to talk about the object-language) statements to the eﬀect that any substitution instance of such and such a schema is logically true. This conclusion, of course, applies not only to the laws of arithmetic and of higher mathematics but also to the basic laws of logic. For example, consider

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What do the Ontological Questions Mean?

the law of medieval logic, “what is true of all is true of any,” which, in symbolic logic, reads as follows: for any y, if all x’s have attribute f , then y has f . A Platonistic logician can refer to one logical truth expressed by the sentence: for any attribute f and for any individual y, if all x’s have f , then y has f . A nominalistic logician must ascend to the meta-language and there declare that every statement of the form “for any y, if all x’s are f , then y is f ” is logically true.14

4.

What Do the Ontological Questions Mean?

To what extent a nominalistic reconstruction of mathematics is feasible is certainly an interesting technical problem that deserves investigation in its own right. But attempts at such reconstruction are usually motivated by a philosophic prejudice about what “really exists.” Being asked why he wants to get rid of variables ranging over attributes, the nominalist is likely to reply that he cannot admit the existence of attributes because the very notion of “attribute” is obscure to him. But even if the notion could be clarified to his satisfaction, he would probably remain reluctant to admit the existence of such entities. Thus, nominalists usually concede that the notion of “class” is quite precise, and yet they cannot countenance classes as “real entities.” The main argument in support of the accusation that the meaning of the word attribute (and of its synonyms in philosophic literature: universal, property, intension of a predicate) is obscure is that it is not clear under which conditions two expressions can be said to designate the same attribute.15 For example, is the attribute (of a number) of being equal to the product by itself the same as the attribute of being the successor of zero? Is the attribute of being a rectilinear closed figure with three sides the same as the attribute of being a rectilinear closed figure with three interior angles? We have here predicates which are interdeducible (logically equivalent), but it seems debatable whether they are synonymous. And the question of the criterion of synonymy is, indeed, diﬃcult and highly controversial in contemporary semantics.16 The same nominalists who frown on attributes find it easier to accept talk about classes precisely

14 Subtle

problems emerge once one asks whether, granted that the object-languages of logical and mathematical systems can be constructed in accordance with nominalistic restrictions, the meta-language also could be nominalistic. For example, surely we mean to assert the logical truth of all possible substitution instances of a given schema, not just of those that are actually written down somewhere and at some time. But it is even doubtful that the method described in the text makes possible a nominalistic rewriting of mathematical object-languages, for quantifiers that refer to attribute variables within the scope of another quantifier cannot be eliminated by such a simple device. The statement “there is no largest number” illustrates this complication if numbers are construed as attributes of attributes: for any attribute f , if f is a number-attribute of an attribute, then there is a number-attribute (of an attribute) g such that g is larger than f. 15 This argument has been used chiefly by W. V. Quine. See Quine 1951 and Quine 1947b. 16 See Goodman 1952b, Mates 1952, Naess 1949, Frege 1949, Carnap 1955, Quine 1951.

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because the condition of identity of classes is clear and uncontroversial: A and B are the same class if, and only if, they have the same members. Now, this particular argument against “Platonistic” discourse seems to me highly illogical. If “attribute” is held to be obscure on the ground that the identity condition for attributes is not clear, then one would expect that all is well with the identity condition for individuals—that is, those concrete entities, whatever they may be (material particles, observable things, qualitatively distinguishable events, space-time points), which alone really exist according to nominalist ontology. Yet, the identity condition for individuals that is usually accepted is Leibniz’s principle of the “identity of indiscernibles,” which in modern logic is formulated thus: x = y if, and only if, for every attribute f , x has f if and only if y has f . Clearly, if such is the definition of identity of individuals, and a given kind of entity is deemed obscure to the degree that its identity condition is not clear, then the alleged obscurity of “attribute” must infect the term individual too. Two replies are open to the nominalist: (i) Since we realize that an explicit definition of identity requires quantification over an attribute variable, we do not explicitly define this logical constant at all. Instead, we lay down an axiom schema for identity which does the same job as the explicit definition. It says, in eﬀect, that identicals may be substituted for each other without change of truth-value of the sentence into which the substitution is made, and contains a schematic predicate letter, not a quantifiable attribute variable: if F x, and x = y, then Fy. (Note that, in accordance with the explanation given earlier, a single axiom or definition is thus replaced by a family of axioms of the same form, namely, those that result from the substitution of a predicate for the schematic letter F). (ii) Identity is explicitly definable with the help of a class variable; hence, the intelligibility of “individual” exceeds that of “attribute” by at least as much as the latter is exceeded by the intelligibility of “class”: x = y, if and only if x and y are members of just the same classes of individuals. My rejoinder to (i) is that an exactly similar axiom schema can be laid down for identity of attributes. That is, it is postulated that from “ f is F” and “ f = g” we may deduce “g is F.” Here F is a schematic letter to be replaced by predicates that are applicable to attributes, not to individuals (for example, “is a color,” “is possessed by exactly two individuals,” “is a desirable attribute”). Consequently, attributes and individuals are still in the same boat with regard to the alleged obscurity of the identity condition. As rejoinder to (ii), obviously just the same kind of explicit definition of identity of attributes can be given. Attribute f is the same as attribute g if, and only if, f belongs to just the same classes of attributes that g belongs to. The argument from the identity condition, therefore, does not establish the superior intelligibility of nominalistic discourse. Independent arguments are needed. Perhaps the basic argument is the simple “common-sense” argument

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that the only way of proving the existence of anything is sense perception or inference from perceived things or events to unperceived things or events, in the manner of the physical scientist. But attributes cannot be perceived by the senses, nor does the postulation of the existence of attributes explain the nature of the perceived world in the way in which the theory of electrons, say, explains observed phenomena. Nevertheless, the assertion that attributes are “abstract,” not given in sense experience, should not be uncritically accepted. There are many simple qualities which recur in our sense experience and which, by this criterion, are universals. We see the same shade of blue in diﬀerent patches or in diﬀerent parts of the same patch, hear the same pitch in diﬀerent sounds, taste the same taste in diﬀerent portions of the milk we drink. It is likewise artificial to deny that we have direct experience of relational universals—for example, of the relation of temporal succession and the relation of spatial proximity. The counterargument that it is not, after all, certain that precisely the same quality recurs presupposes that one understands what it would be like to perceive exactly the same quality in diﬀerent contexts. But since the expressions “exactly the same shade of blue,” “exactly the same pitch,” and so on, can be explained only ostensively17 (you point, say, to adjoining parts of the same expanse and say “see, they, for example, are the same shade of blue—that is what I mean by ‘same shade of blue”’), this argument is really self-defeating. At any rate, the argument from the limits of sensory discrimination is hardly relevant to the question of the existence of mathematical attributes. For (natural) numbers may be regarded as attributes of attributes of individuals; that is, the number n is the attribute (of an attribute) of having n instances, and to deny that two attributes f and g have the same number is to deny that their instances can be matched one by one. But surely there are attributes that have the same number in this sense—for example, being a finger on my left hand and being a finger on my right hand. And if there are attributes that have the same number, does it not follow that there are attributes? What, then, is the argument between the nominalist and the realist all about? Indeed, it is diﬃcult if not impossible to attach any sense to the “ontological” question about whether there are attributes.18 If someone asks me this question, my natural reaction is, “why, of course there are: blueness, round-

17 It may, indeed, happen that a appears the same color as b, b appears the same color as c, yet a is distinguishable in color from c. Such cases suggest the following general definition of qualitative identity in terms of indistinguishability: x and y have the identical quality Q if, and only if, anything which is indistinguishable with respect to Q from x is also indistinguishable with respect to Q from y. But then, “indistinguishable (with respect to Q)” is the term that must be ostensively defined, and it will be self-refuting to deny that sensory appearances (many philosophers speak of “sense-data”) exhibit genuine universals. 18 The chief advocate of the logical positivist view that such ontological questions are “pseudo-questions,” devoid of cognitive meaning (a view I strongly incline to myself), is R. Carnap. See Carnap 1950a.

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ness, hardness, twoness, and millions more.” If he denies that these are examples of attributes, then he cannot mean by attribute what is ordinarily meant by the word, for one explains its meaning in terms of just such examples before entering into the more technical explanation of what, in general, distinguishes attributes from other types of entities, like individuals, or classes, or propositions. To deny that roundness is an attribute would be as absurd as to deny that 5 is a number, or that parenthood is a relation, or that the desk I am writing on is a thing. But if it is not denied, then one cannot deny, either, that there are attributes, since, according to elementary logic, any statement of the form “a is f ” entails “there is an x (at least one) such that x is f .” In the terminology of some contemporary analytic philosophers of language, the ontological statement is really a trivial analytic statement that cannot be denied if one understands the intended meaning of the type designation, like attribute, relation, class, and so on.19 To be sure, if one cannot countenance entities other than particular things because one cannot understand how something can “be” without having a tangible and visible existence at a particular place (one at a time!), then one’s intellectual discomfort is just self-imposed through the fallacy of “reification”: one is just naively associating with the expression “there are” images of spatially located things. One might, for example, be made intellectually uncomfortable by a mathematician’s statement that the number 0 exists. According to the Russellian class conception of numbers, the number 0 is a very peculiar class: the unit class whose only member is the null class. But how can a class be said to exist unless it has an existing member? And is not the null class, by this very criterion of class existence, nonexistent? Yet, in the sense of “there is” which is implicitly defined by the entailment (which was given in the last paragraph), “if a is f, then there is an x such that x is f ,” the statement “there is a class which is the null class” is just a trivial consequence of the logically true statement, “there is nothing which is distinct from itself” (in class terminology, the class of things that are not self-identical is null) as well as a trivial

19 It

is often overlooked by philosophers that a statement which is analytic in the sense that it cannot be seriously denied by one who understands it does not have to express a necessary proposition—that is, a proposition that cannot be conceived to be false. If this is overlooked, then one can easily refute the argument in the text by an analogy: You might as well argue that “there are men” is a trivial analytic statement, since it follows from “John Smith is a man,” which is analytic, since only a man could properly (that is, would conventionally) be named “John Smith.” But although it is a contingent fact, not a logical necessity, that there are men, one could not possibly deny that there are men if one grasps the conventional meaning of man, for this word has acquired its meaning through ostensive definition—that is, through uttering the word man while pointing at a man. Similarly, the meaning of attribute and other type designations must be explained by examples before one can proceed to construct an abstract definition; hence, a disagreement about whether there are attributes is bound to be as verbal as would be a disagreement, if one ever broke out, about whether there are men. (A comprehensive and detailed analysis of the meanings of analytic and necessary is contained in Pap 1958c. For an acute discussion of the meanings of ontological statements, see White 1956).

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consequence of any logically true statement of the form “there is nothing which both is and is not f .” The puzzle about how such shadowy, ghostly entities as the null class, and therewith the number 0, can exist arises only because one associates with the expressions “there is” and “there exists” images of spatial location, although they function purely as logical constants in the context of mathematical existence assertions: “there is something which is f ” means no more nor less than “not everything is not f .” Most existential statements that occur in everyday discourse refer to physical objects or events that exist, or occur, at some place and at some time—or at some place-time, in relativistic language. From this derives the habit of associating with the logical constant “there is” and its variants, images of spatial localization. It is only these associations, and the failure to understand the purely logical nature of such existential statements as “the null class exists,” “there are classes,” “there are attributes,” which produces intellectual discomfort. To be sure, some mathematicians and logicians reject an existential statement “there is an x such that x is f ” as meaningless in a case where it cannot be proved by deduction from a provable corresponding singular statement “a is f.” They are sometimes called “intuitionists,” sometimes “finitists.” They reject merely indirect proofs by the reductio ad absurdum method: there is an x such that x is f , because if you assume that there is no such x you get involved in a contradiction, whence it follows that the assumption is false; and if the contradictory of a proposition p is false, then p is true (law of the excluded middle). A well-known application of this “finitist” methodology of mathematics is the rejection of Zermelo’s axiom of choice. This is the following existence assertion: for any class K of mutually exclusive and non-null classes, there is a class which contains exactly one member from each member of K (“multiplicative class”). Now, in order to prove that a given class is a multiplicative class with respect to K, one must define it in terms of a method of selection of its members from the members of K. For example, let K be the class of all mutually exclusive pairs of integers. Here, a multiplicative class is easily defined as the class to be constructed by picking, say, the smaller integer from each pair. On the other hand, consider the class K which is formed as follows: we first take the class of proper fractions between 0 and 1, then the class of proper fractions between 1 and 2, then the class of proper fractions between 2 and 3, and so on. Obviously, the members of K are classes that have no members in common, since the integers that delimit the intervals are not themselves proper fractions. But now it is a little less obvious that one can describe a method of constructing a multiplicative class. The classes from which the members of the multiplicative class are to be selected have neither a first nor a last member, since between any proper fraction and the “closest” integer, there is another

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proper fraction. One might think of representing each class by its “mid-point,” perhaps—that is, by that fraction which equals the arithmetic mean between the integral limits. But this method of selection becomes unavailable for the following class K: the class of proper fractions between 12 and 32 , limits excluded, then the class of proper fractions between 32 and 52 , limits excluded, and so on. For here the “mid-points” are integers, not proper fractions. Perhaps there is still a method of selection with respect to this K, but do we have any guarantee that for any class of mutually exclusive non-null classes a method of constructing a multiplicative class can be described? Certainly not. Russell mentions, in his Introduction to Mathematical Philosophy (Russell 1950, 126), the class of all pairs of socks in order to illustrate that it is not self-evident that there is a “selector” for defining a multiplicative class with respect to a class of mutually exclusive non-null classes. A multiplicative class, here, could be defined only if there were a property which just one sock in each pair possessed, say being the left sock, or being the red sock, and so on. The finitist concludes that it is meaningless to assert the axiom of choice in complete generality. He holds that it is meaningless to assert that there is a multiplicative class of classes unless one can give a rule for constructing one. And this fits into his general methodological requirement that existence theorems be proved “constructively”—that is, by exhibiting an instance, or a rule for constructing an instance, that satisfies the existence theorem.20 But, however the methodological dispute between finitists and “realistic” mathematicians who wish to continue using nonconstructive proof methods (such as the reductio ad absurdum method by which Euclid discovered the irrationality of the square root of two) may be resolved, it really has no bearing on the ontological pseudo-questions, “are there classes?,” “are there attributes?,” and so on. For suppose we accept the finitist interpretation of “there is an x such that x is f ” as meaning “there is at least one constructively provable statement of the form ‘x is f ’,” where x ranges over abstract entities like numbers or classes. What should prevent one who has a penchant for asking ontological questions from asking whether there really are such abstract entities? The very asking of a question of the form “is there a number with property f ,” he might say, presupposes that there are numbers,21 whatever proof technique one may use to answer it. This presupposition, however, can be established quite eas-

20 See

Fraenkel 1928, par. 14, and Wilder 1965, chapter 10. finitists usually accept the natural numbers as, in some sense, “given”—in accordance with Kronecker’s often cited statement that God himself created the natural numbers, though he left the construction of all the other kinds of numbers and classes to man. Sometimes it is the “rule of construction” of natural numbers (adding one) that is said to be intuitively given rather than the natural numbers themselves. But it is not clear in what sense a natural number can be said to be a mental construction. Acquiring an idea of a number n by “counting up” to it is, of course, a mental process, but when we ascribe to a class a particular number, we ascribe an attribute to it that cannot plausibly be identified with the mental process of counting.

21 Indeed,

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ily by citing examples of numbers. To be sure, no philosopher who regards the existence of numbers and other abstract entities as problematic would be satisfied with the ironical answer: “Of course, there are numbers: 2, 5, 7, and many, many others.” But the trouble with the ontological question is just that no method of finding the answer has ever been described by those who keep asking it. And this means not that we do not know the answer but that we do not understand our own question. There is but one interpretation of the ontological question that makes it both intelligible and interesting, and this is an interpretation that turns it into that semantical-logical question which was touched on earlier in this article. What exactly did Russell mean by the assertion that classes are not real entities but are just “incomplete symbols”? The best way to discover the meaning of a statement is to look at the reasons oﬀered in support of it. Now, Russell’s reason for denying the reality of classes was that statements that contain names of classes or class variables are satisfactorily translatable into a symbolism which does not contain such expressions. (To take a simple example: “the class of dogs is included in the class of animals” means “for any x, if x is a dog, then x is an animal”). I propose to identify this reason for the denial of the reality of classes with the meaning of the denial. Similarly, the nominalist who denies the reality of any kind of abstract entities may be interpreted as aﬃrming that the meanings of statements that contain names of, or variables ranging over, abstract entities can be analyzed by means of a language that satisfies nominalistic requirements. Whether a nominalistic language is clearer, more intelligible, than a realistic language is, of course, debatable; indeed, this may be a matter of taste, of arational preference. But it must be admitted, I think, that the modern semantical interpretation of the time-honored (or time-dishonored) nominalism-realism dispute has the merit of making it scientifically meaningful, although it may be countered that, for this very reason, it is misleading to call this modern semantical-logical issue by the same name because one thus seems to accuse those who prefer the one or the other type of language of a subconscious addiction to realistic or nominalistic “ontology.”

Chapter 13 EXTENSIONALITY, ATTRIBUTES, AND CLASSES (1958)

In Principia Mathematica, an extensional system embodying the theory of types, Peano’s postulate “distinct (natural) numbers have distinct (immediate) successors” is not formally derivable from purely logical axioms. The axiom of infinity (“the number of individuals is infinite”) is required for the proof that there is no finite cardinal n such that n equals n + 1. Russell argued as follows:1 if only n individuals existed, then the number n + 1, being defined as the class of all classes that have n + 1 members (or that would have exactly n members if exactly one element were withdrawn from them), would be equal to the null class; for in that case no classes with n + 1 members would exist. But by parity of reasoning, the successor of n + 1 would also be equal to the null class; therefore n and n+1, which on the hypothesis made are distinct numbers, would have the same successor. The usual reaction to this argument is that, without abandoning Russell’s conception of numbers as classes of similar classes, we fortunately do not need to postulate the axiom of infinity after all. For there are other ways of solving the logical paradoxes besides the theory of types, and once our constructive eﬀorts are unimpeded by the latter, we can construct an infinite sequence of abstract entities without presupposing the existence of a single concrete individual: the null class, the unit class whose only member is the null class, the class whose members are the foregoing two classes, and so on. And once we have an infinite set of such abstract, though typically impure, entities, we can rest assured that no natural number will collapse into the null class. This is the approach of set theory, where such ghostly classes as the one just mentioned can be postulated to exist provided their definitions do not give rise to contradiction. However, I would like to re-examine Russell’s argument in order to see whether it is perhaps possible to get rid of the axiom of infinity without abandoning the (simple) theory of types.2 1 See

Russell 1950, 132.

2 Already in Carnap 1954a Carnap maintained that no axiom of infinity is needed if a coordinate-language is

employed. A coordinate-language, unlike a “thing-language” (or a “substance-language”), uses numerical

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Obviously Russell’s argument depends on (a) the definition of numbers as classes of similar classes and (b) the principle that classes with the same membership, including the degenerate case of no membership, are identical. Condition (b) can hardly be questioned since it is just this identity condition which is characteristic of classes in contrast to the attributes by which they are (intensionally) defined: attributes may be distinct though they determine the same class. But suppose we defined a natural number as a property3 common to all classes that are similar to a given class; for example, the natural number two is defined as the property common to all couples (of type 1) by virtue of which they are couples.4 Clearly, it could not be argued any more that if only 9 individuals existed, then 10 would be equal to 11, etc., for though both the class of all 10-membered classes and the class of all 11-membered classes would be the null class and hence identical, the numbers 10 and 11 would in the new conception be two perfectly distinct null properties, no more to be identified than the property of being a golden mountain and the property of being a mermaid. The validity of Peano’s axiom would then be independent of the contingency whether the world of individuals is finite or infinite. It should be noted that since Russell himself stipulated the formal deducibility from logical truths of Peano’s postulates as a criterion of adequacy for the definitions of the arithmetical concepts in terms of logical constants, and his definitions of numbers as extensions do not satisfy this criterion within a type-theoretical system, Russell was really committed to a rejection of them as inadequate. In The Principles of Mathematics (§110) Russell rejected the conception of numbers as common properties of similar classes on the ground that this kind of definition “by abstraction” would not guarantee the uniqueness of the individual numbers: how do we know that the property of having two members is the only property that is common to all couples?5 Suppose, for example, that all individuals have mass, as I suppose it should be according to materinames of space-time points, like “the 3rd position along the x-axis from position 0,” and ascribes qualities, resp. values of magnitudes, directly to these points, not to substrata by which they are (allegedly) occupied. Carnap shows that we can construct atomic sentences in such a language in such a way that the arguments are names of numbers; for example, instead of “the space-time point numbered 3 is blue” we can say “the number 3 is correlated to a blue space-time point” (cf. Carnap 1956a, 86). But I have never been able to understand how this trick is to make the axiom of infinity dispensable. Of course, the sentence “there are infinitely many natural numbers” is analytic in the Peano system of arithmetic, but this only means that it follows from the postulates of the system, not that it follows from purely logical axioms. It was in order to demonstrate its analyticity in the latter sense that Russell found himself driven to postulate the axiom of infinity; hence Carnap’s argument seems to me to beg the question. 3 I am using “property” and “attribute” synonymously, as is customary among logicians. Perhaps it would be better to avoid using “property” in this technical context, since in ordinary language a property is essentially a property of something, just as a brother is a brother of somebody, and it therefore sounds paradoxical to speak of a “null property.” 4 The symbolic expression of this definition is a property-abstract, not a class-abstract: 2 = (λ f )[(∃x)(∃y) (x y.(∀z)( f z ≡ (z = x ∨ z = y)))]. See Carnap 1956a, 115. 5 Apparently Russell did not suspect at that time that he would not be able to preserve the intuitive uniqueness of the numbers (i.e., there is but one number n, for any value of “n”) anyway, on account of the restrictions imposed by the theory of types on the use of “all.”

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alist ontology; or suppose that all individuals are qualitatively distinguishable space-time points or space-time regions, as presupposed by coordinate languages patterned after space-time theory. Then all couples would have the common property, not only of being couples, but also of containing members of the described sorts. Yet, could not this diﬃculty be circumvented by defining natural numbers as common properties of similar classes which (properties) are not defined in terms of any properties of the members of the classes or by reference to specific individuals that are members of the classes (the latter restriction would rule out such common properties as “containing individual a,” in the case of overlapping classes)? At any rate, no such diﬃculty arises if we speak of the number n as of the property of being an n-membered class, or an n-instanced property; we do not need to say that two is the property common to all couples any more than we need to say that redness is the property common to all red objects. It is sometimes supposed that the distinction between attributes and classes is a distinction that makes no diﬀerence with respect to extensional languages. This is a prima-facie plausible view, for the following reason: if F and G are distinct attributes though the corresponding classes the x’s such that Fx and the x’s such that Gx are identical, then there must be some context in which the substitution of “F” for “G,” or vice versa, leads to a change of truth-value. But if all contexts in which such an interchange might be made are extensional (e.g., “(∀x)(F x ⊃ Hx),” “Fa”), then by definition of “context which is extensional with respect to ‘F’, resp. ‘G,”’ no change of truth-value can occur. It is only in nonextensional contexts, for example, such modal contexts as “it is necessary that (∀x)(F x ≡ F x),” that the diﬀerence of extensionally equivalent attributes manifests itself. Such being the case, how can it make any diﬀerence at all whether we say “n is the class of all n-membered classes” or whether we say “n is the property of being an n-membered class,” as long as we stay within an extensional system like Principia Mathematica? Yet, have we not just shown that it makes an enormous diﬀerence, since it is only the latter analysis that permits the deduction of Peano’s critical axiom from purely logical truths in a type-theoretical system? This apparent contradiction can, however, be solved as follows. It looks as though Principia Mathematica could be shown to be not completely extensional after all. For consider the statement “if no more than n individuals exist, then the class of n + 1-membered classes is identical with the class of all n + 2-membered classes,” which is true. The statement which results from it by substituting for the class names the property names “the property of being an n + 1-membered class” respectively “the property of being an n + 2-membered class” is, however, false. For in order for property F to be identical with property G it must be necessary that F and G have the same extension, and since

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it would be a contingent fact, if a fact at all, that the mentioned properties are empty (and hence coextensive), they would still be distinct. Nevertheless, there is a subtle fallacy in the conclusion that therefore Principia Mathematica is not completely extensional. The fallacy is the tacit confusion of predicates, or sentential functions, with names of attributes. A sentence is said to be extensional with respect to a constituent expression E, if any replacement of E with an expression of equal extension leaves the truth-value of the sentence unchanged (the latter is then said to depend only on the extensions, not the intensions, of the constituent expressions). Now, the extension of a name is the entity named by it. Since the very distinction between names of classes and names of attributes (“ xˆ(F x)” and “F xˆ” in Principia Mathematica) presupposes the distinction between classes and attributes, it follows that a name of a property cannot have the same extension as a name of a class even if the named class is the extension of the named property. In the symbolism of Principia Mathematica,6 the extension of the class-name “ xˆ(F x)” is identical with the extension of the predicate “F”—or of the sentential function “F x”—but not with the extension of the property-name “F xˆ.” Nothing, therefore, prevents an extensional system from containing sentences of the form “ xˆ(F x) F xˆ,”7 even as theorems! A system, then, may be completely extensional in the explained sense (i.e., all its sentences are extensional with respect to all constituent expressions that can be said to have an extension), and nevertheless distinguish explicitly between classes and attributes by employing, like Principia Mathematica, diﬀerent kinds of names for them. Moreover, this diﬀerence is not a diﬀerence that makes no diﬀerence, as shown by my discussion of the question whether we need an axiom of infinity in a type-theoretical system.

6 For

the sake of typographical convenience I replace the Greek letters by Latin letters. confusion between names and predicates seems to me to underlie the statement by Russell and Whitehead (endorsed by Carnap in Carnap 1956a, 148) that the signs of identity and non-identity are not extensional in contexts like “ xˆ(F x) = F xˆ” and “ xˆ(F x) F xˆ.”

7 The

Chapter 14 A NOTE ON LOGIC AND EXISTENCE (1947)

In reference to Mr. E. J. Nelson’s recent article on “Contradiction and the Presupposition of Existence” (Nelson 1946), I want to comment upon a fundamental assumption involved in the author’s arguments which I find highly questionable. The following is the paradox for the solution of which Mr. Nelson elaborates a distinction between “the necessary conditions of the existence of a proposition” and “the necessary conditions of its truth (exclusive of its existence).” f a implies (∃x)[ f x ∨ ∼ f x], but ∼ f a, which is the apparent contradictory of f a, implies the same existential proposition. This common implicate, however, could conceivably be false, viz., in case no individuals at all existed. Hence, by transposition, both f a and ∼ f a could conceivably be false; and since contradictory propositions cannot conceivably be both false, f a and ∼ f a are not contradictories. Mr. Nelson’s solution of the paradox, if I have understood his arguments correctly, is that implication, as a logical relation, must be construed as a relation between what propositions assert (and propositions assert the necessary conditions of their truth); but f a does not, according to Mr. Nelson, assert that any individual (specifically, the individual a) exists, it only presupposes the existence of a as one of its constituents. Mr. Nelson’s conclusion seems to be that f a does not imply the existence of at least one individual in the same sense of “implies” in which, for example, (∃x) f x implies ∼(∀x) ∼ f x. The existence of at least one individual is rather “presupposed” by the existence of the proposition f a; it is not a truth-condition of f a in the sense in which, say, ga would be a truth-condition of f a if f a ⊃ ga were true. Now, as far as I can see, Mr. Nelson has not justified his assumption that, in asserting (∃x)[ f x ∨ ∼ f x], the existence of at least one individual is asserted and that, therefore, the above existential statement expresses a contingent truth which “is not certifiable on the grounds of logic alone.” Let us first consider a most puzzling consequence of Mr. Nelson’s assumption. If “(∀x) f x implies (∃x) f x” is, as in Russell’s logic, accepted as a transformation rule, then the debated existential statement follows from its corresponding universal which is tautologous: (∀x)[ f x ∨ ∼ f x]. But how can a contingent truth follow from a

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tautology? There seem to be three ways in which the consequence that contingent truths may be entailed by tautologies could be avoided: (1) To abandon the stated transformation rule. (2) To interpret logical truths, in Mill’s fashion, as simply the most extensive empirical generalizations. (3) To deny Mr. Nelson’s assumption that (∃x)[ f x ∨ ∼ f x] asserts the existence of individuals, and to maintain that this existential statement, like its corresponding universal, is purely tautologous. It seems to me that (3) is the most plausible and least burdensome alternative to take. We only have to translate the statement “(∃x)[ f x ∨ ∼ f x]” (where “ f ” is supposed to be a predicate constant) into the language of the class calculus to see its purely tautologous character. For the corresponding statement in the class calculus is: the universal class (i.e., the class which includes every class)1 has members. But to say that the universal class has members is simply another way of saying that it is diﬀerent from the null class, which is surely tautologous. The belief in the contingent character of an existential statement of the sort under discussion might arise as follows. Any existential statement can be formally exhibited as the implicate of a singular statement which verifies it. Thus f a ∨ ∼ f a implies (∃x)[ f x ∨ ∼ f x]. The transformation rule here used is: f z implies (∃x) f x (where “z” is the name of some unspecified individual). However, any existential statement may also be exhibited as the implicate of a (universal statement, with the help of the transformation rule: (∀x) f x implies ∃x) f x. Now, in formal logic, the predicate variable “ f ” may take on as values either empirical predicates (such as “red”) or logical predicates (such as “red or not-red”). Obviously, those existential statements whose predicate is vof the empirical variety cannot be known to be true in any other way than by erifying at least one of the singular statements each of which entail it; unless at least one zebra had been observed, the hypothesis “there are zebras” could not be asserted as true. This is the kind of consideration which naturally leads to the translation of existential statements into logical sums (finite or infinite) of singular statements. However, it is equally obvious that an existential state(ment whose predicate is of the logical variety (examples: (∃x)[ f x ∨ ∼ f x], ∃x) ∼( f x . ∼ f x)), is not asserted as true on the ground of the observation of individuals which exhibit it. It is asserted as a mere corollary of the corresponding universal statement which is a logical truth and in which descriptive empirical predicates have a purely vacuous occurrence (i.e., the truth-value of the statement would not change if any other empirical predicate were substituted instead). 1

It should be understood that the reference is to classes of individuals, i.e., classes designated by class symbols of type 1.

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That existential quantification, in formal logic, cannot be regarded as indicative of the assumption that at least one individual exists, is furthermore evident from the following consideration. Suppose we construct the concept of a kind which, as far as we know, has no instances. We may then draw analytic consequences from our arbitrary definition and formulate them as conditional statements. To illustrate, let “ f x” stand for “x is a goblin,” and “gx” for “x is imponderable.” Then we may assert (∀x)[ f x ⊃ gx] as an analytic truth, assuming that imponderability is a defining characteristic of goblins. But now it may occur to us to subdivide the empty class of goblins into two likewise empty classes, viz., the variety of pale goblins, say, and the variety of pink goblins. Let “hx” stand for “x is pale,” so that we can assert: ∼ (∀x)[ f x ⊃ hx]. But this universal statement is equivalent to the existential statement: (∃x)[ f x . ∼ hx]. By the principle of simplification, it follows that (∃x) f x; in other words, from an arbitrary definitional construction of an empty class and likewise empty subclasses thereof, we seem to have proved that the defined class is not empty after all: there are goblins! A clever theologian might exploit arguments of this sort in order to show that there is nothing logically objectionable in the assumption of the “ontological” argument that “essentia” may involve “existentia.” However, the proper conclusion to be drawn is that existential quantification, in formal logic, is a purely formal operation which has no “ontological” import whatever. As the above sample argument shows, “(∃x) f x” cannot be interpreted to assert that the property f has empirical instances. f might be a definitionally constructed property determining a subclass of a class determined by a more abstract or less complex property. Thus, “(∃x) f x” might assert that there are isosceles triangles—besides other varieties of triangles—without carrying the implication that any empirical instances of the concept “isosceles triangle” exist. Hence, if the individual constants “a,” “b,” etc., are regarded as proper names of empirical particulars, “(∃x) f x” can hardly be said to be synonymous with the logical sum “ f a ∨ f b ∨ . . . ∨ f n.” Mr. Nelson argues that “a exists” is a common implicate of f a and ∼ f a. Whether the former proposition be regarded as contingent or necessary, he continues, at any rate it implies “(∃x)[ f x ∨ ∼ f x],” which is itself contingent. Langford’s interpretation of this situation is alleged to have been that f a and ∼f a are not contradictories and that, indeed, singular propositions have no formal contradictories at all. Mr. Nelson, instead, concludes that “a exists” cannot be said to be implied by what “ f a” asserts. There are three final comments I wish to make. (1) Mr. Nelson seems to assume throughout his discussion that the sentence “a exists” expresses a proposition. But which are the constituents of this curious proposition? Presumably the individual a and the property of existence. However, we simply misuse language if we ascribe existence as a property to an individual; we can properly ascribe it only to a kind of individuals. It makes

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sense to say “zebras exist,” but if someone pointed to an individual and said “this zebra exists,” he could at best mean that what he sees is a real zebra, and that he does not suﬀer from an hallucination. But even so, he would not be ascribing a property, named “reality” (which may or may not be taken as identical with existence), to the individual he sees; he would rather be making the meta-linguistic assertion that the assertion “this is a zebra” (where the designatum of “zebra” is, of course, the class of physical zebras, not the class of mental images of zebras) is true. Mr. Nelson does not seem to have considered Russell’s demonstration that the logical analysis of sentences having “existence” as their grammatical predicate leaves you with propositions which do not contain existence as a constituent. Thus, in Russell’s analysis, “zebras exist” would mean “the sentential function ‘x is a zebra’ is true for some values of x” or “the class of zebras is non-empty.” (2) Why not argue that (∃x)[ f x ∨ ∼f x] must be a tautology just because it is implied by contradictory premises? Mr. Nelson aﬃrms that “no two propositions are contradictories if they have a common implicate, regardless of the status of that implicate.” But is it not an established theorem of formal logic that a tautology is implied by any proposition, just as a self-contradictory proposition distinguishes itself from self-consistent propositions by the fact that it implies any proposition? To say that p entails q is to say that p. ∼ q involves a self-contradiction. Hence, if both p entails q and ∼ p entails q, p. ∼ q, as well as ∼ p. ∼ q, must involve contradictions. But if q is a tautology, then ∼ q is a contradiction hence the condition of entailment is satisfied in both cases. (3) Mr. Nelson might reply that, if apparently contradictory premises have a common implicate which is contingent in the sense of being conceivably false, then either they are not really contradictory (Langford’s conclusion which Mr. Nelson is “reluctant” to accept), or they do not imply that contingent proposition in the ordinary sense of “implication,” i.e., the sense in which implication is a logical relation between assertions. This seems, indeed, to be Mr. Nelson’s own conclusion. The alternative, however, which he has not considered, is that the common consequence (∃x)[ f x ∨ ∼ f x] could not be false, being a logical truth. I shall now present my final argument for this alternative. According to Mr. Nelson, the above existential statement implies that individuals exist, which could be false. But what is the logical status of the statement “individuals exist,” what is its logical form? It seems to have the same form as the statement, say, “lions exist,” and since the latter would in Russell’s logic, which does not recognize “existence” as a predicate, be rendered as (∃x)(lion(x)), “individuals exist” might be formalized in the same fashion: (∃x)(individual(x)). It thus looks like a contingent truth, since, just as we might find no individual verifiers for (∃x)(lion(x)), we might, as it seems, find no x’s which have the property of being individuals. Thus, “a is an individual” looks like a contingent truth, of the same character as “a is a lion.” But upon closer inspection it turns out to be an example of Carnap’s “pseudo-object-sentences”; the “paral-

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lel syntactic sentence,” from which it is inferable without any factual inquiry into the extra-linguistic referents of language, is: “a” is the name of an individual. If I assign the name “a” to an individual, it still remains logically possible that a should turn out not to be a lion. But it is logically impossible that it should fail to be an individual. But if “a is an individual” is not contingent, then its implicate “individuals exist” cannot be contingent either. It is, indeed, natural to ask: Is it not logically possible that individual constants should remain without referents? But this question arises from a confusion between individual constants (which are proper names) and predicates (which are general names). Predicates are names of properties (or relations) which may or may not have individual instances, but individual constants are, by definition, names of individuals. The same kind of objection may be urged against Mr. Nelson’s assertion that (∃P)[Pa ∨ ∼ Pa] could be false, on account of asserting the existence of properties. “ f is a property,” like “a is an individual,” is a pseudo-object sentence; its “syntactic parallel” is: “ f ” is a predicate. A world of individuals devoid of properties is a logically impossible world. We may be mistaken in believing that a given individual has a given property; it may turn out that it has a diﬀerent property and, indeed, that no individual at all possesses the first property. But it is logically impossible that an individual should fail to have any properties at all. For “a has no properties” implies that a has the property of having no properties, which is self-contradictory. In other words, if there is no way of describing a given individual, it can at least be described as undescribable.

Chapter 15 THE LINGUISTIC HIERARCHY AND THE VICIOUS-CIRCLE PRINCIPLE (1954)

Most contemporary logicians seem to be in agreement that Russell’s hierarchy of orders of functions and of orders of propositions, dictated by the vicious-circle principle “whatever is defined in terms of a totality cannot be significantly said to belong to that totality,” is unnecessary. For, so the argument goes, the simple theory of types suﬃces for the solution of the logical paradoxes; the rest of the paradoxes, however, which Russell proposed to solve (or avoid, whichever term be more suitable) at one stroke by his comprehensive vicious-circle principle are semantical paradoxes which can be overcome by due observance of distinctions of levels of language. It is, of course, understandable that logicians would eagerly embrace the Tarski-Ramsey-Carnap method of solving the nonlogical paradoxes (liar paradox, Grelling paradox, Berry’s paradox, etc.), since the vicious-circle principle led to the embarrassing dilemma “either accept the axiom of reducibility or reject large portions of classical mathematics.” It is worth noting, however, that logicians who take seriously the distinction between sentences and propositions, in Carnap’s sense of “designata” of sentences, may nevertheless be faced with nonlogical paradoxes that call for some form of a vicious-circle principle. In order to show this, I shall formulate a paradox which is perfectly analogous to the liar paradox except that it refers to propositions, not to sentences. Instead of assuming that the only sentence inscribed in a given space S is false, and that the foregoing sentence is itself the sentence inscribed in S (or that the only sentence uttered by A within a specified time interval t is false, and the foregoing sentence is itself the sentence uttered by A within t), let us assume that the only proposition believed by A is false, and that the proposition thus described is the very proposition that the only proposition believed by A is false. Denoting the proposition in question by “P,” we can prove that P is both true and false, as follows:

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(1) (∀p)[bel(A, p) ⊃ F(p)] H1 (= P) (2) (∀p)[bel(A, p) ⊃ p = P] H2 (3) bel(A, P) H3 (4) bel(A, P) ⊃ F(P) from H1 (5) F(P) from H3 , 4 (6) (∃p)[bel(A, p). ∼ F(p)] from5, H1 (7) (∃p)[p = P. ∼ F(p)] from 6, H2 (8) ∼ F(P) from 7 The distinction of language-levels is not relevant to this paradox since falsity is predicated of propositions, not of sentences. If the bound variable in “P” were a sentential variable, then “P” would belong to a higher language-level than the sentences which are substitutable for the variable. Step (4) would accordingly be a violation of the prohibition of “semantically closed” languages. But the bound variable is meant as a propositional variable; all the sentences constituting the proof belong to the “non-semiotic” part of the meta-language, in Carnap’s phrase. A “non-semiotic” paradox even more closely related to the liar paradox in its usual semantic formulation can be obtained by substituting for the utterance of a sentence an assertion of a proposition. To be sure, one asserts a proposition by uttering a sentence, but nevertheless “A asserts the proposition that . . . ” is not synonymous with “A utters the sentence ‘. . . ’.” The sentence “A asserts that the only proposition which he asserts within time interval t is false” belongs to the same language-level as whatever sentence expresses the proposition described by “the only proposition which . . . ” However, since the distinction between believing a proposition and uttering a sentence expressing the proposition is even more obvious than the distinction between asserting a proposition and uttering a specific sentence expressing the proposition—if someone is so confused as to suppose that “A asserts that p” entails “A says ‘p’,” he still may not be so confused as to suppose that “A believes that p” entails “A says ‘p”’—I have constructed the paradox above about belief. Now, if we accept the vicious-circle principle, the solution of the paradoxes, which are at once nonlogical (in that nonlogical constants, “believing” and “asserting,” occur in them) and non-semiotic, is obvious: step (4) is a violation of this principle since the proposition P, being of higher order than the values of “p,” lies outside the range of significance of the function “bel(A, p).” This is but another way of saying that according to Russell’s principle hypotheses 1 and 2 are significant only if the quantifier “for every p” is restricted to “for every p of a given order.” However, one might object that it is not necessary to question any deductive step leading to the contradiction “F(P). ∼ F(P).” For, since the hypotheses are all of them contingent propositions, what is to be concluded is simply that they are incompatible, that at least one of them is false. (In Hilbert and Ackermann 1949, 120-21 it is claimed that for just this

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reason the liar paradox is no genuine paradox at all.) While this objection may be valid with respect to the belief-paradox, I think that it is invalid with respect to the assertion-paradox. For, as is pointed out by Lewis and Langford in their discussion of the paradoxes (Lewis and Langford 1951, chapter XIII), it is clearly possible that someone should assert, within t, that the only proposition asserted by him within t is false, and should assert no other proposition within t. Such a possibility can be established empirically by actually uttering a sentence expressing this proposition within t and uttering no other sentence within t. It is true that from “A asserts within t that the only proposition he asserts within t is false” it does not follow that A utters within t the sentence “the only proposition I assert within t is false”; he might utter another sentence which expresses the same proposition. Indeed, the whole distinction between the semiotic and the non-semiotic version of the liar paradox, here emphasized, hinges on this fact. But what is here essential is rather that the converse inference is justified: from the utterance of the sentence we can infer the assertion of the proposition. This inference is perhaps not logically valid, since a sentence which is commonly used to express one proposition may in a given instance be used to express a diﬀerent proposition. But if the evidence of linguistic utterance establishes as much as a probability for the hypothesis that the proposition in question was asserted, that is all that is needed to prove the possibility of the hypothesis. If so, the vicious-circle principle seems to be the only acceptable solution of the paradox after all. It should be noted that the vicious-circle principle, interpreted as a general restriction of the significance-ranges of functions, solves all the paradoxes which can be solved by avoiding “semantically closed” languages, while the converse does not hold, as just illustrated in terms of the non-semiotic paradoxes of belief and of assertion. Thus, consider the much discussed Grelling paradox about the heterologicality of “heterological.” The definition of “het” involves universal quantification over a property-variable: het(‘P’) =d f (∀P)[Des(‘P’, P) ⊃∼ P(‘P’)]. In the derivation of “het(‘het’). ∼ het(‘het’),” the property het is taken as a value of the property-variable “P,” which is a clear violation of the viciouscircle principle. In this case, indeed, the solution in terms of stratification of expressions coincides with the solution in terms of stratification of extralinguistic entities: if it is insignificant to say of a designation of a second-order property, like “het,” that it has the second-order property het, then the predicate “het” cannot have the same meaning in the sentences “het(‘het’)” and “het(‘long’).” In Russell’s terminology, the semantic phrase “het” is systematically ambiguous. But it would be a serious mistake to suppose that, just because the stratification of extralinguistic entities (properties, propositions) entails the systematic ambiguity of logical and semantical constants, there-

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fore what can be accomplished by this “ontological” stratification can always be accomplished by the stratification of expressions as well. For the latter is irrelevant to paradoxes that are derivable within what Carnap calls the “nonsemiotic part” of a meta-language. To be sure, the entire problem which is here submitted for further consideration would disappear if statements of belief and assertion were translatable into extensional statements about expressions. But, as particularly Church has shown (Church 1950), it is highly doubtful whether Carnap’s attempts at such translation have succeeded. Further, for reasons partly similar to Church’s, I think that the absolute concept of truth (truth as predicable of propositions) is not reducible to the semantic concept of truth (truth as predicable of sentences).1 At any rate, the foregoing considerations show that there is a connection between the question whether all nonlogical paradoxes are soluble without the vicious-circle principle and the question whether propositions are reducible to sentences.

1 Cf.

the arguments I present in chapter 6.

V

PHILOSOPHY OF MIND

Chapter 16 OTHER MINDS AND THE PRINCIPLE OF VERIFIABILITY (1951)

1.

The Principle of Verifiability as Generator of Philosophical Theories

The principle of verifiability is that famous criterion of propositional significance whose discussion, I venture to guess, has occupied more space in the publications of contemporary analytic philosophers than discussion of any other topic. It is not the purpose of this discussion to add a little more water to the ocean of literature on the precise meaning or formulation and the general implications of this principle, which is often referred to as the very soul of the movement called variously “logical positivism” and “logical empiricism.” A comprehensive review of this kind has, indeed, been made unnecessary by C. G. Hempel’s excellent contribution to a recent issue of Revue Internationale de Philosophie dedicated to the theme “logical empiricism.” My purpose in this paper is to seize upon fairly clear standard formulations of the principle and, without commitment pro or con the principle itself, scrutinize their implications for one of the basic problems of the philosophy of psychology (or “philosophy of mind,” as I dare unashamedly call it): the analysis of statements about other minds. Specifically, I shall attack the wide-spread view that acceptance of the principle logically compels a physicalistic interpretation of statements about other minds. But before proceeding to this specific task, I shall discuss the concept of a “philosophical theory dictated by the principle of verifiability” in terms of a crucial instance from contemporary analytic philosophy. In this way the reader may be led to appreciate the necessity of coming to grips with this semantic principle if one wishes to discuss intelligently any fundamental issue in philosophical analysis, or at least in that branch of philosophical analysis which occupies itself with empirical discourse in everyday as well as scientific language. By the principle of verifiability will here be understood the liberal version laid down in (Carnap 1937): a sentence is cognitively meaningful if and only

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if it is in principle confirmable. The criterion is intended to apply only to those declarative sentences (contrast “declarative” with imperative, interrogative, optative, etc.) which are “putatively factual,” in Ayer’s phrase, i.e., the predicates “confirmable” and “non-confirmable” are so meant that they do not significantly apply to logically decidable (analytic and self-contradictory) sentences. The qualification “cognitively” is used in order to indicate which of the various senses of the ambiguous term “meaningful” is intended as the explicandum: it is the sense of “meaningful” in which “S is meaningful” entails “S is either true or false” or, in the technical terminology of some semanticists, “S expresses a proposition.” Thus it would be an irrelevant test of the verifiability principle if one adduced a sentence heavy with emotive and pictorial meaning and asked rhetorically “well, is it confirmable?” Now, in this formulation the principle purports to tell us whether or not a sentence is cognitively meaningful but it does not tell us how one is to find out what a sentence cognitively means. Nevertheless, once one has assented to the principle one will with little hesitation accept the following rule which may be used to discover what the cognitive meaning of a sentence is: a sentence means the conditions (or states of aﬀairs) whose ascertainment would establish its truth. It is important that “establishing its truth” here be understood in the sense of conclusive verification, not in the sense of (partial) confirmation: the ascertainment of bloodstains on the clothes of a man suspected of murder confirms the proposition “he committed the murder,” but nobody would maintain that this condition is even part of the meaning of the proposition. The distinction between logical entailment and factual implication is of paramount importance in this context. If there is a law by means of which S 1 implies S 2 , we would ordinarily say that the state of aﬀairs described by S 2 is evidence for the truth of S 1 , but no reputable version of the verifiability principle has ever maintained that evidence in this sense has anything to do with the meaning of S 1 . It is only if S 2 is analytically entailed by S 1 , that the state of aﬀairs described by S 2 would be called a truthcondition, in the Wittgensteinian sense, for S 1 and that it, consequently, would be identified with part of the meaning of S 1 . It will be assumed that the reader is familiar with the crucial distinction between theoretical and practical possibility of confirmation (instead of “theoretical confirmability” one also speaks of “confirmability in principle”), and is aware that “confirmable,” in this context, means “theoretically capable of confirmation.” But one further ambiguity should be cleared away before proceeding since it is crucial with regard to the application of the principle to the language of psychology which the ensuing discussion is to focus on. What is meant by an “ascertainable (verifiable) condition”? What constitutes an admissible method of verification? There have been and are opponents of logical positivism (e.g. A. C. Ewing) who gratuitously take the positivists to identify verification with sense-perception and hence arrive at the conclusion that for the positivist sentences which can be conclusively established by introspection only, such as “I remember to have had scrambled eggs for breakfast,” are

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cognitively meaningless—or at least are so unless translated into physicalistic language. But if logical positivism is defined by the liberal principle of verifiability as stated, it is neutral with respect to the issue between introspectionist psychology and methodological behaviorism: introspection is a proper method of verification. As a matter of fact, a positivist might make the following use of his semantic principle, which would show clearly that he does not intend to restrict the term “verification” to the procedure of establishing a statement by sense-perception. A so-called ethical intuitionist claims that an ethical sentence of the form “x is wrong” is either true or false, yet not verifiable or refutable by sense-perception: wrongness, he would say, is an objective property of certain actions, but it is “non-natural,” inaccessible to sense-perception. He likewise disclaims that ethical sentences are psychological. The positivist asks: what experience, then, constitutes verification of such a judgment? Answer: a feeling of disapproval in the mind passing the judgment. Conclusion forced by the principle of verifiability: the ethical sentence means either (a) the speaker disapproves of x, or (b) any person meeting specified requirements (e.g., enlightened about the probable consequences of such an act) would feel that way about x. According to translation (a) the self-proclaimed “intuitionist” is reduced to a disreputable “subjectivist” (according to the somewhat undiscriminating terminology of Ewing, in Ewing 1949); according to (b) he is an “objectivist,” since wrongness is analyzed as a dispositional property of actions whose presence is independent of the actual occurrence of feelings of disapproval (if nobody were properly enlightened about the act, everybody might approve of it even though everybody would disapprove of it were he properly enlightened). But according to either analysis “x is wrong” would be an empirical statement, contrary to the intuitionist claim. The fact that the principle of verifiability can, and has been, used that way shows that the accusation of a “narrow” identification of verification with sense-perception is unfounded: a feeling of disapproval cannot be known to occur through sight, touch, smell etc., but only through introspection. Phenomenalism, as a theory of the meaning of statements about physical objects, is an excellent illustration of a philosophical theory closely connected with the principle of verifiability. What do we mean by a statement like (p) “there is a table here-now”? The principle of verifiability gives us the following direction: determine the verifiable consequences of this statement: the state of aﬀairs expressed by the totality of these verifiable (logical) consequences is what the statement means, no more and no less. But how should we determine the consequences? Is (q) “somebody here-now has a sense-impression of a table”1 for example, a logical consequence of the statement? It cannot be denied that verification of (q) is relevant to the truth or falsity of (p), but 1 This

is intended as a statement in the so-called sense-data language. “Sense-impression of a table,” therefore, should be replaced by a complex phrase containing the names of the kind of sense-data normally produced by tables.

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before we would be justified in saying that q is part of the meaning of p we would have to show that p logically entails q. As a matter of fact, it does not, since it is not self-contradictory to suppose that p is true and q is false. It must be admitted, I think, that this decision, viz. that a certain conjunction of statements (p and not-q) is not self-contradictory, is arrived at by philosophers quite independently of the principle of verifiability. Indeed, this seems to be demonstrably the case, since there are both positivists and non-positivists who hold that p does not entail q and that Berkeley’s “esse est percipi,” therefore, is a sheer semantic mistake. The claim, then, that the principle of verifiability functions as a guide to the meaning of a statement must be taken with a grain of salt. Before I can decide whether a verifiable consequence of p forms part of the meaning of p, or is only indirect evidence for p (in the sense of “indirect evidence” in which the bloodstains, in my previous example, were indirect evidence for the statement “he committed the murder”), I must know whether it is a logical consequence or follows from p only with the help of additional factual premises. And to this question the principle of verifiability seems to be irrelevant. Thus all philosophers except confused disciples of Berkeley will agree that in order to deduce q from p we require the further premise (r) “somebody is here-now, fulfilling such and such conditions (suitably placed, open eyes, perceptually normal etc.).” The only logical consequence of p, then, is the conditional statement joining r as antecedent to q as consequent. Since we are here discussing phenomenalism only by way of illustration, we may pardonably disregard all the subtleties involved in an adequate formulation of this theory, and roughly define it as the claim that p means no more and no less than “if r, then q.” The question before us is: in which sense can it be said that phenomenalism is dictated by the principle of verifiability? We have already seen that this principle does not tell us how to find out what a statement does or does not logically entail and thus what it does or does not (cognitively) mean. The instruction “find the conditions whose ascertainment would establish the truth of the statement” is not helpful, since “ascertainment of C would establish the truth of p” means “the statement expressing C is logically equivalent to p,” and we have not been provided with a method for determining what the logical consequences of a statement are.2 Positivists and nonpositivists alike decide such questions, in their analytic practices, intuitively by asking themselves whether a given conjunction of statements (like “there is a table here-now but it is not perceived by anybody”) sounds self-contradictory. The principle of verifiability exercises only a negative function, in this sense: if a set of statements q1 , . . . , qn is admitted to exhaust the verifiable conse-

2 It

should be kept in mind throughout this discussion that the relevant concept of “logical consequence” is the concept applied to natural languages, not the formal concept applied to deductive systems. (See on this point Pap 1949d.)

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quences of a statement p, and the consequences are held to be logical, i.e., to follow from p alone, without the aid of additional premises, then the principle says: p means what this set of statements means, and no more. In other words, if p is a putatively factual statement, it is held to be synonymous with the total set of its verifiable logical consequences. It follows that if a putatively factual statement has no verifiable consequences, it has no cognitive meaning.3 Now, let us, omitting niceties which certainly ought to be discussed if this were a paper on phenomenalism, define phenomenalism as the thesis that a statement like p is synonymous with a finite set of sense-data statements like “if r, then q.” The correctness of this thesis cannot, for the reasons stated, be deduced from the principle of verifiability. What can, however, be deduced from the latter is that if that set of sense-data statements constitutes the totality of verifiable logical consequences of a statement about physical objects, then phenomenalism is correct. A philosopher could consistently accept the principle of verifiability and reject phenomenalism on the ground that none of those sense-data statements is logically entailed by a physical statement. On the other hand, if he could be brought to admit that if physical statements have any verifiable logical consequences, then these are the mentioned sensedata statements, the verifiability principle would, indeed, force him into the dilemma of either accepting phenomenalism, which he finds intuitively an unsatisfactory meaning analysis, or confessing that he has no way of intelligibly communicating what according to him physical statements mean (I think the latter alternative adequately characterizes the position of many opponents of phenomenalism). But actually he need not make the admission on which this dilemma is contingent: he may simply say that the only logical consequences of physical statements are, not pure sense-data statements, but statements containing likewise names of physical objects and events. The main point of this preliminary discussion was to show that the principle of verifiability by itself can never dictate a specific analysis of a statement or class of statements, but only relatively to the assumption that if the statement or class of statements to be analyzed have a verifiable content at all, they must be synonymous with the analysans proposed. Once this assumption is granted, the principle of verifiability leads to the conclusion that the proposed analysis is a correct analysis of the only cognitive meaning the analysandum has. It will be shown now that one may consistently accept the principle of verifiability and

3 If

the requirement were stated simply in terms of “verifiable consequences” instead of “verifiable logical consequences,” it would be demonstrably so liberal as to be wholly ineﬃcient in the exclusion of nonsense. On the other hand, it has been argued that if the stronger requirement is adopted, then abstract scientific theories would be condemned as meaningless since no testable consequences are deducible from them without the use of additional premises which connect the abstract terms of the theory with observables. The diﬃculty can perhaps be solved by acknowledging reduction sentences, along with explicit definitions, as semantic rules, but a thorough investigation of this problem is beyond the scope of this discussion.

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repudiate physicalism or logical behaviorism, since the assumption, explicitly or implicitly made by all logical behaviorists, that behavioristic translations alone give statements about other minds a verifiable content, is false.

2.

The Behaviorist’s Confusion about the Notion of Verifiability

This section will be opened by a representative quotation from an early defense of physicalism by Carnap (Carnap 1932a). It is only fair to Carnap, however, to remind the reader that the physicalistic thesis maintained by Carnap and his followers at the present time diﬀers from the earlier thesis in a very important respect: the claim that psychological sentences are synonymous with physicalistic sentences, however complex, has been abandoned, and all that is asserted is reducibility of psychological terms to physicalistic terms (“physicalistic” is used broadly to cover both physiological and behavioristic terms). Whether Carnap, after having modified the physicalistic thesis in this fashion, would still agree with his former argument to be quoted, I do not know. I shall argue that if he does, then he is inconsistent with his own confirmability criterion of meaning; and if he does not, then physicalism as asserted by him and his school is not subject to philosophical debate, is neutral with respect to any “theory of mind,” since it is merely a statement analytic of the meaning of “intersubjective language” (see section 4). Arguments from analogy are, indeed, not demonstrative, but they are legitimate as probable arguments. Let us consider an example of an argument from analogy from everyday life. I see a box of definite shape, size, color; I notice that it contains steel-point pens. I find another box of similar appearance; I infer by analogy the probable conclusion that it likewise contains steel-point pens. The objector means that his inference to the other mind has the same logical form. If this were the case, his inference would indeed be justified. But this is not the case; here the conclusion is meaningless, a mere pseudo-statement. For, as a statement about another mind which is not to be interpreted physicalistically, it is in principle untestable. (My translation.)

This quotation shows clearly that the principle of verifiability led to the physicalist analysis of statements about other minds, because the assumption was made, to quote my own statement, “that if the statement or class of statements to be analyzed have a verifiable content at all, they must be synonymous with the analysans proposed.” But it is easy to show that this assumption is erroneous, that the verifiability of statements about other minds, in the relevant sense of “verifiable,” does not require their translatability into physicalistic language. This relevant sense of verifiable is, of course, “in principle capable of confirmation.” In the early days of the history of logical positivism, the principle of verifiability had, indeed, the “intolerant” form: a sentence is cognitively meaningful (i.e., a statement, or proposition) if and only if it is in

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principle capable of conclusive verification. I shall argue later that even if this strong criterion of propositional significance were adopted—and it has largely been abandoned since (Carnap 1937)—it would not follow that the cognitive significance of statements about other minds could be rescued only by logical behaviorism. For the moment I shall simply demonstrate that statements about other minds are, in the “mentalistic” sense intended by old-fashioned, unabashed psychophysical dualists (and those conservative people you can meet in the street even though you will shock them if you call them by that name), confirmable; and that, since the legitimacy of an argument from analogy only requires the confirmability of its conclusion, there is nothing wrong, from the point of view of the present-day positivist theory of meaning, with the traditional argument from analogy as a justification of beliefs about other minds. Let m2 be a mental state of mind M2 which I infer from observation of a physical state b2 of the body B2 , where B2 is the body of the person with the mind M2 . An inductive justification of this inference requires, of course, a psycho-physical law (= any law correlating mental and bodily or behavioral states) for which there is confirming evidence. The probability of the occurrence of m2 relatively to the evidence that b2 occurred is then, roughly speaking, proportional to the probability of the law asserting that an event of the same kind as m2 is either a suﬃcient or a necessary condition for events of the same kind as b2 . Let this psycho-physical law, formulated for M2 and B2 , be denoted by “m2 —b2 ” and the corresponding law, formulated for my own mind M1 and own body B1 , by “m1 —b1 .” Clearly, the only way the logical behaviorist could deny that the inference to m2 , which we may suppose to be drawn by M1 , is capable of inductive justification, would be by claiming that the relevant psycho-physical law which would have to enter into the argument is incapable of confirmation. Superficially, it looks as though his case could be constructed as follows: the only psycho-physical law for which M1 can obtain a confirming instance is m1 —b1 ; the law required as a premise, however, is m2 —b2 , and to suppose that M1 could verify a conjunction of singular statements that would count as a confirming instance of m2 —b2 (like “he is afraid now and his hands tremble now”) is to suppose that a person could be directly aware of a mental state not his own, which is a contradiction in terms. This whole argument, however, breaks down once we make a distinction between direct and indirect confirming evidence for a law. Consider the special case of the general gas law which we obtain by replacing “for any ideal gas” with the more restricted quantifier “for any instance of nitrogen gas under ideal conditions.” Suppose that we had never experimented with nitrogen gas, but nevertheless predicted, from the analogy of experiments with other gases, that the volume of a mass of nitrogen gas kept at constant temperature would be halved if the pressure upon it were doubled. Would anybody deny the induc-

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tive justifiability of this prediction on the ground that we had not, as yet, any confirming evidence for the law “if kept at constant temperature, nitrogen gas behaves in accordance with the formula PV = k”? Rather elementary rules, whether explicit or implicit, of inductive logic would lead one to say that there is indirect confirming evidence for this special law, inasmuch as the latter is a deductive consequence of a universal law for which there is confirming evidence (let us assume that the universal gas law was obtained as an induction from experiments with a series of gases excluding nitrogen). In just the same way the universal psycho-physical law “whenever a person satisfying such and such conditions is in a state of anxiety, his hands tremble, his pulse accelerates etc.” is obtained as an induction (how reliable such an induction may be is an important problem of the methodology of psychology, but irrelevant to the logical issue under debate) from the special case m1 —b1 , for which M1 has direct confirming evidence, and since m2 —b2 is a deductive consequence of the universal psycho-physical law, it is indirectly confirmed by the very evidence which directly confirms m1 —b1 . Surely the following rules of inductive logic, invoked in my defense of the psycho-physical use of the argument from analogy, could not be questioned: if e confirms p, and p entails q, then e confirms q; if p entails q and e confirms q, then e confirms p. These rules are part and parcel of accredited scientific procedure.4 In case the reader should mistrust my argument on account of my speaking of “minds,” a terminology that reeks of disreputable metaphysics, it may be well to add that the argument does not depend on any commitment to any “theory of mind.” The argument would not be aﬀected, for example, by a materialistic translation of “mental event m1 is owned by mind M1 ” into “m1 is directly caused by an event in the cerebro-neural system of body B1 .” The very possibility of such a materialistic translation disposes of an argument against the psycho-physical use of the argument from analogy, put forth by Carnap in the same article, which unlike the argument already quoted is independent of 4 It

cannot be denied that these rules of inductive logic would require considerable refinement before being acceptable as valid in general. Specifically the second rule leads to a paradoxical consequence in case the relation of p to q is that of a conjunction of unrelated conjuncts to one of its conjuncts: e confirms B, (A and B) entails B, therefore e confirms (A and B); actually e may be wholly irrelevant to A, in which case one would not admit that the conjunction had been confirmed. But the case we are considering is diﬀerent: the universal generalization can, indeed, be interpreted as the conjunction of the sub-generalizations referring to the sub-classes of the whole class, but then the conjuncts have predicates in common and are thus not wholly unrelated. In any case, the only rule needed to justify the above argument for the indirect confirmability of propositions about other minds is this: let A1 and A2 be subclasses of A, where A is a suﬃciently proximate genus with respect to these sub-classes (e.g., I would say that the genus “organisms” is not suﬃciently proximate to the sub-class “negroes” to warrant analogical inference from negroes to, say, oak trees); and let e confirm a generalization about A1 ; then e also confirms the corresponding generalization about A2 . To the extent that the expression “suﬃciently proximate genus” is vague, the rule itself is, indeed, vague. But if this vagueness be held as a reason against the use of the rule for purposes of inductive justification, purely physical arguments from analogy will likewise suﬀer, and it could not be said that the psycho-physical use of this type of argument presents a unique diﬃculty.

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the principle of verifiability. It is worth quoting in full (my translation from the original German): The predicative linguistic form ‘I am angry’ does not adequately express the intended state of aﬀairs. It expresses that a given object has a given property. But all that is given is an experienced feeling of anger. A correct linguistic formulation of this feeling would be, say, ‘now anger’. But once this correct formulation is used, the very possibility of an argument from analogy vanishes. For then the premises become: whenever I, i.e. my body, exhibit symptoms of anger, then anger occurs; the other person’s body resembles my body in many respects; the other person’s body now exhibits symptoms of anger. Now the conclusion cannot be formulated any more, since the sentence ‘now (or: then) anger’ contains no more ‘I’ that could be replaced by ‘the other’. If, on the other hand, one wanted to formulate the conclusion by refusing to make a replacement and simply retaining the form of the premise, the resulting conclusion would indeed be meaningful, yet evidently false: ‘therefore now anger’; for this means, put in ordinary language: ‘I am now angry’. (Carnap 1932a, 120)

This argument is vitiated by a curious inconsistency. Carnap begins by claiming that “I am angry” can mean only “anger occurs now” (Russell makes a similar claim in criticizing Descartes for starting from the premise “I think” as though it expressed an experiential datum and nothing more). This claim is obviously untenable since it leads to the consequence that “I am angry” as spoken by myself and “I am angry” as asserted by another person are synonymous statements! But even if Carnap’s initial claim were valid, the conclusion of his argument contradicts that very claim: if Carnap says that the conclusion “therefore now anger” is evidently false, he evidently presupposes that “I am angry” and “you are angry” are not synonymous. It is surely conceivable that a feeling of anger did occur at the time referred to by the conclusion of the analogical argument, hence if Carnap is sure of the falsehood of the conclusion it must be because he clearly distinguishes the sense of “I am angry” from the sense of “you are angry.” And, indeed, this diﬀerence of meaning can be recognized pre-analytically, i.e., independently of any analysis of such sentences. Carnap might reply that the only way the diﬀerence of meaning between “I am angry” and “you are angry” could be analyzed on empiricist ground, i.e., without postulating metaphysical substances (Christian “souls,” Cartesian “res cogitantes”), would be by translating them into physicalistic sentences about diﬀerent bodies. But apart from the circumstance that such translations would obviously be incorrect since I can know that I am angry while being blissfully ignorant of the bodily symptoms by which my feeling is manifested, the reply may be countered by indicating two (vastly diﬀerent) methods of translation which should be equally acceptable to an empiricist as conforming to the maxim “refrain as much as possible from postulating inferred entities.” The first is none other than the materialist analysis mentioned above: “I am angry” (as spoken by body B1 ) = “anger occurs, and this mental event is directly caused by a brain-event in B1 .” The diﬀerence between the

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singular premise and the singular conclusion of the argument from analogy is then perfectly clear; but the translation is not physicalistic since the psychological term “anger” reappears in the analysans without physicalistic interpretation. The essential point is that, as illustrated by this analysis, it is possible to distinguish the sense of the predication “I am angry” from the sense of the existential statement “anger occurs now” without postulating a metaphysical “self” objectionable to empiricists. But actually this materialist analysis is no more acceptable an account of what is meant by “I am angry” than a physicalist analysis which pretends to do away with “mental” events in the sense of events inaccessible to public observation, for this obvious reason: if it is conceivable that I should know about my anger without knowing about the bodily symptoms of my anger, it is still more easily conceivable that I should know about my anger without knowing anything about the connection of feelings with brain-events. And in fact it is not a priori impossible that my feelings should be directly caused by events in another body than the body whose back I cannot see with my unaided eyes. For this reason what has come to be known as the “phenomenalist” or “logical construction” theory of the self, which is equally inspired by the desire to do without metaphysical substances and whose historical ancestor is Hume, is far more plausible. According to this analysis “I am angry now” is a statement of class-membership: the present mental event is asserted to be a member of the class of mental events which is what the pronoun “I” refers to. Just what the relation between two successive mental events (each element of the temporal series, the mental history, may itself be a complex event, just like instantaneous states of a physical system described in terms of several state variables) is by virtue of which they are members of the same series, it is not easy to tell. If we are the kind of philosophers who rest satisfied with analogies, we can compare successive states of a mind to points on a line, the mind to the line, and then say that just as two points belong to the same line if their coordinates satisfy the same equation, so two mental events belong to the same mind if their descriptions satisfy the same “equation.” It is, of course, highly obscure what could be meant by the “equation” defining a mental history, and moreover this analogy leads to the questionable consequence that a mental event could be a state of several minds, a point of intersection of mental histories, as it were. A detailed discussion of the logical construction theory of the self, which Hume adumbrated, would not be to the present purpose. My purpose was merely to show that Carnap was wrong in suggesting that if “I am angry now” is to be considered as a phenomenological statement free from metaphysical commitments it must be translated into the impersonal form “anger occurs now.” Nevertheless I might briefly indicate, before leaving the topic, that the phenomenalist theory of the self can do better than suggest obscure analogies, such as the equation of a line in analytic geometry. One can at least state,

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in literal language, a straightforward suﬃcient condition for the truth of the proposition “x and y are events owned by the same mind (or ‘states’ of the same mind)”: this will be the case if x is the event of remembering y or y the event of remembering x.5 What makes the problem diﬃcult, however, is the fact that this is not a necessary condition; it would normally be held that there are innumerable past experiences of mine which I do not now remember, which statement would be a contradiction if the above criterion were an analysis of the notion of mind-ownership.

3.

Are Statements about Other Minds Conclusively Verifiable?

There is a school among present-day analytic philosophers (consisting mainly of followers of Wittgenstein) according to which it does not make sense to speak of increasing confirmation of a statement if the limit approached, but never reached, by this process is so conceived as to be in principle beyond attainment. They would therefore argue that as long as it is conceded to be logically impossible to verify statements about other minds conclusively, it does not help to prove their confirmability, and we will still have to interpret them physicalistically if we wish to rescue their cognitive significance. Now, in the first place, this semantic principle, that it does not make sense to speak of increasing approach to a limit if it is logically impossible to reach the limit, is highly dubious. The statement, e.g., that certainty, in the sense of maximum probability (probability equal to unity), is the limit approached by the probability of a generalization as the number of confirming tests of the generalization increases without limit (which implies that certainty would be reached only after an infinite number of confirming tests had been completed), is perfectly analogous to limit statements in mathematics and mathematical physics, and it would be interesting to see how the Wittgensteinians would prove their meaninglessness. Second, this principle is radically inconsistent with the central idea of Carnap 1937, which is to construct a meaning criterion that is compatible with the significance of unrestricted generalizations, where unrestricted generalizations are statements that are confirmable, yet in principle incapable of conclusive or complete verification. Third, it is hard to see how the validity of this semantic principle, if it were valid, could be a good reason for adopting a physicalistic interpretation of statements about other minds: for few logical behaviorists would be prepared to say that the meaning of such a statement would be exhausted by a finite set of physicalistic sentences, and if the set is infinite,

5 The

above presupposes that remembering is never directly a relation to a physical object or event even though our language frequently suggests this: we say “I remember that house” but we would accept “I remember to have seen that house” as a translation of the intended meaning.

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then adoption of that very semantic principle would lead to the conclusion that the statement had not been made any more significant by the translation.6 But fourth, the questions just discussed are, within the context of our problem, purely academic. They would be to the point only if statements about other minds, as intended by common sense, were in fact beyond the logical possibility of conclusive verification. And I proceed to give a simple proof of their conclusive verifiability. If Mr. A says with regard to himself “I am angry,” he is, as will readily be conceded, making a statement which is conclusively verifiable, by introspection. Suppose, now, that Mr. B says to Mr. A “you are angry.” Here we have a statement about another mind which is said to be in principle unverifiable if it is intended as a non-physicalistic statement, like Mr. A’s statement “I am angry.” But this position is, as it stands, glaringly inconsistent since the two statements which diﬀer only in the personal pronoun are strictly synonymous. They are synonymous by virtue of asserting the same fact, in the same way in which “it is raining today” and “it rained yesterday,” spoken a day later than the first sentence, assert the same fact. We must not confuse, of course, propositions that are inferable from the occurrence of a given token of such a statement with propositions that are inferable from the meaning of the statement. From the statement “a token of the sentence-type ‘you are angry’ was spoken” we can infer with some probability that the speaker of the token referred to a person other than himself, and from the statement “a token of the sentence-type ‘I am angry’ was spoken” we can infer with some probability that the speaker of the token referred to himself. Which proves that these statements, describing linguistic events, have diﬀerent meanings, but not that the statements which they are about have diﬀerent meanings. Indeed, if the claim that “you are angry” means observable behavior and physiological states were taken seriously, it would follow that when asked “are you angry?” and then answering the question after a moment’s introspection, the answer we give is not an answer to the question we were asked. This proof is so simple that it may easily provoke an accusation of simplemindedness: “You have naively overlooked the crucial distinction between ‘verifiability by me’ and ‘verifiability by anybody (public verifiability)’. What inspired logical hehaviorism was the desire to eliminate from psychology statements which are not publicly verifiable.” The criterion of public verifiability here proposed evidently comes to this: a statement is cognitively meaningful if and only if it is in principle possible for anybody to verify it conclusively. The task to be confronted, then, is to prove that statements about other minds, as intended by common sense, are even publicly verifiable in this sense. I shall

6 Even

if one could argue in favor of translatability into a finite conjunction of physicalistic test-sentences, some of these test-sentences would undoubtedly be themselves unrestricted generalizations, e.g., “if anybody had insulted John at that time, John would have become violent.”

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follow more or less old lines of argument used by Ryle and Ayer7 to the effect that what holds with respect to public verifiability for statements about the past holds also for statements about other minds. The strategy, as it were, of the argument will be this: if statements about the past are conceded to be publicly verifiable, statements about other minds will have to be conceded to be publicly verifiable in the same sense; if statements about the past are held to be publicly unverifiable, the same will have to be said of statements about other minds; but, on this alternative the version of the criterion of verifiability under discussion will simply have to be repudiated, since nobody in his senses would want to lay down a law which leads to the elimination of history along with the elimination of metaphysics; therefore, whichever alternative be true (whether statements about other minds are publicly verifiable or publicly unverifiable), no acceptable criterion of significance can make logical behaviorism inevitable. Consider the statement “exactly 50,000 years ago, it rained where I am now standing.” Is it logically possible that I should have conclusive evidence for this statement? The question reduces to the question: is it logically possible that I should have been alive 50,000 years ago? Now, the statement “I was alive 50,000 years ago” would be self-contradictory only if “I,” in this sentence, were a synonym for the description “the person who was born x years ago” (where x, of course, is less than 50,000) or for a description which entailed information about the person’s birth date. But this could be the case only if a sentence like “I was born 30 years ago” were analytic. But surely such a sentence conveys factual information. One cannot establish its truth by analyzing the meaning of “I,” for “I” is an indicator term like “now,” “this,” that has no intensional (connotative, conceptual) meaning to be analyzed. In this respect a statement of the form “I have property P” is like a statement of the form “this object has property P.”8 In order to get an analytic statement out of the latter statement-form, one would have to substitute for “P” the trivial predicate “being the object referred to by this sentence” (and then one obtains a self-referential sentence which some semanticists would rule out as meaningless). Similarly, to get an analytic sentence out of “I have property P” one would have to substitute for “P” a trivial predicate like “being the per-

7 See

Ryle 1936 and Ayer 1947, chapter III, section 15. first glance this statement may seem to conflict with the possibility of constructing a theory of the self, where “theory of the self” is interpreted, in the formal mode of speech, as a rule for translating sentences of the form “I am in state Q” into sentences not containing the pronoun “I.” But actually there is here no more conflict than there is between the claim that “this” has no connotation and the claim that sentences of the form “this physical object has quality Q” could be analyzed. What is analyzed, in the latter case, is the statement-form “physical object O has quality Q,” and here the analysanda are stated without indicator terms (synonyms for “indicator term” in current literature are “egocentric particulars,” used by Russell, and “token-reflexive terms,” used by Reichenbach). 8 At

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son referred to by this sentence.” But we fortunately need not prove any such general proposition as that all predicate-substitutions of a certain kind in the statement-form “I have property P” yield factual statements. All we need is the admission that a statement of the form “I was born at time t” (where t is a time preceding the time of utterance of the sentence) is synthetic. Then it follows that it is not self-contradictory to suppose that I should have witnessed, and thus conclusively verified, an event which occurred before my birth. At first sight it seems as if the verifiability-by-me of a statement about your mental state constituted a radically diﬀerent problem from the verifiabilityby-me of a statement about an event that preceded my birth. For is it not self-contradictory to suppose that I could be you; and is this not the only condition that would enable me to obtain the required conclusive evidence? Ayer’s answer to this question is worth quoting at some length: although it is a necessary fact that the series of experiences that constitutes my history does not in any way overlap with the series of experiences that constitutes the history of any other person, inasmuch as we do not at present choose to attach any meaning to statements that would imply the intersection of such series, nevertheless, with regard to any given experience, it is a contingent fact that it belongs to one series rather than another. And for this reason I have no diﬃculty in conceiving that there may be experiences which are not related to my experiences in the ways that would be required to constitute them elements in my empirical history, but are related in similar ways to one another . . . This does not mean that any experience can actually be both mine and someone else’s; for I have shown that that possibility is ruled out by the conventions of our language. It means only that with regard to any experience that is in fact the experience of a person other than myself, it is conceivable that it should have been not his but mine. (Ayer 1947, 168-169)

The discussion of this subtle matter may be simplified by the introduction of a few symbols. Let ‘M2 ’ refer to your mind, which according to the logical construction theory of the self, evidently adopted by Ayer, is a series of “experiences” to be denoted by ‘(m2 , ma2 , mb2 , . . . , mn2 )’; and similarly ‘M1 ’ refers to my mind, and is short for the series-symbol ‘(m1 , ma1 , mb1 , . . . , mn1 )’. Ayer interprets “you are in mental state mi2 ” to mean that mi2 belongs to the series M2 by virtue of standing in certain relations to the other elements of that series which it does not bear to any elements of the series M1 . He does not, indeed, tell us what those relations are in terms of which minds or “mental histories” are to be logically constructed, but it is not necessary to work the logical construction theory of the self out in detail in order to deal with the issue at hand. It is a contingent fact, says Ayer, that mi2 should bear these relations to the other elements of M2 rather than to the elements of M1 (presumably in the same sense in which it is a contingent fact that a stone which belongs to a given heap of stones belongs to that heap and not to another heap). And if it had in fact stood in these relations to the elements of M1 , then this very fact, expressed in our

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symbolism by “mi1 occurred,” would have constituted conclusive verification of the statement ascribing this mental state to M2 . I have deliberately restated Ayer’s argument in such a way that the non sequitur, as it were, springs before one’s eyes. If the sentence “mi1 occurred,” which expresses what according to Ayer might conceivably have been the case and would, if it had been the case, have constituted conclusive verification of the statement about the other mind, is translated back into non-symbolic English it reads: “I was in this mental state,” and is thus not a statement about the other mind at all. It remains true, therefore, that to suppose conclusive verification, by the speaker, of a statement about another mind is to suppose the contradiction that a statement be verified which is at once about the speaker’s mind and not about the speaker’s mind. But while Ayer’s argument is invalid, it nonetheless makes the point it was intended to make, viz. that statements about other minds are, with respect to conclusive verifiability by the speaker, in exactly the same boat with statements about the past. What spoils Ayer’s argument is that if the speaker had experienced, say, the toothache which he predicated of the other mind, and at that moment had been asked “which statement is conclusively verified by your present experience?,” he would have replied “the statement ‘I have a toothache now’,” not “the statement ‘he has a toothache now’.” But in just the same way, if I had been standing 50,000 years ago where I am standing now and had observed that it rained, the proposition verified by my observation would then have been formulated by the sentence “It is raining here now,” not by the sentence “It rained here 50,000 years ago.” It could therefore be argued that once a statement about the past was conclusively verified by the speaker it would cease to be a statement about the past. Should we say, then, that public verification, i.e., verification by any observer of any age, including the speaker, of statements about the past is logically impossible? I submit that it is logically arbitrary which way we answer this question. The main point is that, whichever answer we decide upon, we should give the corresponding answer to the corresponding question concerning the public verifiability of statements about mental events not owned by the speaker. The remainder of my argument has already been stated in my “strategic outline” above. Before leaving this question of the logical possibility of direct verification, by the speaker, of a statement about another mind, something may profitably be said about the irrelevance of telepathy to this issue. If it could be shown that instances of telepathy are instances of direct knowledge of other people’s mental states, the possibility of such direct knowledge would of course be established empirically. To take a typical case of alleged telepathic cognition, let A be the telepathic knower, B a person who at this moment thinks of a given card drawn at random from a shuﬄed pack of cards, screened oﬀ from A, and suppose that the frequency of correct answers given by A as to the card B is

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thinking of, is significantly above chance expectation. Suppose we interpret this situation by saying A is directly aware of B’s mental state since his belief evidently could not have been arrived at by inference from observations of B’s body and behavior. What could be meant here by “direct awareness”? If it means that A’s belief about B’s mind is not the result of inductive inference, this fact is irrelevant to this question of the possibility of direct knowledge: A does not have direct knowledge of B’s act of attention in the sense in which B has such direct knowledge. One could, indeed, define a sense of “direct knowledge” in which it would be true to say that A had direct knowledge of B’s mental state: “I directly know that p” = “I believe that p without the use of induction, and p is true.” But this is hardly a relevant sense of “direct knowledge” since in this sense there is no doubt that even direct knowledge of propositions about the past and of unrestricted generalizations is possible: it is, of course, possible that utterly groundless beliefs about the past, about generalizations, about other minds, should be true. The only relevant sense, then, in which it could be said that A has direct knowledge about B’s mind would be: A believes without the use of induction that B’s mind is in this state, his belief is true, and A has grounds for his belief. But once the direct-knowledgeclaim is analyzed in this way, it turns into a self-contradiction! What grounds could A have for his belief if not inductive grounds! The only alternative interpretation of “direct awareness” that remains is: A is directly aware of B’s mental state in the sense in which B is directly aware of it, i.e., introspectively. Evidently this interpretation implies that telepathic cognition is an “intersection of mental histories” since A is thus asserted to own the numerically same mental event which is owned by B’s mind. As Ayer points out, in the quoted passage, such a statement oﬀends against our present linguistic rules according to which the expression “two people owning an identical experience” has no significant usage. It may be added that this interpretation presupposes that at least the mental state of the telepathic cognizer, when telepathic communication occurs, is of the same kind as the mental state cognized; which is not true in general, since the cognized mental state may, for example, be a state of desire, and if the knower at the moment of telepathic cognition shared that desire his mental state could not be described as cognitive at all. If such a coincidence of desires in two minds occurred, and did not admit of ordinary psychological explanation, one might look for a special kind of causal explanation, but one would not describe the coincidence as a case of cognition at all.

4.

Physicalism as an Analytic Thesis

So far we have been concerned exclusively to show that no plausible version of the principle of verifiability could be said to entail physicalism or logical behaviorism, meant as the thesis of the synonymity of psychological sentences

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with sentences mentioning only physical events or states. As was mentioned, however, since the recognition of reduction sentences as indispensable tools of meaning specification, this earlier thesis has been abandoned anyway, and present-day physicalists merely assert the reducibility of psychological terms to physical terms (specifically to terms of the “thing-language,” which is the non-technical part of the physical language chosen as a common reduction basis for the language of psychology and the language of physics).9 The point to be argued in this concluding section is that physicalism in the specified sense is no “theory of mind” at all; that it provides no answer to the question what is meant by “mind” and “mental event”; that it is merely an analytic consequence of what is understood by “intersubjective language”; that it, therefore, cannot be in conflict with any philosophical theories concerning the relation of mind to body; that, in particular, it would be a gross misunderstanding to suppose that physicalism is somehow materialism in the semantic mode of speech and a refutation of psycho-physical dualism. As a preamble to the proof of these assertions, let us again quote Carnap, this time from a later publication with which he is probably still in substantial agreement: We do not at all enter a discussion about the question whether or not there are kinds of events which can never have any behavioristic symptoms, and hence are knowable only by introspection. We have to do with psychological terms, not with kinds of events. For any such term, say, ‘Q’, the psychological language contains a statement form applying that term, e.g., ‘The person . . . is at the time . . . in the state “Q”.’ Then the utterance by speaking or writing of the statement ‘I am now (or: I was yesterday) in the state “Q,”’ is (under suitable circumstances, e.g., as to reliability, etc.) an observable symptom for the state Q. Hence there cannot be a term in the psychological language, taken as an intersubjective language for mutual communication, which designates a kind of state or event without any behavioristic symptom. Therefore, there is a behavioristic method of determination for any term of the psychological language. Hence every such term is reducible to those of the thing-language. (Carnap 1950b)

Carnap here explicitly leaves it an open question whether there might not occur mental events without any behavioristic symptoms, such as, I suppose, thoughts and wishes of the deceased which are not communicable to the survivors, except perhaps in a s´eance. Notice that this admission already involves a repudiation of the sort of logical behaviorism which Gilbert Ryle, in his provocative book The Concept of Mind (Ryle 1949a), plays up as an eﬀec9 Reducibility

of terms, it should be noted, does not insure reducibility of laws. Even if the terms of science S 1 were explicitly definable with the help of terms of science S 2 , it would not follow that the laws of S 1 are deducible from the laws of S 2 . For example, terrestrial mechanics and astronomy operate with the same set of primitive terms, but it does not follow that their respective laws are inter-deducible; as a matter of fact, they would have remained separate sciences if it were not for Newton’s unifying discovery of the law of universal gravitation.

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tive antidote against the dualistic “Cartesian Myth,” and according to which to talk about mental events is to talk about behavioral dispositions if not overt behavior. It is true that the opening sentence of the above quotation could be interpreted as a polite hint that the “question” is regarded by the author, like most philosophical questions formulated in the “material mode of speech,” as a pseudo-question. But this interpretation would be in conflict with Carnap’s explicit statement, made in the same paragraph, that a proposition like “I am angry” is clearly knowable by introspection without observation of behavior. As a matter of fact, I would maintain that the fundamental reason behind the substitution of reduction sentences for explicit definitions as bridges from the psychological language to the physical language is not the reason given by Carnap himself for the use of reduction sentences, but a consideration which brings him into substantial agreement with psycho-physical dualism. If we gave an explicit definition of a dispositional predicate “P,” Carnap argued, of the form (1) “Px ≡ (Ox ⊃ Rx)” (where “O” refers to a test operation and “R” to a test result), we would get the paradoxical result that “P” would be predicable of x provided the test operation is never performed upon x. This paradox derives, of course, from the definition of material implication according to which a material implication is true provided its antecedent is false. This paradox is avoided by using the reduction sentence (2) “Ox ⊃ (Px ≡ Rx)” which, not having the form of an equivalence, does not enable elimination of the predicate “P.” But what if a satisfactory definition of the nomological meaning of “if-then” (Reichenbach’s term for the meaning of this connective in statements of natural laws or causal connections) were at hand which allowed us to restate (1) in terms of nomological implication instead of material implication? Would this destroy the raison d’ˆetre of reduction sentences? It is easily seen that it would still be impossible to construct an adequate explicit definition of psychological sentences in terms of sets of behavioristic test-conditionals (conditional sentences of the form “if stimulus S acts on organism O, then response R occurs”) or any kind of physicalistic sentence at all. For according to the standard method of testing the adequacy of explicative definitions, practiced by analytic philosophers however divergent their theories about philosophical analysis may be, such a definition would imply that at least one physicalistic sentence is logically entailed by a psychological sentence like “I experience now such and such a color image.” But it is evidently a question of fact which behavioral and physiological symptoms are correlated with which mental states of an organism. This is the fundamental reason why reduction sentences, which are sentences with factual content though they have a semantic function,10 have to be employed. If by “psycho-physical dualism” 10 It

is, indeed, a pressing question just what precisely is conveyed by the antithesis factual-analytic, as applied to non-formalized languages, if a reduction sentence is said to be in a sense both factual and a

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is meant the logical (or semantic) thesis that for any mental event m and for any physical event p it is logically possible that m should occur without p and p without m, there is nothing in physicalism to conflict with it. We must now examine the meta-linguistic formulation of physicalism contained in the above quotation from Carnap. Carnap maintains that if “Q” is a psychological term whose meaning is communicable, then there must be at least one reduction sentence connecting the mental state Q with a behavioristic symptom, since “the utterance by speaking or writing of the statement ‘I am now (or: I was yesterday) in the state Q’ is (under suitable circumstances, e.g., as to reliability, etc.) an observable symptom for the state Q.” But it turns out that in order to obtain a plausible reduction sentence of this kind a “suitable circumstance” will have to be specified which makes the reduction circular. The reduction sentence “if x is in state Q at time t, then, if x is asked at t ‘in what state are you now?’, x will (under conditions C1 , . . . , Cn ) reply ‘I am in state Q”’ is halfway plausible only if the set C1 , . . . , Cn contains the condition “x knows that ‘Q’ means Q”! But thus the reduction basis for the term “Q” contains not only a name of the term “Q” but the term “Q” itself, which makes the test-sentence ineﬀective in just the same sense in which a circular definition is ineﬀective as a semantic rule of application. This criticism, however, hits only an unfortunate formulation of the proof of physicalism, not the central idea of the proof. I propose the following alternative formulation. What does it mean to communicate the meaning of a psychological word, say, “happy”? One might think that such communication involves a theoretical diﬃculty which does not confront us if the problem is to communicate the meaning of a physical term, say, “hard.” The latter problem is solved by the method of ostensive definition, of pointing at objects to which the predicate is applicable; but how, so it may be asked rhetorically, can one “point at” a mental state, such as happiness? Little analysis, however, is required to see that essentially one and the same method of conditioning is used in both cases. If I expect that by uttering the word “hard” while you touch hard objects I can get you to understand what I mean by the word, it is because I expect that similar tactile sensations will be produced in you when you touch objects which produce such sensations in me (our psycho-physical argument from analogy again!) and that you will in time associate a memory-image of that kind of sensation with the verbal stimulus. Similarly, in order to communicate what I mean by “happy” I must catch you in, or get you into, a situation in which I have inductive reasons to believe you are in a state of happiness, and produce a corresponding habit of association in you. Now, if there were no reliable correlations between feelings of happiness and behavioral symptoms of partial specification of meaning. In my opinion this is one of the most urgent problems of semantic analysis, stimulated but far from solved by Carnap 1937.

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happiness—if, in other words, observations of a certain kind of behavior would never justify our inference “probably he feels happy now”—we just could not use this method of conditioning in order to communicate our meaning; we would have no way of telling when a feeling of happiness is likely to occur in a child to whom we wish to communicate the meaning of “happy.” It follows that the proposition “there are inductive reasons to believe that ‘Q’ is intersubjectively meaningful”11 entails the proposition “there are inductive reasons to believe that there are reliable behavioral symptoms of the mental state Q.” If there are such behavioral symptoms, their names can be used as a reduction basis for the psychological term “Q.” Hence the proposition “if ‘Q’ belongs to an intersubjective language, then there is a behavioristic reduction basis for ‘Q”’ is purely analytic. And evidently an analytic statement cannot be incompatible with a speculative existential statement like “there are mental states (states for which there are no intersubjectively meaningful designations!) that are entirely uncorrelated with bodily states.” I conclude, with my eye on those who feel that logical positivism, if true, shatters their fondest hopes, that the postulate of the reducibility of intersubjectively meaningful terms to the prosaic terms of the “thing-language,” like the postulate of verifiability, is completely neutral with respect to the traditional speculative question concerning the causal possibility of disembodied minds.

11 To

be quite accurate, we should say “intersubjectively meaningful and unambiguous,” since it is conceivable that I communicate a psychological meaning of “Q” to you but not the meaning I intended to communicate.

Chapter 17 SEMANTIC ANALYSIS AND PSYCHOPHYSICAL DUALISM (1952)

In his highly significant and provocative book, The Concept of Mind, Gilbert Ryle undertakes to show that the Cartesian theory of two worlds, the physical world characterized by publicity and the mental world characterized by privacy, inaccessibility to all but one, is a “myth” created by misunderstandings of language. His method is an excellent example of the Wittgensteinian method of diagnosing the origin of puzzling philosophical theories as pointless, confusing departures from ordinary language. If, for example, a man puzzles how on earth it is possible ever to verify a proposition about the future since, after all, one cannot observe an event that has not yet occurred, his puzzle is of the kind which can be eﬀectively dissolved by the Wittgensteinian treatment. The purpose of the following discussion is not a comprehensive critical review of Ryle’s book. It is rather to show that his crucial arguments against the “privacy theory” of sensations—which is the aspect of psycho-physical dualism most relevant to epistemology—fail to establish what they are supposed to establish; and that they fail mostly because they exhibit a kind of exploitation of ordinary usage for purposes of criticism of philosophical theories which is characteristic of the Wittgensteinian method and which I consider to be an unwholesome feature of an otherwise most sanitary movement of semantic hygiene. Ryle’s basic thesis is that the theory of mental acts like believing, knowing, aspiring, results from the failure to see that sentences containing such psychological verbs are statements about dispositions, not about events or processes. I do not wish to add here to the literature pro and con the theory of mental acts; for, as stated, my specific purpose is to examine Ryle’s arguments, set forth in chapter 7 (“Sensation and Observation”), against the Sense-Datum theory of Descartes, Locke, and Hume, which survives vigorously in the epistemological writings of Russell, Broad, and Moore. I cannot refrain, however, from expressing the opinion that Ryle’s discussion is inconclusive because he has failed to formulate and discuss the basic semantic questions at issue between behaviorism and dualism: (1) Are any behavioristic sentences analytically entailed by such “mentalistic” sentences as “I remember to have seen this person

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before,” i.e. are there any behavioristic sentences with which a sentence of the latter type is logically incompatible; or is the relation between the mentalistic and the behavioristic language only that of reducibility in Carnap’s sense? If the latter is the case, as maintained by the contemporary form of physicalism, the most enlightened semanticist can go on believing in “Cartesian ghosts” since dualism is perfectly compatible with the public confirmability of mentalistic statements. (2) Is the above formal clarification of the issue really so clarifying, considering that the distinction between L-implication and Fimplication is clearly defined for formalized language-systems but not for natural languages? If this skepticism with regard to the very tools of non-formal semantic analysis, which is becoming more and more vociferous, should be unanswerable, it would be anything but clear what Ryle is saying when he says that statements which the dualists interpret as referring to “ghostly” mental acts are really about behavioral events or behavioral dispositions. I think the following is a fair summary of Ryle’s basic criticism of the SenseDatum Theory. Sensations are wrongly regarded as ways of observing, as though to see, to hear, to touch, to smell, were necessarily to observe something. The making of observations involves, indeed, the having of sensations, but it involves, moreover, an active element, the attempt to find something out, and above all, what one can properly be said to observe (where proper usage = ordinary usage) is always a physical object, event, or process—where “physical” is used broadly so as to cover anything capable of being publicly witnessed, including, e.g. purposive behavior. “. . . the ordinary use of verbs like ‘observe’, ‘espy’, ‘peer at’ and so on is in just such contexts as ‘observe a robin’, ‘espy a ladybird’, and ‘peer at a book”’ (Ryle 1949a, 224). Ryle thus implies that the verb “to observe” (or synonyms like “to watch”) is misused if it is used to describe an apparition in a dream or a state of hallucination—a contention surely open to debate though it will not be debated now. We may, I think, formulate Ryle’s central criticism of dualistic epistemology as follows: Knowledge is a state of mind (= behavioral disposition?) terminating a successful inquiry; where there is no attempt to find something out, there is no occasion for knowledge to eventuate (this, of course, is again a statement about proper usage). It follows that the mere having of sensations does not constitute knowledge and that to speak of absolutely certain, immediate knowledge, expressed by such sentences as “I see red,” “I taste a bitter taste,” is to misuse the word “knowledge” (notice that hyper-semanticist Ryle and anti-semanticist Dewey shake hands here). The Sense-Datum Theory postulates a curious kind of performance, called “pure sensation,” which diﬀers from observation ordinarily so-called in that it does not logically involve physical objects. But since an act requires an object (at least if it is expressed by a transitive verb, as are the verbs “seeing,” “hearing,” “touching,” etc., in ordinary usage), and pure sensations by contrast to perceptions have no physical objects, the theory invents an appropriate kind of object called “Sense-Datum.” With the help of

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this device the epistemologist constructs a pure sense-datum language whose sentences have the same subject-verb-object structure as sentences describing perceptions of physical objects: “I taste a sour sense-datum” resembles “I taste a lemon,” “I see a round sense-datum” resembles “I see a round penny,” etc. It must be conceded, I think, that Ryle correctly diagnosed the linguistic origin of one form of the Sense-Datum Theory, viz. the form which splits pure sensations into an act and a non-physical object called variously “sense-datum” or “sensum” or “sensibile.” I am thinking particularly of Broad’s theory of Sensa, which involves the postulation of non-physical entities which, like Platonic essences (except that they are “sensible” rather than “intelligible”) are somehow there to become objects of the kind of awareness called “pure sensation”: “It is quite certain that there is a diﬀerence between the two propositions: ‘This is a red round patch in a visual field’ and ‘This red round patch in a visual field is intuitively apprehended by so-and-so’. Even if as a matter of fact there are no such objects which are not intuitively apprehended by someone, it seems to me perfectly certain that it is logically possible that there might have been” (Broad 1949, 209-10). If all that Ryle had intended to establish by revealing that sensation is not a form of observation had been that it is misleading to speak of sensations as (mental) acts, I would have no further quarrel with him—except remarking what is perhaps obvious, viz. that one can consistently repudiate the act-theory of sensation and still believe in other kinds of mental acts, such as deciding, inferring, etc., as events defying definitional reduction to publicly observable actions. But evidently this accomplishment would not be enough to lay the Cartesian ghosts of traditional dualistic epistemology. Ryle claims moreover to dispose of the myth of private facts (or events), called “sensations,” which constitute the subject-matter of immediate, absolutely certain knowledge. But if it is linguistic confusion to suppose, with Broad, that there are mysterious non-physical objects, called “sensa,” ready to be sensed, it does not in the least follow that it is linguistic confusion to suppose that there are events, called “sensations,” which are not publicly observable in the way physical events are publicly observable. How does Ryle propose to show that it is nonsense to speak of sensations as events whose occurrence can be said to be known, yet never publicly known, i.e. known by anybody except the person owning the sensation? As far as I am aware, he uses two main arguments, which may be called: (1) the argument from the impossibility of a pure sense-datum language, (2) the argument from the impropriety of speaking of observing sensations. (1) Before entering into explicit discussion of the first argument it might be pointed out that by no means all philosophers who speak the language of sense-data are guilty, like Broad and like Moore at the time of (Moore 1903) of the crime commendably lamented by Ryle, to split sensation-events into act and non-physical object. In Ayer’s sense-datum language, for example, “I see a red sense-datum” is simply an artificial way of saying “I seem to see

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something red,” the latter sentence involving the ordinary perceptual use of “to see,” i.e., the use in which “I see x” entails that x is a physical object ready to be seen. (Strictly speaking, what is here called the perceptual use is not the ordinary usage but at best the most frequent usage. It is proper to speak of what one saw in a dream. If the perceptual use were the only proper use, a man describing his dream in the “I saw . . . ” idiom rather than the “I seemed to see . . . ” idiom would be speaking either improperly or falsely—which I doubt.) A philosopher who finds it convenient to speak of sense-data, therefore, is not necessarily committed to the analysis of perceptual situations into a relative product involving over and above the observer and the physical object an intermediate entity: (x perceives y) = (∃z)[(x directly senses z) and zRy], where the interpretation of “R” is the big headache for this kind of Sense-Datum Theory. But assuming that Ryle would grant this point, we may formulate argument (1) as follows. Let “S p ” denote a sentence reporting a perception, like “I see a robin,” a sentence which could not be true, that is, unless a physical fact existed (“there was a robin at that time and at that place”); and let “S s ” denote a sentence which is just like an S p except that it has no physical implication— understanding “implication” here in the sense of “analytic implication,” since by virtue of psycho-physical laws (correlations of sensations with brain-events) any S s may factually imply physical facts. Ryle’s argument then is that no S s can be constructed, but that the moment one attempts to construct such a sentence, it inevitably turns into an S p . “I see blue,” e.g., can only be explained by translating into “I see the color usually presented by such objects as A, B, C,” and thus implies by its very meaning the physical proposition that there are (in the tenseless sense of “there are”) objects of kind A, B, C. Now, it is nonsensical to assert the occurrence of events (events described by S s sentences) which it is in principle impossible to describe: “Whereof one cannot speak, thereof one must be silent” (the concluding sentence of Wittgenstein’s Tractatus (Wittgenstein 1933) which latter, by the author’s own ironical confession, is one big sin against this precept). Now, I shall argue that what Ryle has shown is that the concept of an S s sentence with a communicable meaning is self-contradictory—and in that sense meaningless—but has not shown what needs to be shown in order to refute the theory of private sensations, viz. that the supposition of the occurrence of pure sensations unconnected with physical situations is meaningless. Suppose I taste an apple-like taste (I mean this sentence in a way in which it does not logically imply the possession of taste-organs). The only way I can explain the meaning of “apple-like taste” to another person is by directing him to perform the physical operations which I believe, on the basis of analogical reasoning, will produce a similar taste-sensation in him. If the occurrence of taste-sensations were entirely uncorrelated with such operations as biting into an apple and thus were entirely unpredictable, we simply would lack the means

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of developing an intersubjective language of tastes. But the very fact that this statement concerning the necessary condition of an intersubjective language of sensations is intelligible proves that the hypothesis of a world in which sensations are entirely uncorrelated with definite physical situations, a world in which one would never be in a position to say “it is probable that he is experiencing such and such a sensation now,” is meaningful (cf. on this point, Schlick’s article “On the Relation between Physical and Psychological Concepts” (in Feigl and Sellars 1949, German original in Schlick 1938), especially section VII. That ordinary language contains no “neat” sensation-vocabulary, i.e., a vocabulary usable for constructing sensation-reports wholly devoid of physical implications, simply proves that ordinary language is intersubjective. Naturally, there is no other way of communicating to a child the meanings of such words as “sour” and “red” but to direct him to perform the kind of physical operations (tasting lemon juice, looking at blood) which one expects, be such expectations logically justifiable or not, will cause him to experience the kind of sensations designated by the words.1 Perhaps the following simile will clarify my argument. Suppose there existed a few circles to which I alone had privileged access, and that in my soliloquies I occasionally talked about those circles, using the word “circle.” Assuming that no other circles exist and that I alone could see the circles that exist, how could I explain to other people what I mean by “circle”? Well, I could give a so-called causal definition by saying “a circle is the kind of figure which you would produce if you rotated a stretched thread kept fixed at one end.” Suppose furthermore that a law holds in this hypothetical universe according to which a circle once produced by the described method is visible only to the person who produced it. It is clear that in this universe the meaning of “circle” could not be communicated by pointing to public circles but only by reference to straight-lines in rotational motion (and, incidentally, it would be theoretically impossible to make sure that one had communicated one’s meaning, since, by hypothesis, one could not verify whether the experiment leads to the same result when it is performed by other people). Would it make sense for those people to speak of a possible universe in which there were circles but no straight-lines and no rotational motions? It surely would be a mean-

1 In the “Afterthought,” at the end of Ryle 1949a, chapter 7, Ryle confesses to a bad conscience about having

adopted the epistemologist’s “sophisticated” use of the word “sensation.” Ordinary language, he contends, contains no such expressions as “visual sensation” and “auditory sensation” (where ordinary language is contrasted with both philosophical and scientific language). I confess that this bad conscience impresses me as just another example of that pathological oversensitivity, cultivated in some circles of analytic philosophy, to philosophers’ departures from ordinary language. What philosophers find epistemologically interesting about painful sensations (in one’s leg or throat or eye), viz. the “privacy” of these events, is just a property which likewise characterizes the events described by such sentences as “I seem to see a red patch.” It is for this reason that they extend the term “sensation” to these events. Why should philosophers feel guilty about such departures from ordinary usage any more than scientists?

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ingful hypothesis, I should think, even though they would have to admit that in such a universe the word “circle” would lack a communicable meaning. I think that Ryle’s argument from the impossibility of a neat sensation-vocabulary is parallel to the argument of that apostle of ordinary language who got up and accused his speculating philosopher friend of contravening the ordinary usage of the word “circle” in speaking of circles isolated from straight-lines and rotational motions. It might be added, before proceeding to examination of the second argument, that Ryle’s arguments from ordinary usage against the epistemologist’s language are largely irrelevant because they overlook that the epistemologist is engaged in what has been called logical reconstruction of empirical knowledge, and in this process inevitably and quite consciously develops an artificial language. No philosopher, to my knowledge, has ever claimed that atomic sentences, i.e. sentences which are logically simple in the sense that any two such sentences are logically consistent and logically independent, occur in ordinary language, nor that they describe isolated events—pure sensations actually occurring outside of perceptual contexts. If a sentence on the atomic level, like “I see red now,” is to be construed as a sentence of ordinary English which may be true or false, it must be translated into “I see something red now” and thus ceases to be atomic: it has turned into an existential statement describing a perceptual situation2 involving both sensation and a physical object. True, but irrelevant, for it is the epistemologist’s purpose to reveal a level of language which is implicit in ordinary language though never spoken. A brief analogy may serve to remove the air of paradox from this concept of implicit language. It is impossible to imagine a sound-intensity divorced from any definite soundpitch, and accordingly a sentence describing an experience of sound-intensity isolated from an experience of sound-pitch could hardly be found in ordinary descriptions of sound phenomena. Still, such sentences may have to be constructed if one wanted to make the complexity of the meaning of ordinary sound-language explicit (“a loud tone was heard” would not be suﬃciently atomic, since “tone” already designates a complex of pitch, intensity, and quality). As a matter of fact, it might be conjectured that some epistemologists have fallen into the confusions justly criticized by Ryle just because they did not go far enough in divorcing their artificial language of logical reconstruction from ordinary speech forms: “I taste sour” does not sound grammatical (unless it is a lemon that is speaking), so the epistemologist says “I taste a sour sensedatum” in order to preserve the subject-verb-object form of ordinary sentences

2 My

use of “perceptual situation” in this paper diﬀers from Broad’s usage in that hallucinations, i.e., situations in which one seems to see an object which does not exist, are not perceptual situations at all according to my usage, while according to Broad they constitute a species of perceptual situation, called “totally delusive perceptual situations.”

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describing perceptual situations (“I taste a sour fruit”). And in some instances, as already admitted, these sentences of the epistemologist’s own making have led to the mythical belief in sense-data as sensible, yet non-physical entities. (2) Ryle undertakes to show not only that the mistaken assimilation of sensation to observation led semantically confused epistemologists to the invention of sense-data, but also that it is nonsense to think of sensations as possible objects of observation. “To observe a sensation,” he holds, makes no more sense than “to spell a letter” (Ryle 1949a, 206). If this is so, then the assertion that sensations, unlike physical events, cannot be observed by other minds is, indeed, of the same order as the assertion that letters cannot be spelled. The tacit implication of this comparison is evidently that the theory of absolutely certain knowledge of private sensations constituting the infallible basis of all our fallible beliefs about the physical world and other minds, vigorously defended again by Russell in (Russell 1948), is as devoid of significance as would be the emphatic assertion that only words, not their constituent letters, can be spelled. Now, Ryle’s unquestionably superb knowledge of the English language denies me the right to throw doubt on his assertion that it is not proper usage to speak of “observing” or “witnessing” one’s headache the way one may speak of “observing” or “witnessing” a car accident, while it is proper to use the verb “noticing” in this context (Ryle 1949a, 206). But what of it? The psychophysical dualist, I am sure, will gladly accommodate his language to the rules of good Oxford English and henceforth speak of sensations as events which cannot be noticed by more than one person at a time, unlike such physical events as car accidents. If Ryle allows the propriety of speaking of a “noticed headache,” he should also allow the propriety of the sentence “I know that I have a headache—though you may doubt it”; but this is an excellent instance of private knowledge in the sense in which according to the dualist all knowledge of sensations is private. But in fact Ryle’s subtle investigation of the proper usage of such English verbs as “observing,” “witnessing,” “noticing,” seems to me to be wholly irrelevant to the epistemological problem. The question is whether it makes sense to speak of sensations as events whose occurrence can be known with certainty by one person but not with the same doubt-precluding certainty by other people. If this question is answered aﬃrmatively, the “Cartesian ghosts” have not been eﬀectively banished. And the only relevant semantic question in this connection is whether it is proper to speak of “knowledge” of the occurrence of a sensation. We could easily imitate Ryle’s method of observing carefully the speaking habits of educated English or American people, and cite in support of the propriety of such a usage of “knowledge” such sentences as “I know that these patches look all the same color to me whatever they may look like to you—you can only know what they look like to you.” But it is more important to point out that Ryle is logically committed by his own statements

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to acceptance of the propriety of the expression “knowledge of sensations.” Observation of a robin, we are told, entails the having of sensations, though it involves more than that. Put in the formal mode, this means that the statement “I observed a robin” could not be true unless some statement reporting a sensation were true. But if it makes sense to speak of the truth of statements reporting sensations it must surely also make sense to speak of knowledge of the events reported. I shall make some concluding remarks about the recent tendency of analytic philosophers practicing therapeutic positivism,3 a practice brilliantly exemplified by Ryle’s The Concept of Mind, to regard the theory of the logical privacy of the mental as merely the upshot of arbitrary linguistic conventions. Consider the statement “it is logically impossible for one person to notice directly (notice that ‘notice’ has been substituted for ‘observe’) another person’s headache, the way he could notice directly another person’s sneeze; that’s what makes the former event mental and the latter physical.” The usual reaction of therapeutic positivists to this assertion of the impossibility of direct knowledge about other minds is to ask: “Well, what would it be like to notice (directly) another person’s headache? Would this not amount to noticing a headache which is yours and not yours at the same time? But then the event which you say is impossible is simply a self-contradiction, just like the event of spelling a letter (= decomposing into its constituent letters what is not composed of letters). Your talk about the ‘logical privacy of the mental’, then, is as pointless as would be talk about the impossibility of spelling letters.” I would be the last one to deny that this kind of analysis of the expression “logical privacy of the mental” is immensely clarifying. It is important to realize that it would be self-contradictory, in terms of current meanings of the expressions “thought,” “direct knowledge,” to suppose that A could have direct knowledge of B’s thoughts in the sense in which B could have such knowledge. One important consequence of this semantic fact is that telepathy should not be interpreted as direct knowledge of another mind’s mental states but rather as direct belief, i.e. belief not derived from physical observations, about another mind which stands a high chance of being correct. This may be demonstrated as follows: if we credited A with direct knowledge of B’s present thought in the sense in which we credit B with such knowledge, then we would accept the following two premises as conclusive evidence for the proposition describing B’s mental state: (1) A asserts that B is in this mental state, (2) A’s assertion is honest, i.e. A believes what he says. But the fact is that while we would accept

3 I am not using this label so as to imply the conception of philosophy as resembling closely psychoanalysis in method and purpose. By a therapeutic positivist I mean a philosopher who conceives of (good, worthwhile) philosophy as having primarily a therapeutic purpose, viz. the dissolution of typically “philosophical” perplexity by revealing the linguistic confusions which cause such perplexity.

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these premises as providing conclusive evidence if A and B were one and the same person (not that I mean to imply that we could have conclusive evidence for the truth of these premises in the sense of “conclusive evidence” in which the premises, if true, would be conclusive evidence for the conclusion), we don’t if they are diﬀerent persons: we would first ask B directly about his mental state before we accepted A’s telepathic belief as an instance of knowledge. The point I am arguing can be made particularly evident if we suppose that the mental state of B which A is alleged to have direct knowledge of is a state of belief. For in that case the claim would either mean that A is introspecting his own belief which happens to be shared simultaneously by B, or that the state of belief is literally a common constituent, a point of intersection as it were, of the mental histories of A and B. In the first case there is no telepathic situation at all, in the second case we are violating the ordinary usage of the expression “mental state” in somewhat the same way as we would violate the ordinary usage of “particle” if we spoke of the possibility of the same particle occupying simultaneously diﬀerent places. Yet, I submit that after all the virtues of this kind of semantic analysis have been recognized, one is left with an enormous non sequitur when one turns to the claim that the dualist’s concept of the mental defined by logical privacy is nonsense. All that has been shown is that the assertion of the impossibility of certain, immediate knowledge of unowned mental states is an analytic consequence of the meanings of the expressions “certain, immediate knowledge” and “unowned mental state,” and not a factual thesis like the impossibility of catching a certain thief. The further premise, which would have to be added and substantiated before the argument of Ryle and associates could be accepted, is that those meanings are gratuitously stipulated by dualists in sheer violation of ordinary usage. To illustrate such gratuitous deviationism in linguistic conduct: if a philosopher arbitrarily defines “there is evidence for p” to mean “p is logically demonstrable” and then announces that it is impossible to produce evidence for empirical propositions referring beyond present experience; or, if he defines “seeing a physical object” to mean “seeing simultaneously the whole surface of the object as well as all its past and future states” and then announces that common sense is mistaken in believing that physical objects could be seen; or if he defines “seeing the same object A again” to mean “seeing A again with exactly the same properties it had previously” and derives from his definition the paradoxical consequence that it is impossible to see the same object twice—in cases of this sort philosophers are rightly blamed by therapeutic positivists for building perplexing philosophical theories by means of nonsensical (i.e. self-contradictory) concepts misleadingly associated with meaningful everyday expressions. The question before us is: Is a philosopher similarly guilty of arbitrary departure from ordinary usage when he says “I can

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be aware only of my feelings, not of anyone else’s; I cannot know for certain, though I may conjecture, what other people’s thoughts are”? That we do, in everyday discourse, speak of awareness and certain knowledge of other people’s feelings and thoughts, is admitted. In fact, such sentences as “I am clearly aware of her jealousy; I know he thinks I do not know about his aﬀair with Mrs. X” are, in the sense in which they are intended, frequently known to be true. The philosopher’s claim that such sentences cannot be known to be true or to be false seems, therefore, like the same sort of willful paradox, brought about by surreptitious departures from the ordinary meanings of words. Yet, the semantic situation here diﬀers in an important respect from the kind of “philosophical perplexity” illustrated above. In the cases I adduced, the tricky philosopher convicts the man in the street of cockily holding unjustifiable beliefs, by substituting for his vague but applicable concepts, precise but self-contradictory (and so inapplicable) concepts. But this idle game must be distinguished from the fruitful game of revealing ambiguities in ordinary language which are unnoticed by the man in the street. Suppose I asked the ordinary mortal for whom knowledge of other minds is simply a fact, no problem at all: “Are you aware of your friend’s loneliness in the same sense of ‘awareness’ as you might be aware of such a feeling in yourself?” Surely, Mr. Common Sense need not be analytical above average to come to see, after some reflection, that there is a big diﬀerence between introspective awareness of a feeling, which is in a sense not easy to analyze infallibly, and inference to similar feelings in others, which is in an obvious sense fallible. The essential point is that this concept of direct, introspective knowledge of a mental event is a concept in actual use, not a concept willfully constructed by philosophers intent on belittling common sense. The dualist’s statement “it is impossible to know that another person feels lonely the way one can know the occurrence of such a feeling in oneself” is, indeed, analytic of the relevant meaning of “knowledge.” But the statement’s analyticity would convict it of triviality only if the concept analyzed were one artificially constructed and never used by non-philosophers. If the advocates of common speech whose attack on psycho-physical dualism I have criticized were to use their method consistently, they would, I suspect, approach practically any philosophical analysis of a concept in use with the same cynical attitude. Take, e.g., Hume’s denial of a necessary connection between cause and eﬀect. This denial, of course, is exactly the same kind of “violation of ordinary usage” as the dualist’s denial of the possibility of knowing for certain what another person is thinking or sensing or feeling: we do say, quite often, “this eﬀect must necessarily follow. . . ,” and in the sense in which this kind of statement is intended we frequently have excellent grounds for asserting it. But the ordinary man will quickly cease to be puzzled by Hume’s statement if the latter is clarified in the usual fashion, viz. as the denial

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of the logical demonstrability of causal laws. The fact that the ordinary man now discovers that he never really held the belief just demolished by the subtle philosopher does not convict the philosopher of having launched a Quixotic attack on windmills. For, as Moore has taught us, the purpose of analytic philosophy is not to refute common-sense beliefs nor to declare them as unfounded, but to clarify them. Thus Hume, whatever his own confusions may have been, has the merit of having clarified the diﬀerence between inductive and deductive inference, empirical and logical connection, a diﬀerence obscured by the ambiguous use of the words “reason,” “ground,” “consequence” (even “cause,” in the Aristotelian tradition) in both senses. The concept of the mental as the logically private, in contrast to what is publicly observable, is similarly the result of analysis of confused meanings. And I submit that if by psycho-physical dualism be meant no more than the conception of the mental here defended against Ryle’s ingenious assault, there is no reason to suspect psycho-physical dualists of metaphysical backwardness isolated from the progressive tools of modern semantic analysis.

VI

PHILOSOPHY OF SCIENCE

Editors’ Note: Throughout this part Pap uses a notational convention adopted from Carnap 1937, which is described in full in the Introduction, at page 18.

Chapter 18 THE CONCEPT OF ABSOLUTE EMERGENCE (1951)

I understand the business of the philosophy of science to be painstakingly careful analysis of concepts, principles, and methods used in science. This broad statement fails, of course, to diﬀerentiate such analysis of concepts as inevitably occurs in a developed science itself (e.g. analysis of the concepts “simultaneity,” “absolute motion,” “energy,” etc. in physics) from such analysis as is more likely to be the professional concern of the philosopher of science as philosopher. The most natural object of distinctively philosophical analysis concerned with science would seem to be the very activity, or class of activities, defining science in general, rather than some one specific science. As it goes without saying that one such activity characteristic of science is prediction, the analysis of the concept of predictability is a vital task of the philosophy of science. Ability to predict is a virtue marking a good scientist; ability to analyze clearly the concept of predictability is a virtue marking a good philosopher of science. Now that permanent limits are set to the scientist’s ability to predict by “the very nature of things” (if I may be permitted to use loose language in paraphrasing a doctrine propounded primarily by philosophers who speak without precision) has been one of the inspiring themes of the doctrine known as the theory of “emergent evolution.” The best known emergent evolutionists in the English tradition are probably S. Alexander, the author of Space, Time, and Deity (Alexander 1920), and C. L. Morgan, the author of Emergent Evolution (Morgan 1923) and, more recently, The Emergence of Novelty (Morgan 1933). Their central idea was that the process of evolution produces more and more complex “levels,” like the atomic level, the level of chemical compounds, the biological level, etc.; and that on each level new qualities emerge which are absolutely unpredictable on the basis of the laws applying to the lower levels. Perhaps the best way to impress upon the reader the urgency of analyzing the meaning of the doctrine before either embracing or rejecting it, is to present a sample or two of the language through which Alexander expresses his metaphysical insight:

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The Concept of Absolute Emergence (1951) The higher quality emerges from the lower level of existence and has its roots therein, but it emerges therefrom, and it does not belong to that lower level, but constitutes its possessor a new order of existent with its special laws of behavior. The existence of emergent qualities thus described is something to be noted, as some would say, under the compulsion of brute empirical fact, or, as I should prefer to say in less harsh terms, to be accepted with the “natural piety” of the investigator. It admits no explanation. (Alexander 1920, vol. 2, 46) A being who knew only mechanical and chemical action could not predict life; he must wait till life emerged with the course of Time. A being who knew only life could not predict mind, though he might predict that combination of vital actions which has mind . . . Now it is true, I understand, that, given the condition of the universe at a certain number of instants in terms of Space and Time, the whole future can be calculated in terms of Space and Time. But what it will be like, what qualities it shall have more than spatial and temporal ones, he cannot know unless he knows already, or until he lives to see. (Alexander 1920, vol. 2, 327-328)

I have italicized the phrases which cry out most loudly for analysis. The following semantic discussion of the problem of emergence will not, however, make any further reference to Alexander. My point of departure will be, instead, a similar view expressed with far greater precision by a far more lucid philosopher than Alexander: C. D. Broad, in The Mind and its Place in Nature (Broad 1949). My purpose is to shed some light on the old question of emergent qualities, thrown into prominence mainly through the vitalism versus mechanism issue in the philosophy of biology, by using a semantic line of analysis which, to my knowledge, has been neglected by both parties to the dispute. It is customary to discredit the belief in absolutely unpredictable qualities on the ground that what scientific theories known today do not permit us to predict, scientific theories known tomorrow may well bring within the bounds of predictability. Thus there was a stage in chemistry when no general laws correlating molecular structure and sensible properties of compounds were known which would enable one to predict sensible properties of a hitherto unobserved compound on the basis of its molecular structure; but such laws are now known.1 To speak of absolute unpredictability, unpredictability once and for all, convicts one, in fact, of metaphysical obscurantism, motivated perhaps by a subconscious hostility against the faith in the omnipotence of science. Indeed, I would not wish to deny that those who, following Alexander, recommend “natural piety” in the face of absolute novelty, most likely have no clear idea as to what they mean by such “absolute novelty.” It seems to me, however, that this vague notion of absolute emergence can, with the help of semantic concepts, be explicated in such a way that whether a quality is emergent is independent of the stage of scientific knowledge, but rather depends on the question whether certain 1 For

a precise statement of this “relativistic” theory of emergence or novelty see Henle 1942 and, more recently, Hempel and Oppenheim 1948, §5.

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predicates are only ostensively definable. Specifically, my purpose is to show that a law correlating a quality Q with causal conditions of its occurrence can, without obscurantism, be argued to be a priori unpredictable if the predicate designating Q is only ostensively definable. The concept of “a priori unpredictability” here used will be defined in due time. My starting-point is a distinction elaborated by Broad with great analytical eﬀort though questionable success in the chapter “Mechanism and its Alternatives” of Mind and its Place in Nature: the distinction between an emergent (or “ultimate”) law and a non-emergent (or “reducible”) law. An example by means of which Broad explains his notion of emergent law is the law connecting the properties of silver-chloride with those of silver and of chlorine and with the (presumably molecular) structure of the compound: if we want to know the chemical (and many of the physical) properties of a chemical compound such as silver-chloride, it is absolutely necessary to study samples of that particular compound. It would of course (on any view) be useless merely to study silver in isolation and chlorine in isolation; for that would tell us nothing about the law of their conjoint action . . . The essential point is that it would also be useless to study chemical compounds in general and to compare their properties with those of their elements in the hope of discovering a general law of composition by which the properties of any chemical compound could be foretold when the properties of its separate elements were known. (Broad 1949, 64)

Let us notice that according to the definition of an emergent law, implicit in the quoted passage, at least a necessary condition (but I suspect likewise a suﬃcient condition) of emergence of a law of the form “if C1 , . . . , Cn , then R” (where the antecedent refers to a set of interacting components, and the consequent to a resultant of this interaction) is that instances of R must be observed before the law could be known with some probability. More exactly, if L is an emergent law in Broad’s sense, then it cannot be confirmed indirectly, by deduction from more general laws, before direct confirming evidence is at hand. For short, let us say that an emergent law is deducible only a posteriori, or unpredictable a priori. The meaning of this condition will best be grasped if we consider Broad’s illustration of reducible laws, i.e. laws that could be theoretically predicted with the help of a general composition law before any direct observational evidence exists. Consider the law of projectiles according to which the trajectory of a projectile is, under ideal conditions, a parabola. It is true that direct observational evidence for this law was at hand before Galileo deduced it, with the help of the parallelogram law of forces (Broad’s example par excellence of a general composition law), from the law of freely falling bodies and the law of inertia. But it is clearly conceivable that the law should have been reached by deduction from those premises concerning the eﬀects of isolated force components before instantial evidence was obtained (in fact, this was the case with regard to a special case of the law, namely

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the flight of high-speed cannonballs). This, then, is what Broad would call a reducible law: it is a priori predictable in the sense that it is capable of prior confirmation through deduction by means of a general composition law before any confirming instances are observed. For the present purpose we may be satisfied with a denotative definition of “general composition law” as the kind of deductively fertile composition law illustrated by the parallelogram law. Notice that the general composition law is not claimed to be itself a priori predictable; it is rather claimed to make special composition laws, like the law of projectiles, a priori predictable. Now, this concept of reducible law is no sooner defined than it provokes the question: how could it ever be shown that a given law is absolutely irreducible? In order to show this, one would have to prove that no general composition law could conceivably have been known which would have enabled a skilled scientist to predict the law a priori. Broad himself seems to recognize the relativity of such irreducibility to the stage of scientific knowledge at least in the case of chemistry, for the quoted passage concerning the properties of silver chloride is followed by the statement “so far as we know, there is no general law of this kind.” Indeed, it is easily describable what such a general composition law of chemistry might be like: if a metal combines with an acid in solution, there results a salt and free hydrogen. This law may have been inductively derived by observing interactions of metals M, M , M with acids A, A , A respectively, and then be used to predict that if M should react with A a salt will result of which no instances have yet been observed. Broad indeed admits that “mechanistic” progress in chemistry is possible to the extent that a deduction of R from C1 , . . . , Cn with the help of general composition laws might be accomplished if R is a physical disposition of compounds, like ready solubility in water. But he holds such deduction to be in principle impossible if R is a secondary quality, i.e. a disposition to produce a sensation of a certain kind, like a pungent smell (Broad 1949, 71). Does Broad have a point? I shall concede that in claiming absolute emergence for laws correlating secondary qualities with microscopic physical conditions he is inconsistent with his own definition of emergence; but I shall nevertheless argue that he came close to making a valid point overlooked by the “relativists.” Broad claims that not even the “mathematical archangel” (that is, the Laplacian calculator turned to physical chemistry) could predict what NH3 would smell like unless someone (not necessarily himself) had smelled it before. If Broad asserts the impossibility of theoretically certain prediction, his assertion is true but trivial: even the probability which is conferred on a special law of dynamics by deduction from the parallelogram law falls short of a maximum, since the parallelogram law itself is still capable of falsification as long as not all of its deductive consequences have been tested. But if he asserts the impossibility of “prediction” in the only sense in which prediction is ever possible, he is

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clearly wrong: just suppose that chemists had evidence suggesting the generalisation “whenever two gases chemically combine in the volume proportion 1:3, the resulting compound has the smell S .” If the original evidence for this general composition law does not include observations upon the formation and properties of ammonia, the special composition law “NH3 has smell S ” could well have been a priori predicted by somebody less than a mathematical archangel before anybody had smelled that gas. It is conceivable, incidentally, that Broad confused the proposition he did assert, and which has been shown to be false, with the undeniable but irrelevant proposition that “this gas has smell S ” cannot be logically deduced from the premise “this gas has such a molecular structure” alone, without the use of an additional premise asserting the correlation between structure and secondary quality. However, there is a logical diﬀerence between the hypothetical general composition law just mentioned and the parallelogram law, which will prove to be crucial for the problem of emergence. The deduction of a special law made from the former took simply the form of deriving a substitution-instance and the predicate referring to the predicted quality was explicitly contained in the general premise. But the parallelogram law does not contain the concept of a specific type of compound motion, such as circular motion or motion along a parabola; it only contains the concept of a specific form of functional dependence of the direction and magnitude of a compound motion upon the directions and magnitudes of the component motions. A simply way of putting the diﬀerence is this: one could understand the parallelogram law without thinking of the determinate quality of motion it may be used to predict, and therefore without ever having witnessed an instance of the predicted quality. But since the general law correlating microscopic conditions with sensations of quality Q contains the very same concept of Q as the derived substitution-instance, and Q is a simple quality of which, in Hume’s language, one cannot have an “idea” without “antecedent impression,” the law cannot even be understood unless an instance of the predicted quality has at some time been witnessed.2 In this sense the deductions made from such a general law do not lead to “novelty”; when we test the deduction empirically we do not encounter a new quality the way physicists would acquaint themselves with a new quality of motion if they tested their prediction of circular motion from the parallelogram law which latter, as we might suppose, they had inductively derived from observations of rectilinear motions only. Notice that if a predicted quality Q fails to be novel in the specified sense, it does not follow that the special law “if C1 , . . . , Cn , then Q” must be directly confirmed before it could be indirectly confirmed by deduction from a general composition law; it only follows that instances of Q,

2 The

assumption, here involved, of only ostensively definable predicates is discussed in the sequel.

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which may be associated with other complexes than C1 , . . . , Cn as well, must be observed before indirect confirmation is possible. Let me clarify the point in terms of the laws correlating wave motions of the air with sound phenomena. One might be inclined to think that a general law correlating frequencies and pitches could easily be formulated which would enable a priori prediction of hitherto unexperienced sound phenomena in just the way in which the parallelogram law enables the prediction of so far unobserved forms of motion. Thus, let X be the highest pitch so far heard, which we shall assume to be not the highest audible pitch, and suppose that comparisons of various pitches led to the well-known law “the higher the frequency, the higher the pitch.” With the help of this simple law we can easily predict that the pitch corresponding to a frequency higher than the frequency corresponding to X will be higher than X, before ever having heard such a pitch. If now the question should be raised whether this deduction is just like the discussed case of deduction by simple substitution, namely, a prediction of a quality which must already have been observed if the general premise is to be intelligible, the answer will have to be somewhat qualified. Strictly speaking, the quality which is being predicted is “higher in pitch than X,” which quality is not explicitly mentioned in the general premise and therefore need not have been experienced in order for that general premise to be understood. Yet, the reason why such a priori prediction is possible is that the predicate designating the quality is complex and made up of parts whose meanings are understood through ostensive definition: the meaning of the relational predicate “higher pitch” is understood because some, though not all, instances of this relation have been experienced, and the meaning of the proper name “pitch X” is understood, let us say provisionally, because pitch X has been heard. We might generalize from this example, and lay down the following principle: if a novel (that is, so far unobserved) quality Q is to admit of a priori prediction, then it must be complex in the sense that the expression describing it contains sub-designators (predicates and/or proper names), and these sub-designators, being understood through ostensive definition only, designate old qualities. If this principle is correct, then it follows that if there are qualities which admit of a priori prediction, there must also be qualities, less complex ones, which do not admit of a priori prediction. In terms of our illustration: one would, indeed, make a perfectly defensible claim if one said, a` la Broad, that no amount of physical and physiological information could enable one to predict that a frequency increase would produce a sensation of rising pitch, if the relational predicate “higher pitch” admitted only of ostensive definition; since in that case one would not know what quality one is predicting and the deduced statement would acquire its meaning only after verification, which is absurd. It is, however, of the utmost importance to realize that the concept of a priori unpredictability, as here analyzed, is absolute only relatively to the assumption

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that certain descriptive predicates admit only of ostensive, not of verbal, definition. If this semantic premise fails in a given instance, the claim of a priori unpredictability likewise breaks down. Take, for example, the question whether it could be a priori predicted that a definite frequency would produce that definite pitch named “X.” According to the present analysis, this question reduces to the question whether the meaning of the proper name “X” could be understood, in other words, whether the designated quality could be imagined, by someone who had never experienced the quality. Conceivably the pitch might be described in terms of interval-relations to already heard pitches (say, as the pitch one third higher than pitch Y, where the expression “one third higher” would itself be defined as meaning “such that if Y and X occur simultaneously, the interval named ‘third’ is heard”). If such a relational description enabled one to imagine the as yet unheard pitch, one would know what one was predicting before verification of the prediction in terms of immediate experience. A “relativist” with regard to the problem of emergence might now think that his position remains unconquered after all, since it is always conceivable that a given quality-designation be understood by description rather than by ostentation. Whether a given proper name be only ostensively definable or verbally definable by means of a synonymous definite description is, indeed, not a logically decidable question but a question of psychology, specifically concerning possibilities of imagination. Just as in one logical calculus the logical constants C1 and C2 may be primitives and the logical constants C3 , C4 , C5 defined, while in an alternatively constructed calculus, say, C4 and C1 are taken as primitives and the rest defined; so in a descriptive language, say, phenomenological acoustics, the set of ostensively defined proper names3 (what Russell calls “logically proper names,” contrasted with proper names introduced as abbreviations for descriptions) is not uniquely determined. Thus, using “C” as ostensively defined proper name, and “higher pitch” (and its converse “lower pitch”) and “third” as ostensively defined relational predicates, we could introduce the proper names “E,” “G,” “B,” by verbal definition (for the sake of simplification, I assume the C-major scale as the field of the relation so as to be able to neglect the distinction between major and minor intervals). This is assuming the psychological possibility of imagining an as yet unsensed quality in the field of a relation R some instances of which have been sensed, on the basis of sensed qualities in the same field. If we further provide an ostensive definition for the relational predicate “equidistant” (pitches), we might even be able to introduce the names of all the missing pitches in the C-major scale

3 In

calling such quality-designations as “B-flat” proper names, I do not mean to imply, of course, that the designated pitches are particulars. I am using “proper name” as a term relative to a given language-level, such that the descriptive terms of lowest order in language L (terms occurring as grammatical subjects but not, in L, as grammatical predicates) are called “proper names” relatively to L.

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by description. But an alternative construction of the language is clearly conceivable: we might take “second” and “higher pitch” as ostensively defined relational predicates, “G” as ostensively defined proper name, and then all the other proper names and interval-designations might be introduced by description without the use of the relational predicate “equidistant.” It may perhaps be doubted whether the description “the complex pitch resulting if a pitch a second higher than the pitch a second higher than G is sounded simultaneously with G” would enable one to get an auditory image of a third if one had never heard a third before; but this is a psychological question of fact. If we call our first model of a language of phenomenological acoustics ‘L’ and our second model ‘L ’, we can now make the following assertions: a law correlating the pitch G with a frequency is a priori unpredictable in L , and so is the law correlating a definite frequency-ratio with the pitch-interval called “second”; but those same laws are a priori predictable in L, if we assume that the meanings of verbally defined expressions in such languages are intelligible in the sense that the verbal definition can produce an image of the quality or relation defined, independently of any previous experience of the latter. If to say that quality Q (or relation R) is absolutely emergent is to say that the law correlating Q (or R) with quantitative physical conditions is a priori unpredictable, it follows that absolute emergence is relative to a system of semantic rules. In this respect the concept of absolute emergence turns out to be surprisingly analogous to the concepts of indefinability and indemonstrability. Is the relativist, then, wrong in denying the existence of absolutely emergent qualities? He is wrong if he denies the semantic truism that some descriptive terms must be given meaning by ostensive definition if it is to be possible to give meaning to any descriptive terms by verbal definition. Perhaps he is right, on the other hand, in his claim that no descriptive term is, by some obscure kind of necessity, definable by ostentation only. Even Hume, whose principle that every simple idea must be preceded by a corresponding impression is equivalent to the semantic principle that predicates designating simple qualities can become meaningful only through ostentation, allowed for the famous exception, the missing shade of blue—in fact, it could be argued that the same logic which forced him to admit this one exception should consistently have led him to allow for an infinite class of similar exceptions. However, I would like to conclude with the tentative suggestion that for every sense-field there is a general ordering relation, instances of which could not possibly be imagined antecedently to being sensed. I am referring to the relation “higher pitch” for the auditory sense-field, the relation “brighter color” for the visual sensefield, and analogous transitive and asymmetrical ordering relations for other sense-fields or other dimensions of the same sense-fields. Whenever a verbal definition is given for a term designating an element in the field of such an ordering relation R, the relational predicate “R” is itself used in the definiens,

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together with one or more names of other elements in the field. Thus Hume’s missing shade of blue, which has not been seen yet, would be verbally defined as the shade equidistant from, say, b4 and b6 , where these are separated by a larger distance than the other consecutive elements in the series of increasingly dark shades. To say that b5 (the missing shade) is equidistant from b4 and b6 evidently means that it is just as much darker than b4 as b6 is darker than it. But then the meaning of “darker than” must be understood by ostentation, and, on pain of circularity, the method of relational description by which “b5 ” was verbally defined is unavailable. Indeed, I do not have the faintest notion what an analysis of such a simple relational concept could be like. If so, then a law correlating quantitative changes in physical conditions with such changes in sensed qualities as are expressed by the terms “darker,” “louder,” “higher in pitch,” etc., is absolutely emergent after all. And limits would be set to the possibility of a priori prediction, not by the stage of scientific progress, but by the limits of semantic analysis.

Chapter 19 REDUCTION SENTENCES AND OPEN CONCEPTS (1953)

Since the time when Carnap published his classical contribution to the analysis of scientific language, “Testability and Meaning” (Carnap 1937), it has come to be universally recognized by competent philosophers of science that the concept of explicit definition is inadequate for the analysis of specifications of meaning going on in science. Instead the term reduction sentence has been incorporated into the essential terminological furniture of the meta-language in which scientific procedures are described. In Carnap 1937, Carnap initially explained the need for reduction sentences in connection with the problem of defining dispositional predicates. His argument was that if an explicit definition of dispositional predicate “P,” of the form “P = (if Q, then R)” (where “Q” refers to a test operation, “R” to the corresponding test result) is used, one is faced with the paradoxical consequence that “P” would be vacuously predicable of any substance upon which the test operation is never performed. As this paradox altogether depends upon the interpretation of “if, then” in the sense of material implication, Carnap’s argument might conceivably fail to be persuasive, on account of the following objection. Employment of reduction sentences is necessary only as long as we fail to analyze the meaning of “if, then” in nomological conditionals, in other words, fail to analyze the concept of necessary connection involved in contrary-to-fact conditionals. For, as among others Reichenbach has emphasized (Reichenbach 1947, chapter 8), one of the necessary conditions of adequacy of an analysis of nomological conditionals is just that the truth of the conditional should not follow merely from the falsity of its antecedent. In view of such an objection, it would certainly be lamentable if Carnap 1937 should have conveyed the impression (in all probability contrary to the author’s intention) as though the mentioned paradox of material implication were the basic reason, or even the only reason, for the construction of reduction sentences as means of specifying the meanings of scientific terms. It is conceivable that, once we have a correct analysis of the nomological conditional, we could analyze dispositional predicates of the thing-language (like “soluble,” “fragile”) in terms of nomological conditionals,

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or sets of such, connecting test operations and test-results, so that we could revert from the bilateral reduction sentence “if Q, then (if and only if P, then R”) to the explicit definition “P = (if Q, then R).” Carnap’s argument merely shows the need for reduction sentences in a language of logical reconstruction which suﬀers from the shortcoming that no causal statements of ordinary language and scientific language are translatable into it; and in such a language we had better not attempt to talk about dispositions and causal connections in the first place. However, I shall argue in the following that reduction sentences will remain indispensable, whatever the outcome of the logic of nomological conditionals now in the making may be. The other argument for reduction sentences, likewise to be found in Carnap 1937, but whose cogency is independent of whether our language of logical reconstruction is extensional or not, may be called the argument from surplusmeaning. It is often asserted that the meaning of a construct like electrical current cannot be exhausted by a single test-conditional, or even a finite conjunction of such, in the language of observables; and for this reason explicit definition of a construct in terms of observable predicates is said to be impossible. Carnap points out that the physicist implicitly uses the method of partial meaning-specification by means of reduction sentences since he wishes to leave his concepts “open” for application to new contexts, contexts which the reduction sentences so far stated do not legislate for, as it were. I believe, however, that the characterization of reduction sentences as application-rules for open concepts conflicts to some extent with their characterization as introductions of terms without antecedent meaning. And since I wish to make the argument from surplus-meaning as convincing as possible, it will be useful to reveal the mentioned inconsistency in Carnap’s own discussion of reduction. A reduction pair [(p1 ⊃ (p2 ⊃ p3 )); (p4 ⊃ (p5 ⊃∼ p3 ))] is said to have factual content, expressible by a sentence which does not contain the introduced term, viz. ∼ (p1 .p2 .p4 .p5 ). A point of great importance which will be emphasized in the sequel might as well be touched on right now. Carnap and the logical empiricists use the expression “p has factual content” in such a way that if and only if p has factual content (or is “synthetic”), then p is empirically refutable. Now, since the conjunction of the two members of a reduction pair has the mentioned factual consequence, it must be regarded as itself a factual statement. But a conjunction cannot be factual unless at least one conjunct is factual. Yet, if p3 is first given a meaning by a given reduction sentence then that reduction sentence is empirically irrefutable. Hence a reduction pair could have no factual content if both members were meaning rules in a sense in which this implies empirical irrefutability. We could, of course, arbitrarily designate one member of the pair as a meaning rule and the other member as a factual statement in the “object-language”: it is not self-contradictory to suppose that an instance of p1 .p2 . ∼ p3 be found if ∼ p3 can be inferred from p4 .p5 , and it is not self-contradictory to suppose that an instance of p4 .p5 .p3 be

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found if p3 can be inferred from p1 .p2 . But this way of splitting the reduction pair into a refutable factual statement of the object-language and a (partial) semantic rule of the meta-language (with regard to which it does not make sense to speak of “refutation”), is clearly contrary to the spirit of Carnap’s theory of reduction. If so, then the concepts “having factual content” and “being factually empty” which logical empiricists, including Carnap, have always intended as contradictories are not applicable in the original sense to sentences whose predicates acquire meaning through reduction instead of explicit definition; and it is, in that case, misleading to apply the same semantic concepts to such sentences. It is true that Carnap explicitly extends the meanings of “analytic” and “synthetic” in such a way as to make these semantic concepts applicable to sentences whose predicates admit of reduction but not of explicit definition. Yet, this extension of the meaning of “analytic” is confusing since according to the original meaning of the term it is characteristic of analytic sentences that they are empirically irrefutable, while according to the generalized meaning an analytic sentence may be empirically disconfirmable. This will become clear as we turn our attention to the special case of reduction pairs called by Carnap “bilateral” reduction sentences. In the case of a bilateral reduction sentence (Rb ) the “representative” sentence degenerates into ∼ (p1 .p2 .p1 . ∼ p2 ), which is a tautology, and hence this kind of reduction sentence is said to be factually empty. Now, this sounds plausible enough if we think of an isolated Rb ; for to refute p1 ⊃ (p2 ≡ p3 ) we require a case of p1 .p2 . ∼ p3 or of p1 . ∼ p2 .p3 , and if the only basis for predicating p3 is p1 .p2 , and the only basis for predicating ∼ p3 is p1 . ∼ p2 , then neither of those cases can occur. However, Carnap himself recognizes that, apart from the paradox of material implication which precludes explicit definability of disposition concepts in an extensional language, the main reason for reduction sentences is the desire the leave concepts “open” for application in new contexts. Isolated occurrence of a reduction pair or of a Rb is therefore the exception, and occurrence within a system of convergent R-sentences (converging, that is, to the open concept) is what the very purpose of R-sentences would lead one to expect. As a matter of fact, it would seem that if an Rsentence occurs isolated, like the famous R-sentence for “soluble” and similar R-sentences for dispositional predicates of the thing-language, this indicates that we have to do with a closed concept serving merely the purpose of shorthand, which would be explicitly definable were we only permitted to use the concept of causal implication in the definiens. Carnap sees clearly that what may be called the “systemic” occurrence of R-sentences is the rule, as the following quotation shows: in most cases a predicate will be introduced by either several reduction pairs or several bilateral reduction sentences. If a property or physical magnitude can be determined by diﬀerent methods then we may state one reduction pair or one bilateral reduction sentence for each method. The intensity of an electric current

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Reduction Sentences and Open Concepts (1953) can be measured for instance by measuring the heat produced in the conductor, or the deviation of a magnetic needle, or the quantity of silver separated out of a solution, or . . . We may state a set of bilateral reduction sentences, one corresponding to each of these methods. The factual content of this set is not null because it comprehends such sentences as e.g. ‘If the deviation of a magnetic needle is such and such then the quantity of silver separated in one minute is such and such, and vice versa’ which do not contain the term ‘intensity of electric current’, and which obviously are synthetic. (Carnap 1937, 444-45)

Like a reduction pair, such a set of convergent Rb -sentences is said to have factual content because factual statements not containing the “introduced” term are deducible from it (if the set consists, e.g., of the two Rb -sentences: p1 ⊃ (p2 ≡ p3 ), and p4 ⊃ (p5 ≡ p3 ), then such a consequence is: ∼ (p1 .p2 .p4 . ∼ p5 )). But it should now be obvious that the “theorem” of the factual emptiness of Rb sentences holds only for isolated Rb -sentences, and so only for those of them that really involve closed concepts. If “p1 ⊃ (p2 ≡ p3 )” is accompanied by “p4 ⊃ (p5 ≡ p3 ),” it makes good sense to say “even though p1 .p2 was verified, it is probable that ∼ p3 ,” for the probability judgment could be supported by the evidence p4 . ∼ p5 . And since a set of convergent Rb -sentences is never constructed (I did not say reconstructed) all at once, but, as Carnap himself observes, new R-sentences for the same term are laid down as new discoveries are made, reduction sentences are rarely introduction-sentences at all. Indeed, it is just because the reduced term does have antecedent meaning that the Rsentence can without paradox be said to function both as meaning rule and as expression of an empirical law. Before proceeding with the elaboration of the positive thesis of this paper, I wish to clear away a possible objection to the view that an isolated Rb -sentence, such as the one for “soluble,” is not a partial meaning rule at all, and hence could easily be replaced by a (nominal) explicit definition in a non-extensional descriptive language. Let such an explicit definition have the form: S (x, t) ≡d f if O(x, t), then R(x, t). Then “S ” is clearly predicable if, on performing O, R occurs, and “non-S ” is clearly predicable if, on performing O, R fails to occur. But what about the x’s upon which O has not been performed? Do they or don’t they have the defined disposition? According to the argument which I wish to refute, the definition leaves the question whether those x’s have the disposition undecidable; in other words, the definition does not permit us to apply the law “(∀x)(S x ∨ ∼ S x)” since, on the contrary, we have: (∀x)(∼ O(x, t) ⊃ (∼ S (x, t) . ∼ (∼ S (x, t))). Thus Carnap, referring to the Rb sentence for “soluble,” writes: “If a body b consists of such a substance that for no body of this substance has the test-condition—in the above example: ‘being placed in water’—ever been fulfilled, then neither the predicate nor its negation can be attributed to b” (ibid.) Carnap is of course right since he is referring to an extensional reduction sentence, not to an explicit definition in terms of causal implication. From O(x, t) we can deduce, with the help of

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the Rb -sentence, the equivalence S (x, t) ≡ R(x, t), but since it is impossible to decide whether R is predicable without performing O, it is on the mentioned supposition strictly undecidable whether S is predicable. But in the case of the contemplated explicit definition no such undecidability arises, for the following reason. If O has not been performed upon x, then it is either conceivable that it might be or it is inconceivable that it might be performed on x. If, e.g., x is the wooden match, mentioned by Carnap, which was just burnt up, then the first alternative holds: it is clearly conceivable that the match had not been lit and had instead been thrown into water. “Conceivable” is here used in the sense of “it is meaningful to suppose that O be (or had been) performed on x” (no commitment to any theory of meaning is necessary in the context of this argument). If so, then we simply do not know whether or not “S ” is predicable of this x, at least not with certainty—an obvious qualification in view of the possibility of indirect confirmation by means of laws of the form “if one instance of kind K has disposition D, then all instances of K have D.” On the second alternative, it does not make sense to suppose that O be performed upon x. For example, the sentence “a hydrogen atom is thrown into water” is presumably meaningless. If so, the sentence “hydrogen atoms are soluble in water” is itself meaningless, and in that sense neither true nor false. But thus it appears that the argument according to which the explicitly defined dispositional predicate does not fall under the law of the excluded middle merely shows that the definiendum has a limited range of significant application. Since this holds presumably for any predicate, it would follow that complete meaning-specification is in all cases impossible. I proceed now to formulate the argument from surplus-meaning against explicit definability of such concepts as “electric current,” “mass,” “temperature” (and it will turn out that a similar argument holds for qualitative concepts of kinds of things). If only we take a close look at the thesis of explicit definability, at the consequences entailed by it, we shall convince ourselves of its untenability. Suppose, then, that out of the n laws of functional dependence1 involving the quantity Q as variable, one be selected as definition of Q (and thus shifted from the object-language to the meta-language), while the others are interpreted as empirical propositions about Q (not about the symbol “Q”). To fix our ideas, suppose we accepted Mach’s definition of mass in terms of the ratio of the accelerations mutually produced in two interacting particles, which turns the third law of motion, via the second law, into a definitional truth. Let “p” stand for a predication of the defined functor (such as “the

1 It

might be noted that such functional laws have precisely the form of bilateral reduction sentences since on the condition which is not expressed in the equation itself a determinate change of x is a necessary and suﬃcient condition for a determinate change of f (x). E.g., on the condition of constant gas pressure, a doubling of gas temperature is a necessary and suﬃcient condition for a doubling of gas volume.

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mass of particle A is m”), and “E” for the evidence which, by the definition, is logically equivalent to p (such as “the ratio of the acceleration imparted to A by unit particle B to the acceleration imparted by A to B is equal to m”). Then Prob(p/E) = 1. This means that the outcome of any further tests of p, based on contingent laws connecting mass with other quantities, would be irrelevant to the question of the truth of p if only E is accepted as certain. And if all these other tests yielded a value for the mass of A inconsistent with m, then all the contingent laws would have to be abandoned while p would remain unshaken. Thus, suppose that A and B had equal mass by Mach’s definition (i.e. m = 1), but that we found that A and B produced unequal strains on a spring scale, further that they did not balance on a beam-balance, further that they exerted unequal gravitational attractions on a given mass at a given distance (since the law of gravitation is an independent assumption of mechanics, not deducible from the third law, this is a logical possibility), and so forth. We still would be logically compelled to stand by our hypothesis. For the method of explicit definition here criticized implies that, where E1 , . . . , En is the sum-total of evidence confirming p by virtue of contingent laws, Prob(p/E) = Prob(p/E.E1 . . . . . En ) = Prob(p/E. ∼ E1 . . . . . ∼ En )! I submit that this is highly counterintuitive, and that no scientist would act in accordance with such peculiar equations.2 The following illustration should clarify the point at issue. Suppose a scientist, finding that each metal has a unique melting-point, decides that it is best to define metals by their melting-points. The question before the philosopher of science is how to interpret the meaning of “definition” as used in this context. My thesis is that it should not be interpreted as a declaration of synonymy (such as “ ‘iron’ is a synonym for ‘substance with melting-point M’ ”) but rather as an assignment of great probability-weight to a selected reduction sentence for the name of the metal. Suppose, then, that while according to past experience 2 In

a recent article, Bergmann 1951b, Gustav Bergmann argues that a psychological concept like “seeing green” is definable (in use) in terms of a stimulus-response sequence in just the same way in which a physical construct like “electrical field” is definable (in use) in terms of a subjunctive conditional “if an electroscope were placed at P at t, then the leaves of the electroscope would diverge.” Bergmann is aware of the argument from surplus-meaning against such explicit definitions of physical constructs, the argument “. . . that it (the definition) does not do justice to all we mean by ‘electric field’ . . . ; that there are many other ‘tests’; that the description of any one of these could equally well serve for R (read: the definiens); and that this very fact, together with other laws about electric fields, belongs to the meaning of ‘electric field’ ”(Bergmann 1951b, 99). But his answer is anything but convincing: the opposition is simply accused of the elementary confusion between a (meta-linguistic) statement about the meaning of “electric field” (that’s what the definition is) and physical statements about electric fields. Bergmann gives no evidence of being aware of the really serious reason behind the objection from the “surplus-meaning” of constructs: that, on the formalization of the theory of electrical fields proposed, the degree of confirmation of the existential hypothesis about the electric field relatively to the electroscopic evidence would be a maximum, so that a negative outcome of all other possible tests could have no tendency whatever to diminish the credibility of the existential hypothesis but would instead logically compel us to abandon all the physical laws about the electric field defined as proposed.

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any substance with melting-point M had properties P1 , . . . , Pn (for which reason M was selected as a reliable indicator of these properties), the scientist is suddenly confronted with a specimen having M but lacking all of these properties. Would he really insist that the specimen is an instance of the metal in question and that the generalization “all instances of this metal have properties P1 , . . . , Pn ” had simply been refuted? Should such anomalous specimens turn up frequently, I think it likely that he would be frankly “illogical” and say “probably all these specimens ought to be classified as the metal in question, and we’d better give up the definition in terms of melting-point”: such a statement, be it noted, sounds illogical if we attach to the word “definition”3 the meaning customarily attached to it by logicians who like things to be neat, but not if we interpret it as a reduction sentence. For, as we have seen, the very systemic occurrence of a reduction sentence makes it possible for it to be corrected through its associates. If it be asked whether, according to the view here presented, a generalization about a natural kind, like “all iron has melting-point M,” is analytic or synthetic, it must be replied that this dichotomy is inapplicable to propositions involving open concepts. If an explicit definition of “iron,” in the sense of a statement of synonymy, were at hand, then the question would be appropriate; but this is precisely the presupposition which is here denied. Suppose the question were raised “is it self-contradictory to suppose that a lemon tasted just like an apple, in other words, does ‘x is a lemon’ analytically entail ‘x does not taste like an apple’?” Clearly the question cannot be answered since the line between the defining properties of lemons and those properties which lemons have contingently just is not clearly drawn. This is what I mean by calling such a concept open; an alternative terminology would be to say that such class-names as “lemon” or “iron” are intensionally vague4 . It is not de-

3 The

fact that not only the admittedly unprecise class-concepts of everyday language, but likewise the precisely “defined” class-concepts of science are really open concepts, was clearly seen by Waismann, in (Waismann 1945). Waismann makes the point so well that he deserves to be quoted at some length: “The notion of gold seems to be defined with absolute precision, say by the spectrum of gold with its characteristic lines. Now what would you say if a substance was discovered that looked like gold, satisfied all the chemical tests for gold, whilst it emitted a new sort of radiation? ‘But such things do not happen’. Quite so; but they might happen, and that is enough to show that we can never exclude altogether the possibility of some unforeseen situation arising in which we shall have to modify our definition. Try as we may, no concept is limited in such a way that there is no room for any doubt. We introduce a concept and limit it in some direction; for instance, we define gold in contrast to some other metals such as alloys. This suﬃces for our present needs, and we do not probe any farther. We tend to overlook the fact that there are always other directions in which the concept has not been defined. And if we did, we could easily imagine conditions which would necessitate new limitations.” (Waismann 1945, 122-23) 4 I contrast intensional vagueness with extensional vagueness, the sort of vagueness besetting terms like “hot,” “bald,” “blue,” since intensional vagueness is predicable only of complex predicates, predicates whose meaning can be explained by verbal description. Intensional vagueness entails, I suppose, extensional vagueness, since if we are not certain about the intension of a term we cannot be certain about its extension either, but the converse entailment does not hold since unanalyzable predicates may be extension-

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nied that the semantic rules for a word like ‘lemon’ are determinate to some extent; a man who applied the word “lemon” to, say, a glass could promptly be accused of semantic sin. But while it may be easy to agree on the genus of an explicit definition, trouble will quickly come when the diﬀerentia is to be specified. If a man should confidently declare, e.g., that sourness forms part of the meaning of the term, his confidence could easily be shaken by asking him “would you unhesitatingly, then, classify a thing as a non-lemon if it resembled lemons in all respects except that it tasted like a sweet apple?” The point is that the semantic rules governing such class-names are not suﬃciently determinate to force a clear-cut decision between the classification “non-lemon” and the classification “apple-tasting lemon.” It may be asked at this point how this admitted phenomenon of intensional vagueness that characterizes a qualitative thing-language bears on the question, here primarily at issue, of the explicit definability of scientific concepts. Prima facie, it is diﬀerent with a scientific language, like the language of quantitative physics, since here we do find explicit definitions, carefully constructed by the scientists. But as pointed out before, if a physicist like Mach speaks of a “definition” of “mass” in terms of the third law of motion he does not attach to the word “definition” the meaning underlying the analytic-synthetic distinction. I would like to make the point quite clear by adding as illustration physical definitions of such a fundamental concept as “time-congruence.” As is well known, such a definition consists in the selection of a standard-clock (where “clock” has the generalized meaning of “physical system in periodic motion”), such as the rotating earth, and if interpreted as a declaration of synonymy, turns the statement that the selected standard clock goes at a uniform rate into a tautology. Thus, relatively to the convention to designate as equal time intervals during which the earth rotates through equal angles, the statement that the earth rotates uniformly would be a tautology. Now, that such a definition was never intended by physicists as a declaration of synonymy should be evident from the fact that quite early in the history of post-Copernican physics it was recognized that the earth’s rotation cannot be strictly uniform, for various physical reasons. It might be replied that when such a statement was made a tacit shift to a new standard-clock must already have occurred, such that “uniform motion” in the statement “the earth does not rotate uniformly” had a diﬀerent meaning from its meaning in the tautology (relatively to the original convention) “the earth rotates uniformly,” and that therefore an interpretation of the definition as a declaration of synonymy does not really force us into the absurd claim that subsequently physicists contradicted a tautology. But the reply is unconvincing since recognition of the non-uniformity of the original standard-clock ally vague and it does not make sense to distinguish between the defining and the accidental properties in the case of simple concepts.

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motivates the physicist to look for a better standard-clock, which presupposes that the original standard-clock is rejected as non-uniform before a better one has been selected. Actually, what leads to such a rejection is the fact that the original “definition” together with physical principles implies physical consequences that are not borne out by observational evidence and it is found more convenient to change the “definition” and retain the physical principles. Thus it turns out that if time is measured in terms of the earth’s rotation, then the centripetal acceleration of the moon as calculated by the inverse square law is discrepant with the same acceleration as calculated on the basis of the law of 2 uniform circular motion: a = vr , where v is the linear velocity with which the particle revolves around the center, and a is the acceleration directed toward the center. Since the latter law is a direct consequence of the general laws of motion, this discrepancy might have led Newton to abandon the law of universal gravitation. Instead, realizing that the calculation of the moon’s velocity of revolution was based on the use of the earth’s rotation as time-measure, he suspected that consistency could be restored by assuming suitable variations in the earth’s velocity of rotation. The moral of the illustration is that a statement like “the earth rotates uniformly” or “light rays move, in vacuo, with constant speed” actually functions like a physical hypothesis subject to correction even though it is called a “definition” or “convention.” By Einstein’s definition of “distant simultaneity,” it would be meaningless to question whether two light-rays emitted from places A and B toward one another, and reaching an observer placed midway simultaneously, really were emitted simultaneously from A and B—the question, that is, would be just as meaningless as the question “is this husband really married” if the definition were a true statement of synonymy. And since the question is equivalent to the question whether the two light-rays really moved toward the observer with the same speed, it would also be meaningless to question the constancy of the speed of light in this context. But suppose, now, that the same observers who perceive the light-signals emitted from A and B simultaneously perceived sound-signals, emitted from A and B simultaneously with the light-signals, in succession, and suppose that this discrepancy occurred regularly. Suppose furthermore that the most refined measurements led to the conclusion, long ago, that the velocity of sound relatively to the sound-source in a given still medium at constant pressure and temperature is constant. Assuming these ideal atmospheric conditions and further strict simultaneity of the emissions of light- and sound-signals from a given place, the hypothetical discrepancy could be resolved in two and only two ways: either we must assume that the sound-waves were propagated from A and B with unequal speeds, or we must make the analogous assumption for the light-rays. Now, according to the theory of physical definition here criticized, it is meaningless to make the latter assumption, while the former assumption makes perfectly good sense. I con-

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tend that if one assumption makes sense so does the other. Such a contention conflicts, indeed, with a famous statement made by the great Einstein himself, but I derive courage from the reflection that this is a statement he made qua philosopher of physics, not qua physicist: That light requires the same time to traverse the path A → M as for the path B → M is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own free will in order to arrive at a definition of simultaneity. (Einstein 1931, 27-28)

The fact is that, as just illustrated, situations are conceivable which would make the question significant whether the velocity of light is really constant, and since the possibility of encountering facts which throw doubt on p is just what marks p as an empirical hypothesis, there is no reason why what Einstein calls a “stipulation” should be sharply distinguished from a physical hypothesis. Now, if such a “definition” of distant simultaneity be interpreted as a (systemic) reduction sentence for an open concept, the paradox of the statement “the physical law itself defines the physical concept” vanishes. Suppose we constructed a pair of Rb -sentences for “distant simultaneity” as follows: (1) if two light-rays are emitted from places A and B when events Ea and Eb happen at those places (so far only contiguous simultaneity has been mentioned!), then Ea and Eb are simultaneous if and only if the light-rays reach an observer stationed midway simultaneously. (2) if two sound-waves are emitted from A and B when events Ea and Eb happen at those places, and if the medium is still (i.e., at rest relatively to the sound-sources) and uniformly dense, then Ea and Eb are simultaneous if and only if the sound-waves reach an observer stationed midway simultaneously. It is, now, conceivable that this pair of convergent laws should be inconsistent in the sense that there are distant events which are simultaneous according to (1) but not according to (2), or vice versa. And if careful experiments revealed such an inconsistency, the physicist would reject either (1) or (2) as probably false even though he may have declared them as operational “definitions.” The designation, therefore, of one out of several convergent laws, call it L, as a “definition” is best interpreted as a declaration to the eﬀect that L will be maintained if inconsistencies of the described kind turn up until all other remedies fail. The theory of open concepts will now be further supported by turning to the tricky problem of the fields of significant application of physical concepts. Carnap points out (Carnap 1937, 449) that what motivates the scientist to specify meanings by reduction sentences instead of laying down explicit definitions is the desire to leave open the field of application of the term beyond the field already investigated. For example, if we laid down an explicit definition by which statements about temperature are synonymous with statements about reactions of mercury thermometers when brought into contact with the sub-

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stance to which temperature is ascribed, we would preclude application of the term “temperature” to, say, the sun. Carnap’s technical explanation, however, of the disadvantage of such explicit definitions seems inadequate. Now we might state one of the following definitions: Q3 ≡ (Q1 .Q2 ) (D1 ) Q3 ≡ (∼ Q1 ∨ Q2 ) (D2 ) If c is a point of the undetermined class, on the basis of D1 ‘Q3 (c)’ is false, and on the basis of D2 it is true. Although it is possible to lay down either D1 or D2 , neither procedure is in accordance with the intention of the scientist concerning the use of the predicate ‘Q3 ’. The scientist wishes neither to determine all the cases of the third class positively, nor all of them negatively; he wishes to leaves these questions open until the results of further investigations suggest the statement of a new reduction pair; thereby some of the cases so far undetermined become determined positively and some negatively. (Carnap 1937, 449).

Carnap’s argument is clearly based on the concept of material implication, and thus the impression is again created as though reduction sentences were a temporary device which could be disposed of once we have an analysis of nomological conditionals. If a physicist were to lay down the definition (temp(x, t) = y◦ ) ≡d f if a mercury thermometer is brought into contact with x at time t, then the top of the mercury will coincide with the mark y at t,

he would obviously intend “if, then” in the sense of nomological implication. If so, the predicability of the defined functor cannot be inferred from the nonfulfilment of the antecedent of the definiens. If for x we substitute the sun and for t any time at all, the antecedent turns into a description of a physical impossibility; but this does not convert the conditional into a true statement (nor, of course, into a false statement, since there is no diﬀerence between material and nomological implication as far as the falsity-conditions are concerned). What would follow from such a definition is rather that it is insignificant to apply the defined term in a situation in which the condition described by the antecedent cannot, by virtue of logical or physical laws, be fulfilled. Perhaps a more telling illustration would be a definition of “the temperature of gas G at t is uniform” in terms of “if V1 and V2 are any two volume elements of G, then the average molecular velocities in V1 and V2 at t are equal.” If now the question were raised whether all the individual molecules of G have the same temperature at t, it would surely be dismissed as meaningless. At any rate, the situation is this: when the physicist speaks of the temperature of the sun, he surely does not mean to predict the result of hypothetical operations with mercury- or gas-thermometers; this will be admitted regardless of whether or not one holds it to be meaningless to speak of what would happen if such a thermometer were brought into contact with the sun. It follows that if such an operational definition explicates the meaning with which “T ” is used in contexts of limited T -ranges, then in extrapolating beyond those limited ranges one uses the same symbol either without meaning or with a diﬀerent meaning.

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This dilemma becomes especially apparent if we turn from extrapolation to high temperatures to extrapolation to low temperatures, and take a fresh look at the old question whether it is meaningless to speak of a temperature below the absolute zero. If the operational definition of temperature in terms of the gas thermometer were a true statement of synonymy, then the hypothesis that a substance has a temperature lower than −273◦ C would assert that if thermal contact were established between the substance and a gas thermometer the latter would be found in a state of negative volume and negative pressure—which is surely nonsense. But on the other hand, strong reasons can be adduced for the meaningfulness of the hypothesis of temperatures lower than the absolute zero. It is, after all, a contingent fact that the pressure coeﬃcient (or the coeﬃcient of volume expansion, which has the same value) for gases has the value it 1 ) has been verified only for a limited range has. And at any rate this value ( 273 of temperature variation. It would seem significant, then, to entertain the possibility that complete shrinkage of a gas has not yet occurred when it reaches −273◦ C.5 But then again, one might plausibly argue that in entertaining such a possibility we must be thinking of some other way of measuring T than with the gas thermometer as presently calibrated. Therefore a strict operationalist who holds, with Bridgman, that diﬀerent operations define diﬀerent concepts, would be justified in saying that in the hypothesis “T might sink below the absolute zero” the symbol “T ” must have a diﬀerent meaning from the meaning explicated by the gas laws. But that this second horn of the dilemma, the view that in extrapolating a physical law beyond the experimental range of its variables we change the meanings of the variables and thus really talk about diﬀerent physical quantities, is no easier to take than the first, is not diﬃcult to show. Consider the hypothesis about the sun’s temperature again. This temperature is usually calculated by means of the Stefan-Boltzmann law relating the absolute temperature of a surface to its intensity of heat-radiation (I) in conjunction with the law that I is inversely proportional to the square of the distance of the surface from the heat-source. That is, after the value of I at the sun’s surface has been calculated by means of the latter law, the Stefan-Boltzmann law is applied to calculate the value of T at the sun’s surface. But as this value of T is outside the range to which the “operational” definition of “T ” refers, the “T ” in the conclusion of the mathematical deduction has a diﬀerent meaning from the “T ” in the premises (both the proposition of measurement about the temperature of the earth’s surface and the Stefan-Boltzmann law) of the mathematical deduction. Now, since the terms in the conclusion of a valid argument must

5 We

may disregard in this discussion the fact that there would be no gas anyway at such a low temperature, on account of condensation; for the question is what suppositions are meaningful, and it is surely meaningful to suppose that a substance could survive the gaseous state at such a low temperature.

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have the same meaning which they have in the premises, we are led to this consequence: if the conclusion has any physical significance at all, then it does not follow from the premises which are the sole ground of its credibility! Our problem, then, is to formulate a criterion of identity of meaning of a physical functor in diﬀerent contexts of application; in other words, a criterion for determining whether a symbol like “T ” stands for the same physical quantity in diﬀerent contexts of application. Carnap’s theory of reduction sentences suggests this solution: if the symbol is a “nodal point,” to use a suggestive metaphor, of a system of convergent reduction sentences, then it has the same meaning in all the contexts described by the various members of the system provided the system is consistent. Thus, suppose that on the basis of measurements of the pressure of a thermometric gas, we obtain the result: T (A, t0 ) > T (A, t1 ), and that on the basis of measurements by means of a mercury thermometer we obtain the same result. Then we have confirmed the consistency of the reduction-system converging to “T ,” and thus we have (partially) justified our identification of the meanings of “T ” in these diﬀerent contexts, or our claim that what is measured by these diﬀerent methods is the same quantity. Similarly, if a physicist finds in terms of Mach’s actionreaction test (measurement of accelerations) that two bodies have equal mass, and subsequently verifies that they produce equal strains on a spring-scale kept at constant height (which proves equality of the gravitational forces acting on the bodies), he confirms the hypothesis that the quantity measured in these two ways (in terms of impact-forces in one experiment, in terms of gravitational forces in the other experiment) is one and the same. Now, the method of extrapolating numerical laws involves the employment of the same physical functor in contexts of measurement and in contexts of calculation. But how could it be maintained that the functor has the same meaning in the two kinds of contexts? It must be conceded that to say, with Bridgman, that it still has “operational” meaning in the contexts of calculation since calculations, after all, are also operations, is tantamount to reducing the operationalist theory of meaning to a truism. Nevertheless, if we closely observe the reasons for which a physicist believes that in calculating the values of the sun’s temperature and mass he comes to know the values of the same quantity as is measurable in other contexts (though these calculations cannot be verified by measurements), we shall find that the same criterion of physical consistency is presupposed. If the extrapolated law leads to consistent calculations, the extrapolation is considered justified, and the variables of the equations are said to represent the same quantity no matter whether their values can be determined by measurement or by calculation only. In this connection it is important to distinguish two kinds of test of a physical equation which may be called the correspondence-test and the consistency-test respectively. The correspondence-test of the physical equation y = f (x) consists in

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the three steps: measurement of x, calculation by means of the equation of the corresponding value of y, measurement of y for the purpose of testing the calculation. But if the calculated value of y lies outside the experimental range of y, the correspondence-test is obviously unavailable. All that can be done is to determine whether diﬀerent calculations of y from diﬀerent bases yield consistent results. Thus the mass of the sun was calculated, in Newtonian astronomy, on the basis of the centripetal acceleration of a revolving planet, according to the equation (deducible from the second law in conjunction with the law of . Now, it is logically conceivable that as the calculation gravitation): a = G·M r2 is repeated on the basis of diﬀerent values of a and r, corresponding to the diﬀerent orbits and periods of revolution of the various planets, inconsistent values of M are obtained. If this happened, the law of gravitation would have failed the consistency-test (provided, indeed, that the trouble is not blamed on the general laws of motion), and the extrapolation of the concept of mass to celestial bodies would be suspect. If, on the other hand, calculations of M on the basis of optical data, viz. spectral displacements of the light emitted by atoms on the surface of the sun (which according to the general theory of relativity are due to the sun’s gravitational field), should corroborate the calculations based on mechanical data, the extrapolation would be further justified. The same holds, of course, for extrapolations to microscopic contexts, as when we speak, e.g., of the mass of an atom, or of an electron (think of the extrapolation of the law of conservation of mechanical energy to the motion of electrons, which led to the calculation of the mass of an electron). The main point is that a concept like mass is, by the physicist, left open not only for further experimental contexts but also for calculational contexts in which no correspondence-test is feasible; and that the attribution of an identical property (uniformly called “mass”) is justified to the extent that the extrapolation of experimentally confirmed laws survives the described consistency-test. In recent years the suspicion has been growing on the part of some analytic philosophers that all is not well with the analytic-synthetic distinction as applied to natural, unformalized languages. Whether or not the reasons by which their skepticism is nourished are cogent, the sketched theory of open concepts clearly implies that the analytic-synthetic distinction is not applicable to propositions involving open concepts. This is overlooked by many philosophers of science who conceive it as a major task of “logical reconstruction” to make the language of science logically tidy by sharply segregating object-linguistic sentences which are empirically refutable statements “about reality” and metalinguistic sentences which express stipulations concerning the use of symbols and thus cannot significantly be said to be refuted by facts. They are the ones who say “either ‘F = ma’ is a definition of force, in which case it expresses no physical law; or it expresses a physical law, in which case the meaning of ‘force’ must be determined independently of this equation (say, in terms of

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the spring-balance)”; and since either alternative seems unsatisfactory to some people some of the time, we have the interminable controversy about the logical status of the second law of motion. In order to leave no doubt that this pattern of semantic analysis is ill fitted to the language of physics, let us take a look at one more illustration. The principle of the conservation of heat says, qualitatively, that the amount of heat lost by one part of a thermally isolated system is equal to the amount of heat gained by the remaining part, and quantitatively: m1 c1 (t2 − t1 ) = m2 c2 (t1 − t0 ); for example, the system may consist of a hot piece of iron immersed in a volume of water, in which case the c’s are the specific heats of water and iron respectively, m1 the mass of the piece of iron, m2 the mass of the water, t2 the initial temperature of the piece of iron, t0 the initial temperature of the water, and t1 the temperature at which equilibrium is reached. Since c2 here is unity, by the definitions of specific heat and of the unit of heat (the calorie), c1 can easily be calculated in terms of this equation on the basis of measurements of the masses and the temperatures. But if we adhere to the principle that an equation does not express a physical law unless each quantity involved in it is measurable independently, i.e., without the calculational use of that equation, it becomes doubtful whether the heat-equation could be interpreted as a physical law. For how could c1 be determined without presupposing conservation of heat? The usual definition of specific heat as amount of heat required to raise the temperature of unit mass by unit degree hardly suggests a non-circular method of measurement: “quantity of heat” absorbed by m grams of a substance S as S is heated one degree C is itself defined as the product of m times the specific heat of S . Indeed, the elementary experimental method of determining specific heats is just the method of mixtures, a method altogether based on the assumed validity of the conservation-equation. Most physicists, however, would regard the latter as expressing a physical law even if they had to admit that the method of mixtures is the only method for determining specific heats. For though no correspondence-test could be carried out, the consistency-test would remain possible and this possibility is suﬃcient for bestowing physical content on the equation. Conceivably thermal equilibrium is reached at a diﬀerent temperature the next time the mixture experiment is performed with the same masses of the same substances at the same initial temperatures, and in that case the method of mixtures would fail to yield unique values of specific heat. And the physicist is inclined to say that the equation has physical content inasmuch as its consistency, in the sense illustrated, can be established only by experiment. Those who are relentless in enforcement of the analytic-factual dichotomy will no doubt reply: The analytic statements of physical theory can be disentangled from its factual statements easily enough. The heat-equation is nothing more than a definition of specific heat relatively to a conventional unit of specific heat. But the gen-

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Reduction Sentences and Open Concepts (1953) uinely factual proposition is the proposition that the defined property is a constant for a given substance. Mach made just this logical point in connection with the concept of mass. According to his definition of mass, the equation of conservation of momentum (which, by the second law of motion, is equivalent to “action is equal and opposite to reaction”) is analytic. But that the same value of the mass of a given body is obtained no matter how the conditions of the interaction-experiment are changed, this is a contingent fact, not deducible from the definition.

This argument would, indeed, be convincing if the equations which it is proposed to move into the meta-language of physics6 could be interpreted as explicit definitions of closed concepts. But since, as has been shown, the word “definition” refers in this context to nothing but a reduction sentence selected as reliable indicator, it only looks as though the tidy pattern of analysis had won out. We have seen that a systemic reduction sentence has factual content inasmuch as it is open to correction through the other members of the system, but at the same time is a (partial) meaning rule. It is surprising that so little cognizance has been taken of this unorthodox implication of Carnap’s theory of reduction sentences by the advocates of the either-or pattern of semantic analysis. It will be shown, now, that a close look at the meaning of the “if, then” occurring in reduction sentences brings into sight a hitherto neglected concept of semantics which I choose to label “quasi-semantic probability implication.” Consider again the simple implication “if x is a lemon, then x is sour.” It is not analytic, for, as we have seen, circumstances are conceivable under which we would be strongly inclined to classify x as a lemon even though x is not sour. According to what I called the either-or pattern of semantic analysis, one would conclude that “sour” forms no part of the meaning of “lemon” nor is deducible from the definition of “lemon”—in short, that the implication is synthetic. But the obvious trouble with this conclusion is that it presupposes that “lemon” is a precisely defined class-term, that we can draw a sharp line between those properties diﬀerentiating lemons from other kinds of fruit that are “logically connoted” by the class-term and those which lemons have contingently. Since, on the contrary, such class-terms are intensionally vague, the sharp concept of logical connotation had better be replaced by a continuous concept “term T connotes property P to degree x.” We might ask, for example, whether people would more readily refuse application of the term “lemon” to a fruit which, though mature, is not yellow than to a fruit which is not sour; 6 This

type of reconstruction of the language of physics seems to be recommended, e.g., by Philipp Frank, in Frank 1946. Thus he says that the second law of motion could be regarded as an operational definition of mass: “the ratio of two masses is inversely proportionate to the accelerations which they get from one and the same force,” but adds: “This definition is only unambiguous if we obtain the same value of mass whatever force we apply” (Frank 1946, 14). The physical law, then, is not the equation, but the statement that the defined quantity is constant with respect to a specified group of variations.

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and if the answer is aﬃrmative, “lemon” might be said to connote yellowness to a higher degree than sourness.7 But if it is possible for a lemon to be non-sour, in the sense that the complex of qualities which is usually compresent with sourness might be present in some instance without sourness, then “if x is a lemon, then x is sour” ought to be interpreted as a probability implication. Similarly, a term like “furious” stands for a correlation of an emotion with bodily symptoms of the emotion, such as fiery eyes, but situations are conceivable in which we would apply the term “furious” to a smiling person; a definite kind of facial expression may be connoted by this psychological term to a high degree, yet it is not connoted to the maximum degree. Therefore, “if x is furious at time t, then x does not smile at t” ought to be interpreted as the same kind of probability implication. Yet, it is only by means of such probability implications that the diﬀerential meanings of such terms can be explicated (“diﬀerential” meanings are contrasted with “generic” meanings, such as “fruit” in the case of “lemon,” “emotion” in the case of “furious”). I call such implications “quasi-semantic,” in analogy to Carnap’s term “quasi-syntactic,” since they convey semantic information and yet belong to the object-language. If the highly pertinent question should be raised how the concept of probability involved in such implications which state, not the truth-conditions, but the probability-conditions of a sentence, is to be explicated, negative answers can, at this stage, be given more readily than positive answers. It can be identified neither with Reichenbach’s frequency concept nor with Carnap’s logical concept of degree of confirmation, and for exactly the same reason: If “Prob(F x/Gx) = p” is an empirical statement about a frequency, then it can presumably be determined whether an object falls into the reference-class G independently of ascertaining whether it has property F. Even if it should not be possible to state a suﬃcient condition for the truth of “Gx” and only criteria could be specified which make it more or less probable that the predicate “G” applies, those criteria surely could not involve a reference to the very property whose relative frequency in class G is to be determined (substitute, for illustration, “suﬀering from tuberculosis” for “G,” and “living another ten years” for “F”). If the above formula, on the other hand, expresses an assignment of a degree of confirmation in Carnap’s sense, it is still more obvious that it 7 The

idea, here advanced, of replacing the clear-cut relation of meaning with a continuum of degrees of meaning may be regarded as an extension of the theory of vagueness put forth long ago by Max Black in the study Black 1937. Black there attempts to generalize the laws of logic (specifically, the law of the excluded middle) so as to make them applicable to vague predicates. In order to do so, he replaces the function “L is applicable to x” with the more complicated function “L is applicable to x with degree mn ” which is defined as follows: the limit approached by the ratio of the number of applications of L to x(m) to the number of applications of non-L to x(n) as the number of discriminations of x with respect to L and the number of observers increases indefinitely, is mn . This gradation of the relation of denotation (predicability) naturally suggests a similar gradation of connotation as here called attention to.

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diﬀers from the kind of probability implication here presented for further examination. For Carnap’s degree of confirmation is a measure of the amount of overlap of the ranges of two sentences, where the range of a sentence is defined as the class of state-descriptions in which the sentence would be true. These ranges, however, are determined by the semantic rules of the language, and the semantic rules for sentences have the form of statements of truth-conditions. The idea of probability-conditions as meaning-specifications seems, therefore, to be foreign to Carnap’s treatment of logical probability.8 As just remarked, a probability implication “Ax −− Bx” in Reichenbach’s sense (or, for classes of successive events, “Axi −− Byi ”) cannot be regarded as a meaning-specification of “A” in terms of “B,” since Reichenbach defines the concept of probability in such a way that it must be possible to determine whether an element belongs to the reference-class A independently of determining whether it, or the corresponding element, falls into the attribute-class B. However, if we take the definition of “meaning” in terms of psychological (or behavioral) dispositions seriously, we might nevertheless analyze the statement “if A, then with probability p, B,” intended as a meaning-specification, as a statistical generalization about verbal behavior, and thus, to coin a new term for a hitherto neglected concept, as a quasi-semantic probability implication: If a person (of specified description, such as “making a truthful statement”) verbalizes “A,” then with probability p he expects, or is disposed to believe, B—where p is a relative frequency. Notice that the reference class of this statistical law of behavioristics is the class of persons of specified description verbalizing “A,” and not the class which “A” designates; it can therefore be maintained without contradiction that membership of this reference-class can be determined independently of membership of the corresponding attributeclass. At least this tentatively suggested analysis works in some cases, if not in all. Consider, e.g., the question whether “having a back” is part of the meaning of “chair,” so that it is self-contradictory to call a stool a “chair,” or whether stools may be regarded as a species of chair. The fact is, of course, that the relevant linguistic habits are by no means fixed. All we can say is that the fact that x is called a chair makes it probable to some degree that x is believed to have a back, where the probability in question may be interpreted as a frequency. But notice that the quasi-semantic probability implication in the object-language 8A

sketch of the program of generalization of the concept of connotation (designation, intension), somewhat analogous to the generalization of the concept of implication in Reichenbach’s probability-logic, was presented by A. Kaplan, in Kaplan 1946. The sketch has recently been worked out in formal details, see A. Kaplan and H. F. Schott (Schott and Kaplan 1951). The central idea is the replacement of the concept of “necessary and suﬃcient condition” for membership in a given class with the continuous concept of indicators, of varying probability-weights, of membership. Probability implications are to replace analytic implications as means of specifying the meanings of class-terms. However, the above considerations make it doubtful whether the concept of probability involved in such partial meaning rules could be identified either with Reichenbach’s frequency concept or with Carnap’s concept of logical probability.

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“if x is a chair, then with probability p, x has a back” diﬀers in kind from a probability implication like “if x is a human male, then with probability p, x is white.” In the first place, it would be incorrect to interpret the latter probability implication as semantic, since its validity obviously does not depend on the existence of language-habits: if no language at all existed, it might still be true that a certain proportion of human males are white. Second, there is no connection at all between the meanings of “human male” and “white,” even though the corresponding classes overlap. According to the sketched analysis, “‘S ’ means D with probability (or degree) p” is a statistical generalization of descriptive semantics. It would therefore be meaningless, on the basis of this analysis, to ask to what degree “S ” means D in a single instance of verbalization. For this reason, it may be prudent to give consideration to an alternative analysis which is compatible with the meaningfulness of such questions. “Meaning to degree p” might be interpreted as designating a given strength of expectation (or, of conditioned response, if “mentalistic” terms must be avoided), where strength of expectation is measured in some appropriate way to be determined by psychometricians. Assuming that “S ” represents a declarative sentence and “D” a state of affairs, degree of meaning D would be related to reluctance of withdrawing S as follows: the greater the degree with which “S ” means D, the smaller the reluctance with which “S ” will be withdrawn (or “not-S ” will be asserted) in case D is disbelieved. To illustrate, consider the following possible properties of a chair: (a) having at least three legs, (b) capable of seating just one person, (c) having a back. Correspondingly let us construct the functions: “chair” means (for a fixed individual) (a) to degree p1 ; “chair” means (b) to degree p2 ; “chair” means (c) to degree p3 . It is reasonable to conjecture that p1 is the highest and p3 the lowest degree. In that case the reluctance with which the individual will call an object “chair” if it is believed to lack (a) is greater than the reluctance with which he will call it “chair” if it is believed to lack (b), and still greater than the reluctance with which he will predicate the word if it is believed to lack (c). It is evident that systemic reduction sentences are just such quasi-semantic probability implications, a fact which is concealed by their formulation as material implications. For, suppose that, using “if p1 , then (if p2 , then p3 )” as a rule of inference, we conclude p3 on the basis of verification of p1 and p2 . If this inference were deductive, then, if all the negative reduction sentences for the same open concept (i.e., those formulating conditions under which ∼ p3 may be asserted) led to the contradictory conclusion, we would simply have to throw them out on the ground that p3 had already been established conclusively. This would be tantamount to treating the negative reduction sentences as refutable and the positive ones as analytic—an absurd asymmetry. A system of reduction sentences is rather a system of reciprocally confirming hypothe-

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ses: the evidence p1 .p2 confirms the hypothesis p3 , and the other members of the system point to further tests which may further confirm or disconfirm the hypothesis. The very applications of the method of reduction of terms which Carnap mentions show that the connections between the reduced terms and the terms of the reduction basis are probability connections and not deductive ones. This emerges clearly from his discussion of behavioristic reduction of psychological terms, in Foundations of the Unity of Science. Speaking of the possibility of behavioristic reduction of a term like “angry,” Carnap writes: It is suﬃcient for the formulation of a reduction sentence to know a behavioristic procedure which enables us—if not always, at least under suitable circumstances— to determine whether the organism in question is angry or not. And we know indeed such procedures; otherwise we should never be able to apply the term ‘angry’ to another person on the basis of our observations of his behavior, as we constantly do in everyday life and in scientific investigation. A reduction of the term ‘angry’ or similar terms by the formulation of such procedures is indeed less useful than a definition would be, because a definition supplies a complete (i.e., unconditional) criterion for the term in question, while a reduction statement of the conditional form gives only an incomplete one. But a criterion, conditional or not, is all we need for ascertaining reducibility. (Carnap 1934, 419)

Carnap here says that we know behavioristic symptoms which are suﬃcient conditions, but none which are suﬃcient and necessary conditions, for a state of anger and a state of non-anger respectively. But clearly we cannot assert with any greater confidence that the presence of a given behavioristic symptom is invariably accompanied by the presence of anger than that the absence of such a symptom is invariably accompanied by the absence of anger. Since the correlation between behavioral states and mental states is neither one-many nor many-one (i.e., many-many), probability implications are all we can establish no matter whether a suﬃcient condition or a necessary condition for a given mental state is in question. Just as the possibility of strong self-control, mentioned by Carnap, makes it diﬃcult to find a behavioristic term which expresses a strictly necessary condition for anger, so the possibility of putting on an act makes it diﬃcult to find one that expresses a strictly suﬃcient condition. The theory of open concepts has a particularly noteworthy implication for the psycho-physical problem, which may just be touched upon in conclusion. It was hinted above that just in the way class-terms like “lemon” connote a correlation of qualities, so names of mental states like “furious” connote a correlation of a feeling with bodily expressions and behavior. According to the tidy pattern of semantic analysis, we ought to distinguish clearly the meaning of “furious,” which is an introspectable emotion, from the logically contingent accompaniments of the mental state, such as a red face and a trembling voice. The psycho-physical law “whenever x is in mental state M, then x exhibits bodily and behavioral expressions B” is, on this view, clearly a contingent proposition. But the question whether this is a contingent proposition may be

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like the question whether “all lemons are sour,” or “all lemons are yellow” is a contingent proposition. As we saw, instances are conceivable which we would be inclined to classify as lemons even though they are not sour, or not yellow: in this respect the propositions are contingent. But if we were pressed to mention diﬀerentiating defining properties of lemons, we would be likely to mention just such properties as these: in this respect the propositions are analytic. The trouble, of course, comes from treating “lemon” as a closed concept and hence attempting to apply the dichotomy analytic-contingent whereas some such continuous concept as “degree of connotation” ought to be applied. Similarly, the question about the logical status of the psycho-physical law presupposes that a term like “furious” either is, or is not, definable in terms of behavior. But clearly such terms are not “introduced” into a natural language through verbal definition. They come to mean what they mean by learned association with both introspectable (mental) states and bodily and behavioral expressions. We should not therefore ask whether the latter are logically connoted but instead to what degree they are connoted by the psychological terms. It may be that the degree to which the introspectable feeling is connoted is much higher than the degree to which any given “expression” of the feeling is connoted. Thus it may be that I would not retract the statement “I am furious” even if it were pointed out to me that I exhibit all the symptoms of a happy, contented man, since the emotion is undeniably present. On the other hand, there are names of emotions, like “love,” which connote public symptoms to a higher degree; this means that if one is convinced by others that one’s behavior diﬀers from the characteristic behavior of people who are “in love,” one would more readily retract the introspective judgment. Thus the inconclusiveness of the behaviorism-dualism dispute is ultimately due to the uncritical application to a natural language involving open concepts of a semantic meta-language, involving the Carnapian dichotomy “Limplication-F-implication,” which is suitable only to language-systems. If space permitted it, I would argue that the uncritical use of this dichotomy lies also at the root of another time-honored inconclusive controversy, that of phenomenalism vs. realism: do statements about physical reality mean sensedata? Perhaps analytic philosophy has come to the stage where it is ripe for a “Copernican revolution” analogous to the one credited to Kant: we have to scrutinize the semantic tools with which we explore semantic reality; before proceeding with asking questions of the sort “does p really entail q, and is r really self-contradictory,” we ought to give more serious attention to the question “what do we mean by ‘entailment’ as applied to statements involving open concepts?”

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Appendix [Editors’ note: We include in this appendix Pap’s reply to a brief discussion of chapter 19 by A. Caracciolo and V. Somenzi. Both the discussion and reply follow the article reprinted here as chapter 19 in Methodos vol. V, no. 17. (1953), at 43-4.] In chapter 19 I made the point that a physicist’s selection of a law as a “definition” of a physical quantity (as e.g., Mach’s definition of mass in terms of the third law of motion) is best interpreted as the assignment of a great probability-weight to one member of a system of reduction sentences. This interpretation was contrasted with the interpretation of such a definition as a rule of substitution of symbols which is devoid of “factual content,” i.e., such that it is meaningless to speak of its empirical confirmation or disconfirmation. Now, I am quite uncertain whether my critics really disagree with me on this point. Their explicit denial that such a selection “entails attribution to the selected reduction sentence of a greater probability-weight than to other possible indicators” suggests that they do. But they are silent about their reasons for denying it. My reason for aﬃrming this is, briefly, the following: suppose we consider, for simplicity’s sake, a reduction-system for “mass” consisting of just two members, viz. the third law of motion and Hooke’s Law in the special form “bodies of equal mass suspended successively on a spring at constant level produce equal strain,” and suppose that this system is discovered to be inconsistent in the sense explained in chapter 19. Assuming that the source of the inconsistency is theoretical error and not experimental error (there is no doubt, we may suppose, that the bodies imparted equal accelerations to each other and that they produced unequal strains), one will have to conclude that one or the other of the two laws is false. But once this inevitable conclusion has been drawn, the question will arise: which is more likely to be false? Now, on the interpretation of such a “definition” (like Mach’s definition of mass) as a logical equivalence— or more exactly, a meta-linguistic rule of substitution relatively to which the object-linguistic statement “equal masses impart to each other equal accelerations” is a tautology—it does not even make sense to say that on the experimental evidence the definition is probably false. Since I have repudiated this interpretation (for reasons to be found in chapter 19), the only alternative interpretation that remains is that the law which is called the “definition” is that law which would be considered less likely to be false in case such an inconsistency were to arise. The concept of “probability-weight” here involved is a pragmatic concept (in the terminology of Morris and Carnap) since it refers to the attitudes of scientists toward their theories when faced with the necessity of revising one or the other of them; although a scientists’s degree of reluctance to abandon a given law is, of course, closely related to the objective degree of confirmation (Carnap’s semantic concept) which he assigns to it on the basis of past experience. My critics seem to prefer what I chastisingly referred to as the “tidy (either-or) pattern of semantic analysis,” according to which a given law is in a given context of inquiry either analytic (definitional) or synthetic, though, as they point out, it is up to the inquirer to make the choice. But thus they simply dodge the main argument of chapter 19, which was that since physical laws involve open concepts the analytic-synthetic dichotomy is not applicable to them; that a reduction sentence is, by virtue of its systemic occurrence, both a (partial) meaning rule and a factual statement; and that accordingly the analytic-factual dichotomy should be replaced by a continuous concept (concept admitting of degrees) which I choose to call “quasi-semantic probability implication.”

Chapter 20 EXTENSIONAL LOGIC AND LAWS OF NATURE (1955)

By an extensional logic I mean a logic satisfying the following two requirements: a) all connectives are extensional, i.e., the truth-values of compound statements formed by means of them depend only on the truth-values, not the meanings, of the component statements; b) any two predicates of equal extension, no matter how diﬀerent their meanings, are mutually substitutable salva veritate in any context. The example par excellence of such a logic is, of course, Principia Mathematica, where the concept of “material” implication satisfies the first, and the concept of “formal” implication the second requirement. The insuﬃciency of this logic to deal with the logical modalities (in particular the concept of deducibility, since “q is deducible from p” is obviously not a truth-function of p and q) has been realized for a long time, ever since C. I. Lewis constructed his system of “strict implication,” and in recent years logicians have been intensively occupied with the problems of modal logic. At the same time, however, much attention has been given, especially by the analytic philosophers in the United States and in England, to the problem of interpreting an important class of contingent statements, which prima facie violate the postulate of extensionality, by means of the extensional language of Principia Mathematica (supplemented, perhaps, by a meta-language containing the syntactic concept of derivability): I am referring to the subjunctive conditionals and statements of laws of nature, which abound in both everyday language and scientific language. Among these analysts there are some who continue to believe that this language (or language-structure) is adequate for the expression of all genuine propositions. As against them, I am going to argue that these eﬀorts to reconstruct the mentioned type of statements in terms of material implication plus derivability are bound to fail, and that some kind of intensional implication must be accepted. As the subjunctive conditional first raised its ugly head when the problem of explicitly defining disposition predicates was first investigated (by Carnap, in Carnap 1937), it will be convenient to begin with a brief consideration of

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this problem. Since material conditionals are capable of vacuous truth (truth due to the falsity of the antecedent), the reconstruction of “if x were immersed in water, then x would dissolve,” which is a natural analysis of “x is soluble in water,” as a material conditional leads of course to the paradoxical consequence, pointed out by Carnap, that a match which is kept dry until it is burnt up, is soluble in water. But the reduction sentence: (∀x)(∀t)(Q1 (x, t) ⊃ (Q3 (x) ≡ Q2 (x, t)))

(R)

which avoids this paradoxical consequence of the explicit definition: Q3 (x) ≡d f (∀t)(Q1 (x, t) ⊃ Q2 (x, t)

(D1 )

has in turn counterintuitive consequences, in the face of which the desire to return to explicit definitions, of a more complicated character, which avoid the shortcomings of both (D1 ) and (R), is quite natural. That (R) has counterintuitive consequences is easy to show. As Carnap explicitly pointed out, it attributes a meaning to Q3 only within the class determined by Q1 ; for individuals outside this class the question whether or not they have the dispositional property is undecidable, and since this undecidability is due to the fact that the disposition predicate is not defined for non-members of that class, it is a theoretical, not just a practical, undecidability. In other words, with respect to such individuals it is meaningless to say either that they have or that they do not have the disposition in question. But thus it appears that Carnap’s device throws us from Scylla into Charybdis: explicit definitions were discarded because they entail that any individual upon which the relevant kind of experiment is never performed has the disposition, but now we are faced with the no more palatable result that it is meaningless to attribute the disposition to such an individual. Carnap tried to escape from this consequence by conceding that it is still meaningful to attribute a disposition Q3 to an individual b outside the class determined by Q1 , provided that b belongs to a natural kind (such as wood, in the case of the match mentioned above) which overlaps that class. Thus we have confirming evidence for the law “all wood is insoluble in water,” and therefore the question “is this match, which has never been, and never will be, immersed in water, soluble in water?” is still significant on Carnap’s theory. But two serious objections remain. First, assume a disposition which is unique to a particular individual. For example, a particular human being may have the disposition to feel nauseated when exposed to the smell of an orange, and it is logically possible than no other organism has that disposition and that the disposition is never actualized for the simple reason that the individual concerned is never exposed to the fatal smell. To be sure, as long as the class determined by Q1 (in this example, the class of organisms exposed at some time to the smell of an orange) is not empty, confirming or disconfirming evidence with respect to the statement “if this individual were exposed to the

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smell of an orange, he would feel nauseated” is still obtainable. But what if no oranges existed? Then the class Q1 would be empty and hence the above statement would definitively be meaningless by Carnap’s theory. And this is strange, since intuitively the truth, and a fortiori the significance, of the nonexistential conditional “for any x and y, if x were an orange and y smelled x, then y would feel nauseated” is compatible with “there is no x such that x is an orange.” Second, (R) is constructed on the assumption that solubility is a permanent disposition, as evidenced by the universal quantification over the time variable. But there are, of course, also transient dispositions, like electrical charge. Suppose, now, that x is the very first body whose electrical charge has been ascertained by a human observer. Then the statement that x is electrically charged at t1 , the time of our supposed first experiment, would be meaningful, yet the statement that x is electrically charged at some earlier time t0 would be meaningless.1 An adequate explicit definition of a disposition predicate Q3 should have the advantage over (R) that it rescues the significance of the statement Q3 (b), where b is an object upon which the test operation has never been performed and possibly belongs to a natural kind K such that the test operation has not been performed upon any member of K, and over (D1 ) that it does not allow vacuous predication of Q3 . The following explicit definition satisfies these requirements:2 Q3 (x) ≡d f (∃ f )[ f (x).(∃y)( f (y).Q1 (y)). (∀z)( f (z).Q1 (z) ⊃ Q2 (z))]

(D2 )

Unfortunately (D2 ) is likewise open to serious objections. First, it entails that it is self-contradictory to say “a is soluble but there are no liquids” (if there are no liquids, the class of immersed objects—ˆyQ(y)—is empty); but the intuitive sense of “a is soluble” is “if there were a liquid, and a were immersed in it, then a would dissolve.” Second, (D2 ) involves a confusion between the grounds for believing a proposition and the analysis of the proposition believed. For, according to (D2 ), the subjunctive conditional which is condensed into Q3 (a) asserts not only the existence of a law in accordance with which 1 In

connection with such time-dependent disposition predicates (“elastic, “irritable” are other examples) there arises a further diﬃculty for Carnap’s theory. Let Q3 (x, t) represent such a disposition with respect to which an individual may change, and let (∀x)(∀t)(Q1 (x, t) ⊃ (Q3 (x, t) ≡ Q2 (x, t))) be the bilateral reduction sentence by which this predicate is introduced. Carnap argues that this sentence is analytic if the disposition predicate has no independent meaning, specified by other reduction sentences. If so, then the singular sentence Q3 (a, t0 ) can be analytically inferred from the singular sentences Q1 (a, t0 ) and Q2 (a, t0 ). But this result is irreconcilable with the plausible view that to assert a subjunctive conditional is to make an implicitly general statement, an assertion of causal connection which goes beyond a purely descriptive “post hoc” report. 2 This definition is copied from Wedberg 1944, 237, who cites it for purposes of criticism from Kaila 1967. Kaila later improved the definition, in his book Kaila 1941 (in answer to an objection by Carnap), and finally, by the time he wrote Kaila 1945, abandoned this kind of definition altogether.

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Q2 (a) is predictable from Q1 (a), but moreover the existence of confirming evidence—(∃y)(Q1 (y).Q2 (y))—for that law. Yet, to suppose that “there exist no grounds for believing p” entails “p is false,” is to be guilty of confusing truth with knowledge of the truth.3 The third objection leads us straight to the line of analysis of subjunctive conditionals whose futility has been demonstrated especially by Chisholm and Goodman4 . It is that values of the predicate-variable f can be constructed such that objects to which we do not want to ascribe Q3 will again, as in the case of D1 , vacuously satisfy the definiens. Thus, we might set f (x) = (x = a ∨ x = b), where Q1 (b) is true yet Q1 (a) false.5 If, to avoid this impasse, f is restricted to general properties (i.e. properties whose definition does not involve reference to a particular individual), trivialization is still possible in terms of such predicates as Q1 ⊃ Q2 .6 And if this variable is still further restricted, to something like “intrinsic” properties, properties determining natural kinds (“wooden,” “golden,” “metallic,” etc.), we run the danger of defining per obscurius. For, apart from the consideration that the concept of “natural kind” stands in at least as much need of analysis as—and may even logically presuppose—the concept of “disposition,” to say of a property P that it is intrinsic to an individual x is meaningless unless it is elliptical for P is intrinsic to x qua member of class K. Thus even color could be intrinsic to a given individual relatively to a description of the individual in terms of color predicates; on the other hand, if a given match is described as an instrument for lighting cigarettes which I now hold in my hand, being wooden may be argued to be an extrinsic property of it, since it is not entailed by the description by which it was referred to. If the intuitive notion of a property without which an individual “would not be what it is” is analyzed in this way, it appears that one cannot significantly divide properties into intrinsic and extrinsic ones. Now, the program of precluding trivial satisfaction of the definiens by imposing suitable restrictions on the range of its bound major variable, is precisely what Chisholm and Goodman have shown to face insuperable diﬃculties. If we analyze “if it were the case that p, then it would be the case that q” (regardless of whether it is in fact the case that p)7 as meaning “there is a true

3 This

objection applies to Thomas Storer’s explicit definition of “soluble,” which is very similar to D2 (Storer 1951). I think that the analysis of counterfactual conditionals given by B. J. Diggs (Diggs 1952), subtle and careful as it is, is likewise open to it.—For a lucid warning against the confusion of truth with knowledge of the truth, see Carnap 1949. 4 Chisholm 1946 and Goodman 1947. 5 This counterexample is due to Carnap. 6 Cf. Wedberg 1944 7 “The Problem of Subjunctive Conditionals” is a better terminology than “The Problem of Contrary-to-Fact Conditionals,” precisely because in asserting subjunctive conditionals one just as often implies nothing with respect to the truth-value of the antecedent as one implies that the antecedent is false.

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proposition R such that q is entailed by p.R (and by no less than that),”8 we must exclude material or formal implications which are vacuously true from the range of “R”; otherwise incompatible subjunctive conditionals could be both true.9 But this restriction has been shown to be insuﬃcient because of the possibility of taking as R a true, yet not vacuously true, formal implication such that again incompatible subjunctive conditionals turn out to be both true: such formal implications are the so-called accidental universals. Thus one could prove both “if you were to go to the concert, you would spend an enjoyable evening” and “if you were to go to the concert, you would not spend an enjoyable evening” from the falsity of the antecedent, by taking as R alternatively the accidental universal “(∀x)( f (x) ⊃ (x goes to the concert ⊃ x spends an enjoyable evening))” or “(∀x)( f (x) ⊃ (x goes to the concert ⊃ x does not spend an enjoyable evening)),” provided the person denoted by “you” is the one person satisfying “ f (x).”10 It was pointed out by both Chisholm and Goodman that accidental universals are intuitively distinguished from laws of nature by their inability to support subjunctive conditionals: it is unreasonable to infer from “all the people in this room are tall” that if my little boy were in this room he would be tall; it would be more reasonable to infer from this hypothesis that some people in this room would not be tall. On the other hand, it is alleged to be reasonable to infer from “all ravens are black” that if my little boy were a raven he would be black. Clearly, if this were all that could be done to distinguish these two types of universal statements, the explored analysis of subjunctive conditionals would be circular. Let us see, however, whether we can explain this intuitive diﬀerence in terms of a characterization of accidental universals which makes no use of the notion of the subjunctive conditional. It may seem that the diﬀerence is simply that between a universal statement about an extensionally defined (and hence “closed”) class and a universal statement about an intensionally defined (and hence “open”) class. This is actually the position taken by Popper (Popper 1949), who blames the whole “problem” here under review on the confusion of extensional and intensional interpretation of class-terms. However, this simple solution of what Popper calls a “modern riddle” misses the mark for two reasons. First, if x ∈ A meant x = a ∨ x = b ∨ . . . ∨ x = n (extensional definition), then the inference from “if y (which is not an A) were an A” to “then y would be a B” would be perfectly valid, for the simple reason that its premises would

8 It

may be remarked en passant that even if this analysis were adequate it is doubtful whether it accomplishes the aimed at reduction to extensional concepts, since entailment itself is prima facie an intensional function. 9 Notice that “if it were the case that p, then it would be the case that q” diﬀers from p ⊃ q precisely in that it is incompatible with “if it were the case that p, then it would not be the case that q,” whereas p ⊃ q is compatible with p ⊃∼ q. 10 Cf. Chisholm’s example, Chisholm 1946, 491.

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be self-contradictory (inferences from self-contradictory premises are bound to be valid). Therefore the proposed interpretation does not account for the specified intuitive diﬀerence. Second, the class-terms in accidental universals have intensional meanings, just like the class-terms in laws, since one cannot deduce from an accidental universal about A, by mere analysis of the meaning of “A,” which individuals belong to A (e.g. we can all understand the statement “all the advisers of Eisenhower speak English” without knowing who the advisers of Eisenhower are). But while this simple consideration, viz. that one can understand an accidental universal without knowing which objects fulfill it, is a strong argument against the interpretation of accidental universals as finite conjunctions of singular statements, we still are driven to recognize the distinction between closed and open classes as crucial, as the following dialectic will reveal. Why is it that from the (counterfactual) hypothesis “my boy is in this room” I am entitled to infer “not all people in this room are tall?” Evidently because I tacitly entertain the further premise “my boy is not tall.” But if I similarly make use of the premise “my boy is not black,” then the statement “if my boy were a raven, then not all ravens would be black” is as defensible as the statement “if my boy were in this room, then not all people in this room would be tall.” On the other hand, if we are not permitted to make any assumptions about the individual which we are considering as a possible member of A, then the statement “if that individual were an A, then some A’s would not be B’s” is never defensible, no matter whether “all A are B” be accidental or lawlike. Now, if in raising this vexing question what would be the consequence if an individual, call it c, which as a matter of fact has properties incompatible with being an A, were diﬀerent from what it is and were an A, one is not to suppose anything about c except membership of A, how can one be inhibited from inferring “c would be B” if supplied the information “all A are B”? And the simple answer is of course: if A is a closed class excluding c, such that c just is not an instance to which the generalization applies. I wish to show, however, that the distinction between closed and open classes, here invoked, can itself be explained only in terms of the subjunctive mood. In the first place, the distinction in question surely is not the distinction between finite and infinite classes. Even if we include in a biological species, e.g., individuals yet to be born, we consider it unlikely (though logically possible) that their number is infinite, and nevertheless a true generalization about such a species is regarded as a law. To this one might reply that even though the classes determined by the predicates are finite in either case, the diﬀerence is that in the case of laws only the variables of quantification have an infinite range. And since one could not plausibly maintain that in asserting a law we commit ourselves to the belief in an infinite universe of individuals, this interpretation amounts to populating the range of the variable in “for any x, if

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x ∈ A, then x ∈ B” with possible individuals. And once we permit ourselves this way of speaking—which is reminiscent of Meinongian ontology—we are led to conceive of “open” classes as of classes of actual or possible individuals. We have come here upon Lewis’ distinction between the “denotation” and the “comprehension” of a class-term: the comprehension of “raven,” e.g., includes not only all actual ravens, but also all “consistently thinkable” ravens, and insofar as “all ravens are black” is a law it asserts an inclusion of comprehensions. But while Lewis was no doubt on the track of an important distinction, his formulation of the distinction is most unfortunate. For, in the first place, the most natural interpretation of the statement “there are no consistently thinkable ravens that are not black” is “it is not consistently thinkable that something should be a raven without being black.” But on this interpretation the supposed law of nature turns into an analytic truth! Second, Lewis overlooks that whether or not it is consistently thinkable that a given individual have a given property depends on our way of referring to the individual. For example, if the individual who is now speaking is described as the individual who was born in Z¨urich on Oct. 1 1921, then he is a consistently thinkable raven, for it involves no contradiction to suppose that one and only one individual was born at that time and place and that it happened to be a raven. Indeed, in defining the comprehension of Φ as the totality of the x’s such that it is consistently thinkable that Φx (cf. Lewis 1946, 63), he implicitly ascribes to all self-consistent predicates the same, viz. universal, comprehension. For, for any individual a, some description “( x)ψx” can be found such that “Φ( x)ψx” is self-consistent (as a last resort, “( x)(x = a)” could be taken). These diﬃculties seem to me to arise from the conception of an “open class” as a totality comprising, along with actual entities, possible entities. But to say that there exist, besides the actual members of A, also possible members of A, can only mean that there are individuals which might have property A although they actually do not have it. Thus A, in the universal implication “for any x, if x ∈ A, then x ∈ B,” may be an open class even if the range of x is a finite set a, b, . . ., n. The openness of A can mean nothing else than that the connective “if-then” is meant as a subjunctive connective. In other words, in saying that the universal implication refers to an open class, we are saying that it involves statements about unactualized possibilities, such as “if Arthur Pap, instead of being a man, were a raven, he would be black.” This entails, of course, that the extensional conception of universal statements as being conjunctions of purely descriptive singular statements11 is valid only for accidental universals.

this is not meant the equivalence of (∀x)(x ∈ A ⊃ x ∈ B) to (a ∈ A.a ∈ B).(b ∈ A.b ∈ B).. . ..(n ∈ A.n ∈ B), where a, b, . . ., n are all the members of A, which must be rejected not only on the ground that the class A may be empty, but also on the ground that one cannot deduce from the universal statement which 11 By

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I hope to have shown that it is futile to think one can avoid the subjunctive “if-then,” as an irreducibly non-extensional connective, by defining a law as a true, and synthetic, formal implication about an open class. But further, in order to answer the obvious objection from the incompatibility of subjunctive conditionals with common antecedent and incompatible consequents (e.g. “all sugar is soluble” and “all sugar is insoluble”), one would have to qualify that “(∀x)(x ∈ A ⊃ x ∈ B)” expresses a law only if either A is nonempty or it is deducible from a true formal implication which is not vacuously true.12 By virtue of this qualification one could maintain, e.g., that “(∀x)(sugar(x).immersed(x) ⊃ dissolves(x)))” expresses a law even relatively to a universe devoid of liquids, because it is deducible from the non-vacuous implication “(∀x)(sugar(x) ⊃(immersed(x) ⊃ dissolves(x))).” But since relatively to such a universe “(∀x)(sugar(x) ⊃ (immersed(x) ⊃∼ dissolves(x)))” would likewise be true, one would have no basis for defending the plausible opinion that, even relatively to such a hypothetical universe, “all sugar is soluble,” and not “all sugar is insoluble,” expresses a law. Perhaps this diﬃculty, and related ones, can be solved by first defining a “fundamental law of nature” as a true, synthetic proposition expressed by a formal implication none of whose predicates are empty and which contains no individual constants, and then defining a “law of nature” as a synthetic universal proposition deducible from a fundamental law of nature. But I cannot refrain from adding at once three objections to this proposal: 1. Notice that relatively to a universe devoid of liquids, the sentence “all sugar is soluble” would be meaningless and hence would not express a law; for, “soluble” can be introduced into an extensional language only by a reduction sentence, but, as we have seen, if there are no immersed objects, the reduction sentence endows “soluble” with no meaning at all; hence we could not meaningfully say “even in a universe devoid of liquids, it would be a law that all sugar is soluble.” Mutatis mutandis, this holds for laws concerning such dispositions as vaporization points, melting points, freezing points, relatively to universes lacking those temperatures at which in this universe substances change their states of aggregation. 2. While it is, indeed, a characteristic feature of accidental universals that they contain individual constants essentially, and it is therefore tempting to stipulate that a sentence expresses a law only if no individual constants occur essentially in it, this stipulation is nonetheless counterintuitive. Suppose, for example, that Galileo’s law of freely falling bodies and Kepler’s three laws, which contain such individual constants as “the earth” and “the sun,” had never

individuals belong to A (cf. Lewis 1946, 65); but the usually accepted equivalence to (a ∈ A ⊃ a ∈ B).(b ∈ A ⊃ b ∈ B).. . ..(n ∈ A ⊃ n ∈ B), where a, b, . . ., n are all the individuals over which the variable ranges. 12 For an analysis of the concept of law along this line, cf. Reichenbach 1947, chapter VIII.

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been explained by deduction from such (apparently) purely general propositions as the law of gravitation and the principles of Newtonian mechanics. Would it not be perfectly proper to refer to them as “laws,” albeit unexplained laws (what Mill called “empirical” laws) just the same? 3. Think of all the laws of mathematical physics that refer to ideal conditions and entities, like motions under the sole influence of gravity, ideal gases, frictionless engines, point-particles, etc. Would it not be bold to claim that they are deducible from statements whose predicates denote empirically given objects and situations? Yet, if these be not “fundamental laws of nature,” which are? However, my main argument against the possibility of expressing laws of nature by means of the symbolism of extensional logic is best stated by returning to the inquiry, commendably started by Lewis, into the range of the individual variables. It may be obscure to say that it consists of all actual individuals, since this involves, by contrast, the conception of “possible individuals.” But the idea can be expressed in the “formal mode of speech” by saying that the substitutable constants a, b, etc. are what Russell calls “logically proper names,” i.e. names in the sense in which “Pegasus,” “Hamlet,” etc. are not names. If a formal implication, then, is logically equivalent to the conjunction of its substitution-instances, it makes an assertion only about observed cases (as was emphasized by Lewis in Lewis 1946).13 It may seem that a predictive content can nevertheless be secured for formal implications by allowing descriptions as substituends for “x”: as Russell pointed out, by means of ( x)xRa we can make significant statements about unobserved individuals. But in the language of Principia Mathematica descriptions in turn are defined (contextually) in terms of statements involving variables, and thus we are led back to a, b, etc. as the primitive names of the entities over which the variables range. If, therefore, we want to avoid the presupposition of “possible entities” as included in the range of the variables, we are driven to the subjunctive mood after all: for any (at some time actual) x, if x were an A, then x would be a B. But is it significant to suppose that an individual which is, say, a stone, might instead be, say, a raven? Clearly, such a supposition is significant only if the notion of a substratum is significant. A name a denotes a substratum if P(a) is a self-consistent, contingent statement regardless of what property P is, provided only that it is a self-consistent property of the first level (notice that this condition would not be fulfilled if a were a description). Curiously, then, the substratum (or a multitude of numerically distinct, though indistinguish-

13 Even before Lewis, Russell became aware of this diﬃculty, in Russell 1940, 255: “There is thus a hypothetical element in any general proposition; ‘ f (x) is true of every x’ does not merely assert the conjunction f (a). f (b). f (c). . . where a, b, c, . . . are the names (necessarily finite in number) that constitute our actual vocabulary. We mean to include whatever will be named, and even whatever could be named.”

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able substrata) lurks behind the individual variables of the usual algorithm of symbolic logic, and makes its embarrassing presence felt especially if we try to express subjunctive conditionals in that algorithm. Let me call attention, in conclusion, to the alternative type of language which contains only names of properties, no names of substrata, and which is advocated by Russell himself as an alternative to the usual subject-predicate type of language.14 In such a language we could presumably express the law “all ravens are black” without subjunctive mood, by asserting simply the compresence of the property being a raven with the property being black.15 This would, of course, be a realistic language which nominalistically-inclined logicians, like Quine and Goodman, who want to admit only names of individuals and treat all other descriptive constants as syncategorematic, will repudiate. But it is worth thinking about the question whether the desire to avoid the obscure notion of the substratum, and its symbolic counterpart, the Russellian “logically proper names,” does not force such a realistic language upon us after all.

14 Russell

1940, chapter VI. See also Ayer 1952a. Chisholm 1946: “This suggests that the terms of ‘non-accidental’ connections are the properties of things. And if we cannot get rid of the subjunctive by any other means, we can define it in terms of these ‘connections’.” 15 Cf.

Chapter 21 DISPOSITION CONCEPTS AND EXTENSIONAL LOGIC (1958)

One of the striking diﬀerences between natural languages, both conversational and scientific, and the extensional languages constructed by logicians is that most conditional statements, i.e., statements of the form “if p, then q,” of a natural language are not truth-functional. A statement compounded out of simpler statements is truth-functional if its truth-value is uniquely determined by the truth-values of the component statements. The symbolic expression of this idea of truth-functionality, as given in Principia Mathematica, is p ≡ q ⊃ ( f (p) ≡ f (q)). That is, if “ f (p)” is any truth-function of “p,” and “q” has the same truth-value as “p,” however widely it may diﬀer in meaning, then “ f (q)” has the same truth-value as “ f (p).” Clearly, if I am given just the truthvalues of “p” and “q,” not their meanings, I cannot deduce the truth-value of “if p, then q”—with a single exception: if “p” is given as true and “q” as false, it follows that “if p, then q” is false, provided it has a truth-value at all. On the contrary, the knowledge that matters for determination of the truth-value of a “natural” conditional—let us call them henceforth “natural implications,” in contrast to those truth-functional statements which logicians call “material conditionals” or “material implications”—is rather knowledge of the meanings of the component statements. In the case of simple analytic implications like “if A has a niece, then A is not an only child” such knowledge of meanings is even suﬃcient for knowledge of the truth of the implication; at any rate knowledge of the truth-value of antecedent and consequent is irrelevant. In the case of those synthetic natural implications which assert causal connections, knowledge of meanings is not, indeed, suﬃcient, but it is necessary, and knowledge of the truth-values of the component statements is not presupposed by knowledge of the truth-value of the implication.1 Consider the conditional (which 1 In

this context “knowledge” is used in the weak sense in which “p is known to be true” entails that there is evidence making it highly probable that p, not the stronger claim that there is evidence making it certain that p.

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may or may not be “contrary-to-fact”): if I pull the trigger, the gun will fire. It would be sad if belief in such an implication were warranted only by knowledge of the truth of antecedent and consequent separately, for in that case it would be impossible for man to acquire the power of even limited control over the course of events by acquiring warranted beliefs about causal connections. Notice further that frequently presumed knowledge of a causal implication is a means to knowledge of the truth, or at least probability, of the antecedent; if this is an acid then it will turn blue litmus paper red; the reaction occurred; thus the hypothesis is confirmed. Knowledge of the consequences of suppositions is independent of knowledge of the truth-values of the suppositions, no matter whether the consequences be logical or causal. The diﬀerence between material implication and natural implication has been widely discussed. The logician’s use of “if p, then q” in the truthfunctional sense of “not both p and not-q,” symbolized by “p ⊃ q,” is fully justified by the objective of constructing an adequate theory of deductive inference, since the intensional meaning of “if, then,” be it logical or causal connection, is actually irrelevant to the validity of formal deductive inferences involving conditional statements. This is to say that such conditional inference-forms as modus ponens, modus tollens, and hypothetical syllogism would remain valid in the sense of leading always to true conclusions from true premises if their conditional premises or conditional conclusions were interpreted as material conditionals asserting no “connections” whatever. The so-called, and perhaps misnamed, paradoxes of material implication, viz., that a false statement materially implies any statement and a true statement is materially implied by any statement, are not logical paradoxes. The formal logician need not be disturbed by the fact that the statements “if New York is a small village, then there are sea serpents” and “if New York is a small village, then there are no sea serpents” are, as symbolized in extensional logic, both true; for since this is due to the falsity of their common antecedent, modus ponens cannot be applied to deduce the contradiction that there both are and are not sea serpents. No contradiction arises. However, it is in the application of extensional logic for the purpose of precise formulation of empirical concepts and propositions that serious diﬃculties arise. The “paradoxical” feature of material implication that the mere falsehood of the antecedent ensures the truth of the implication leads, not to formal inconsistency, but to grossly counterintuitive factual assertions when extensional logic is applied to the language of empirical science. This becomes particularly evident if one tries to formalize so-called operational definitions by means of extensional logic. For the definiens of an operational definition is a conditional whose antecedent describes a test operation and whose consequent describes a result which such an operation has if performed upon a certain kind of object under specified conditions. A concept which is operationally defined in this sense may be called a “disposition concept.” Suppose, then, that a disposition concept is defined by a material conditional as follows:

Disposition Concepts and Extensional Logic (1958)

D(x, t) ≡d f (O(x, t) ⊃ R(x, t))

329 (1)

The question might be raised whether the time-argument could be omitted from the disposition predicate, so that the definition would look as follows: Dx ≡ (∀t)(O(x, t) ⊃ R(x, t)). Which form of definition is suitable depends on inductive considerations. If the disposition is “intrinsic” in the sense that a generalization of the form (∀t)(∀x)[x ∈ K ⊃ (O(x, t) ⊃ R(x, t))] has been highly confirmed (where K is a natural kind), a time-independent disposition predicate is appropriate. Examples of such intrinsic dispositions are solubility and melting point (the latter is an example of a quantitative disposition whose operational definition accordingly would require the use of functors, not just of qualitative predicates). On the other hand, the symbol “D(x, t)” is appropriate if D is such that for some objects y both “(∃t)(O(y, t).R(y, t))” and “(∃t)(O(y, t). ∼ R(y, t))” holds; for example, being electrically charged, elasticity, irritability. Now, as Carnap pointed out in “Testability and Meaning,” a definition of the form of (1) has the counterintuitive consequence that any object has D at any time at which it is not subjected to O, and that any object on which O is never performed has D at all times.2 There is a close analogy between the interpretation of the Aristotelian A and E propositions as generalized material implications (or “formal implications,” in Russell’s terminology) and the extensional interpretation of operational definitions, in that both have the consequences that intuitively incompatible statements are compatible after all. If “all A are B” means “(∀x)(Ax ⊃ Bx)” and “no A are B” means “(∀x)(Ax ⊃ ∼ Bx),” then both may be true, since both would be true if nothing had the property A, which is logically possible. Thus the student introduced to extensional symbolic logic learns to his amazement that both “all unicorns live in the Bronx zoo” and “no unicorns live in the Bronx zoo” are true statements—for the simple reason that there are no unicorns, from which it follows that there are no unicorns of any kind, neither unicorns that live in the Bronx zoo nor unicorns that don’t live in the Bronx zoo. Similarly, suppose a physical functor like “temperature” were operationally defined as follows: temp(x, t) = y ≡d f a thermometer is brought into thermal contact with x at t ⊃ the top of the thermometric liquid coincides with the mark y at t + dt. Then the clearly incompatible statements “temp(a, t0 ) = 50” and “temp(a, t0 ) = 70” would both be true, on the basis of this definition, if no thermometer were in contact with a at t0 ; indeed a would have all temperatures whatsoever at any time at which its temperature is not measured.3 2I

have slightly changed Carnap’s way of putting the counterintuitive consequence, in accordance with my using “D(x, t)” instead of “Dx.” 3 There seems to be fairly universal agreement now among philosophers of science that the simple kind of explicit definition of disposition concepts in terms of material implication is inadequate, precisely because we want to be able to say of an object which is not subjected to the test operation by which a disposition D is defined that it does not have D. One exception to this trend might, however, be noted: Gustav Bergmann maintains (Bergmann 1951a) that such explicit definitions nevertheless provide adequate analyses of the

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Some philosophers have suggested that the reason why counterintuitive consequences result if material implication is substituted for natural implication is that a material implication is true in cases where the corresponding natural implication has no truth-value. If the antecedent of a natural implication is false, they suggest, then the natural implication is “undetermined”; it is true just in case both antecedent and consequent are true, and false in case the antecedent is true and the consequent is false.4 Now, the combinations F F and F T do, indeed, leave the truth-value of a natural implication undetermined in the sense that they leave it an open question which its truth-value is. But the same holds for the combination T T. It is not the case that every true statement naturally implies every true statement. If it should be replied that nevertheless the joint truth of antecedent and consequent confirms a natural implication, it must be pointed out that if so, then the joint falsehood of antecedent and consequent likewise confirms it, by the principle that whatever evidence confirms a given statement S also confirms whatever statement is logically equivalent to S :5 if “p and q” confirms “if p, then q,” then “not-q and not-p” confirms “if not-q, then not-p,” and therefore confirms “if p, then q.” Or, to put it diﬀerently but equivalently: if “p and q” confirms “if p, then q” then it also confirms “if not-q, then not-p,” but this is to say that a natural implication is confirmable by an F F case. To illustrate: suppose I say to a student “if you study for the course at least one hour every day, then you will pass the course.” If this conditional prediction is confirmed by the fact that the advised student put in at least one hour for the course every day and passed the course, then the same fact ought to confirm the equivalent prediction formulated in the future perfect: “if you will not pass the course, then you will not have studied for it at least one hour every day.” But further, it just is not the case that no truth-value is ordinarily assigned to a natural implication whose antecedent is false. Everybody distinguishes between true and false contrary-to-fact conditionals. In particular, the belief that disposition concepts—in a sense of “adequate analysis” which is obscure to me. Referring to Carnap’s example of the match which is burned up before ever being immersed in water and therefore would be soluble by the criticized definition of “soluble,” he says “I propose to analyze the particular sentence ‘the aforementioned match is (was) not soluble’ by means of two sentences of the ideal schema, the first corresponding to ‘This match is (was) wooden,’ the second to the law ‘No wooden object is soluble.”’ In what sense do these two sentences provide an analysis of “soluble”? Bergmann is simply deducing “the match is not soluble” from two well-confirmed premises, and is therefore perhaps giving a correct explanation of the fact described by the sentence, but since “soluble” reappears in the major premise—as it must if the syllogism is to be valid!—its meaning has not been analyzed at all. It is one thing to give grounds for an assertion, another thing to analyze the asserted proposition. 4 See O’Connor 1951, 354. Also, the Finnish philosopher E. Kaila once attempted to escape from Carnap’s conclusion that disposition concepts are not explicitly definable by proposing that “Dx” be taken as neither true nor false in case x is not subjected to O (which proposal, incidentally, is consonant with Carnap’s proposal of introducing dispositional predicates by reduction sentences, as we shall see later); see Kaila 1945. 5 This has been called the “paradox of confirmation.” See Hempel 1945a, Hempel 1945b, and Carnap 1950b, section 87.

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an object has a certain disposition may motivate people to subject it, or prevent it from being subjected, to the corresponding test operation; we are, for example, careful not to drop a fragile and valuable object because we believe that it would break if it were dropped. What we believe is a proposition, something that is true or false; to say that it only becomes true when its antecedent and consequent are confirmed, is to confuse truth and confirmation.6 Let us see, now, whether perhaps a more complicated kind of explicit definition of disposition concepts within the framework of extensional logic can be constructed which avoids the shortcoming of (1): that an object upon which test operation O is not performed has any disposition whatsoever that is defined by means of O. Philosophers who follow the precept “to discover the meaning of a factual sentence ‘p’ reflect on the empirical evidence which would induce you to assert that p” might arrive at such a definition by the following reasoning. What makes one say of a wooden object that it is not soluble in water even before testing it for solubility in water, i.e., before immersing it in water? Obviously its similarity to other objects which have been immersed in water and were found not to dissolve therein. And what makes one say of a piece of sugar that it is soluble before having immersed it? Evidently the fact that other pieces of sugar have been immersed and found to dissolve. In general terms: the evidence “(x1 ∈ K . Ox1 . Rx1 ) . (x2 ∈ K . Ox2 . Rx2 ) . . . . (xn ∈ K . Oxn . Rxn )” led to the generalization “(∀x)(x ∈ K ⊃ (Ox ⊃ Rx))” from which, together with “x0 ∈ K,” we deduce “Ox0 ⊃ Rx0 .” The latter conditional is not vacuously asserted, i.e., just on the evidence “∼ Ox0 ,” but it is asserted on the specified inductive evidence. Such indirect confirmability7 of dispositional statements seems accurately reflected by the definition schema:8 Dx ≡d f (∃ f )[ f x.(∃y)(∃t)( f y.O(y, t)). (∀z)(∀t)( f z.O(z, t) ⊃ R(z, t))]

(2)

If we take as values of “ f ” alternatively “being wooden” and “being sugar,” then it can easily be seen that on the basis of such a definition, involving application of the higher functional calculus to descriptive predicates, wooden objects that are never immersed in a liquid L are not soluble in L, whereas pieces of sugar can with inductive warrant be characterized as soluble in L even if they are not actually immersed in L. 6 For

a lucid warning against this confusion, see Carnap 1949. confirmation of a conditional is distinguished from (a) direct confirmation, consisting in the verification of the conjunction of antecedent and consequent, (b) vacuous confirmation, consisting in the verification of the negation of the antecedent. 8 D is here assumed to be an intrinsic disposition in the sense explained above. The above schema is, with a slight alteration, copied from Anders Wedberg’s “The Logical Construction of the World” (Wedberg 1944, 237), who cites it for purposes of criticism from Kaila 1967. A variant of this definition schema has more recently been proposed by Thomas Storer: Storer 1951. 7 Indirect

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Unfortunately, however, the undesirable consequences of (1) reappear if certain artificial predicates are constructed and substituted for the predicate variable “ f .” Thus Carnap pointed out to Kaila that if “(x = a) ∨ (x = b) ,” where a is the match that was burned up before ever making contact with water and b an object that was immersed and dissolved, is taken as the value of “ f ,” “Da” is again provable. This seemed to be a trivial objection, since evidently “ f ” was meant to range over “properties” in the ordinary sense of “property”: who would ever say that it is a property of the match to be either identical with itself or with the lump of sugar on the saucer? But if “ f ” is restricted to general properties, i.e., properties that are not defined in terms of individual constants, the undesirable consequences are still not precluded. As Wedberg pointed out (Wedberg 1944), vacuous confirmation of dispositional statements would still be possible by taking “O ⊃ R” as value of “ f .” Nevertheless, I doubt whether the objection from the range of the predicate variable is insurmountable. To be sure, it would lead us to a dead end if we defined the range of “ f ” as the class of properties that determine natural kinds. For our philosophical objective is to clarify the meaning of “disposition” by showing how disposition concepts are definable in terms of clearer concepts. But I suspect that we need the concept of “disposition” for the explication of “natural kind,” in the following way: if a class K is an ultimate natural kind (an “infima species,” in scholastic terminology), then, if one member of K has a disposition D, all members of K have D. If “ultimate natural kind” could be satisfactorily defined along this line, “natural kind” would be simply definable as “logical sum of ultimate natural kinds.” To illustrate: would a physicist admit that two samples of iron might have a diﬀerent melting point? He would surely suspect impurities if the two samples, heated under the same standard pressure, melted at diﬀerent temperatures. And after making sure that the surprising result is not due to experimental error, he would invent names for two subspecies of iron—that is, he would cease to regard iron as an “ultimate” kind—and look for diﬀerentiating properties other than the diﬀerence of melting point in order to “account” for the latter. But be this as it may, it seems that vacuous truth of dispositional statements could be precluded without dragging in the problematic concept of “natural kind” by the following restriction on the range of “ f ”: we exclude not only properties defined by individual constants, but also general properties that are truth-functional compounds of the observable transient properties O and R.9 There remains, nevertheless, a serious objection relating to the second conjunct in the scope of the existential quantifier: there is a confusion between the meaning of a dispositional statement and the inductive evidence for it. To

9 The

latter restriction has been suggested to me by Michael Scriven.

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see this, just suppose a universe in which the range of temperature is either so high or so low that liquids are causally impossible in it. If so, nothing can ever be immersed in a liquid, hence, if “O(y, t)” means “y is immersed in L at t,” “(∃y)(∃t)( f y.O(y, t))” will be false for all values of “ f .” But surely the meaning of “soluble” is such that even relative to this imaginary universe “sugar is soluble” would be true: in using the dispositional predicate “soluble” we express in a condensed way the subjunctive conditional “if a sample of sugar were immersed in a liquid, then it would dissolve,” and this just does not entail that some sample of sugar, or even anything at all, is ever actually immersed in a liquid. True, no mind could have any evidence for believing a proposition of the form “x is soluble” if nothing were ever observed to dissolve; indeed, it is unlikely that a conscious organism living in our imaginary universe (for the sake of the argument, let us assume that the causal laws governing that universe are such that conscious life is possible in it in spite of the prevailing extreme temperatures) would even have the concept of solubility. But it does not follow that the proposition could not be true just the same. The idea underlying (2) is obviously this: the evidence on which a contraryto-fact conditional is asserted—if it is a confirmable, and hence cognitively meaningful, statement at all—is some law that has been confirmed to some degree; therefore the conditional is best analyzed as an implicit assertion of the existence and prior confirmation of a law connecting O and R.10 Now, I agree that the existence of some law in accordance with which the consequent is deducible from the antecedent is implicitly asserted by any singular counterfactual conditional, though the asserter may not be able to say which that law is (formally speaking, he may not know which value of “ f ” yields a universal conditional—the third conjunct of the definiens—which is probably true). To take an extreme example: if I say, “if you had asked your landlord more politely to repaint the kitchen, he would have agreed to do it,” I have but the vaguest idea of the complex psychological conditions that must be fulfilled if a landlord is to respond favorably to a tenant’s request which he is not legally obligated to satisfy, yet to the extent that I believe in determinism I believe that there is a complex condition which is causally suﬃcient for a landlord’s compliance with such a request.11 But that there is confirming evidence for the law whose existence is asserted—and more specifically instantial evidence—is causally, not logically, presupposed by the assertion of the dispositional state-

10 For

example, the painstaking attempt made by B. J. Diggs, in Diggs 1952, to achieve an extensional analysis of the counterfactual conditional is guided by this idea. 11 One might, though, take the more moderate view that warranted assertion of counterfactual conditionals merely involves statistical determinism, i.e., belief in the existence of a statistical law relative to which the consequent is inferable from the antecedent with a probability suﬃciently high to warrant practical reliance on the conditional. But on either view singular counterfactual conditionals derive their warrant from a law, whether causal or statistical.

334

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ment. A proposition q is causally presupposed by an assertion of proposition p, if p would not have been asserted unless q had been believed; in other words, if the acceptance of q is one of the causal conditions for the assertion of p;12 whereas q is logically presupposed by the assertion of p, if p entails q. To add an illustration to the one already given: consider the singular dispositional statement “the melting point of x is 200◦ F,” which means that x would melt at any time at which its temperature were raised to 200◦ F (provided the atmospheric pressure is standard). Surely this proposition is logically compatible with the proposition that nothing ever reaches the specified temperature. That there should be instantial evidence for a law of the form “any instance of natural kind K would, under standard atmospheric pressure, melt if it reached 200◦ F” is therefore not logically presupposed by the dispositional statement, though very likely it is causally presupposed by its assertion. Could our schema of explicit definition, then, be salvaged by extruding the existential clause? Only if “(∀z)(Fz.Oz ⊃ Rz)” (where “F” is a constant predicate substituted for the variable “ f ”) were an adequate expression of a law. But that it is not follows from the fact that it is entailed by “∼ (∃z)(Fz.Oz).” Thus, if “F” means “is wooden,” and it so happens that no wooden thing is ever immersed in a liquid, it would be true to say of a match that it is soluble. It may well be that to ascribe D to x is to ascribe to x some intrinsic property f (however “intrinsic” may be explicated) such that “Rx” is deducible from “ f x.Ox” by means of a law; but this, as most writers on the contrary-tofact conditional have recognized, leaves the extensionalist with the tough task of expressing laws in an extensional language. The view that every singular counterfactual conditional derives its warrant from a universal conditional is sound—though one cannot tell by a mere glance at the predicates of the singular conditional which universal conditional is presupposed13 —but it should not be overlooked that universal conditionals that are accorded the status of laws by scientists may themselves be counterfactual. There are no finite physical

12 Notice

that while “A says ‘p”’ does not entail, but at best confers a high probability upon “A believes that p,” the latter proposition is entailed by “A asserts that p,” according to my usage of “assert” as an intentional verb. I am not denying, of course, that there may be a proper purely behavioristic sense of “assert”; nor do I deny that “A asserts that p” may properly be so used that it is compatible with “A does not believe that p.” My usage may be explicated as follows: A believes that p and utters a sentence expressing the proposition that p. 13 This seems to be overlooked by O’Connor, who, following Broad, concludes his analysis of conditional sentences (O’Connor 1951) with the claim that “a particular contrary-to-fact conditional has exactly the same meaning as the corresponding universal indicative statement.” The examples given by him indicate that by the universal statement corresponding to the “particular” contrary-to-fact conditional he means the universal conditional of which the latter is a substitution instance. Obviously, it might be true to say “if the trigger of the gun had been pulled, the gun would have fired” though there are exceptions to the generalization “any gun fires if its trigger is pulled.” The singular conditional is elliptical; in asserting it one presupposes the presence in the particular situation of various causal conditions which the antecedent does not explicitly mention. (See, on this point, Pap 1952b; Pap 1955a, chapter IV A; and Chisholm 1955.)

Disposition Concepts and Extensional Logic (1958)

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systems that are strictly closed, isolated from external influences, but the law of the conservation of energy says that if there were such a system its total energy would remain constant; there are no gases that are “ideal” in the sense that their molecules do not exert “intermolecular” forces on one another, but the general gas law says that if there were such a gas it would exactly satisfy the equation “PV = RT ;” there are no bodies that are not acted on by any external forces (indeed, the existence of such bodies is incompatible with the universal presence of gravitation), but the law of inertia says that if there were such a body it would be in a state of rest or uniform motion relative to the fixed stars.14 If such laws were formulated extensionally, as negative existential statements, like “there are no ideal gases that do not satisfy the equation: PV = RT ,” they would be vacuously true; anything whatsoever could be truly asserted to happen under the imagined unrealizable conditions. And it could hardly be maintained that, in analogy to the process of validation of singular contrary-to-fact conditionals, such laws are asserted as consequences of more general laws that have been instantially confirmed. If the general gas law, for example, is asserted as a deductive consequence of anything, then it is of the kinetic theory of gases, whose constituent propositions are surely not the kind of generalizations that could be instantially confirmed.15 But further, there is the much-discussed diﬃculty of distinguishing extensionally laws from universal propositions that are but accidental. It may be the case that all the people who ever inhabited a certain house H before H was torn down died before the age of 65. The statement “for any x, if x is an inhabitant of H, then x dies before 65” would then be true, yet nobody would want to say that it expresses a law. As Chisholm and Goodman have pointed out, if it were a law, then it would support a counterfactual conditional like “if Mr. Smith (who is not one of the inhabitants of H) had inhabited H, he would have died before 65.” Now, an extensionalist might try the following approach: what distinguishes laws from accidental universals is, not an obscure modality of existential necessity (as contrasted with logical necessity), but their strict universality. That is, a universal statement expresses a law only if either it contains no individual constants or else is deducible as a special case from well-confirmed universal statements that contain no individual constants. The predicates of the fundamental laws, i.e., those that contain no individual constants, should be purely general.

14 It

might be objected that the law of inertia can be formulated in such a way that it is not contrary to fact: if no unbalanced forces act on a body, then it is at rest or in uniform motion relative to the fixed stars. But when the law is used for the derivation of the orbit of a body moving under the influence of a central force, it is used in the contrary-to-fact formulation since the tangential velocities are computed by making a thought experiment: how would the body move at this moment if the central force ceased to act on it and it moved solely under the influence of its inertia? 15 For further elaboration of this argument against the extensional interpretation of laws, see Pap 1963b.

336

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However, a serious criticism must be raised against this approach. Just suppose that H were uniquely characterized by a property P which is purely general in the sense that it might be possessed by an unlimited number of objects.16 P might be the property of having a green roof; that is, it might happen that H and only H has P. In that case the accidental universal could be expressed in terms of purely general predicates: for any x, if x is an inhabitant of a house that has a green roof, then x dies before 65.17 It may be replied that although the antecedent predicate is purely general it refers, in the above statement, to a finite class that can be exhausted by enumeration of its members, and that it is this feature which marks the statement as accidental. Admittedly, so the reply may continue, it sounds absurd to infer from it “if y were an inhabitant of a house that has a green roof, then y would die before 65,” but this is because we tacitly give an intensional interpretation to the antecedent predicate. If instead it were interpreted extensionally, viz., in the sense of “if y were identical with one of the elements of the actual extension of the predicate,” the inferred subjunctive conditional would be perfectly reasonable. To cite directly the proponent of this explication of the distinction under discussion, Karl Popper: “the phrase ‘If x were an A . . . ’ can be interpreted (1) if ‘A’ is a term in a strictly universal law, to mean ‘If x has the property A . . . ’ (but it can also be interpreted in the way described under (2)); and (2), if ‘A’ is a term in an ‘accidental’ or numerically universal statement, it must be interpreted ‘If x is identical with one of the elements of A’ ” (Popper 1949). But this just won’t do. For “x is one of the elements of A” would, in the sense intended by Popper, be expressed in the symbolism of Principia Mathematica as follows: x = a ∨ x = b ∨ . . . ∨ x = n, where a, b, . . ., n are all the actual members of A.18 But if Popper were right, then, if “all A are B” is accidental, it could be analytically deduced from it that such and such objects are members of B, which is surely not the case. To prove this formally for the case where the actual extension of “A” consists of just two individuals a and b: (∀x)(x = a ∨ x = b ⊃ x ∈ B) is equivalent to (∀x)[(x = a ⊃ x ∈ B).(x = b ⊃ x ∈ B)], which is equivalent to the simple conjunction: a ∈ B.b ∈ B. But surely it can be supposed without self-contradiction that, as a matter of accident, all the inhabitants of houses with green roofs die before 65, and yet individual a, or individual b, survives the age of 65. What is logically excluded by the accidental universal is only the conjunctive supposition that a is an inhabitant

16 Notice

that a property may fail to be purely general in this sense even if it is not defined in terms of a particular object, e.g., “being the highest mountain.” 17 This criticism applies to C. G. Hempel and P. Oppenheim’s explication of “law” relative to a simplified extensional language system, in Hempel and Oppenheim 1948. R. Chisholm makes the same criticism, in Chisholm 1955. 18 He could hardly mean it just in the sense of “(∃y)(y ∈ A.x = y) ,” for this says nothing else than “x ∈ A,” and so does not amount to one of alternative interpretations of “x ∈ A.”

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of a house with a green roof and survives the age of 65. It is not denied that “a ∈ B.b ∈ B,” where a and b happen to be the only objects that have property A, is the ground, indeed the conclusive ground, on which the accidental universal “all A are B” is asserted; what is denied is that any atomic statements, or conjunctions of such, are analytically entailed by a universal statement, regardless of whether it is accidental or lawlike. The same confusion of the meaning of the universal statement with the ground on which it is asserted is involved in the following interpretation: (a ∈ A).(a ∈ B).(b ∈ A).(b ∈ B).. . ..(n ∈ A).(n ∈ B).(∀x)(x ∈ A ≡ x = a ∨ x = b ∨ . . . ∨ x = n). For clearly none of the atomic statements are entailed by the universal statement “all A are B.” But suppose that accidental universals were characterized pragmatically rather than semantically, in terms of the nature of the evidence which makes them warrantedly assertible. Thus P. F. Strawson suggests that only the knowledge that all the members of A have been observed and found to be B constitutes a good reason for asserting an accidental universal “all A are B.”19 Now, Strawson cannot mean conclusive evidence by “good reason,” since as long as there remain unobserved members of the subject class the evidence for a lawlike20 generalization is not conclusive either. He must therefore be making the more audacious claim that observations of a part of the subject class of an accidental universal cannot even make it probable that its unobserved members are likewise positive instances. He is then taking the same position as Nelson Goodman, who holds that if “all A are B” is accidental, it does not make sense to say that the evidence that observed members of A are B’s confirms the prediction that unobserved members of A are likewise B’s. But this criterion is highly counterintuitive. If 10 apples are picked out of a basket filled with apples and are found to be rotten without exception, it will be inductively rational to predict that the next apple that will be picked is likewise rotten. Yet, it may be just an accidental fact that all the apples in the basket are rotten. It is not necessary to assume that somebody deliberately filled the basket with rotten apples, though the circumstances may make this hypothesis plausible. It is possible, for instance, that somebody who made random selections (with closed eyes) of apples from a larger basket in order to fill up a smaller basket had the misfortune to get nothing but rotten ones though there were quite a few good specimens in the larger basket. An attempt to define the law-accident distinction in pragmatic rather than semantic terms, i.e., in terms of the kind of evidence leading one to assert the respective kinds of propositions, while retaining extensional logic for the formulation of the asserted propositions, has likewise been made by R. B. Braithwaite (Braithwaite 1953, chapter 9). He says as much as that the assertion of a 19 Strawson 20 A

1952, 199. lawlike statement is a statement which expresses a law if it is true.

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contrary-to-fact conditional causally presupposes acceptance of an instantially confirmed law from which the conditional component (the other component is the negation of the antecedent) is deducible, but that the truth-condition21 of the contrary-to-fact conditional is expressible in extensional logic: (p ⊃ q). ∼ p. There are two major objections to this approach: In the first place, contraryto-fact conditionals with identical antecedents and contradictory consequents (e.g., “if he had come, he would have been shot,” “if he had come, he would not have been shot”) are logically compatible on this analysis; whereas one should think that their logical incompatibility is a guiding criterion of adequacy for the semantic (not pragmatic) analysis of contrary-to-fact conditionals.22 Braithwaite in fact is saying that all contrary-to-fact conditionals whatever are true, though not all of them would be asserted by people confronted with the choice between asserting or denying them. But if a person honestly denies “p” and is familiar with the conventional meaning of “p,” then he does not believe the proposition expressed by “p”; yet, if the proposition expressed by “if A had happened, B would have happened” is simply t

SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE

Editor-in-Chief:

VINCENT F. HENDRICKS, Roskilde University, Roskilde, Denmark JOHN SYMONS, University of Texas at El Paso, U.S.A.

Honorary Editor: JAAKKO HINTIKKA, Boston University, U.S.A.

Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands THEO A.F. KUIPERS, University of Groningen, The Netherlands TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES, Stanford University, California, U.S.A. JAN WOLEN´SKI, Jagiellonian University, Kraków, Poland

VOLUME 334

THE LIMITS OF LOGICAL EMPIRICISM SELECTED PAPERS OF ARTHUR PAP with an Introduction by Sanford Shieh Edited by

ALFONS KEUPINK University of Groningen, The Netherlands

and

SANFORD SHIEH Wesleyan University, Middletown, U.S.A.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-4298-1 (HB) 978-1-4020-4298-0 (HB) 1-4020-4299-X (e-book) 978-1-4020-4299-7 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

Contents

ix xi

Preface Acknowledgments Part I Themes in Pap’s Philosophical Writings Introduction Sanford Shieh 1. Overview of Pap’s Philosophical Work 2. Necessity as Analyticity 3. Necessity as (Implicit) Linguistic Convention 4. The Analytic-Synthetic Distinction: Hypothetical or Functional Necessity 5. The Analytic-Synthetic Distinction: Dispositional and Open Concepts 6. The Limits of Hypothetical Necessity: Formal or Absolute Necessity 7. Logical Consequence and Material Entailment 8. The Method of Conceivability 9. Comparison with Necessity in Contemporary Analytic Metaphysics 10. Logicism 11. Concluding Remarks

3 3 10 12 15 16 23 24 27 30 33 42

Part II Analyticity, A Priority and Necessity 1 On the Meaning of Necessity (1943)

47

2 The Different Kinds of A Priori (1944)

57

3 Logic and the Synthetic A Priori (1949)

77

4 Are all Necessary Propositions Analytic? (1949)

91

v

vi 5 Necessary Propositions and Linguistic Rules (1955) 1. Are there Necessary Propositions? 2. The Confusion of Sentence and Proposition 3. Are Propositions “Logical Constructions”? 4. Necessary Truth and Semantic Systems 5. Implicit Definitions

Contents

109 109 117 122 132 137

Part III Semantic Analysis: Truth, Propositions, and Realism 6 Note on the “Semantic” and the “Absolute” Concepts of Truth (1952) Appendix: Rejoinder to Mrs. Robbins (1953)

147 154

7 Propositions, Sentences, and the Semantic Definition of Truth (1954)

155

8 Belief and Propositions (1957)

165

9 Semantic Examination of Realism (1947) 1. Universals in Re and the Resemblance Theory 2. Platonism and the Existence of Universals

181 181 187

Part IV Philosophy of Logic and Mathematics 10 Logic and the Concept of Entailment (1950)

197

11 Strict Implication, Entailment, and Modal Iteration (1955)

205

12 Mathematics, Abstract Entities, and Modern Semantics (1957) 1. Traditional Problem of Universals 2. Modern Semantics and the Traditional Dispute 3. Classes, Attributes, and the Logical Analysis of Mathematics What Do the Ontological Questions Mean? 4. 13 Extensionality, Attributes, and Classes (1958)

213 213 216 219 226 233

14 A Note on Logic and Existence (1947)

237

15 The Linguistic Hierarchy and the Vicious-Circle Principle (1954)

243

THE LIMITS OF LOGICAL EMPIRICISM Part V

vii

Philosophy of Mind

16 Other Minds and the Principle of Verifiability (1951) 1. The Principle of Verifiability as Generator of Philosophical Theories 2. The Behaviorist’s Confusion about the Notion of Verifiability 3. Are Statements about Other Minds Conclusively Verifiable? 4. Physicalism as an Analytic Thesis

249 254 259 264

17 Semantic Analysis and Psycho-Physical Dualism (1952)

269

249

Part VI Philosophy of Science 18 The Concept of Absolute Emergence (1951)

285

19 Reduction Sentences and Open Concepts (1953) Appendix

316

20 Extensional Logic and Laws of Nature (1955)

317

21 Disposition Concepts and Extensional Logic (1958)

327

22 Are Physical Magnitudes Operationally Definable? (1959) 1. Operational Definition as Contextual Definition of Classificatory Predicates 2. Operational Definition in the Form of Reduction Sentences 3. Physical Magnitudes and the Language of Observables 4. Theoretical Definition and Partial Interpretation The Breakdown of the Analytic-Synthetic Distinction 5. for Partially Interpreted Systems

295

351 351 352 355 358 360

Part VII Arthur Pap’s Life and Writings 23 Arthur Pap (1921-1959) : Intellectual Biography of Arthur Pap

365

Alfons Keupink 24 Arthur Pap: Biographical Notes Pauline Pap

369

viii

Contents

25 A Bibliography of Arthur Pap

375

Alfons Keupink 1. Main Publications 2. Editions 3. Translations 4. Articles, Papers and Reviews

375 375 375 376

References

381

Index

393

Preface

We would like to begin by telling a bit of the somewhat complicated story of how this volume came into being. Several years ago, one of us—Keupink—stumbled across some of Arthur Pap’s major publications1 in a secondhand bookstore in Groningen. As a Ph.D. student in philosophy of science with a special interest in the history of logical positivism, he was taken by the fecundity of Pap’s thought. Here was someone who, to him at least, seemed equally well-versed in all kinds of diﬀerent philosophical traditions (notably ordinary language philosophy and logical empiricism), yet always with something original to say. Keupink quickly made a decision to compile a list of Pap’s writings. He discovered, to his surprise, that Pap had written well over fifty papers during an extremely productive but all too short life. Gradually Keupink formed a plan to edit this material, in order to make it more accessible to a wide philosophical audience. He contacted Kluwer and in the Spring of 2003 submitted a manuscript entitled Arthur Pap: Collected Papers. It contained all of Pap’s papers and Keupink hoped that they could be published in two volumes. Kluwer asked Shieh to review Keupink’s manuscript. Nowadays Pap’s work is relatively unknown in Anglo-American analytic philosophy. Mostly he is read only by philosophers interested in the history of the analytic tradition. Indeed, it is because one of Shieh’s main research interests is in the development of modal logic and the concept of necessity in analytic philosophy that he knew parts of Pap’s magnum opus, Semantics and Necessary Truth, before Kluwer contacted him. As he read through the papers in the proposed collection, he became more and more impressed by the historical and philosophical significance of Pap’s work. The principal criteria that the two of us share for historical-philosophical assessment are: 1 Did the work play an important role in the development of the tradition to which it belongs? 1 Pap

1946b, Pap 1949b, Pap 1955a, Pap 1958c, and Pap 1962.

ix

x

Preface

2 Did it anticipate prominent subsequent developments? 3 Did it provide distinctive solutions or perspectives on problems of contemporary concern, perhaps by pointing out unnoticed problems in contemporary work, perhaps by proposing arguments that are more cogent than contemporary ones, perhaps by allowing us to see the significance of the problems diﬀerently? Shieh’s suggestion was that the best way of making Pap’s work more visible and easily available to the philosophical community is to make a smaller selection of Pap’s papers that best satisfy these criteria, and provide an introduction that clarifies their significance. Shieh was very happy to be asked by Keupink (via a letter written by Prof. Theo Kuipers and Dr. Jeanne Peijnenburg of the University of Groningen), to make the small selection and write the introduction. We hope that the present edition will have the eﬀect of furthering interest in Arthur Pap’s thought and contributing towards a reevaluation of his highly original and stimulating contribution to the development of 20th -century (analytic) philosophy (of science). A K Groningen, the Netherlands, 2005 S S New York, USA, 2005

Acknowledgments

We would like to thank Arthur Pap’s wife, Mrs. Pauline Pap, for her confidence and support; Keupink would like also to thank her for her help with his Intellectual Biography of Pap. We are also very grateful for the help and advice of Prof. Dr. Theo Kuipers, and Dr. Jeanne Peijnenburg. Our other colleagues in Groningen made various constructive remarks on earlier versions of the Intellectual Biography of Pap and the Introduction. In addition, Dr. Peijnenburg carried out the unenviable task of tracking down the copyright holders of the essays herein reprinted, and securing permission for this reprinting; without her help this volume would not have been possible. Acknowledgments are also due to the Netherlands Organization for Scientific Research (NWO), which partly sponsored the research for this project. We would like to thank Andrew Catalano of Wesleyan University and Julia Perkins of History and Theory and Wesleyan University for their invaluable editorial assistance.

Origin of the Essays All permissions granted for the previously published essays by their respective copyright holders are most gratefully acknowledged. There are instances where we have been unable to trace or contact the copyright holder. If notified the publisher will be pleased to rectify any errors or omissions at the earliest opportunity. 1 “On the Meaning of Necessity,” in The Journal of Philosophy 40 (1943), pp. 449-58. Reprinted with permission from The Journal of Philosophy, Columbia University, New York. 2 “The Diﬀerent Kinds of A Priori,” in Philosophical Review 53 (1944), pp. 465-84. Copyright 1944 Cornell University. Reprinted by permission of the publisher. 3 “Logic and the Synthetic A Priori,” in Philosophy and Phenomenological Research 10 (1949), pp. 500-14. Reprinted with permission from

xi

xii

Acknowledgements

the editors of Philosophy and Phenomenological Research, Brown University. 4 “Are all Necessary Propositions Analytic?,” in Philosophical Review 58 (1949), pp. 299-320. Copyright 1949 Cornell University. Reprinted by permission of the publisher. 5 “Necessary Propositions and Linguistic Rules,” in Semantica (Archivio di Filosofia), Roma: Fratelli Bocca (1955), pp. 63-105. Reprinted with permission from the Lodetti family of Libreria Bocca. 6 “The‘Semantic” and the ‘Absolute” Concepts of Truth,” in Philosophical Studies 3, 1952, pp. 1-8. Appendix: Rejoinder to Mrs. Robbins, in Philosophical Studies 4 (1953), pp. 63-4. Reprinted with permission from Springer-Kluwer, Dordrecht. 7 “Propositions, Sentences, and the Semantic Definition of Truth,” in Theoria 20 (1954), pp. 23-35. Reprinted with permission from Theoria, Royal Institute of Technology, Stockholm and Mrs. P. Pap. 8 “Belief and Propositions,” in Philosophy of Science 24 (1957), pp. 12336. Reprinted with permission from The University of Chicago Press, Permissions Department. 9 “Semantic Examination of Realism,” in The Journal of Philosophy 44 (1947), pp. 561-75. Reprinted with permission from The Journal of Philosophy, Columbia University, New York. 10 “Logic and the Concept of Entailment,” in The Journal of Philosophy 47 (1950), pp. 378-87. Reprinted with permission from The Journal of Philosophy, Columbia University, New York. 11 “Strict Implication, Entailment, and Modal Iteration,” in Philosophical Review 64 (1955), pp. 604-13. Copyright 1955 Cornell University. Reprinted by permission of the publisher. 12 “Mathematics, Abstract Entities, and Modern Semantics,” in Scientific Monthly 85, (29 July 1957), pp. 29-40. Reprinted with permission from the American Association for the Advancement of Science, Washington. 13 “Extensionality, Attributes, and Classes,” in Philosophical Studies 9 (1958), pp. 42-6. Reprinted with permission from Springer-Kluwer, Dordrecht. 14 “A Note on Logic and Existence,” in Mind 56 (1947), pp. 72-6. Reprinted with permission from Oxford University Press.

Acknowledgements

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15 “The Linguistic Hierarchy and the Vicious-Circle Principle, in Philosophical Studies 5 (1954), pp. 49-53. Reprinted with permission from Springer-Kluwer, Dordrecht. 16 “Other Minds and the Principle of Verifiability,” in Revue Internationale de Philosophie 5 (1951), pp. 280-306. Reprinted with permission from Revue Internationale de Philosophie, Universit´e Libre de Bruxelles. 17 “Semantic Analysis and Psycho-Physical Dualism,” in Mind 61 (1952), pp. 209-21. Reprinted with permission from Oxford University Press. 18 “The Concept of Absolute Emergence,” in British Journal for the Philosophy of Science 2 (1951), pp. 302-11. Reprinted with permission from Oxford University Press. 19 “Reduction Sentences and Open Concepts,” in Methodos 5, 1953, pp. 3-30. The editors were unable to trace or contact the copyright holder. 20 “Extensional Logic and Laws of Nature,” in Actes du deuxi`eme congr`es international de l’Union International de Philosophie des Sciences, Z¨urich 1954. Neuchˆatel: Editions du Griﬀon, Paris: Dunod (1955), pp. 116-27. Reprinted with permission from Editions du Griﬀon. 21 “Disposition Concepts and Extensional Logic,” in Concepts, Theories and the Mind-Body Problem: Minnesota Studies in the Philosophy of Science, Volume II, University of Minnesota Press (1958), pp. 196-224. Reprinted with permission from The University of Minnesota Press. 22 “Are Physical Magnitudes Operationally Definable?,” in C. West Churchman & P. Ratoosh (eds), Measurement: Definitions and Theories. New York: John Wiley, London: Chapmann & Hall (1959), pp. 177-91. (Publications of contributions to the symposium of the American Association for the Advancement of Science, 1956.) The editors were unable to trace or contact the copyright holder.

A Note on this Edition Minimal editorial changes have been made to the originally published versions of the essays reprinted in this volume. Typographical mistakes are corrected; references, spelling, and punctuation are made uniform throughout the volume. Details of original publication of the essays are given both in chapter 25, a bibliography of Pap’s publications, and in the relevant entries for Pap in the References, starting on page 387 below. All citations of Pap’s works in this volume are exclusively by the entries in the References.

I

THEMES IN PAP’S PHILOSOPHICAL WRITINGS

Introduction Sanford Shieh

1.

Overview of Pap’s Philosophical Work

There are, of course, many styles of philosophizing; but it is sometimes illuminating to think of philosophers as divided into two broad types: the Socratic and the Platonic. The former are the critics, the ones who never cease questioning the grounds of accepted opinions and turning over the details of arguments; the latter are the systematizers, the ones who have a sweeping vision that they take more seriously than the details. On this division Arthur Pap is a Socratic philosopher. As he himself says in describing Semantics and Necessary Truth (Pap 1958c, cited in this Introduction as SNT), “It will hardly escape the reader’s notice that very few definitive conclusions are reached in this book, that perhaps more problems have been formulated than have been solved” (SNT, Preface, xiv). Although Pap was very much an analytic philosopher, a good part of his most philosophically rewarding work are critiques of two prominent strands of analytic philosophy in the 1940s and 1950s. One of these is logical positivism or logical empiricism, the movement stemming from the Vienna Circle of Moritz Schlick and Rudolf Carnap and the Berlin school of Hans Reichenbach, which dominated analytic philosophy, especially in the USA, from the end of World War II through most of the 1960s. The other is ordinary language philosophy, deriving from the work of Gilbert Ryle and the followers of Ludwig Wittgenstein,1 which played a significant and in certain ways oppositional role, in post-war British analytic philosophy until the end of the 1960s. There were many diﬀerences and disputes between positivism and ordinary language philosophy, yet in certain respects their central, as it

1 Interestingly,

J. L. Austin’s work never made much of an impression on Pap. Of course, Austin’s work also failed to make much of an impression on the rest of analytic philosophy until after the period in which Pap was active, when Austin’s ideas about speech acts were taken up in philosophy of language.

3

4

Overview of Pap’s Philosophical Work

were popular, ideas converged. In order to understand Pap’s work and appreciate its relevance to contemporary philosophy, it will be useful to begin with a brief sketch of some major preoccupations of the analytic philosophy of this period that included this convergence. The doctrines I will outline characterize logical empiricism and ordinary language philosophy as philosophical movements, and so should not be confused with the much more nuanced views of, e.g., Carnap and Wittgenstein.

1.1

Intellectual Background: Logical Empiricism and Ordinary Language Philosophy

One of the fundamental motivations of positivism is a perceived contrast between traditional philosophy and the natural sciences. Whereas the sciences seem to display steady if not uninterrupted progress and continued general agreement on results, philosophy seems mired in endless irresolvable disputes. Positivism thus started from the idea that the paradigm of genuine knowledge is empirical scientific knowledge, and from a distrust of metaphysical concepts and theories outside the natural sciences. Positivism pursued two closely related projects. One was an attempt to arrive at criteria for distinguishing genuine procedures for assessing claims to knowledge from spurious ones. Since natural science was the model, these criteria all turned on the existence of impersonal standards, formulated in terms of objective experience, for acceptance and rejection of statements. The other was an attempt to oﬀer a diagnosis of where traditional metaphysics had gone wrong. Perhaps the most well-known thesis of positivism, the verificationist theory of meaning, subserved both of these projects. The principle of verification is a formulation of the shared and objective standards governing the acquisition of knowledge in the sciences: a statement is accepted only on the basis of sensory experiences that establish it as true. This principle goes naturally with an account of meaning: the cognitive meanings of statements are given by the sensory experiences that would verify or falsify them; a statement for which there is no method of verification has thus no objective meaning at all. The language of science was taken to abide by this principle, while that of traditional metaphysics did not. Thus science deals only with cognitively significant statements, while metaphysics traﬃcked in (cognitive) nonsense. The principle of verification came to grief in all sorts of ways,2 of course, but one important problem spurred the formulation of another central tenet of positivism. (Yet another crucial problem will be discussed in section 5 below.) This is the problem of the a priori, as A. J. Ayer called it (Ayer 1952b). Logic and mathematics are indispensable to the practice of modern science; yet nei-

2 See

Hempel 1951 for a detailed account of the problems faced by this principle.

Introduction

5

ther the laws of logic nor the axioms and theorems of mathematics seem to be justified on the basis of experience. The solution adopted by positivism was to hold that logic and mathematics are not based on experience for their truth because they rest, at bottom, on the meanings of our words, the rules governing the use of our language. These meanings or rules are conventional, at least in the sense that their adoption is not constrained to reflect empirical reality, and this explains why it is possible for us to know the truth of whatever is based solely on them without reference to experience. Necessary truth, then, is just truth based on features of language. It should be stressed that, for the most reflective positivists, such as Carnap, meaning rules cannot be constrained by empirical reality, because it is only when the rules are in place that we have a conception of what it is for our acceptance of statements to be responsible to empirical evidence. Thus, not only is necessity of logic and mathematics consistent with the empirical status of science, but they have to have this modal status in order for there to be such a thing as empirical science at all. The solution also represents the principal point of convergence between positivism and ordinary language philosophy. The latter diﬀered from positivism in rejecting science as the paradigm of knowledge and of philosophical method. But, ordinary language philosophers, especially the followers of Wittgenstein, agreed with positivism in taking traditional philosophy to be deficient in (cognitive) meaning. In addition, they went beyond the positivists’ characterization of traditional metaphysics as nonsense. The grand metaphysical theories of traditional philosophy from Plato to Hegel all claimed to provide insight into necessary structures of reality. Armed with its version of the linguistic theory of necessity, ordinary language philosophers took themselves to have exposed these pretensions as mistaking merely conventional features of our language, the rules (of philosophical grammar) that regulate what it is to make sense, for the ultimate truth underlying reality. The linguistic theory of necessity goes naturally with a sharp distinction between the analytic and the synthetic. The meanings of our words, the rules governing their sensical use, are supposed to be unconstrained by empirical reality. As Pap puts it, if these rules are formulated in the guise of statements, “they express stipulations concerning the use of symbols and thus cannot significantly be said to be refuted by facts”3 In contrast, the sentences governed by these rules “are empirically refutable statements ‘about reality’ ” (19, 308). Ordinary language philosophers tend to be suspicious of the notion of analytic truth, since, in their view, a rule of language is strictly speaking neither true nor false, but defines what it is for something to be true or false. But they nevertheless maintain an analytic-synthetic distinction, primarily in the form 3 Chapter

19 below, page 308. In this Introduction, unless otherwise noted, references to Pap’s papers in the present volume will be given in the text, by chapter number and page number.

6

Overview of Pap’s Philosophical Work

of Wittgenstein’s distinction between criteria and symptoms. In the end, their disagreement with the positivists over whether there are analytic truths is a merely verbal one. Both accept a distinction between that which sets the rules for empirical justification and that which, given the rules, are susceptible of being empirically justified. When the shared doctrine is formulated in this way, we can see that what underlies these two major strands of postwar analytic philosophy is a deeplyrooted conception of rationality. The fundamental idea is that participation in rational inquiry presupposes acknowledgment of shared and objective standards for what investigatory results counts in favor of accepting or rejecting what statements. Without this prior agreement on standards, there is no way to separate genuine agreements and disagreements from merely verbal ones; it is unclear whether participants in inquiry even understand one another. Thus a distinction between the rules for conducting empirical inquiry and the results of such inquiry appears inescapable if one is to be rational; this is why the analytic-synthetic distinction appears inescapable.4

1.2

Two Main Themes of Pap’s Work

Although Pap is a Socratic philosopher, it would be false to say that his work consists only of piecemeal and incisive criticisms with no overarching philosophical unities. Two broad themes stand out in his papers, call them anti-reductionism and intuitionism. The first theme is connected to Pap’s critical relation to the analytic philosophy of his time. Pap’s critiques attempt to show, over and over again, the failure of positivism’s and ordinary language philosophy’s attempts to reduce or explain away the categories of traditional metaphysics. Below I will discuss a number of philosophically significant instances of this type of criticism. The second theme emerges from the moral that Pap repeatedly finds in his critiques of reductionism. For example, from his critique of the linguistic theory of necessity he concludes that “the concept of necessity involved in such statements as ‘the conclusion of a valid syllogism follows necessarily from the premises’ and ‘the relation of temporal succession is necessarily . . . asymmetrical’ is not analyzable at all” (SNT, Preface p. xv; emphases mine). Rather, we have to accept that we have some form of irreducible intuition of these concepts and their applications. Another example of Pap’s intuitionism is his argu-

4 In

his “Intellectual Autobiography” Carnap explicitly associates the lack of progress in traditional philosophy with the lack of shared standards therein: “most of the controversies in traditional metaphysics appeared to me sterile and useless.. . . I was depressed by disputations in which the opponents talked at cross purposes; there seemed hardly any chance of mutual understanding, let alone of agreement, because there was not even a common criterion for deciding the controversy” (Carnap 1963, 44-5, emphases mine). My view of the deeper motivations underlying the analytic-synthetic distinction is indebted to Ricketts 1982.

Introduction

7

ment that the relation of logical consequence or entailment cannot be reduced either to syntactic logical form or to model theoretic semantics, so that we have to accept that we have irreducible intuition of material (i.e., non-formal) deductive validity. As we will see, Pap takes this to support the ineliminability of synthetic a priori truths. (As this reference to Kant’s views suggests, the name “intuitionism” refers, not to the mathematical principles of Brouwer, but to the philosophical ideas which he claims to be the basis of his revision of mathematics.) This last point brings me to a general observation. These two broad themes in Pap’s work can be traced to his early training. Pap was a student of Cassirer at Yale, and although his main philosophical works do not fall within the tradition of Cassirer’s neo-Kantianism, they nevertheless manifest an aﬃnity with the kind of pragmatically oriented Kantianism developed by Cassirer. Pap’s most important works applied the best ideas of neo-Kantianism to the logical empiricist tradition, resulting in an internal critique of the latter that revealed its limits.

1.3

The Contemporary Significance of Pap’s Work

In order to indicate the contemporary significance of this work, let me begin by noting that, even to this day, positivism is often taken by many scholars, outside and inside philosophy, to be the very core of analytic philosophy. And, although the identification of analytic philosophy and positivism is a mistake, it is a revealing mistake that contains a substantial grain of truth. Many of the themes of positivism are still with us, in all sorts of surprising ways. For example, perhaps the most prominent contemporary interpreters of Wittgenstein are the British scholars Peter Hacker and Gordon Baker. (See their four volume commentary on the Philosophical Investigations, Baker and Hacker 1980, Baker and Hacker 1988, Hacker 1990, Hacker 2000, as well as Hacker 1986 and Hacker 1996.) Although they strenuously deny that Wittgenstein was a positivist, the views, especially about necessity, that they attribute to Wittgenstein are in the end essentially no diﬀerent from the views of the positivists that Pap criticized. And, to my mind at any rate, Pap’s criticisms apply equally forcefully to the views they attribute to Wittgenstein.5 For another example, the positivists’ distrust of claims outside the natural sciences survive in contemporary attempts to “naturalize” all sorts of things: the mind, ethics, mathematics, truth. Here again, Pap’s criticisms raise serious doubts that fund a plausible skepticism about these naturalization programs. In addition, as we will see in more detail below, Pap’s arguments anticipate a number of contemporary philosophical positions. Pap’s version of Russellian

5I

should say that I take this to cast doubt on Hacker’s and Baker’s interpretations of Wittgenstein.

8

Overview of Pap’s Philosophical Work

logicism, I will argue in section 10, may be understood as a version of the neo-Fregean philosophy of mathematics championed by Crispin Wright and Bob Hale. (See, inter alia, the articles collected in Hale and Wright 2001.) Pap’s critique of the semantic account of logical consequence and his argument for the indispensability of material entailment arrive at positions held by John Etchemendy (Etchemendy 1988a, Etchemendy 1988b, and Etchemendy 1990) and Robert Brandom (Brandom 1994), but by rather diﬀerent routes. Finally, as I will argue in section 9, Pap’s positive views of necessity constitute a defensible alternative to the conceptions of necessity prevalent in contemporary analytic metaphysics.

1.4

Pap in the Analytic Tradition

Let me now say a word about the place of Pap’s work in the history of analytic philosophy. In contemporary Anglo-American analytic philosophy (perhaps especially in America), it is generally thought that two fundamental changes occurred in this philosophical tradition after World War II. First, in the 1950s, W. V. Quine’s critique of the analytic-synthetic distinction led to the waning of logical positivism. Second, in the mid to late 1960s, Saul Kripke made modality philosophically respectable. Pap’s work complicates this picture considerably. As we will see in section 4, Pap had rejected the analytic-synthetic distinction nearly a decade before Quine’s “Two Dogmas of Empiricism” (Quine 1951), for reasons very similar to Quine’s. In addition, Pap’s Semantics and Necessary Truth, published in 1958, was something of a standard reference on necessity in analytic philosophy. This shows how far the analytic tradition had already gone away from the positivist/empiricist reduction or denigration of necessity, some 10-15 years before 1972, when Kripke gave the lectures later published as Naming and Necessity (Kripke 1980). Of course, Pap’s work is not the only significant lacuna in the standard picture of modality in analytic philosophy. Another is Carnap’s Meaning and Necessity, first published in 1947 (Carnap 1947, second edition Carnap 1956a). (It should be noted that Carnap’s interest in modal logic was in many ways a reductionist one, whereas Pap’s views of necessity are just the opposite.) Yet another is the work on the logic, semantics, metaphysics, and epistemology of modality that Ruth Barcan Marcus, Pap’s fellow student at Yale, published from the mid-1940s to the late 1960s (Marcus 1946b; Marcus 1946a; Marcus 1947; and Marcus 1993, chapter 1, esp. appendix 1A, 3, and 4), which advanced many of the theses for which Kripke argued in Naming and Necessity. Finally, there is the work of Jaakko Hintikka, also published in the same period as Marcus’s papers.

Introduction

9

Indeed, the historical picture appears even more perplexing if we ask: what is the basis of Kripke’s view of necessity? Kripke explicitly says that it is intuition. For example, When you ask whether it is necessary or contingent that Nixon won the election, you are asking the intuitive question whether in some counterfactual situation, this man would in fact have lost the election.. . . [S]ome philosophers think that something’s having intuitive content is very inconclusive evidence in favor of it. I think it is very heavy evidence in favor of anything, myself. I really don’t know . . . what more conclusive evidence one can have about anything. (Kripke 1980, 41-2; second and fourth emphases mine)

Indeed, Kripke’s fundamental argument for the coherence of modal notions proceeds by appealing to our intuitive judgment that statements such as “Bush might have won the popular vote in the 2000 US presidential election” and “Weapons of mass destruction might have existed in Iraq in 2003” make sense, even if we’re not sure how we would establish their truth or falsity.6 How, then, is this diﬀerent from Pap’s intuitionism and anti-reductionism? Clearly more work needs to be done if we are to have an accurate history of the analytic tradition.

1.5

Plan of the Introduction

In the remainder of this Introduction, I will discuss the most philosophically interesting and historically significant aspects of Pap’s work represented in the papers of this volume. They are the following: Necessity: sections 2-7 The Analytic-Synthetic Distinction: sections 4 and 5 Dispositional and Open Concepts in Science: section 5 Logical Consequence and Material Entailment: section 7 Logicism: section 10 The Semantic Concept of Truth: section 8.1 The Problem of Other Minds: section 8.2 This is by no means an exhaustive list of the topics of philosophical interest treated by Pap in the papers of this volume. Some other topics which I do not have the space to discuss in this Introduction are listed below in the concluding section 11, at page 42. 6 For

a more detailed account of the intuitive bases of Kripke’s arguments, see Shieh 2001.

Necessity as Analyticity

10

2.

Necessity as Analyticity

In this section I will begin a discussion of Pap’s views of necessity. I will focus on Pap’s critique of one version of the positivists’ linguistic theory of necessity, the thesis that necessity is reducible to analyticity. An extremely familiar formulation of the linguistic theory holds that, e.g., All sisters are female

(1)

is necessarily true because it is analytic, i.e., true by definition. Specifically, since ‘sister’ is defined as, and so synonymous with, ‘female sibling’, and since the mutual substitution of synonymous expressions in a sentence surely doesn’t change its meaning, (1) is synonymous with All female siblings are female

(2)

This sentence is logically true; it is a substitution instance of a valid schema of first-order logic: (∀x)(F x.S x ⊃ S x)

(3)

Thus the obscure notion of a statement’s being more than simply true, but necessarily true, is clarified by reducing it to two notions with no metaphysical baggage: that of meaning, and that of logical truth. The first of Pap’s criticisms of this theory that I will discuss is given in chapters 3 and 4 below. Pap claims that this account of necessity simply fails to cover all truths which we intuitively recognize as necessary. For Pap the basic notion of necessity is independence from empirical facts: “a necessary condition which any adequate analysis of ‘necessary’ must satisfy is that the truth-value of a necessary proposition does not depend on any empirical facts” (4, 92). Pap asks us to consider the following statement: If A precedes B, then B does not precede A

(4)

He argues, I assume that few would regard this statement as factual, i.e., such that it might be conceivably disconfirmed by observations.. . . [But t]o see that [it] is not analytic, in the sense defined, we only need to formalize it, and we obtain “(∀x)(∀y)[xRy ⊃ ∼ yRx],” which is certainly no principle of logic. This statement, then, is not deducible from logic; hence if we want to call it necessary (nonfactual), we have to admit that there are necessary propositions which are not analytic. (4, 96)

It will not do, of course, to respond to this argument by claiming that in such cases we simply don’t know the definitions of the key terms, such as ‘precede’. Such a response presupposes that any necessary truth must be analytic,

Introduction

11

but this presupposition is precisely the linguistic theory of necessity that is being challenged by a putative counter-example. The burden of proof is thus on the defender of the linguistic theory. Note, moreover, a point Pap does not make: if indeed we know that truths such as (4) are necessary without knowing the definition of ‘precede’, then it is plausible that analyticity is not the epistemic basis of necessity; hence the linguistic theory should, at least, oﬀer some explanation of why it is that while ontologically, as it were, necessity is based on analyticity, our epistemic access to necessity does not go through analyticity. Pap’s first argument is clearly not conclusive; it amounts in the end to a challenge to the linguistic theory to meet the burden of proof by coming up with the required definitions. Pap goes further and argues that attempts to meet this burden of proof must overcome a general diﬃculty. One could, of course, give any definition one pleases to any expression. But, if there are no constraints on what definitions may be given for demonstrating analyticity, then it is not clear that there are any true statements that cannot be shown to be analytic and so necessary according to the linguistic theory. It follows that, in meeting the burden of proof, one must abide by criteria of adequacy for definitions. The question then becomes, what are these criteria? What makes a definition a good one? What, in the case of (4), would be an adequate definition of ‘precede’? According to Pap, “most philosophers would agree that no definition of ‘xPy’ (to be used as an abbreviation for ‘x precedes y’) could be adequate unless it entailed the asymmetry of P” (4, 97). There are clearly other commonly agreed constraints: P should, e.g., also be irreflexive and transitive, and it can hold only of events and of whatever other entities there may be with temporal location, etc. Suppose that these et ceteras are suﬃcient to guarantee that a unique relation satisfies them. Let’s write “Q(R)” for the claim that a relation R satisfies these further constraints. Then, one might oﬀer this definition of P: xPy ≡d f (∃R)[(∀x )(∀y )(x Ry ⊃ ∼ y Rx ) . Q(R) . xRy]

(5)

From this definition it is obvious that in second-order logic we can derive (∀x)(∀y)(xPy ⊃ ∼ yPx). But, Pap writes, is it not obvious that acceptance of the definition from which the asymmetry of temporal succession has thus been deduced presupposes acceptance of the very proposition ‘Temporal succession is asymmetrical’ as self-evident? This way of proving that the debated proposition is, in spite of superficial appearances, analytic, is therefore grossly circular. (4, 98)

More generally, the problem for finding definitions to demonstrate analyticity is that if a definition is selected only because it allows for the derivation, from logic alone, of the necessary to be established as analytic, then the analysis of necessity in terms of analyticity is circular. Hence, in order for the

12

Necessity as (Implicit) Linguistic Convention

linguistic theory to work, the criteria of adequacy for definitions must not be based on the necessity of truths expressed using the defined terms. Pap is skeptical of the prospects of avoiding this problem. This is not, it seems to me, because he has an argument which establishes the impossibility of finding adequate definitions independently of recognizing necessary truths. Rather it is because, over and over again, he finds proposed definitions and analyses, especially those of the positivists, turning out, upon examination, to rest on prior recognition of necessary truths. The reader will notice this pattern of argument throughout Pap’s work. When faced with a reductive explanation of X in terms of Y, Pap examines the basis on which we identify Ys, and finds that we do so by recognizing Xs. Pap’s conclusion is then that we have a primitive and irreducible capacity for intuitive knowledge of Xs. It should be clear that a crucial assumption underlying the foregoing considerations is that there are clear instances of truths which we unproblematically recognize as non-factual. One may well ask here: why should an empiricist accept this assumption? Indeed, one way of seeing how Pap diﬀers from Quine is to see that for the latter there are no clear cases of non-factual truths. This thesis is, of course, closely connected with Quine’s rejection of the analyticsynthetic distinction. But the interesting point here is that Pap, as we will see in section 4, also rejects the analytic-synthetic distinction. We will be in a position to understand why Pap’s empiricism goes with the recognition of non-factual truths only by section 6.

3.

Necessity as (Implicit) Linguistic Convention

In this section I turn to Pap’s attack on another version of the linguistic doctrine of necessary truth, one which persists in contemporary philosophy in Hacker and Baker’s version of Wittgenstein. I focus on Pap’s paper “Necessary Propositions and Linguistic Rules” (chapter 5 below, cited in this Introduction as NPLR), which summarizes the major themes of SNT. The problem that Pap raises here follows on the one that we discussed in section 2 above, and applies even if it is possible to overcome the problem of successfully attaining definitions or analyses for certifying the analyticity of intuitively necessary truths. As we saw on page 10 above, part of the linguistic theory’s explanation of the necessity of (1) is the fact that (2) is a logical truth, an instance of the valid schema (3). But, then, even if the notion of analyticity is unproblematic, the linguistic theory doesn’t yet suﬃce as an explanation of necessity, since so far nothing has been said about why a logical truth is necessary or what makes instances of valid schemata necessarily true. So the linguistic theory requires supplementation. Moreover, this supplement had better demonstrate that the

Introduction

13

necessity in question is a matter of linguistic conventions; otherwise necessity will depend on factors other than language. The account of logic that Pap attacks holds that the “primitive” truths of logic are necessary because they implicitly define the meanings of the logical constants. For example, the schemata corresponding to the elimination rules for ‘and’—p . q ⊃ p and p . q ⊃ q—partly define the meanings of ‘and’ and ‘if . . . then’. But now the question is, what is implicit definition? Since these implicit definitions are supposed to be conventional, are they not proposals to use the words ‘and’ and ‘if . . . then’ in certain ways? But are proposals even candidates to be true or false? Do we not think that we can propose any way of using words we like? Surely in making proposals we are not constrained to be faithful to anything. Now, of course this argument by Pap is hardly conclusive. A supporter of the linguistic theory of necessity could reply that Pap has missed the point, because the proposal is precisely that we accept certain statements as true. Moreover, it is precisely because this proposal is unconstrained by empirical evidence that the truth in question is a priori and so necessary. What makes the statement true is just that we have agreed to take it to be true, and nothing we discover about the world need force us to give up this agreement. For the time being, let’s leave this reply and go on to a second version of the linguistic theory. This version holds that the linguistic conventions underlying necessary truths describe, or are based on, the ways in which speakers of a natural language use the words involved in expressing those truths. But then, Pap argues, these conventions are empirical truths about, say, speakers of English. It might be replied that these conventions are not facts about English speakers but rules that constitute what it is to speak English, so failure to follow them results in speaking a language other than English. But, Pap argues, this can’t be right, since natural languages undergo changes. I have heard it said that in Elizabethan England, ‘meat’ was more or less synonymous with ‘groceries’. But if natural languages change, then, intuitively, people in Elizabethan England still spoke English, even though they don’t use the word ‘meat’ in the same way that it is used in contemporary English. It follows then that the way in which words are used at any given time can’t constitute or define what it is to speak a language, and so how a term of the language is used at some given time is an empirical question. The problem that Pap raises for this version of the linguistic theory is that if linguistic conventions are empirical facts about usage then it becomes very unclear how they “can be a reason for the necessity of any statement” (5, 114) As Pap puts the point: “Nobody would say ‘any father must be a parent because there are nine planets’, although undoubtedly ‘there are nine planets’ entails ‘any father is a parent’. Would it be any less absurd to say ‘any father must

14

Necessity as (Implicit) Linguistic Convention

be a parent because, at least according to present linguistic conventions, it is incorrect to apply “father” to an object to which “parent” is inapplicable’?” (5, 114).7 This rhetorical question is unlikely to move adherents of the linguistic theory. Consider again the analytic truth ‘All sisters are female’. Given our present linguistic usage, no experience is relevant to its truth or falsity because the rule is a sister’ to anything in experience is for correctly applying the predicate ‘ is a sibling’ and ‘ is female’ that it is also correct to apply the predicates ‘ to it. There could be no empirical counterexamples to this statement because to recognize anything correctly as a sister requires recognizing it correctly as female. So, to revert to Pap’s example, if it is incorrect to apply ‘father’ to an object to which ‘parent’ is inapplicable, how could anything that is a father fail to be a parent? The only way seems to be that being a father is diﬀerent from being correctly described as a father. But how could that be? The answer to these replies, and the heart of Pap’s argument, is contained in the following passage: Consider [the] necessary proposition: there are no fathers that are not male. If we are a priori certain of it, i.e. in advance of having observed all fathers, past, present, and future, it is surely because we see that fatherhood entails malehood. This is to say that we see a priori the truth of the modal proposition ‘father(x) male(x)’ . . . and hence derive the corollary ‘(∀x)(father(x) ⊃ male(x))’. . . . The point is that since it is our knowledge of the entailment which is the ground of our certainty with regard to the universal proposition, the universal proposition would itself have only the status of an inductive generalization if the entailment were not known a priori. The argument ‘there are at no time fathers who are not male, because fatherhood entails malehood’ has the form ‘p, because N(p)’: for ‘p q’ is definable as ‘N(p ⊃ q).’ Therefore our acceptance of p would be based on empirical evidence if our acceptance of N(p) were based on empirical evidence. Therefore one cannot consistently hold that p is necessary and N(p) contingent. (5, 116)

The crux of this argument is that there is a tension between characterizing a truth as a priori and what the linguistic theory, according to Pap, must take to be its ultimate grounds. Whether the theory takes necessity to be based on proposals or on facts about norms of usages, it implies that one cannot be 7 It’s

worth pointing out that sometimes the terms in which Pap formulates this argument are misleading. For example, he characterizes a linguistic convention by stating that “people who understand English never apply” certain predicates in certain ways. One might take this to be a description of actual language use, and then object to it on the ground that the conventions or rules that make up a language are normative— they state how words ought to be used, not how they invariably are used. But acknowledging the normative character of rules of usage (“grammar” in Baker’s and Hacker’s Wittgensteinian terminology) doesn’t invalidate Pap’s argument. For the standards of correct use in force for a single language can change over time. Moreover, surely it still makes sense to say that what norms are in force in a language at a given time is an empirical fact, or, at the very least, it is a contingent feature of that language which we cannot ascertain merely by reflecting on, e.g., the concept of the English language.

Introduction

15

justified in asserting a statement that is supposed to rest on a priori grounds, unless one is sure that certain facts obtain about the language one is speaking— facts about what proposals for usage have been accepted, what conventions are in force, what norms of usage govern the language. But then whatever knowledge one might express by the statement is not arrived at by reflection on meanings alone. How, then is it a priori? And if it is not a priori, has the linguistic theory explained how it is necessary? This line of argument is the fundamental basis underlying Pap’s objection to the linguistic theory of necessity. None of this criticism, of course, answers the question what makes the laws of logic necessary, or indeed whether they are necessary. Pap does maintain that they are necessary; I discuss his reasons in sections 6 and 7.

4.

The Analytic-Synthetic Distinction: Hypothetical or Functional Necessity

The rejection of the analytic-synthetic distinction is a constant theme in Pap’s philosophical work. It appears as early as 1943 and 1944, in chapters 1 and 2 respectively, and as late as 1963 in the posthumously published “Reduction Sentences and Disposition Concepts” (Pap 1963b). The dates of the early papers points to their significance for the history of analytic philosophy: they are eight years earlier than Quine’s “Two Dogmas of Empiricism” (Quine 1951) and seven years earlier than Morton White’s less well-known “The Analytic and the Synthetic: An Untenable Dualism” (White 1950). In the early papers, which derive from the first part of his dissertation, Pap develops a theory of a type of necessity and apriority which he calls “hypothetical” or “functional” necessity. (I will use these two terms interchangeably.) This is a pragmatic notion of necessity—necessity of means for accomplishing certain purposes—traced ultimately back to Aristotle. The fundamental idea is that, in the course of empirical inquiry, certain statements are taken to be, or treated as, necessary, for the purpose of “systematizing facts, of rendering the body of factual knowledge coherent” (1, 49). Underlying this idea is an abstract picture of scientific inquiry: If a conjunction of traits a-b is found to be repeated without exception, we generalize it into a “universal,” a definitional connection: if A, then B. If, then, experience should one day disclose a contradictory instance, viz., ‘a and not-b,’ we will have the choice between refusing to identify a as an instance of A . . . and considering our law (“if A, then B”) as refuted . . . . (1, 54)

Thus in science we are “free . . . to make empirical truths . . . necessary and thus to deprive them of their intrinsic contingency” (1, 54). But if an empirical law that we have made into a “prescriptive definition” or “rule” comes into conflict with experience, we are equally free to decide it is no longer “good for conducting inquiry,” and start looking for a rule of inquiry “better than it.” In

16

The Analytic-Synthetic Distinction: Dispositional and Open Concepts

Pap’s picturesque language, “experience is free to unmake our makings again” (1, 55). Anyone at all familiar with Quine’s much better-known “Two Dogmas” will have been reminded of its concluding section. Pap’s idea that the a priori is made, in order to perform certain functions, finds an echo in Quine’s claim that “the conceptual scheme of science [i]s a tool . . . for predicting future experience in the light of past experience. Physical objects are conceptually imported into the situation as convenient intermediaries . . . irreducible posits comparable, epistemologically, to the gods of Homer” (Quine 1951, 41). In addition, Quine also sketches an abstract picture of scientific inquiry: [T]otal science is like a field of force whose boundary conditions are experience. A conflict with experience at the periphery occasions readjustments in the interior of the field.. . . [T]he total field is so undetermined by its boundary conditions, experience, that there is much latitude of choice as to what statements to re-evaluate in the light of any single contrary experience. . . . Any statement can be held true come what may . . . . Even a statement very close to the periphery can be held true in the face of recalcitrant experience by pleading hallucination or by amending certain statements of the kind called logical laws. Conversely, by the same token, no statement is immune to revision. Revision even of the logical law of the excluded middle has been proposed as a means of simplifying quantum mechanics; and what diﬀerence is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristotle? (Quine 1951, 39-40)

Similarity is not identity, and there are two important diﬀerences. First, one way in which, in my view, Pap’s critique of the analytic-synthetic distinction is more persuasive than Quine’s is the fact that from his dissertation to his later papers Pap engages much more deeply with actual examples of scientific practices, and so oﬀers more than a highly abstract account of science. These later papers focus on dispositional and what Pap calls “open” concepts in science; I discuss Pap’s views in the next section. The second diﬀerence is that unlike Quine Pap does not hold that the laws of logic are open to revision;8 I discuss this diﬀerence in section 6.

5.

The Analytic-Synthetic Distinction: Dispositional and Open Concepts

As I noted above, one of the fundamental ideas of positivism is that what distinguishes the sciences from metaphysics is acknowledgment of shared empirical standards for the acceptance and rejection of statements; in the verificationist phase of positivism, these standards are given by methods for establishing the truth of statements on the basis of sensory experience. I mentioned 8 Of

course in writings later than “Two Dogmas” Quine was a staunch defender of classical logic. It is, to my mind, a very diﬃcult question whether this represents a genuine change of mind.

Introduction

17

earlier the challenge for the verificationist criterion of meaning posed by the problem of the a priori. Dispositional concepts in science present a less obvious, but more radical, challenge. Indeed, giving a satisfactory account of these concepts is one of the main motivations of Carnap’s “Testability and Meaning” (Carnap 1937; cited in this Introduction as TM), a paper which constituted a pivotal development in positivism.

5.1

Carnap’s Theory of Dispositional Predicates

As Carnap puts it, the verificationist theory of meaning “led to a too narrow restriction of scientific language, excluding not only metaphysical sentences but also certain scientific sentences having factual meaning,” and he argues in TM for “a requirement of confirmability or testability as a criterion of meaning” (TM, 421). Let’s begin by saying what dispositional concepts are. Carnap characterizes them as expressed by “predicates which enunciate the disposition of a point or body for reacting in such and such a way to such and such conditions, e.g. ‘visible’, ‘smellable’, ‘fragile’, ‘tearable’, ‘soluble’, ‘indissoluble’ etc” (TM, 440). The question about these predicates, for a positivist, is: on the basis of what experiences do we correctly ascribe them, i.e., should we accept that something has such a property? Clearly if an object, say a lump of sugar, is observed to be placed in water, and we see that it dissolves, then we would be correct in claiming that it is soluble. This might lead one to think that the meanings of disposition terms can be specified conditionally. In the case of solubility, for instance, we might take the meaning of ‘soluble’ to be given by: is soluble’ is correctly applied to x just in case if x is placed in a liquid, ‘ then x dissolves. But there are two problems with this idea. First, intuitively we think that lumps of sugar which at no time are placed in a liquid are still soluble, and, to use Carnap’s example, a match burnt up completely before it is ever placed in water is “rightly” said to be “not soluble in water” (TM, 440). Second, if the conditional expressions ‘if . . . then’ used to specify the meanings of dispositional predicates are understood truth-functionally, then any forever-untested object would falsify the antecedent of the conditional, rendering the entire conditional true, and the dispositional predicate correctly ascribed to it. This does imply that the forever-untested lump of sugar is soluble; but, at the same time, the forever-untested match would also be soluble. This problem is sometimes called “Carnap’s paradox” in the literature on dispositions. Carnap’s response to these problems is a retreat from the principle of verification. Dispositional predicates are meaningful, but not in virtue of being associated with necessary and suﬃcient observable conditions of application. Rather, they are meaningful in virtue of being associated with a set of suﬃ-

18

The Analytic-Synthetic Distinction: Dispositional and Open Concepts

cient conditions of application, and a set of conditions suﬃcient for applying its negation. For example, associated with solubility in water are If x is placed in water and x dissolves, then x is soluble If x is placed in water and x does not dissolve, then x is not soluble

(6) (7)

Carnap calls statements of these conditions “reduction sentences.” Two reduction sentences which, as in the present example, provide suﬃcient conditions for applying the predicate and its negation is called a “reduction pair.” In general, a dispositional predicate ‘Q3 ’ is associated with the following reduction pair: (8) Q1 ⊃ (Q2 ⊃ Q3 ) Q4 ⊃ (Q5 ⊃∼ Q3 )

(9)

(TM, 441. Note Carnap’s notational convention at TM, 434: “If the sentence ‘(∀x)[− − −]’ is such that ‘− − −’ consists of several [open] sentences which are connected by [truth-functional connectives] and each of which consists of a predicate with ‘x’ as [variable], we allow omission of the [quantifier] and the [variables]. Thus e.g. instead of ‘(∀x)(P1 (x) ⊃ P2 (x))’ we shall write shortly ‘P1 ⊃ P2 ’.”) The present example has the feature that the reduction pair involves the same test condition, i.e., Q1 is the same as Q4 , and the defining reaction for the negative ascription Q5 is just the negation of the defining reaction of the positive ascription, Q2 , i.e., Q5 is ∼ Q2 . In this case the reduction pair is jointly equivalent to (10) Q ⊃ (R ≡ Q3 ) where Q = Q1 = Q4 , R = Q2 and ∼ R = Q5 . This is called a “bilateral reduction sentence.” For our example, the bilateral reduction sentence for solubility is: If x is placed in water, then, x dissolves just in case x is soluble

(11)

The move from verification conditions to reduction sentences does not disturb the basic positivist idea that cognitively significant scientific language is governed by impersonal experiential standards of correctness. The diﬀerence is that reduction sentences do not provide standards of application for objects which don’t fulfill the test conditions. That is, there is an area of indeterminacy where ascriptions of dispositional predicates are not governed by reduction sentences, and so have no truth-values. For this reason, Carnap holds that in these cases reduction sentences do not give the empirical meaning of the dispositional predicate; it follows that reduction sentences are partial meaningspecifications. We will see shortly that this is only one sense of ‘partial’.

Introduction

19

In TM, Carnap takes the specification of the meaning of dispositional predicates by (chains of) reduction sentences to be a model for the way in which scientific predicates in general acquire empirical meaning. Pap’s critique of the analytic-synthetic distinction is based on the attempt to interpret actual scientific practices in terms of partial meaning-specifications by reduction sentences. Before turning to that critique, I pause to discuss Pap’s criticisms of Carnap’s theory of dispositional predicates.

5.2

Critique of Carnap’s Theory of Dispositional Predicates

Pap makes two principal criticisms of Carnap’s theory of dispositional predicates. First, Carnap’s paradox depends on the truth-functional conditional, so one could avoid it by formulating the conditional specification of meaning of dispositional terms using, say, a causal conditional, or Lewis’s strict conditional. Pap’s own preferred account of dispositional concepts uses a non-truth-functional conditional, which he calls “quasi-semantic probability implication” (19, 312). Second, let’s go back to the never tested lump of sugar and the burnt match. They fall in the area of indeterminacy left open by reduction sentences; on the basis of reduction sentences ascriptions of solubility to these objects are neither true nor false. But this means that partial meaning-specifications still fail to account for our judgments that the sugar is soluble, and the match is not. Carnap was quite aware of this point, and he attempts to meet it as follows:

If we establish one reduction pair (or one bilateral reduction sentence) . . . in order to introduce a predicate ‘Q3 ’, the meaning of ‘Q3 ’ is not established completely, but only for the cases in which the test condition is fulfilled. In other cases, e.g. for the match in our previous example, neither the predicate nor its negation can be attributed. We may diminish this region of indeterminateness of the predicate by adding one or several more laws which contain the predicate and connect it with other terms available in our language. . . . In the case of the predicate ‘soluble in water’ we may perhaps add the law stating that two bodies of the same substance are either both soluble or both not soluble. This law would help in the instance of the match; it would, in accordance with common usage, lead to the result ‘the match c is not soluble,’ because other pieces of wood are found to be insoluble on the basis of the first reduction sentence. Nevertheless, a region of indeterminateness remains, though a smaller one. If a body b consists of such a substance that for no body of this substance has the test-condition—in the above example: “being placed into water”—ever been fulfilled, then neither the predicate nor its negation can be attributed to b. This region may then be diminished still further, step by step, by stating new laws. (TM, 445)

20

The Analytic-Synthetic Distinction: Dispositional and Open Concepts

Pap’s criticism of this proposal has become a standard objection in the literature.9 Carnap’s response amounts to claiming that we can extend the conditions given by reduction sentences to never-tested objects, provided that these are of the same kind as some tested objects. Pap asks, in reply, isn’t it conceivable that on some uninhabited planet there exists a species of metal non-existent on this planet which, like the metal species the human race has experimented with, has the disposition called ‘electrical conductivity.’ This question seems to be perfectly meaningful, yet it seems to be condemned to meaninglessness by Carnap’s theory. Carnap might deny this on the ground that our hypothetical bodies are, after all, metals, and that we can avail ourselves of the law ‘all metals are electrical conductors’ in order to make the question whether they also have this disposition significant, just as he availed himself of the law ‘all wood is insoluble in water’ in order to make the question ‘is this match, which has never been placed in water, soluble in water’ significant. But . . . assume a disposition which is unique to a particular individual. For example, a particular human being might have the disposition of feeling nauseated when exposed to the smell of an orange, and it is logically possible . . . that no other organism has that disposition and that the disposition is never actualized for the simple reason that this individual is never exposed to the fatal smell. To be sure, as long as . . . the class of organisms exposed at some time to the smell of an orange . . . is not empty, confirming or disconfirming evidence with respect to the statement ‘if this individual were exposed to the smell of an orange, he would feel nauseated’ is still obtainable. But what if no oranges existed? Then the class . . . would be empty and hence the above statement would definitely be meaningless by Carnap’s theory. And this is strange, since intuitively the truth, and a fortiori the significance, of the non-existential conditional ‘for any x and y, if x were an orange and y smelled x, then y would feel nauseated’ is compatible with ‘there is no x such that x is an orange.’ (Pap 1963b, 561-2)

That is to say, in the end, Carnap’s proposal fails to account for our intuition that objects can have dispositions that are unconnected with observable manifestations, either directly, or via membership in a kind.10

5.3

Open Concepts and the Rejection of the Analytic-Synthetic Distinction

Although he does not state so explicitly, in TM Carnap seems to regard his account of partial meaning-specifications as what he elsewhere calls a “rational reconstruction” of scientific practice. Of particular importance for Pap are two aspects of scientific practice that Carnap interprets in terms of partial meaningspecifications. First, “a property or physical magnitude can be determined by 9 See,

e.g., Mumford 1998, 60 kind-membership revision of Carnap’s original proposal has been worked out formally by Eino Kaila and Thomas Storer (Kaila 1967, Kaila 1941, Storer 1951). These are open to a number of objections, some of which Pap originated. See Pap 1963b for details. 10 The

Introduction

21

diﬀerent methods,” for example, “[t]he intensity of an electric current can be measured . . . by measuring the heat produced in the conductor, or the deviation of a magnetic needle, or the quantity of silver separated out of a solution, or the quantity of hydrogen separated out of water etc.” (TM, 444-5). It follows that no single reduction sentence or reduction pair defines such scientific concepts; this is a way in which they are “partial” distinct from that discussed earlier in which they do not apply to objects not fulfilling the test conditions. Second, Suppose that we introduce a predicate ‘Q’ into the language of science first by a reduction pair and that, later on, step by step, we add more such pairs for ‘Q’ as our knowledge about ‘Q’ increases with further experimental investigations. In the course of this procedure the range of indeterminateness for ‘Q’, i.e. the class of cases for which we have not yet given a meaning to ‘Q’, becomes smaller and smaller. Now at each stage of this development we could lay down a definition for ‘Q’ corresponding to the set of reduction pairs for ‘Q’ established up to that stage. But, in stating the definition, we should have to make an arbitrary decision concerning the cases which are not determined by the set of reduction pairs. A definition determines the meaning of the new term once for all. We could either decide to attribute ‘Q’ in the cases not determined by the set, or to attribute ‘∼ Q’ in these cases. [Suppose Q3 is introduced by a reduction pair, D1 settles the area of indeterminacy negatively, D2 positively.] Although it is possible to lay down either D1 , or D2 , neither procedure is in accordance with the intention of the scientist concerning the use of the predicate ‘Q3 ’. The scientist wishes neither to determine all the cases of the third class positively, nor all of them negatively; he wishes to leave these questions open until the results of further investigations suggest the statement of a new reduction pair; thereby some of the cases so far undetermined become determined positively and some negatively. If we now were to state a definition, we should have to revoke it at such a new stage of the development of science, and to state a new definition, incompatible with the first one. If, on the other hand, we were now to state a reduction pair, we should merely have to add one or more reduction pairs at the new stage; and these pairs will be compatible with the first one. In this latter case we do not correct the determinations laid down in the previous stage but simply supplement them. (TM, 448-9, emphasis mine)

Pap calls scientific concepts with this feature “open concepts.” That it is always possible to add new reduction sentences to the set that defines a particular open concept gives the third sense in which any single reduction sentence is “partial.” Pap’s criticism of Carnap begins, in eﬀect, with a question. When a scientist settles on a set of procedures for determining a physical quantity, and thereby, according to Carnap, implicitly adopts a partial meaning-specification, has she implicitly adopted an analytic truth? In other words, is a partial meaningspecification, as used in science, analytic or synthetic? Pap’s fundamental claim here is that, with respect to open concepts, neither classification is in accord with actual scientific practice. Consider the following two examples.

22

The Analytic-Synthetic Distinction: Dispositional and Open Concepts Suppose a scientist, finding that each metal has a unique melting-point, decides that it is best to define metals by their melting-points. The question before the philosopher of science is how to interpret the meaning of ‘definition’ as used in this context.. . . Suppose, then, that while according to past experience any substance with melting-point M had properties P1 , . . . , Pn (for which reason M was selected as a reliable indicator of these properties), the scientist is suddenly confronted with a specimen having M but lacking all of these properties. Would he really insist that the specimen is an instance of the metal in question and that the generalization ‘all instances of this metal have properties P1 , . . . , Pn ’ had simply been refuted? Should such anomalous specimens turn up frequently, I think it likely that he would be frankly ‘illogical’ and say ‘probably all these specimens ought to be classified as the metal in question, and we’d better give up the definition in terms of melting-point’ . . . . (19, 300) [S]uppose we accepted Mach’s definition of mass in terms of the ratio of the accelerations mutually produced in two interacting particles, which turns [Newton’s] third law of motion, via the second law, into a definitional truth. Let ‘p’ stand for a predication of the defined functor (such as ‘the mass of particle A is m’), and ‘E’ for the evidence which, by the definition, is logically equivalent to p (such as ‘the ratio of the acceleration imparted to A by unit particle B to the acceleration imparted by A to B is equal to m’). Then[, if the definition is not open to empirical refutation,] the outcome of any further tests of p, based on contingent laws connecting mass with other quantities, would be irrelevant to the question of the truth of p if only E is accepted as certain. And if all these other tests yielded a value for the mass of A inconsistent with m, then all the contingent laws would have to be abandoned while p would remain unshaken. Thus, suppose that A and B had equal mass by Mach’s definition (i.e. m = 1), but that we found that A and B produced unequal strains on a spring scale, further that they did not balance on a beam-balance, further that they exerted unequal gravitational attractions on a given mass at a given distance (since the law of gravitation is an independent assumption of mechanics, not deducible from the third law, this is a logical possibility) and so forth. We still would be logically compelled to stand by our hypothesis.. . . I submit that this is highly counterintuitive, and that no scientist would act in [this way]. (19, 299)

The conclusion that Pap draws about the first example is this: If it be asked whether, according to the view here presented, a generalization about a natural kind, like ‘all iron has melting-point M,’ is analytic or synthetic, it must be replied that this dichotomy is inapplicable to propositions involving open concepts. If an explicit definition of ‘iron,’ in the sense of a statement of synonymy, were at hand, then the question would be appropriate; but this is precisely the presupposition which is here denied. (19, 301)

He makes a similar point about the second example: “if a physicist like Mach speaks of a ‘definition’ of ‘mass’ in terms of the third law of motion he does not attach to the word ‘definition’ the meaning underlying the analytic-synthetic distinction” (19, 302). The reader will have recognized, in these examples, concrete illustrations of the abstract description of science underlying Pap’s concept of functional

Introduction

23

necessity. The scientist selects a statement to make into a definition; but this action does not insulate the statement from empirical evidence. In fairness to Carnap, it should be pointed out that in the passage quoted above, he explicitly denies that reduction sentences are definitions. But Carnap’s reason for this is that a definition must cover all cases. So if one wanted a definition for an open concept at any stage of inquiry, one would have to make arbitrary decisions about the area of indeterminacy left by the reduction sentences discovered by that stage. These decisions, however, may conflict with further reduction sentences that might be discovered in the future. What Carnap fails to consider, and Pap’s examples bring out, is the possibility of conflict among reduction sentences, as inquiry progresses. For Carnap open concepts are open only to future narrowings of the area of indeterminacy. Pap shows, however, that they are also open to the possibility that future inquiry could reveal internal inconsistencies among reduction sentences and thereby force us to give up previously adopted procedures for determining physical quantities. Thus, Pap’s point is independent of whether reduction sentences are definitions or not. His point is that if scientific practice can be reconstructed as progressive adoptions of reduction sentences, then these adoptions can be both meaning-fixing and sensitive to empirical evidence. This characteristic of reduction sentences is inconsistent with a sharp analytic-synthetic distinction.

6.

The Limits of Hypothetical Necessity: Formal or Absolute Necessity

For Pap, the freedom that we have in making and unmaking necessities has limits. Quine, as we see from the passage quoted on page 16 above, held that even the laws of logic may be open to revision in response to experience. This is the point at which Pap would part company with Quine. For Pap there is a kind of necessity not reducible to functional necessity; he calls it “formal” or “absolute” necessity: A formally necessary . . . judgment . . . is a judgment whose contradictory is inconsistent. . . . [T]his . . . means that it contradicts the laws of logic, and the latter are themselves functionally necessary in the highest degree, insofar as their rejection would force us to abandon all laws whatever . . . . It is to be noted, though, that formal necessity is not thereby ‘reduced’ to functional necessity. Inconsistency with the laws of logic remains sui generis, although it can be interpreted as, so to speak, the upper limit of material inconsistency. The laws of logic themselves cannot be said to diﬀer merely quantitatively from functionally necessary laws, such as to be defined by the property of having functional necessity in the highest degree. For functional necessity itself is defined in terms of the logical relation of inconsistency, which is itself defined in terms of the laws of logic. (2, 73-74)

I hope that, by now, I have suﬃciently prepared the reader to see the basic line of thinking underlying this distinction in kind between formal and func-

24

Logical Consequence and Material Entailment

tional necessity. Formal necessity is based on the laws of logic, and it is only by reference to these laws that we have a conception of inconsistency; but consistency and inconsistency between experience and statements is the standard by which we judge whether to accept or reject statements. In short, the laws of logic have to be acknowledged in order for us to have any conception of empirical evidence. The intuitive basis of Pap’s view is thus an instance of the conception of rationality wherein there have to be commonly shared rules governing empirical inquiry, and these rules cannot themselves be accepted or rejected on the basis of experience because if they were not in place, there would be no such thing as an objective practice of empirical belief formation. Thus, for Pap, empiricism is itself not possible without the absolutely necessary truths of logic. I note in passing that this line of thinking anticipates significant aspects of Michael Dummett’s criticism of Quine (Dummett 1974). From Dummett’s and Pap’s perspective, Quine’s openness to the possibility of revising the laws of logic bespeaks the abandonment of rationality in science.11 A natural question that arises here is this. We saw that functional necessity is made by us; hence, we know what truths are functionally necessary (at some point in inquiry) by finding out about our decisions. It would seem that to know what truths are formally necessary, we use the laws of logic to determine if their negations are inconsistent. But how do we know what are the laws of logic? And how do know when some statement is inconsistent? There is a clue to the view that Pap eventually develops in this early paper, where he says that “Inconsistency with the laws of logic remains sui generis” (2, 74); this suggests that this inconsistency is unanalyzable and our knowledge of it irreducibly intuitive. We turn now to the topic of logical consequence for an account of Pap’s view of our knowledge of logic, and thus of absolute necessity.

7.

Logical Consequence and Material Entailment

The laws of logic, for Pap, are not simply a set of statements of determinate syntactic forms. But neither does he accept the view that there is an explanation of what makes a statement a law of logic, in terms of the meanings of the logical constants. This view, which continues to be widely accepted, holds that deductive implication or entailment is truth preservation in an argument in virtue of the occurrence of the logical constants in that argument. Laws of logic are forms of statements that are logical consequences of any statement whatsoever. The view is closely connected to the one discussed in section 3 (page 13 above), that the “primitive” truths of logic are necessary because

11 For

a more detailed discussion Dummett’s criticisms see Shieh 1997.

Introduction

25

they implicitly define the meanings of the logical constants. Pap’s objections to these ideas are scattered throughout his writings; I will focus on chapter 10, “Logic and the Concept of Entailment” (cited in this Introduction as LCE). The question he sets out to address is: what is a logical constant? Of course we can, and in fact do when teaching elementary logic, identify the logical constants by displaying arguments that are intuitively correct and then showing that intuitive validity is preserved when we vary certain expressions but not when we vary others. We then tell our students that the latter are the logical constants. Now, Pap’s argument about this procedure is very simple. If it constitutes the fundamental ground for classifying expressions as logical, then the notion of logical constant presupposes the notion of (intuitive) deductive validity. But then deductive validity cannot be based, ultimately, on an independent notion of the syntactic forms of statements characterized through the occurrences of logical constants—i.e., cannot be based on the notion of logical form. Pap’s conclusion is this: no adequate epistemological account of the construction of semantic systems of logic can be given without countenancing the concept, held in disrepute by many logicians and philosophers, of material (=non-formal) entailment. It is widely held that entailment is essentially a formal relation, i.e., ‘ “p” entails “q” ’ is held to be equivalent to ‘ “if p, then q” is true by virtue of its logical form’.. . . But . . . judgments of entailment are presupposed by the very process which leads to the definition of the meta-logical concept logical form. For the only way in which this concept can be defined (if it be proper to call such a procedure “definition” at all) is to exhibit logical forms by the use of logical constants. (LCE, 201)

This argument does not, at first, seem to be eﬀective against the dominant account of logical consequence in contemporary philosophy of logic. This semantic account of logical consequence rejects the claim that deductive validity is ultimately based on logical form. It oﬀers, instead, a general explanation of the validity of arguments that also shows why arguments with certain logical forms are valid. The explanation is in terms of the meanings of the logical constants. For example, modus ponens ponendo is a valid form of argument for the following reasons. Given the meaning of the words ‘if . . . then’, a statement of the form p ⊃ q is true just in case p is false or q is true. If the sub-statement p is true, then it is not false. It follows that the truth of p ⊃ q requires the truth of the sub-statement q. That is to say, given the meanings of the logical constants, statements in which they occur have certain truth conditions—in the case of the sentential constants these conditions are given by truth tables. An argument is valid because the occurrence of the logical constants implies that the truth conditions of the premises cannot be fulfilled without those of the conclusion being fulfilled. This just happens to be the case in modus ponens. In this account, intuitive judgments of deductive validity play only the role of identifying certain expressions whose meanings are to be given an analysis.

26

Logical Consequence and Material Entailment

This analysis, rather than any particular set of independently identified syntactic forms, explains why our intuitive judgments hold.12 But an answer to this reply can be gathered from NPLR. Pap writes, the usual way of proving that the ponendo ponens rule has the ‘truth-preserving’ character which any acceptable rule of deduction must have, is to prove that the corresponding calculus formula “if p and (if p, then q), then q” is a tautology. Since a tautology is a truth-function of propositional variables which is true for all values of the variables, such a proof cannot get started until “if, then” is interpreted as a truth-functional connective, specifically in the sense of the symbol of Principia Mathematica ‘⊃’ (material implication). But what is the justification for the truth-functional interpretation of “if, then”? None other than that it preserves that common core of meaning in the various uses of “if, then” . . . which enables us to justify those and only those methods of deduction which we intuitively accept as valid, i.e. as corresponding to entailments. Thus the interpretation of “if, then” in the sense of material implication makes it easy to prove that arguments of the form “p; if p, then q; therefore q” and “if p, then q; not-q; therefore not-p” are always truth-preserving . . . . In that case, however the proof of the truth-preserving character of the ponendo ponens rule is, if not formally circular, based on our intuitive knowledge that any proposition expressed by “p and (if p, then q)” entails the corresponding proposition expressed by “q.”. . . The apprehension of logical [entailment] is thus prior to the adoption of . . . the definition of “if p, then q” as “not both p and not-q,” that render formal proofs possible. Such apprehension alone can explain why we accept just this analysis of the logical constant “if, then” as adequate, and not, say, the analysis “not both q and not-p.” (NPLR, 142-143)

We may thus formulate Pap’s reply to the semantic account of logical consequence thus. The semantic account fails to acknowledge that the analysis of the meanings of the logical constants does not take place in a vacuum, but, rather, is constrained precisely to accord with our intuitive judgments of the validity of arguments. The indispensability of material entailment appears also in considering what is actually accomplished by the logicist “reduction” of arithmetic that I will discuss in section 10. Pap writes, What . . . is the attitude of logicians when they are faced with an evidently nonempirical . . . statement which, though expressing an a priori truth, does not seem to be demonstrable with the sole help of the definitions of specified logical terms? Their endeavor may be described as the analysis of the non-logical concepts that seem to occur essentially by means of logical concepts, so that what appeared as a material entailment reduces to a formal entailment once the proposition is fully analyzed. The obvious illustration of this procedure that comes to one’s mind is the reduction of arithmetic to logic. To take a very simple example, since the relational predicate of arithmetic, ‘being greater than’, does

12 I

give a more detailed account of the contemporary semantic conception of logical consequence in Shieh 1999, sections 2-3, 82-7.

Introduction

27

not belong to the vocabulary of logic, the sentence ‘if G(x, y), then not-G(y, x)’ could not, oﬀhand, be said to express a formal entailment. This postulate of arithmetic, however, can be reduced to pure logic by defining numbers as properties of classes and defining the mentioned arithmetical relation in terms of the concepts ‘similarity’ . . . and ‘proper subclass’. But the definition which enables such a reduction is obviously not an arbitrary stipulation; rather it expresses an analysis of a primitive concept of arithmetic. And I would like to know what else I judge, in judging this analysis to be correct, but that the proposition ‘the number of A is greater than the number of B’ entails and is entailed by the proposition ‘B is similar to a proper subclass of A (where A and B are finite classes)’. But since the non-logical term ‘greater’ occurs essentially in the corresponding conditional statement this is definitely not a formal entailment. We have succeeded in formalizing the entailment from ‘G(x, y)’ to ‘not-G(y, x)’ only by accepting the material entailment just mentioned. (LCE, 201-202)

This leads Pap to a surprising Kantian conclusion: [I]f by a synthetic proposition you mean a proposition not deducible from logic alone, and by an a priori proposition you mean one that is not empirical, and if you define logic by means of an enumeration of a set of concepts called ‘logical constants’—to which there is no alternative in the absence of a satisfactory general definition of ‘logical constant’—then you have to accept the conclusion that synthetic a priori propositions are acknowledged whenever the territory of logic expands. And it appears, then, that in ‘reducing’ the non-geometrical parts of mathematics to logic, the logisticians have not eliminated the synthetic a priori from mathematics; they have merely dislocated it to those regions where mathematical and logical concepts make definitional contact. (LCE, 203)

It should be clear that this conclusion is Kantian not merely because Pap uses the words ‘synthetic a priori’. After all, Russell at one point held that logic was synthetic a priori; but upon examination it turns out that he meant by ‘synthetic’ any proposition that is not an instance of the law that everything is self-identical—clearly not Kant’s view of syntheticity. For a contemporary example, Michael Potter claims that Frege’s logic, in contrast to Kant’s, is ampliative and so synthetic (Potter 2000, 64); Potter means, it turns out, that Frege’s logic is “object-involving,” but gives no explanation of what this means or why this is the case.13 Pap’s view is genuinely Kantian because his point is that we can never eliminate from epistemology propositions that we come to know not on the basis of experience, nor by (discursive) reflection on concepts, but in virtue of intuitive insight into their truth.

8.

The Method of Conceivability

Intuitive insight into entailments and contradictions is for Pap a methodical enterprise. And the method is none other than the one practiced by the great Rationalists such as Descartes and Leibniz: 13 For

more on Potter’s accounts of Kant and Frege see Shieh 2002.

28

The Method of Conceivability In order to be sure that ‘if A, then B’ is a necessary proposition, you must . . . get a negative reply to the question: ‘Can you conceive of an object to which you would apply “A” and would refuse to apply “B’?”’ . . . [W]e . . . discover the necessity of the proposition in an intuitive manner, viz., by trying to conceive of its being false, and failing in the attempt. (2, 106; all emphases in original)

The procedure outlined is continuous with the epistemology of necessity in contemporary analytic philosophy; it is the basis, for instance, of Stephen Yablo’s influential “Is Conceivability a Guide to Necessity?” (Yablo 1993). The reader will see the method of conceivability repeatedly in action in this volume. I sketch here two philosophical interesting applications.

8.1

The Semantic Concept of Truth

In Chapters 6 and 7, Pap argues against the version of the semantic concept of truth adopted by Carnap from Tarski. In Carnap’s formulation, quoted by Pap, it goes: to assert that a sentence is true means the same as to assert the sentence itself ; e.g., the two statements ‘the sentence “The moon is round” is true’ and ‘The moon is round’ are merely two diﬀerent formulations of the same assertion. (Carnap 1942, 26)

Pap’s criticism, it should be noted, is not the standard one that the theory is not capable of handling generalizations using a truth predicate. That criticism, of course, engendered the current cottage industry on deflationary theories of truth which, in one way or the other, argue that something like the use of a truth predicate in generalizations is all there is to the notion of truth. Pap’s criticism is this. The semantic conception has the consequence that predicating truth of a sentence is the same thing as making a statement with that sentence. “Is the same” Pap understands as necessary equivalence. Hence, the semantic concept also implies that, e.g., It is not the case that ‘the moon is round’ is true, and the moon is round

(12)

is self-contradictory. This is where the method of conceivability comes in. If (12) is self-contradictory, then we must fail when we try to conceive of circumstances which are correctly described by it. But, A universe which is just like ours except that it does not contain language, and thus contains no sentences, is surely logically possible. In such a universe it would still be the case that the moon is round, but nothing could be the case in such a universe which logically presupposed the existence of sentences, hence it would not be the case that the sentence “the moon is round” is true. (6, 149)

The conclusion is this. In general, there is no contradiction in conceiving that, by using a sentence, one could make a true description of conditions in

Introduction

29

which there are no sentences or users of them. This means that, intuitively, we cannot avoid the idea of something like a proposition expressed by a sentence which is available to be true, independently of the existence of sentences that express it.

8.2

The Mind-Body Problem

In Chapters 16 and 17, Pap oﬀers a critique of logical behaviorism and physicalism concerning the mental. His principal argument is that the principle of verification does not, contrary to widespread doctrine among positivists, imply either of these doctrines. In order to grasp Pap’s argument, let’s put on the board a formulation of a standard verificationist argument for behaviorism physicalism about the mind: 1 Meaningful statements must be verifiable. 2 If statements about the mental states or processes of another person are made true by non-behavioral or non-physical properties of that person, then they would not be verifiable. 3 Such statements are meaningful. 4 Therefore, such statements are not made true by non-behavioral or nonphysical properties of that person. Pap’s critique begins with a clarification of the principle of verification. On his reading, it must be understood as asserting that the meaning of a statement is the set of conditions whose verification would establish its truth. This has two consequences. First, the circumstances whose verification give the meaning must be entailed by the statement. They can’t be merely good evidence for the statement, which makes its truth more likely, but is compatible with its falsity. Second, the principle by itself has no consequences for what methods of verification are allowable. In particular, the principle does not require that verification is by sensory perception. The principle is consistent with proof, introspection, or ethical intuition as methods of verification. Pap now advances two criticisms. First, statements about mental states do not entail statements about behavior or about physical constitution. Here is where the method of conceivability is used: it is conceivable that I can know that I am angry while being blissfully ignorant of the bodily symptoms by which my feeling is manifested . . . . (16, 257) if it is conceivable that I should know about my anger without knowing about the bodily symptoms of my anger, it is still more easily conceivable that I should

30

Comparison with Necessity in Contemporary Analytic Metaphysics know about my anger without knowing anything about the connection of feelings with brain-events. (16, 258)

It follows that establishing the truth of statements about behavior or physiology does not count as verifying statements about other minds. Second, Pap argues that the possibility of verifying statements of the mental state of another is exactly on a par with the possibility of verifying statements about the past. In both cases the possibility of verification requires a(n indexical) change in the statement being verified: if the speaker had experienced . . . the toothache which he predicated of the other mind, and at that moment had been asked ‘which statement is conclusively verified by your present experience?,’ he would have replied, ‘the statement “I have a toothache now,” ’ not ‘the statement “he has a toothache now”.’ But in just the same way, if I had been standing 50,000 years ago where I am standing now and had observed that it rained, the proposition verified by my observation would then have been formulated by the sentence ‘It is raining here now,’ not by the sentence ‘It rained here 50,000 years ago.’ It could therefore be argued that once a statement about the past were conclusively verified by the speaker it would cease to be a statement about the past. Should we say, then, that public verification, i.e., verification by any observer of any age, including the speaker, of statements about the past is logically impossible? . . . . The main point is that, whichever answer we decide upon, we should give the corresponding answer to the corresponding question concerning the public verifiability of statements about mental events not owned by the speaker. (16, 263)

It follows that if statements about the past are held to be publicly unverifiable, the same will have to be said of statements about other minds; but on this alternative the version of the criterion of verifiability under discussion will simply have to be repudiated, since nobody in his sense would want to lay down a law which leads to the elimination of history along with the elimination of metaphysics . . . . (16, 261)

9.

Comparison with Necessity in Contemporary Analytic Metaphysics

In this section I will be a bit more speculative. I will discuss the relation between Pap’s views of necessity and what has become received wisdom since Kripke. As noted above, Pap’s fundamental conception of necessity is non-factuality. It follows, then, that for Pap there is no radical distinction between necessity and apriority. So, one may well think that Pap’s theory of necessity has been definitively refuted by Kripke’s examples of contingent a priori and necessary a posteriori truths, and so can be of no more than “historical” interest. In this section I will argue for a diﬀerent conclusion; I will show that Pap has the resources to give an interpretation of Kripke’s examples quite diﬀerent from

Introduction

31

those Kripke gives them. Most of my discussion will be on Kripke’s purported contingent a priori truths. Let’s recall the well-known case of the standard meter (Kripke 1980, 54-7). The reference of the expression ‘meter’ (or ‘one meter’) is fixed by stipulating that it is the length of a stick, S , at some time t0 . After making this stipulation, it is possible to know that the statement, S is one meter long at t0

(P)

is true without basing this belief about S on any empirical evidence. So (P) is a priori. But, intuitively, at t0 S could have had a diﬀerent length, if, e.g., it had been heated. In such counterfactual circumstances, the length of S would have been diﬀerent from one meter. Hence S , though true, is only contingently so. Pap can redescribe this case using his concepts of hypothetical and absolute necessity. The apriority of (P) for Kripke is its hypothetical or functional necessity for Pap. The stipulation of the reference of ‘meter’ as the length of S at t0 is equivalent to the introduction of a (possibly new) expression, ‘meter’, by a decision to treat (P) as a rule, and so conventionally true. Obviously, the purpose of making (P) a rule is to provide the basis of a system of measurement. Thus, this decision to treat (P) as true does not occur in a vacuum; it is not, to use Wittgenstein’s words, merely a “ceremony” (Wittgenstein 1968, §258). Rather, in order for the decision to fulfill its function, it has to be accompanied by a (again conventional) practice for using the physical object S as a standard of measurement. Such a decision, together with its accompanying practice, clearly seems consistent with thinking of the object S in ways other than as a standard of measure. In particular, one can apply the method of conceivability and consider whether it is self-contradictory to conceive of S as having a diﬀerent length at t0 than it in fact had. Indeed, Kripke’s argument obviously turns precisely on such conceivability “thought-experiments.” Now, the contingency of (P) rests on the contingency of the length of S . But length is a measurable, i.e., determinable, physical property whose determination requires a standard of measurement. Thus the claim that the length of S is a contingent property is equivalent to the claim that, relative to a standard of measurement, it is contingent what is determined as S ’s (determinable) property of length. But S itself, according to our conventions, is treated as the standard for the metric system of measurement. It follows, then, that the contingency of the length of S can be expressed using S itself. But if we do so, we can use the expression, ‘meter’, that S was used to introduce, in which case we would say The length of S at t0 might not have been one meter

(13)

32

Comparison with Necessity in Contemporary Analytic Metaphysics

i.e. S might not have been one meter long at t0

(14)

And so (P) is (absolutely) contingent. Here, then, we have a description of Kripke’s case in Pap’s terms: it is a case of a statement that is functionally necessary but absolutely contingent. This redescription, prima facie, does not require abandoning the equivalence of necessity with apriority. Moreover, it is easy to see that the apriority of (P) is only pragmatic. Suppose, what is in fact the case, that the length of S varies more than we can tolerate in our metric system of measurement. That would be a pragmatic reason to stop using S as a standard of measurement, i.e., stop taking (P) to be a rule of using the language of the metric system of measurement, and thereby stop taking it to be necessarily true, for this purpose. I should note that anyone familiar with contemporary discussions of twodimensional modal logic would recognize some similarities between my Papian account of the standard meter case and that of Davies and Humberstone in Davies and Humberstone 1980. On their account, there are two notions of necessity: true no matter what world is actual, as against true with respect to all alternatives to a given actual world.14 I will end with just a few brief comments on Kripke’s examples of necessary a posteriori truths. One of Kripke’s examples (Kripke 1980, 102-5) is the statement Hesperus is Phosphorus (15) It is an a posteriori truth because it was established by astronomical investigations. But it is necessary because all identities are instances of a law of logic and so necessary. Now, clearly the claim is that empirical evidence is relevant to establishing the truth of (15). But is this empirical evidence relevant to establishing the truth of an instance of the law of identity? Surely not. Indeed, if we appeal to any evidence at all for laws of logic, we appeal only to intuitive evidence: we try, and fail, to conceive of any object not being identical to itself. Now, do we appeal to such (failed) attempts to conceive of a non-self-identity to establish that (15) is true? Of course not, otherwise astronomy would be a much easier endeavor than it is. Kripke’s argument depends on claiming that the proposition expressed by (15) is the same as that expressed by ‘Hesperus is Hesperus’ or ‘Phosphorus is Phosphorus’, i.e., an identity. As he puts the point,

14I discuss the relation between Kripke’s modal

arguments and two-dimensional modality in Shieh 2001.

Introduction

33

[I]t is only a contingent truth . . . that the star seen over there in the evening is the star seen over there in the morning . . . . But that contingent truth shouldn’t be identified with the statement that Hesperus is Phosphorus . . . . (Kripke 1980, 105)

But, from Pap’s perspective, the important point is that in scientific inquiry, we come to accept or reject (15) on the basis of whether “the star seen over there in the evening is the star seen over there in the morning.” This implies that, in science, we do not take the sort of justification relevant to establishing identities to be relevant to establishing (15). That is, from the perspective of science, (15) and identities do not express propositions with the same modal status. Thus, from this perspective, the burden of proof is on Kripke, to show why his theory of propositions should trump scientific practice.

10.

Logicism

Next I turn to Pap’s work in the philosophy of mathematics, specifically, his papers “Mathematics, Abstract Entities, and Modern Semantics” (chapter 12, cited in this Introduction as MAS) and “Extensionality, Attributes, and Classes” (chapter 13, cited in this Introduction as EAC). Pap begins MAS by recalling the classic philosophical problem posed by mathematics. Philosophers from Plato to Kant have been puzzled by the fact that while geometric proof proceeds by construction on particular, concrete figures, the propositions proved by means of these constructions are taken to hold universally for all geometric figures. Indeed the theorems proved don’t even hold exactly for the concrete figures of the demonstration, since no actual physical figure is, e.g., exactly triangular. Something similar can be said of arithmetic. In order to find the sum of, say, 7 and 5, one might proceed by counting out a set of 7 oranges and a set of 5 apples, and then count all these pieces of fruit. But this establishes more than that 7 oranges and 5 apples make 12 pieces of fruit; it shows that 7+5=12 holds for any classes of entities, not just for fruit. And, of course, should we perform this counting procedure on some other two disjoint sets of 7 and 5 objects, and fail to obtain 12, we see no alternative but to conclude that we miscounted, or that special physical circumstances obtained. As Pap sees it, these features of mathematics suggest that mathematical theorems are not about particular geometrical figures or particular sets of objects, but about what these particulars have in common. And, what these have in common are abstract entities that don’t have spatio-temporal location. For the positivists the idea that mathematics concerns non-spatio-temporal entities is problematic. One problem is that it’s not clear how such entities could be the objects of sense experience, and so of empirical knowledge. But the more serious problem goes beyond this. After all, the positivists could and did accept and account for knowledge of theoretical entities of science that

Logicism

34

are not directly experienceable. The deeper problem is rather that, as we have seen, mathematical theorems seem not to be open to empirical disconfirmation or confirmation. It is this that makes it hard for the positivists to see how mathematical theorems are instances of genuine cognition. The solution adopted by positivism is a logicist account of mathematics,15 an account taken from Frege and Russell. It is the counterpart, for mathematics, of the linguistic theory of necessity. The basic idea of such an account is that, e.g., an arithmetical equation such as ‘3+2=5’ is true because the symbols ‘2’, ‘3’, ‘5’ and ‘+’ are “defined (or tacitly understood) in such a way that [the equation] holds as a consequence of the[ir] meaning[s]” (Hempel 1983, 379). Since mathematical truths hold in virtue of the meanings of the key terms involved, they are not any more open to confirmation or disconfirmation than is the analytic truth ‘All sisters are female’. But now what about the idea that mathematical theorems are about abstract objects? One positivist view is that if mathematical statements are true by virtue of the (conventional) meanings of mathematical vocabulary, then we need not assume that this vocabulary refers to any entities at all. This is for the same reason that, on the linguistic theory of necessity, the truth of the analytic statement ‘all sisters are female’ does not depend on there being entities such as the attributes of sisterhood or femaleness denoted by the words ‘sister’ or ‘female’. Indeed, it does not even depend on there being anything in experience correctly described as sisters or siblings or females. All we need to assume to account for its truth is that there are norms governing the correct use of these words. The main question for Pap is: does logicism really show that mathematics is not about abstract entities? He argues first that, in spite of appearances to the contrary, neither Russell’s or Frege’s logicism aimed to eliminate abstract entities from mathematics. Indeed, Pap proposes that, in order to resolve a central problem of Russell’s logicism, one could take arithmetic to be about nothing other than attributes, entities of traditional metaphysics which the positivists insisted are spurious. Second, Pap argues that the ontological commitments of arithmetic are not settled by this proposal, since statements apparently about attributes can be interpreted nominalistically, by semantic ascent, as meta-linguistic statements about schemata, rather than generalizations quantifying over attributes. I will focus on just the first part of this central argument. It is certainly the case that Frege’s and Russell’s accounts of arithmetic include a set of contextual definitions of number words occurring in statements of applied arithmetic. These definitions allow an interpretation of such statements

15 Or

at least the non-geometric parts of mathematics.

Introduction

35

from which numerals are eliminated, an interpretation avoiding even apparent reference to numbers. These are definitions of numerically definite quantifiers, numerical quantifiers for short. For example, the statement (schema) There are (exactly) two Fs is reinterpreted as the quantified statement (schema) (∃x)(∃y)(x y . (∀z)(Fz ≡ z = x ∨ z = y)) More generally, we define by recursion a sequence of quantifiers, ∃0 , ∃1 , ∃2 , . . . ∃n , . . . :16 ‘There are exactly 0 Fs’ ‘There are exactly n + 1 Fs’ ‘There are exactly n Fs’

(∃0 x)F x (∃n+1 x)F x (∃n x)F x

−(∃x)F x (∃x)(F x . (∃n y)(Fy . y x)), given

But logicism cannot be based only on such definitions. This, Pap points out following Frege and Russell, is because they do not allow us to eliminate numerals from statements such as “2 is an even prime.” These contextual definitions only apply when numerals appear as adjectives, not when they appear as singular terms. They tell us how to interpret phrases of the form ‘There are n . . . ’ as wholes, but do not treat the numeral, n, as an independent part of the phrase. The more serious diﬃculty arising from this problem, which Pap does not make explicit, is that, as Frege in eﬀect argues in Grundlagen der Arithmetik (Frege 1884), such definitions do not allow us to give an interpretation of general statements (apparently) about numbers. For example, it is unclear how we could use these definitions to interpret the Euclidean statement, “For every prime number there is a larger prime.” Such a statement contains a quantifier over numbers; but, as just noted, the subscripts of the numerically definite quantifiers are not independent terms, so we can no more quantify over them then we could replace ‘cat’ in ‘cattle’ with a variable. Since most of the theorems of arithmetic are general statements, the contextual definitions in terms of numerical quantifiers are incapable of giving an account of arithmetic. Russell gives a non-contextual definition of numbers as classes of similar classes, where classes are similar just in case there is a one to one correspondence between the elements of each. More precisely, x and y are similar, x ≈ y,

16 Strictly

speaking no expression is a quantifier unless it occurs together with a variable of quantification. Below I will sometimes be strict and use syntactic variables ‘u’ and ‘v’ to indicate occurrences of a variable of quantification together with a quantifier.

Logicism

36 just in case (∃R){(∀a)(∀b)(∀c)(Rab . Rbc ⊃ b = c) . (∀a)(∀b)(∀c)(Rac . Rbc ⊃ a = b) . (∀a)(a ∈ x ⊃ (∃b)(Rab . b ∈ y)) . (∀a)(a ∈ y ⊃ (∃b)(Rba . b ∈ x))}

For example, the number 2 is the class of all classes similar with two-membered classes. This of course looks circular. But circularity is avoided by use of the numerically definite quantifiers. Thus 2 =d f {x : (∃a)(∃b)(a b.(∀c)(c ∈ x ≡ c = a ∨ c = b))}. Unfortunately Pap does not in these articles make clear that such a definition is not enough for Russell’s logicism. The problem is precisely that which we just mentioned: with these definitions in hand, we can at least begin to interpret statements such as ‘3 is a prime number’ (one needs, of course, a definition of is a prime number’ that yields conditions for it to be applicable to classes ‘ of similar classes), but we still have no device for interpreting arithmetical generalizations. What Russell actually did in order to get around this problem is as follows. He gave definitions of the numeral 0, the relation of (immediate) successor, and the predicate of being a number, all in terms of classes and predicates and relations of classes. And then, using these definitions and the laws of logic, Russell proved the 5 axioms of Peano’s axiomatization of arithmetic. This axiomatization crucially includes the principle of induction, the basis for proving arithmetical generalizations in actual mathematical practice. Here is a formulation of Russell’s definitions in a simple (i.e., not ramified) theory of types. In this theory entities are divided into types; the lowest type, 0, consists of individuals, and the entities of type n + 1 are classes of entities of type n. In order to avoid having to specify types all the time, let’s call type 1 classes sets, type 2 classes classes and type 3 classes Classes. Subscripted variables, xn , yn , zn range over entities of type n. But, again in order to improve readability, we adopt some conventions for variables: a, b, c, . . . x, y, z . . . m, n, . . . X, Y, Z, . . .

range over individuals range over sets range over classes range over Classes

We will be defining the successor relation and the ancestral. Strictly speaking, in Russellian type theory binary relations over type n entities are classes

Introduction

37

of ordered pairs of those entities and so of type n + 3. But we will dispense with this and add binary relation variables over each type greater than 0. In particular, R, S . . .

range over binary relations of classes

There is an extensionality axiom for each type: (∀xn+1 )(∀yn+1 )[(∀zn )(zn ∈ xn+1 ≡ zn ∈ yn+1 ) ⊃ xn+1 = yn+1 ], and a comprehension schema for classes and relations of each type, for any formula φ in which yn+1 does not occur free: (∃yn+1 )(∀xn )(xn ∈ yn+1 ≡ φ) Here are the relevant definitions: 1 Λ = {a : a a},

∅ = {x : x x}

[Λ is the empty set, ∅ is the empty class.]

2 0 = {Λ} 3 x − a = {b : b ∈ x . b a} 4 S (m, n) ≡d f {x : (∃a)(a ∈ x . x − a ∈ m) } ∈ n

[n immediately succeeds m]

5 Her(R)(X) ≡d f (∀n)(n ∈ X ⊃ (∀m)(R(n, m) ⊃ m ∈ X))

[X is hereditary in the series of entities related by R]

6 In(X, R)(n) ≡d f (∀m)(R(n, m) ⊃ m ∈ X) ∗

7 R (m, n) ≡d f (∀X)(Her(R)(X) . In(X, R)(m) ⊃ n ∈ X) 8 R∗= (m, n) ≡d f R∗ (m, n) ∨ m = n 9 n ∈ Z ≡d f S =∗ (0, n).

[n induces X in the R-series] [The strong ancestral of R] [The weak ancestral of R] [n is a number]

Still, even with these definitions Russellian logicism is not free of problems. One of the Peano axioms, in the form adopted by Russell, states that distinct numbers have distinct successors. But, as Pap formulates the problem, given the definition of numbers as classes of similar sets, if only n individuals existed, then the number n + 1, being defined as the class of all classes that have n + 1 members (or that would have exactly n members if exactly one element were withdrawn from them), would be equal to the null class; for in that case no classes with n+1 members would exist. But by parity of reasoning, the successor of n + 1 would also be equal to the null class; therefore n and n + 1, which on the hypothesis made are distinct numbers, would have the same successor. (EAC, 233)

Russell’s way out is to adopt an axiom of infinity, a “law of logic” asserting that there are an infinite number of individuals. This move, of course, seems

38

Logicism

rather ad hoc; to use Russell’s own words, it looks more like theft than honest toil. Moreover, this axiom hardly seems a law of logic. Surely logic consists of those propositions that are true no matter what things we may be talking about and no matter what our (nonlogical) words mean. But then surely laws of logic must be true regardless of what things there are in the world, or, for that matter, of whether there are any things in the world at all. So, while Russell’s axiom of infinity allows him to derive Peano arithmeic, the derivation is only dubiously from the laws of logic. It is thus unclear that Russell has established logicism. Pap’s proposal for saving Russell’s logicism is ingenious. He argues that, even if there are no classes with, say more than 100 individuals, this is quite compatible with there being an attribute or property of having 101 members which just happens not to be instantiated. He writes, Now, we have seen that n+1 = n+2 follows from the assumption that the number of individuals is n, together with the Russellian definition of numbers as classes of similar classes. But if the number n + 1 is, instead, defined as the attribute (of a class) of having n + 1 members, Russell’s conclusion does not follow. For though, on that assumption, both attributes would be empty (inapplicable), they would remain just as distinct as, say, the attributes of being a mermaid and of being a golden mountain, and for just the same reason: they are defined as incompatible attributes. For example, the expressions ‘A has two members’ and ‘A has three members’ are so defined in Principia Mathematica . . . that it would be contradictory to suppose that the same class had both (exactly) two and (exactly) three members. Therefore, even if just one individual existed, the number 2 would retain its conceptual distinctness from the number 3; and this argument, obviously, can be generalized for any finite number n and its successor. (MAS, 223)

As it stands, this proposal is not suﬃcient for accomplishing Pap’s aim of showing how Russell’s logicism is possible without an axiom of infinity. The reason, obviously, is that the proposal does not show how to derive the Peano axioms. What we need is an account of the principles governing attributes, either attributes in general, or those attributes that are the numbers. Of course, if the proposal is to constitute a full defense of Russellian logicism, we will also need an argument to show that the principles governing attributes are logical ones, or, at any rate, are more plausibly logical than is an axiom of infinity.17 I will argue that from Pap’s papers we can derive two conceptions of attributes. One of these receives more textual support, but unfortunately it does not suﬃce for a derivation of the Peano axioms without an axiom of infinity. The other 17 There is an issue here that I’ll come back to. Russell’s own formulation of the axioms of infinity is not the statement that there are infinitely many individuals, or that there exists an inductive class of individuals, as is usual in standard formulations of Zermelo-Fraenkel set theory. Rather, Russell’s formulation is: −∅ ∈ Z, i.e., the empty (type 2) class is not a number. This hardly seems like an assertion that there are infinitely many individuals, although it is equivalent to the latter assertion. Of course, it is equally unclear that Russell’s formulation yields a statement of logic.

Introduction

39

conception is less clearly what Pap had in mind; but it does arguably come closer to the idea of providing a derivation of arithmetic from logical laws alone. What do similar sets have in common? Of course the obvious thing to say is that all such sets have the same number of members. That is, for some finite number n, each such set has the property of having n members. So, we might think of the attribute common to members of a class of similar sets as the property of having n members, for some specific n. This idea, of course, oﬀers no definition of numbers, because the attributes in question are specified using a quantifier over numbers, or a quantifier whose substitution instances are numerals. But, we have seen above that for any finite number n, there is a numerical quantifier, “there are exactly n . . . ” which is definable without using numerals. Moreover, a quantifier can be thought of as expressing a property of the class of entities satisfying the open sentence it binds. E.g., ‘(∀x)F x’ asserts that the extension of the predicate F is identical to the domain of individuals. So, ‘∀’ expresses the property of sets of being identical to the domain of individuals. (Note that, as Pap recognizes very well, for Russell all the talk of classes here should be eliminated in terms of talk about propositional functions; hence, the last claim can be reformulated for Russell as the claim that quantifiers over individuals are properties of propositional functions taking individuals as arguments. From now on I will use a higher order logic that more closely mimics Russell’s logic of propositional functions.) Thus, one way of understanding Pap’s proposal is that the finite cardinal numbers just are those properties expressed by numerical quantifiers. We can formulate a condition for a quantifier (Qv) to be a numerical quantifier, namely, that it holds of a predicate only if it holds of all predicates with similar extensions: (∀F){(Qx)F x ⊃ [(∀G)((Qx)Gx ≡ F ≈ G)]} In the remainder of our discussion of the first interpretation of attribute, I will use the numeral n as the symbol for the numerical quantifier (∃n v) In order to implement this proposal along Russellian lines, we have to define 0, successor, and number in terms of numerical quantifiers. The quantifier that is 0 is, of course, −(∃v). The definition of being a number is also simple if we have succeeded in defining the successor relation; we can take it, as before, as holding of just those quantifiers which stand in the ancestral of the successor relation to 0. The crucial question, however, is, how do we define the successor relation among numerical quantifiers? Pap alludes to Russell’s definition “the number n + 1 [is] defined as the class of all classes that have n + 1 members (or that would have exactly n members if exactly one element were withdrawn from them)” (EAC, 233), and this suggests the following idea. We want the numerical quantifier (nv) to be an immediate successor of a given numerical quantifier (mv) just in case (nv) is a numerical property of all those classes

Logicism

40

with exactly 1 more member than any classes which possess the property expressed by (mv). So, consider any class A. This class would have the property expressed by (nv)—i.e., (nx)(x ∈ A)—just in case the class of y’s distinct from some single fixed element a of A has exactly n members. That is, just in case this class has the property expressed by the numerical quantifier (mv). That is, just in case (mx)(x ∈ A . x a). Thus, a natural Russellian definition of ‘n is the immediate successor of m’, S (m, n), using second-order quantification instead of classes, is this: S (m, n) ≡d f (∀F)[(nx)F x ⊃ (∃x)(F x . (my)(Fy . y x))] But a problem immediately arises. Suppose that there are no classes with n individuals. Then (nx)F x is false, no matter what is assigned to the secondorder variable F. But then the antecedent of (nx)F x ⊃ (∃x)(F x . (my)(Fy . y x)) is true no matter what is assigned to F, and, no matter whether (∃x)(F x . (my)(Fy . y x)) is true. This implies that the second-order generalization defining successor is true no matter what the truth-value of (∃x)(F x . (my)(Fy . y x)). From this it follows that if, say, there are only 100 individuals, then every numerical quantifier (∃n v) for n > 100 is the successor of every other numerical quantifier. So, at least with this definition of successor, we can’t prove Peano’s third axiom without assuming an axiom of infinity.18 All is not lost. The interpretation that I have been pursuing identifies attributes with quantifiers over individuals. But nothing in the intuitive idea that if two classes are similar then they have the same numerical attribute requires this interpretation. One could, instead, take numerical attributes to be entities that are not individuals but are of the same logical type as individuals. And we 18 One might here ask, how is this conclusion related to Pap’s claim, quoted above, that “the expressions ‘A has two members’ and ‘A has three members’ are so defined in Principia Mathematica . . . that it would be contradictory to suppose that the same class had both (exactly) two and (exactly) three members. Therefore, even if just one individual existed, the number 2 would retain its conceptual distinctness from the number 3” (MAS, 223)? The obvious way to understand Pap’s claim is that from (∃2 x)F x and (∃3 x)F x we can derive a contradiction, and this is certainly correct. Indeed, for any distinct numbers m and n, from (∃m x)F x and (∃n x)F x we can derive a contradiction. But this is quite consistent with the conclusion we arrived at above. As we know, the claim that there are exactly n individuals can be expressed quantificationally; for example, (∃x)(∃y)(∀z)(z = x ∨ z = y) expresses the claim that there are exactly two individuals. Moreover, from this claim and (∃3 x)F x we can derive a contradiction. Hence from these two claims we can derive the conjunction (∃2 x)F x . (∃3 x)F x.

41

Introduction

can take the notion of class similarity to be the criterion of identity for these entities. Formally, we introduce a term-forming operator, (Nv), which binds an open sentence with one free individual variable, Φ(v), to form a term denoting the entity that is the number of Φs: “(N x)Φ(x).” Call this the cardinality operator. It is analogous to Russell’s definite description operator ( v); the important point is that the terms that can be formed from this operator are values of a first-order quantifier. Note that I said “a” first-order quantifier, not “the” first-order quantifier. This is because we now use a first-order logic with two sorts of variables, x, y, z . . . for individuals and m, n, . . . for numbers. In addition, we use a second-order logic with predicate and relation variables, each with just one sort. The cardinality operator is governed by axioms based on the intuitive criterion of identity for numbers discussed above: the number of Fs is identical to the number of Gs just in case there is a 1-1 correspondence between the Fs and the Gs. And this holds just in case the classes of Fs and of Gs are similar: (Nu)Fu = (Nv)Gv ≡d f F ≈ G, where u and v are first-order variables of any sort. This specification of identity conditions has been christened Hume’s Principle by George Boolos. It is also, of course, closely related to Cantor’s theory of cardinality. Indeed, some people who don’t like Boolos’s interpretation of Frege call it Cantor’s Principle; most of us now compromise and call it the Cantor-Hume Principle. In terms of the cardinality operator we define 0, immediate predecessor, immediate successor, and the predicate of being a number, in ways that by now will seem very familiar: 1 0 ≡d f (N x)(x x) 2 S (m, n) ≡d f (∃F)(∃x)[F x . (n = (Nz)Fz) . (m = (Nz)(Fz . z x))] (m is the immediate predecessor of n; n is the immediate successor of m) 3 Zn ≡d f S =∗ (0, n) (n is a number just in case 0 is a weak ancestor of n in the immediate successor series)

We also use the following abbreviations: m < n for S ∗ (m, n)

Concluding Remarks

42 m ≤ n for S =∗ (m, n)

I leave out the definitions of the ancestrals; they are straightforward rewritings of the type theoretic definitions. In order to show that an axiom of infinity is not needed, we first prove the principle of induction from these definitions, and then prove, by induction, that every number has an immediate successor. That is, we show that the predicate “ξ has an immediate successor” holds of 0, and is hereditary in the immediate successor-series, restricted to numbers. This predicate is formalized as (∃y)[Zξ . S (ξ, y)] The critical move of the proof is to take the successor of a number to be the number of the members of the immediate successor-series up to that number, i.e., we show that (Nm)[m ≤ k] satisfies the predicate Zξ . S (k, ξ) So the strategy is to prove, by induction on S (ξ, (Nm)(m ≤ ξ)) that (∀n)[Zn ⊃ S (n, (Nm)(m ≤ n))].19 Now, is this logicism? One way to argue that it is is that the only nonlogical axioms express, in part at least, our concept of number. Anyone who understands this concept would have to accept that, no matter what entities one is thinking of, if there is a one to one correspondence between entities satisfying one condition and entities satisfying another, then the same number of entities satisfy each condition. Thus, we can claim that the infinity of the natural numbers follows by the laws of logic from our grasp of the concept of number.20

11.

Concluding Remarks

I hope I have managed in this Introduction to convey something of the interest and importance of Pap’s philosophical work. There are, indeed, quite a few more intriguing discussions in the paper collected here than I could discuss in this space. Here are just some of them: 19 For

full details of the proof see the articles in Demopoulos 1995. it is highly implausible that part of the concept of number is that the empty class is not a number.

20 This is in contrast to Russell’s axiom of infinity:

Introduction

43

Pap’s defense of a Humean theory of causation Pap’s defense of Russell’s ramification in type theory as a solution to the semantic paradoxes, Pap’s semantic account of the realism-nominalism debate, Pap’s critique of the notion of emergent property, and Pap’s formulation of the axioms of a logic of belief. Let me conclude by oﬀering a general characterization of Pap’s philosophical work. Mary Hesse once described Arthur Pap as a “logical empiricist with a bad conscience” (Hesse 1966, 456). To my mind, this is true as far as it goes, but its emphasis is not quite right, nor does it go far enough. Much of Pap’s bad conscience derives, as I have suggested, from allegiance to Cassirer’s neo-Kantianism. Pap wouldn’t give up this allegiance because he saw a deep tension in logical empiricism at its very best, namely, in the work of Carnap. The tension is between Carnap’s adherence to the picture of rational inquiry underlying his continued insistence on an analytic-synthetic distinction, and his attempt to be thoroughgoingly pragmatic, as manifested in his adoption of the Principle of Tolerance (Carnap 1954a, 51ﬀ). What Pap, along with Quine, saw, was that a truly thoroughgoing pragmatism cannot countenance any standards not open to empirical revision, and, equally, a truly thoroughgoing commitment to rationality in inquiry cannot make sense of the possibility that the rules defining inquiry could themselves be changed in response to the empirical evidence they make possible. Quine went with pragmatism, thereby giving up a deeply entrenched conception of rationality. Pap took the other horn of the dilemma, and came to hold that logical empiricism has limits beyond which empiricism cannot go, where there lies nothing other than intuitive knowledge of logic itself.

II

ANALYTICITY, A PRIORITY AND NECESSITY

Chapter 1 ON THE MEANING OF NECESSITY (1943)

In this paper I am mainly concerned with an analysis of the Aristotelian concept of “hypothetical necessity.” It will be defined as a functional synthesis that avoids both the Platonistic reduction of necessity to abstract or mathematical necessity (what the scholastics called “simple” necessity, as contrasted with necessity “secundum quid”) and the empiricistic reduction of necessity (cf. John Stuart Mill) to genetically explicable, yet logically ungrounded, generalization of contingent conjunction. What is characteristic of the Platonistic interpretation of necessity as a formal relation between intensions or essences, is that it involves the banishment of necessity from existence: “Whatever is, might not be,” as Hume said. The empiricist, then, emphasizes that, insofar as a necessary judgment is existential in reference, it represents a generalization of a contingent “conjunction,” which generalization will have a psychological cause, viz., the “generalizing propensity,” in Mill’s phrase, or the “gentle forces” of association, in Hume’s phrase, but no logical ground, and will never represent a necessary connection. The concept of hypothetical necessity helps, as I shall endeavor to show, to avoid the exclusive disjunction, advocated by Hume and his positivistic followers: either existential or necessary, but not both. Something is hypothetically necessary if it is a necessary condition or, functionally speaking, a necessary means for something else. Hypothetical necessity, then, is a matter of consequences rather than a matter of antecedents: antecedents derive their hypothetical necessity from consequences. Metaphorically speaking, hypothetical necessity is prospective. Mathematical or abstract necessity, on the other hand, rather is retrospective; it is a matter of antecedents rather than a matter of consequences: propositions derive their mathematical necessity from the antecedents which they follow from. Hypothetical necessity is predicable of hypotheses or postulates or leading principles; mathematical necessity is predicable of theorems or “reasoned facts.” To explain by hypothetical necessity, in other words, is to explain in terms of “in order to” (“worum”), to explain by abstract necessity is to explain in terms of “because” (“warum”).

47

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On the Meaning of Necessity (1943)

It might seem, prima facie, that the distinction between hypothetical necessity and mathematical necessity merely reflects opposite ways of reading logical sequences. If “if p, then q” is such a sequence, then we can say “q, because p,” and thus declare q as mathematically necessary; but if we, instead of reading backwards to the antecedent, read forwards to the consequent, then we can say “p, in order that q,” i.e. p is hypothetically necessary with respect to q; it is, in other words, a conceptual means to render q intelligible. One might also point out that, since any proposition is, at least potentially, in different respects both conclusion and premise, it is, in diﬀerent respects, both hypothetically and “simply” necessary. Yet the distinction between these two types of necessity amounts to more than that: if p is merely a suﬃcient condition for q, then, the hypothetically necessary being defined as a necessary condition, it is not hypothetically necessary with respect to q. In order, then, for a condition to be interpreted as a hypothetical necessity, a necessary “conceptual means,” it must be placed within a series of alternative conditions. If hypothetical necessity were an intra-logical concept, a concept, that is, that can be defined without reference to extra-logical notions or operations, then that series of alternative conditions would have to represent an analytical, exhaustive disjunctive set: for p would have to be demonstrated to be the only possible condition for q, and in order to demonstrate that, we must suppose ourselves to know the totality of possible conditions for q, and must, moreover, show the alternative conditions to be impossible or self-contradictory. However, the disjunctive sets of alternative hypotheses or conceptual means which the inquirer has to select from, are in most cases neither exhaustive nor analytic. Hence the choice of one alternative hypothesis rather than another cannot be logically grounded. The hypothesis selected, in other words, cannot be itself logically or mathematically necessary in the sense of representing the conclusion of a disjunctive syllogism whose major premise is an exhaustive and analytic disjunction. In Leibniz’s terms, in such a selection of conceptual means, not the principle of identity but the principle of “what is best” is operative. The selection of one hypothesis rather than another cannot be logically, it can only be teleologically, grounded. A hypothetical necessity is not the only possibility—i.e., a “simple” necessity, a self-evident axiom that stands without alternatives—but it is the best possibility, “best” relatively to the teleological or functional context in which it arises as a hypothetical necessity. If p is suﬃcient to explain q, it is merely good for q; r and s and x may be just as good for q. If p is a hypothetical necessity with respect to q, it is not only good for q, but better than any other hypothesis we know of, and thus, within the context of our limited knowledge, best. Thus, the Ptolemaic Hypothesis was good for explaining the astronomical facts or phenomena; its rejection in favor of the Copernican Hypothesis had no logical ground, but only a functional or pragmatic ground: the latter did the job of explanation better than it, being simpler and hence more convenient.

On the Meaning of Necessity (1943)

49

Such hypothetical necessities, or leading principles, bridge, so to speak, the gap between the contingency of empirical conjunctions and the “simple” necessity of formal connections, for they function essentially as means of systematizing facts, of rendering the body of factual knowledge coherent. It is by their instrumentality that facts acquire representative or signifying capacity, and thus evidential value.1 That is, in any statement of fact there is implicit an analytic or formal statement which determines the evidential context of the former, viz., what the stated fact is evidence for and what is evidence for it. For, as the pragmatic theory of meaning (cf. C. I. Lewis) contends, to understand what f (a) (i.e., this is of such and such a kind) means, one must know what would empirically verify f (a), say g(a). But the criterion which determines that g(a) is evidence for f (a), is the “syntactical” or formal premise (∀x)[ f (x) ⊃ g(x)]: whatever has property f has property g. But, one might object, are not those “syntactical” propositions that serve as criteria of the evidential value of facts independent of experience or formally a priori? Yes and no. Yes, insofar as those formal premises are concerned whose interrelated terms are mathematical concepts, and which, hence, express a priori connections that hold irrespective of whether they are exemplified in experience or not. No, insofar as those formal premises are concerned that do not express formal and necessary, but empirical and contingent, implications. For those syntactical statements represent inductive generalizations, and were in the process of inquiry selected as criteria of evidential value. They are hence, so to speak, only functionally formal, being adopted as criteria instrumental to empirical verification. Logically, such criteria, whose interrelated terms are empirical traits (Dewey’s “generic” propositions) are merely contingent or probable, being dependent on the inductive principle. But functionally, they are necessary; i.e., they are best for such-and-such purposes. The number of empirical properties which a given empirical property is conjoined with, is indefinite; hence there is an indefinite series of properties from which we can select a needed evidential property; there is an indefinite number of competitive potential criteria or hypothetical necessities. The determination of the essential or definitory properties of kinds, therefore, is not a discovery, but a choice. Logically, all the alternative criteria are on the same footing, in that they are all, prior to their adoption as criteria, contingent empirical laws. Adoption of one criterion rather than another, therefore, has pragmatic reasons; it is determined by the “principle of what is best.” Hypothetical necessity, then, we may state in paradoxical language, is necessity qua contingent upon freedom of choice or evaluation. Hypothetical necessity presupposes free choice, for the hypothetically necessary is the best one of alternative means, and nothing is better or best except with respect to 1 The principle of causality can be said to be the leading principle of leading principles, insofar as it abstracts

from any particular content that may be signified, and merely prescribes that facts be treated as signs. Cf. Cassirer 1937.

50

On the Meaning of Necessity (1943)

an act of preference or selection. The tendency to ignore this practical element of choice that enters into theoretical explanation, and hence to convert hypothetical necessities into simple necessities—into possibilities that have no conceivable alternatives, self-evident axioms, that is—is a characteristic trait of rationalism. Thus it is characteristic of the seventeenth-century mathematical determinists—Spinoza, Descartes, Galileo—that they interpreted hypotheses as necessary explanations, i.e., as statements of formal causes or laws discovered by “intellectual intuition” rather than selected. Hypotheses, in other words, were for them rationes essendi, not rationes cognoscendi merely. The heliocentric hypothesis, e.g., is not true in the sense of rendering astronomical phenomena intelligible in terms of simpler formulas than those implied by the geocentric hypothesis, but it is true by correspondence, i.e., the sun, as a matter of fact is at rest, and the earth in motion, not the other way around. The identification of a ratio cognoscendi with a ratio essendi, of a principle of intelligibility with a principle of being, however, is valid only on the assumption that the ratio cognoscendi, the hypothesis which explains the facts, has no alternatives: a possibility can be said to be an actuality only if it is known to be the only possibility; while, as long as our explanation is a logically contingent postulate, we must be content to consider it as a ratio cognoscendi. The hypostatization of rationes cognoscendi into rationes essendi rests therefore on the typically rationalistic assumption that formal causes or hypotheses are not selected as alternative and hence contingent explanations of the facts, but are discovered, by intellectual intuition, as necessary and hence real explanations of the facts—that they are, in other words, not merely logically, but ontologically, prior to the facts. The history of philosophy and science presents us with many instances of this hypostatization of logical priority or hypothetical necessity into ontological priority or simple necessity. Thus, Spinoza, having found a God, i.e., a logical structure, that explains the world, goes on to assert that the world as we find it is necessary simpliciter, i.e., could not be diﬀerent from what it is. Which implies the assumption that God is a ratio essendi, not merely a ratio cognoscendi, which in turn implies the assumption that He is the only possible explanation. In Spinoza’s language: God is substance, and substance is that whose essence (possibility) involves its existence, which definition is satisfied by any possibility that has no alternatives, no “compossibilities.” Leibniz, the mathematician well acquainted with the intrinsic arbitrariness of sets of postulates, was, in that respect, wiser when he asserted that there are many possible worlds, and thus recognized the logical contingency of explanations. But he supplemented this statement of the logical contingency of the actual world by a statement of its teleological necessity: this is the only one among many possible worlds, but it is the best of all possible worlds: it is contingent in terms of the law of identity, but it is necessary in terms of the “principle of what is best.”

On the Meaning of Necessity (1943)

51

He thus reintroduced into rationalistic philosophy the Aristotelian concept of hypothetical necessity. For, Leibniz’s theological statement that this world is necessary if there is to be a maximum of goodness or harmony has the form which defines the concept of hypothetical necessity: “Such and such is necessary, if such and such end is to be reached, i.e., as a means to such and such end.” To adduce another instance of the hypostazation of logical priority into ontological priority: in the Newtonian scheme, “absolute space” is a hypothesis necessary to make Newton’s first law compatible with his second law. For inertial motion, i.e., uniform motion in a straight-line, has a meaning only in terms of an inertial reference-entity. But, according to the second law, the law of gravitation, all bodies are in accelerated motion relative to each other, hence no reference-body could be found that satisfies the condition of inertia. Therefore, inertial motion must be given meaning in terms of a non-material reference-entity, and as such a non-material reference-entity absolute space is introduced as a hypothesis. This hypothesis then, becomes regarded as a “vera causa,” something which “really exists.” This fallacy of conversion of logical priority into ontological priority, of ascribing ontological status of independent substances to factors that have functional status in the context of inquiry, appears in the empiricistic search for existential archai just as clearly as in the rationalistic search for formal archai. The fallacy I am discussing is due to lack of contextual analysis, to neglect of the intrinsic reference of factors to the total fact—the “situation”—within which they are discriminated, whence flows their hypostatization into atomic antecedents. Factors derive their necessity from the fact within which they are discerned or from which they are abstracted; it is, therefore, self-contradictory to declare them, after having them abstracted, as absolute necessities or ontological priorities, and then to render the total fact from which they were abstracted a problem, a “quaesitum.” The factors a, b, c are (hypothetically) necessary, assuming D is a fact; D, that is, implies a, b, c as its necessary conditions: if D, then a, b, c. But I cannot, then, assuming the absolute necessity of a, b, c go on to ask whether D is a fact or even a possibility: for it is only on the hypothesis that D is a fact that a, b, c were declared as necessities. In other words, we can only ask how D is possible—as Kant asked “how is experience possible,”—we cannot ask whether it is possible. Atomistic empiricism, however, in its search for existential archai, “real archetypes,” “pure” particulars that are “given” to “immediate” perception, commits just that fallacy of mistaking hypothetical necessities for existentially prior data. Thus, Locke starts with the fact of complex ideas and analyzes it into its factors, “simple” ideas. He then ascribes genetic or psychological priority to these logically prior elements of the psychologically prior continuum, thus confusing structural analysis with genetic analysis.

On the Meaning of Necessity (1943)

52

The fallacious conversion of hypothetical necessities into “simple” necessities results, as we saw, from disregard of the selective activity involved in explanation: a “simple” necessity is, by definition, without alternatives; but a selection has, by definition, alternatives. Since Kant’s philosophy purports to oﬀer a rationale for an absolute separation of theory and practice, one might expect Kant to have succumbed to the very same fallacy. Are Kant’s “synthetic a priori principles” hypothetically necessary, or simply necessary? As C. I. Lewis shows (cf. Lewis 1956), in connection with the development of a functional interpretation of the a priori, it is meaningless or self-contradictory to declare a categorial scheme, i.e., in Kant’s own language “den Inbegriﬀ der Bedingungen der M¨oglichkeit der Erfahrung,” as logically necessary in the sense of standing without alternatives. The a priori, in order to be knowable as such, must have alternatives. For, how could we know that the structural limits of experience are due to the mind rather than to objective reality? They could be validly ascribed to the mind only if a modification of the structure of our mind—i.e., of our system of meanings—would entail a modification of the nature of experience. Hence the assumption of absolute or permanent mindstructure makes the Kantian thesis unverifiable and in that sense meaningless. Now Kant certainly is not guilty of the rationalistic habit of “hypostazation” or of what I called “conversion of logical priorities into ontological priorities.” Indeed, as Cassirer shows, hypostatization or “der allgemeine Hang des Denkens, die reinen Erkenntnismittel in ebensoviele Erkenntnisgegenst¨ande zu verwandeln” (also cf. Dewey’s phrase: “conversion of a function in inquiry into an independent structure” (Dewey 1938, 149)) is the very object of Kant’s critique of “metaphysics.” What was a “metaphysical fact” for the dogmatic rationalist, becomes through Kant’s “transcendental method” a mere conceptual condition of experience or inquiry. However, these “conceptual conditions,” i.e., the categories and the a priori synthetic principles to which they, through the mediation of “schemata,” give rise, have, for Kant, no alternatives, and are thus simply, not hypothetically, necessary. “Die Einheit des Bewußtseins [the highest principle, from which all the particular ‘principles of experience’ are derived] verm¨ogen wir nur dadurch zu erkennen, daß wir sie zur M¨oglichkeit der Erfahrung unentbehrlich brauchen.”2 The emphasis on the functional nature of the categorial scheme, on the fact, that is, that it is meaningless and non-existent apart from its use or application, is clear. But just as clear is the emphasis on the “Unentbehrlichkeit,” the simple necessity, of just that scheme, and this emphasis is fatal from the functionalist point of view. The method by which we prove the indispensability of something to something else, is the method of diﬀerence. But how could this method be applied to prove the in-

2 Cassirer

1922, 587.

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dispensability of Kant’s mind-structure to experience? The mind would have to change its own structure, and then see whether experience would still be as it was. Whether Kant falls within the rationalistic philosophies of science, the tradition of “simple” necessity of self-evident axioms and principles, or whether his “criticism” is closer to the functional-pragmatic interpretation of principles as methodological rules whose choice is pragmatically determined, is a historical question for a Kant scholar to decide. Cassirer tends to assimilate Kant’s doctrine of the a priori to the functional-pragmatic interpretation of the a priori, its interpretation as a methodological rule: “Das a priori muß in rein methodischem Sinne verstanden werden; es ist nicht auf den Inhalt eines bestimmten Axiomensystems festgelegt, sondern es bezieht sich auf den Prozess, in dem, in fortschreitender theoretischer Arbeit, das eine System aus dem andern hervorgeht” (Cassirer 1937, 93). Whether this is a reading into, or a reading out of, Kant, at any rate the typically Kantian conjunction of “a priori” with “synthetic” is essentially an attempt to overcome the dualistic separation of the a priori and the empirical, and is thus opposed to the rationalistic identification of “a priori” with “analytic” or “logically necessary.” For the rationalists the “a priori” or “axiomatic” has intra-logical significance, i.e., its meaning is not defined in terms of empirical application; while the “synthetic a priori” is synthetic just insofar as it is essentially a procedural means, to use Dewey’s term, in existential inquiry. Insofar as it has no alternatives, it is, indeed, axiomatic, “simply” necessary; insofar, however, as it is nothing but a conceptual tool of existential inquiry, a “universal” in Dewey’s sense, it is hypothetically necessary. Kant is still a dualist insofar as those “a priori synthetic” principles are fixed once and for all, and are imposed upon experience ab extra, not being themselves derivable from experience. He is still a rationalist insofar as he ignores the temporal character of inquiry. What is a priori at one time, may have been a posteriori at an earlier time; rules or criteria are themselves derived from, generated by, existence. Something is a priori, in other words, not simpliciter, but secundum quid, i.e., for one phase of the continuum of inquiry; it may be a posteriori for another phase of the same. As Dewey puts it (cf. Dewey 1938, 14): Norms of inquiry are “operationally a priori with respect to further inquiry.” Putting this in Dewey’s language of the “universal” and the “generic”: the universal is a rule operative in the establishment of generic propositions, i.e., empirical laws. But it could not be thus operative if it did not itself represent an empirical law that has been transformed into an a priori or prescriptive law. This is what Dewey means when he emphasizes that inquiry is an immanent or self-contained or—non-viciously—circular process, that it generates its own standards or norms, that the latter are not imposed upon it ab extra: Logical forms “originate out of experiential material, and when constituted introduce

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new ways of operating with prior materials, which ways modify the material out of which they develop” (Dewey 1938, 103). If a conjunction of traits a-b is found to be repeated without exception, we generalize it into a “universal,” a definitional connection: if A, then B. If, then, experience should one day disclose a contradictory instance, viz., “a and not-b,” we will have the choice between refusing to identify a as an instance of A—submitting to the rule set up by ourselves which prescribes the incompatibility of not-b with a—and considering our law (“if A, then B”) as refuted, i.e., considering it as no good any more, considering, that is, that it is not logically necessary but only hypothetically necessary, good for conducting inquiry as long as there is no rule better than it. Suppose, e.g., that we found a man talking, although his heartbeat had stopped. Problem: is he dead or alive? If, as is probable, our conception of death is defined as incompatible with power of speech, while, on the other hand, it is defined as a necessary and suﬃcient condition of absence of heartbeat, this instance would prove our conception of death to be inadequate as a rule for identifying phenomena as cases of death, since it would, on the basis of that rule, have to be identified as a case of both life and death. Thus rules or hypothetical necessities are open to modification by experience. Of course, since, given an incompatibility between a theory and a fact, we are always free to reject the fact as a mere appearance rather than changing our theory, it lies within our power to make our theory final and absolute. Given the law “A is B” and the supposedly contradictory fact “a x is not b,” we are free to say that, if a x is not b, it simply is not a representative case of A, A being defined by B. We are free, in other words, to make empirical truths logically necessary and thus to deprive them of their intrinsic contingency. But thus to conventionalize theory is to cut it oﬀ from its actual interaction with factual knowledge, which interaction is what the progressiveness of science consists in: “La physique progresse parce que, sans cesse, l’exp´erience fait e´ clater de nouveaux d´esaccords entre les lois et les faits” (Duhem 1914, 269). Generic judgments presuppose some a priori knowledge, a posteriori knowledge presupposes some universal judgments, since no object can be identified as being of a kind, unless the kind itself is first defined. But, if the a priori is, as a variable, thus logically necessary, as particular value of that variable it is only hypothetically necessary: it is best for inquiry as long as no rival turns up that proves to do the job better. To quote from C. I. Lewis: “If the criteria of the real are a priori, that is not to say that no conceivable character of experience would lead to alteration of them” (Lewis 1956, 263). To recognize the modifiability by experience of a priori principles is to recognize their empirical origin. In order validly to subsume a percept under a concept, I must be in possession of a hypothetical which defines the concept; but if that hypothetical can serve as a tool of identification and in this sense have existential import, it is because its generating antecedent is itself a categorical all-proposition sta-

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ting an empirical conjunction (and hence being logically an I-proposition).3 In order to buy existential import, or hypothetical necessity, “universals” have to pay the price of contingency; they have to abandon the privilege of logical necessity, the privilege of being eternally analytic and/or irrefutable. If judgments are to be both existential in import and analytic, they must be synthetic in origin. We can make synthetic judgments analytic, convert empirical laws into prescriptive definitions; indeed, as Bradley said, “what is added to-day is implied to-morrow. A synthetic judgment, as soon as it is made, is at once analytic.” But we have, then, to recognize that experience is free to unmake our makings again. Hypothetical necessity is, as it were, the mediating link between empirical contingency and logical necessity. Both the empiricist and the rationalist fail to account for the interaction of empirical and formal knowledge. They both subscribe to the Humean disjunction: either existential or necessary, but not both. For the empiricist (cf. Mill 1851) universal judgments are existential, hence, being problematic generalizations, they are not necessary; for the rationalist, on the other hand, universal judgments are logically or “simply” necessary, such as to have no existential reference. Both of these reductionisms leave out of account the functional enterprise of using universal judgments as conceptual tools for the acquisition of factual knowledge. They ignore, that is, the peculiar logical status of principles that are to be existentially applied, of methodological rules, of “synthetic a priori principles,” viz. hypothetical necessity. The positivistic—allegedly exhaustive—disjunction of judgments into empirical (contingent) and analytic (necessary) judgments takes no account of “synthetic a priori principles,” which are, in Cassirer’s words, “Regeln, gem¨aß denen nach Gesetzen zu suchen und nach denen diese zu finden sind.” The empiricist walks on the plane of particulars and contemns the “high priori roads”; the rationalist walks on the “high priori roads” and contemns the plane of particulars. The functionalist, however, recognizes the functional correlation of plane and high-road: “Die H¨ohenwege sind f¨ur unsere Orientierung in dem Gel¨ande, das wir zu durchschreiten haben, unerl¨aßlich” (Cassirer 1937, 67).

3 If

such a hypothetical represents the definition or analysis of a mathematical concept, then, indeed, it is a priori in the sense of logical necessity. But then its applicability to experience, its hypothetical necessity, is contingent upon Nature’s exemplifying, with a tolerable degree of approximation, its ”interrelated characters.”

Chapter 2 THE DIFFERENT KINDS OF A PRIORI (1944)

I am going to distinguish three kinds of a priori: the formal or analytic a priori, the functional a priori, and the material a priori. With these three kinds of a priori there are associated three types of necessity: formal or logical necessity, as characterizing logical truths, whether the latter be called “tautologies,” as by logical positivists, or “truths of reason,” as by Leibniz; functional necessity (Aristotle’s “hypothetical necessity”), predicable of conceptual means in relation to objectives or ends of inquiry; and the kind of necessity that one might call psychological, if it were not the case that the chief proponents of this kind, the kind of necessity that is traditionally defined by self-evidence or the inconceivability of the opposite, are explicitly opposed to “psychologism” in logic (I am referring to the school of phenomenology, or “Gegenstandstheorie,” as founded by Husserl and Meinong). I Kant defined an analytic a priori judgment as a judgment whose predicate forms part of the meaning of the subject. I am quite aware that this is not what Kant literally said. Kant said, in order for a judgment to be analytic, the concept of the predicate must be contained in the concept of the subject. This formulation is to be avoided, however, because it lends itself to a psychological interpretation, and as a matter of fact in some passages Kant uses explicitly psychological language, as when he says in the concept of body we already think the concept of extension, and therefore the judgment “all bodies are extended” is independent of experience.1 Seizing upon this ambiguity of Kant’s terminology, philosophers have since argued that the distinction between analytic and synthetic, as formulated by Kant, is purely psychological, such that the analytic or synthetic character of a judgment varies with the context of its utterance and cannot be determined apart from that context. Indeed, if the an1 It

should be mentioned, though, that in the paragraph entitled “Of the highest principle of analytic judgments,” in the Critique of Pure Reason, Kant improves on his initial definition of “analytic,” by giving a purely logical definition in terms of the principle of non-contradiction. (Translations from the German are by the author, unless otherwise noted.)

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alytic character of a judgment is defined by the fact that we cannot think of the subject without thinking of the predicate, then the formal a priori would hardly be distinguishable from the material a priori, defined by the inconceivability of the opposite. The formal a priori, then, is not to be defined by the psychological predicate “inconceivability of the opposite,” but by the logical predicate “selfcontradictoriness of the contradictory.” “A is B” is analytic, if B forms part of the definition (and meaning in this logical sense only!) of A, such that “A is not B” reduces to the contradictory judgment “XB is not B,” where X stands for the rest of the defining predicates of A. Such a definition of “analytic” is not exposed to the objection of psychological relativity. For one defines a term— thus converting a floating representation or image into a fixed concept—just in order to render its meaning invariant with respect to the psychological idiosyncrasies of the people who use it. Also, one cannot say that one and the same proposition may be taken as analytic or as synthetic, according to the stage of inquiry and the acquired knowledge of the judging person. For if the predicate B forms part of the meaning of the subject A, the concept denoted by A is diﬀerent from the concept denoted by A when B is not definitory of A; hence the respective propositions are diﬀerent, although the verbal sentences are identical. (A proposition may be said to be a logical entity, which has neither physical existence, like a sentence, nor psychical existence, like an act of judgment; it is the “Sachverhalt” expressed by a sentence.) A “realist” would presumably say that to escape from the psychological relativity of logical necessity by substituting for the thought of the concept the conventional definition of the concept, is to fall from Scylla into Charybdis. If the logical necessity of a proposition depends on the way we define the subjectterm, then logical necessity has an extralogical origin. By merely changing the definitions of our terms, we can destroy and create logical truth; and this kind of relativism is just as detrimental to the dignity of logical truth as the psychological relativism it is intended to amend for. Must we, then, say that “A is B” is logically necessary, only if B forms part of the real definition of A? This, indeed, seems to correspond to Kant’s meaning. Surely, Kant could not have insisted that extension is a defining attribute of bodies while weight is not, if he had recognized the conventional character of definitions. Psychologically, there is no ground for regarding the feeling of pressure or resistance as secondary, and the sense of extension as primary, in the formation of the concept of matter. For the Cartesian school, extension was the essential attribute of matter, in the sense that dynamics was considered as reducible to kinematics and kinematics to analytic geometry. Within Newtonian physics, on the other hand, force is a fundamental concept; masses are idealized to such an extent that they are treated as points, and what constitutes them as physical rather than geometrical points is not that they have extension, but that they are subject to gravitational forces. In this context, therefore, weight would seem

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to be more essential to masses than extension.2 Kant, then, must have distinguished the concept of matter from alternatively definable concepts of matter. In Aristotelian fashion, he must have made an ontological distinction between essential and accidental predicates. As Locke distinguished between “real” and “nominal” essence, so Kant distinguishes, in his lectures on formal logic, between “Realwesen” and “logisches Wesen.” Only the latter is relative to the selective definitions of the inquirer, while the former is absolute, something to be discovered. It is, indeed, true that for Kant not only empirical substances, “material archetypes,” are “Realwesen,” but also the genetically or synthetically defined concepts of mathematics. But such “synthetic” definitions are, for Kant, not altogether arbitrary; they are subject to the laws of “pure intuition.” That the straight line is the shortest distance between two points is, within the domain of pure intuition, just as good a discovery as the solubility of gold in aqua regia is a discovery in the domain of empirical intuition. Every definition is conventional insofar as it involves selectivity. It is no longer possible to ignore the pragmatic element in inquiry, and so far one cannot believe in Aristotle’s “real” definitions. At any rate, one cannot base the formal a priori on necessities of the intuitive kind, like Aristotle’s intuitive discrimination between “essence” and “property,” however considerable a part pure intuition may play in the domain of the material a priori, the a priori of the phenomenologists. As Dewey has shown, “‘substance’ is a logical, not an ontological determination.” That is, “essential” predicates cannot be defined as standing for “inherent” properties of independent, “given” substances, without reference to the objectives of inquiry; they are to be regarded as predicates selected as definitory of a concept. For logic, in other words, substance is always a “logisches Wesen,” not a “Realwesen”; “essential” predicates are definitory predicates, and if we select diﬀerent predicates as definitory, the “substance” will be diﬀerent. The analytic nature of a proposition, therefore, must be recognized as independent of whether the definition of the subject-term be “real” or “nominal.” It is quite true that a definition of a term already in use presupposes (in the genetic order) in most cases—or perhaps always—a synthetic judgment. If the term to be defined had, e.g., already a denotation or extension, the definition will formulate the discovery of a common property of the denotata. As we shall see later, it is highly important to recognize this genetic dependence of analytic truth upon synthetic truth (cf. Kant’s statement: “Where the understanding analyzes, there it must first have synthesized, because it is only as synthesized by it that anything can be given to the faculty of representation” (Kant 1912, 113)). But hence to infer that there is no such thing as a purely 2 Historically

speaking, though, Newton and Kant are in the same boat, in that they both distinguish, in scholastic manner, essential predicates from empirically universal predicates; for Newton, as for Kant, weight is, though empirically universal, not an essential predicate of matter (cf. Cassirer 1922, II, 679-80).

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formal or analytic a priori, would be very confused indeed. Whether a proposition of the form “A is B” is analytic depends on the definition of A, no matter what be the reason for adopting just that definition. And insofar as definitions are conventional, analytic truth or logical necessity is, in part,3 conventional. No state of aﬀairs is logically necessary or logically impossible, a priori the case or a priori not the case, per se, but only relatively to definitions. Insofar as definitions, provided the terms that occur in them have empirical reference, may themselves formulate synthetic, empirical truths, it does not seem to me to be fortunate to say, with the logical positivists, that a formally a priori sentence “says nothing about the world.” In the procedure of science empirical laws are used for the definition of its concepts. For example, mass is defined in terms of Newton’s third law (cf. Mach 1904, 231), or heat-capacity in terms of the principle of the conservation of the quantity of heat (cf. Mach 1919, 1868), or internal energy in terms of the first law of thermodynamics. Of course, once an empirical law has been adopted as an implicit definition of a concept in terms of which it is stated, it ceases to be a contingent truth and becomes, qua definition, irrefutable by experience. All that can happen is that future experience will call for revision or abandonment of the definition. But the reason for such a change of convention is itself a non-conventional state of aﬀairs: it is the fact that the empirical law corresponding to the definition fails to be verified. Provided this concomitance between change of convention and change of empirical truth, between changes in the “meta-language” and changes in the “object-language,” is recognized, there may be no harm in maintaining that what is formally a priori, in the sense of being definitional, says nothing about the empirical world; and we have ourselves insisted that, in order to determine whether a given proposition is formally a priori one need not inquire into the reasons that led to the adoption of just that definition of the subject-term. However, neglect of such genetic considerations easily leads to a futile separation between logical and causal or empirical possibility. If science defines its concepts in terms of causal laws, then causal possibility or impossibility is itself the raison d’ˆetre of logical possibility or impossibility. If mass, e.g., is defined in terms of the dynamic relations expressed by Newton’s third law, or by the law of gravitation, then it will be causally impossible and therefore logically impossible that only one mass should exist. Only if logical possibility should be defined in terms of the psychological concept of conceivability could such a state of aﬀairs be maintained as logically possible. One certainly can perform a “Gedankenexperiment” to the eﬀect of picturing just one mass and nothing

3 The

import of the qualification “in part” will become manifest in the discussion of the material a priori. It is meant to indicate that logical necessity depends on definitions and principles of logic, the latter being themselves not formally a priori, since the formal a priori is defined in terms of them.

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else in the world. But the pictorial or phenomenal meaning of terms must be distinguished from their conceptual, relationally defined meaning. We have so far arrived at a definition of the formal a priori which renders it independent of both psychological context and metaphysical presuppositions like Aristotle’s concept of “real” definitions, based on the distinction between “essence” and “property.” It is furthermore desirable to render it independent of the subject-predicate schema of Aristotelian logic. The standard form of analytic truths is formal implication; as Leibniz pointed out, the “truths of reason” are always conditional in character. In many cases it is, indeed, possible to translate a formal implication ((∀x)[φ(x) ⊃ ψ(x)]) into a categorical A-proposition of subject-predicate form (every φ is a ψ). But such a translation easily leads one to overlook the non-existential character of analytic truths: φ and ψ may define null-classes, while the relation of subsumption, in Aristotelian logic, holds between ontological classes. That the Aristotelian logic of subsumption was not formal at all is evidenced by the fact that subalternation is, in Aristotelian logic, a valid mode of immediate inference. If A-propositions are formulated in categorical form, then, indeed, subalternation seems to be valid; if the classes defined by φ and ψ are interpreted extensionally, then a mere inspection of the meaning of the A-proposition “all φs are ψs” induces us to draw the inference: “some φs are ψs.” But, from (∀x)[φ(x) ⊃ ψ(x)] it by no means follows that (∃x)[φ(x).ψ(x)]; “constant conjunction” cannot be inferred from “necessary connection,” and the above formal implication may, though it need not, express a “necessary connection.” Again, the existential or ontological assumptions implicit in Aristotelian logic are transparent in the so-called square of opposition. As Meinong (cf. Meinong 1907, 43) points out, the negation of the particular negative may be formally equivalent to the universal aﬃrmative; but insofar as the universal aﬃrmative expresses non-existential knowledge (“daseinsfreies Wissen”), while a particular statement is about existents, it cannot be said to be identical in meaning (in a non-formal sense of “meaning”) with a particular statement. According to Meinong’s “Gegenstandstheorie” we must distinguish between the “Objektiv” and the “Objekt” of a judgment. The “Objektiv” of a universal necessary judgment (I disregard, for the moment, the fact that the kind of necessity which Meinong talks about is the necessity of the material a priori, which will be discussed later, not the necessity of the formal a priori) is non-existential; the universal necessary judgment is not concerned with empirical objects at all. The meaning of the judgment “all equilateral triangles are equiangular” is not: “there do not exist any equilateral triangles which are not equiangular”; for we come to deny an existential judgment, whether aﬃrmative or negative, by examining instances and finding that a certain conjunction of traits does not hold in any one instance. But the above universal proposition, using Dewey’s terms, is not a generic proposition: it is not about a conjunction of “characteristics,”

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but about a connection of “characters.” For the purposes of formal logic, indeed, a universal aﬃrmative may be said to be derivable from the negation of the corresponding particular negative by a mere syntactical transformation: ∼ (∃x)[φ(x). ∼ ψ(x)] ⊃ (∀x)[φ(x) ⊃ ψ(x)]. But this transformation is possible only because the symbol ‘⊃’ in the implied statement stands for extensionally defined material implication, the logistic analogue of Hume’s “constant conjunction” (in respect of the absence of logical necessity). If the implied statement, however, is analytic, expressive of “daseinsfreiem Wissen,” the above implication represents an idle syntactical rule—that is of no use in the drawing of inferences: no mathematician would attempt to prove, e.g., that all differentiable functions are continuous by examining instances of diﬀerentiable functions and seeing whether there are any discontinuous ones among them; and if the domain of the quantified variable is infinite, such a procedure would anyway lead only to verification, not to proof. Every universal aﬃrmative is anti-existential, in that it analytically entails negation of an existential statement: (∀x)[φ(x) ⊃ ψ(x)] ⊃ ∼ (∃x)[φ(x) . ∼ ψ(x)]. But to maintain the validity of the converse implication is to confuse material implication and analytic entailment, extensional and intensional universality. It is, e.g., certainly true that no unicorns dislike cake. For, let φ stand for the predicate “being a unicorn” and ψ for the predicate “liking cake.” Then “no unicorns dislike cake” is to be formalized as follows: ∼ (∃x)[φ(x). ∼ ψ(x)]. But in order to prove a conjunction false, all we have to do is to prove the falsity of at least one conjunct. Now, (∃x)[φ(x)] is false, since there are no unicorns. Thus we have proved that no unicorns dislike cake. If we accept the equivalence of the negation of the particular negative to the universal aﬃrmative, we have therefore demonstrated that all unicorns like cake. Surely, it needs the routine of a formal logician not to be puzzled by such startling discoveries! If we define, then, the formal a priori as characteristic of implications rather than of categorical statements, we must explicitly rule out material, extensionally defined implications. It is a necessary condition for the implication (∀x)[φ(x) ⊃ ψ(x)] to be valid, that ∼ (∃x)[φ(x). ∼ ψ(x)]. This condition is at the same time suﬃcient to define material implication. But in order to define analytic implication, we must introduce the modality “self-contradictory.” (∀x)[φ(x) ⊃ ψ(x)] is valid as an analytic implication, and hence formally a priori, if (∃x)[φ(x). ∼ ψ(x)] is not only false, but self-contradictory. The process of converting empirical laws into conventions or implicit definitions (which will in the sequel be illustrated by examples) is just this step from material implication to analytic implication. If analytic implications were based on “real” definitions beyond possibility of revision, then (∃x)[φ(x). ∼ ψ(x)], in the face of (∀x)[φ(x) ⊃ ψ(x)], would not even be a possibility; from the implication and the fact ∼ ψ(a), we could, without further empirical investigation, infer the fact ∼ φ(a). But the above conjunction may occur and induce us to abandon

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the implication as formally a priori or definitional, instead of taking it as a rule prescriptive of what is a fact and what is mere appearance, because the implicatory sentence did formerly stand for a material implication, expressing an empirical law, which we now find to be contradicted by experience and hence unworthy of being any longer used as a definitional rule. One often finds analytic statements defined as statements whose truth follows from the very meaning of the terms; they say nothing about the empirical world in the sense that the recognition of their truth does not presuppose any sort of empirical inquiry. This view is expressed, e.g., by Schlick, in an essay entitled “Gibt es ein materiales Apriori?” (Schlick 1932): “An analytic sentence is a sentence which is true in virtue of its mere form; he who has understood the meaning of a tautology, has at the same time recognized its truth; therefore it is a priori. As regards a synthetic sentence, however, one must first understand its meaning, and thereafter find out whether it is true or false; therefore it is a posteriori” (Schlick 1938, 22). However, it is not quite accurate to say that all we need in order to recognize an analytic statement as true, is to understand the meaning of the terms involved; for, if its truth follows from the meaning of the terms, there are, on the other hand, required logical principles, by which it follows. The foremost of these principles is, of course, the law of non-contradiction itself. “A is B” is true, in virtue of the meaning of A; for, by definition, “A is XB” (where X stands for the defining predicates other than B); then, by the principle of the substitutability of equivalents, “A is B” transforms into “XB is B”; then, by the principle of simplification, this statement is equivalent (in the extensional sense of equivalence, defined by reciprocal implication, or identity of truth-values) to “B is B”; finally, the principle of non-contradiction says “X is X” (where X is a variable, standing for any term); by a rule of substitution, “B is B” is then derivable from the law of non-contradiction, and thus the original statement “A is B” is seen to be true by the law of identity. This pedantic analysis is intended to reveal that the formal a priori can be defined only with reference to principles of logic, and the latter certainly cannot be said to be a priori in the same sense in which statements whose analytic character is determined by these very principles are a priori. The truth of these principles of logic cannot follow from the meaning of their terms, simply because their terms have no meaning at all: they are variables. II Thus the very analysis of what is meant by the “formal a priori” reveals the existence of another kind of a priori without which the formal a priori could not even be defined. I shall call this the material a priori, avoiding the more familiar term “synthetic,” because the latter has been ambiguously applied, by Kant, to both the material and the functional a priori. The principles of logic themselves, which we just saw to be essentially involved in the definition of the formal a priori, are materially a priori. Their truth is a matter neither of de-

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duction nor of induction. We are then left with two alternatives: either they are self-evident, “seen” to be true in pure “Wesensanschauung,” as the phenomenologists would say, or they are conventions. Even though it would certainly be more emancipated to accept the latter alternative, and to dismiss the former alternative as mystical Platonism, I venture to suggest that we are not, here, really confronted with mutually exclusive alternatives. Just as empirical laws of nature are used as conventional definitions of empirical concepts, because they are true in a non-pragmatic sense, so the principles of logic can be used as implicit definitions of logical concepts, like negation, implication etc., because they possess some kind of evidence that is independent of the use that can be made of them. To argue that the law of excluded middle, or the equivalent law of non-contradiction, is analytic because it implicitly defines the meaning of negation, such that its denial would necessarily presuppose it, is no better than to argue that Newton’s third law of the equality of action and reaction is analytic, because it implicitly defines the meaning of mass, or that the first law of thermodynamics is analytic because it implicitly defines internal energy. The principles of logic can, indeed, by Wittgenstein’s truth-table methods, be shown to be “tautologies”; but this method presupposes definitions of the “logical constants” which amount to a recognition of those very principles of logic. Thus, presupposing the following definition of the negation sign ‘∼’: p ∼p T F

F T

one can prove the law of non-contradiction: ∼ (p. ∼ p) to be tautologous, i.e., true no matter what be the truth-values of the components. But what else does the above definition of negation express but the couple of inferences: if p is true, then ∼ p is false, and if p is false, then ∼ p is true? And how do these implications diﬀer from the law of non-contradiction? Just as empirical laws have to be recognized as synthetic truths before they can be used as definitions of empirical concepts in terms of which they are stated, and thus made analytically true, so logical laws have to be recognized as synthetic (“synthetic” in the sense of non-definitional) truths before they can be used as definitions of logical concepts in terms of which they are stated. Once, e.g., we have intuitively recognized the validity of the modus ponens (if ‘p ⊃ q’ is true, and ‘p’ is true, then ‘q’ is true), we can conventionally adopt it as an implicit definition of the symbol denoting implication; thereafter, of course, the validity of the modus ponens will follow from the very meaning of implication, and nobody could deny it unless he used the symbol ‘⊃’ in a diﬀerent sense. Or, to take another example, the principle of mathematical induction can be used as an implicit definition of the concept “finite integer,” because of its intuitive evidence. It would hence be absurd to think one refutes

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the claim of Poincar´e and the intuitionists that the principle of mathematical induction is synthetic a priori by pointing out that it “merely” defines what is meant by a finite integer. This type of consideration applies generally to the axiomatic method of modern mathematics, the method of defining the “primitive notions” by a set of axioms or postulates which they satisfy. The postulates give rise to, are the source of, analytic truths; but they themselves must be regarded as synthetic. As Kant said: “One can indeed recognize a synthetic sentence as true by the law of non-contradiction, but only by presupposing some other synthetic sentence from which it can be inferred, yet never in itself” (Kant 1912, 42). Kant thus admitted that the formal implications that constitute the theorems of pure mathematics are analytic; what is synthetic, according to him, are the basic axioms that give rise to those analytic truths. Of course, as I mentioned already, Kant’s usage of the term “synthetic a priori” is not unambiguous: in insisting on the synthetic nature of the axioms of geometry, he means to point out, first, that their denial is logically possible, and second, that it is intuitively impossible. That Kant was right on the first point—the non-analytic nature of the axioms of Euclidean geometry—has been definitely proven by the development of non-Euclidean geometries. If the parallel-axiom were logically necessary or analytic, its denial could not have led to consistent systems of geometry. But does not an axiom like “the straight line is the shortest distance between two points,” claimed as synthetic by Kant, merely amount to a definition of straight line? Again the answer is: it is a definition of this geometrical concept in the same sense in which physical laws are definitions of physical concepts. Euclid did not arrive at it by an analysis of the concept “straight line” any more than Newton arrived at his second law of motion by an analysis of the concept “force.” The point to notice is that concepts do not “exist” at all prior to the judgments which construct and define them. Axioms do not analyze the meaning of symbols; rather they create meanings, or concepts, and once this creation, this “synthesis,” has occurred, a symbol can be attached to those conceptual creatures; then one can say the axioms “merely” define the meaning of these symbols. In talking that way, one forgets that analysis presupposes synthesis. “La place de la synth`ese a priori n’est pas dans la liaison des termes du jugement, ou dans la d´emonstration de telle ou telle formule num´erique particuli`ere; elle est dans le processus g´en´eral dont d´erive tout nombre particulier, dans la cr´eation des notions elles-mˆemes” (Brunschvicg 1922, 207). If I prefer the term “material a priori” to the Kantian term “synthetic a priori,” for the description of the kind of independence from experience that accrues to axioms, whether logical or mathematical, it is in order to avoid the connotation of intuitive necessity, which at once convicts one of “psychologism.” A materially a priori judgment is such that its contradictory is consistent; in this respect it diﬀers toto caelo from a formally a priori judgment.

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But when Kant described the axioms of geometry as synthetic a priori,4 he furthermore implied the intuitive inconceivability of the opposite; and it is this implication which I would not want to take upon myself, not because I regard it as conceivable that it should be conceivable to anyone that parallels intersect, or, that, in a two-dimensional plane, a line can have more than one parallel, or that two straight lines can enclose a finite space, given the Euclidean intuitive meanings of the terms “straight line” and “parallel,” but simply because “je n’ai pas besoin de cette hypoth`ese” in order to defend the material a priori as a kind of a priori distinct from, and presupposed by, the formal a priori. The only advantage I can see in calling materially a priori judgments conventional is that this term involves recognition of the absence of logical necessity, the logical conceivability (though possibly psychological inconceivability) of alternatives. Indeed, if logical necessity is defined in terms of the principles of logic, it would be absurd to ascribe logical necessity to the principles of logic. Still, being, in their capacity as normative principles, logically independent of experience, they are a priori principles. Shall we, then, call them conventions? Is, in other words, the material a priori perhaps reducible to the functional a priori? No, not reducible to, but compatible with it. Why should utility and a priori truth be incompatible? The principles of logic, it would seem, are useful conventions, not in spite of, but because of their a priori truth. It seems to me to be making a rather cheap use of Occam’s razor, if one dismisses the claim of the apostles of “Wesensanschauung” that there are materially a priori truths, as “psychologism,”5 or, even worse, mysticism. It must, of course, be admitted that some of the allegedly “eidetic truths” resolve, upon analysis, into analytic statements, and are confusedly called materially a priori, just because descriptive terms occur in them. However, the really interesting cases of synthetic a priori truths are aﬀorded by Judgments of Interpretation. 4 It seems to me that the logistic demonstration of the analytic nature of the propositions of pure mathematics by no means conflicts with Kant’s doctrine of the synthetic a priori in mathematics. Metamathematical research has, indeed, revealed that what is asserted, in mathematics, are formal implications between axioms and consequents, and that these implications are valid by the mere rules of logic, being entirely independent of intuition. But, as mentioned already, Kant admitted the analytic nature of these implications. What is synthetic, for him, are the axioms, and hence their consequents, which is quite compatible with the analytic character of the consequences. One might point out that, as Hilbert has shown, the axioms of mathematics are purely formal, and hence not synthetic in the sense of depending on intuition. But the axioms whose synthetic character was defended by Kant are not Hilbert’s formal, uninterpreted ones, but their Euclidean interpretations, and nobody would deny that the interpretation of axioms, the establishment of “Zuordnungsdefinitionen,” requires intuition. If it be said that the axioms qua interpreted do not form part of mathematics at all, why this is a declaration of the way in which the term “mathematics” is used nowadays, which is not the way Kant used the term. Kant, one might say, was not concerned with “pure” mathematics at all, but with “applied” mathematics. His mistake lies only in this, that he regarded Euclidean geometry as the only interpretation of formal geometry that is of any use to physics. And presumably he would not have committed this error if he had lived in the age of Einsteinian physics. 5 Notice that Husserl’s phenomenology started out as an explicit revolt against psychologism! Cf. Husserl 1913, I.

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Consider the judgment: “time is a series,” where a series is defined by the properties of asymmetry and transitivity. Once I have intuited that temporal instants are substitutable as proper values for the variable relata of an asymmetrical and transitive relation, I can, of course, use the formal properties of asymmetry and transitivity to define time; and thereafter, indeed, judgments like “time is irreversible,” or “if Tuesday comes before Wednesday, and Wednesday comes before Thursday, then Tuesday comes before Thursday,” will turn out to be analytic. But the synthetic judgment, presupposed by these analytic judgments, lies in the substitution of temporal instants for the variable relata. This act of interpretation cannot be reduced to the abstraction of a structural property from material instances, and thus to an inductive process. For time is one, it does not have instances in the sense in which a class-concept has instances. Logically speaking, I cannot abstract the structural properties of asymmetry and transitivity from an examination of times, i.e., parts of time. For, in the first place, the very meaning of those properties requires at least 2, or 3, “times,” i.e., elements of time. And, second, if within these minimally extended parts of time those properties do hold, it still does not follow that they would hold within the parts of time that are compounded out of those minimal parts. And to reason from the fact that parts of time have those structural properties to the fact that time as one has those properties, is not at all analogous to inductive extrapolation, the inference from some instances to all instances. For, as Kant pointed out, the relation between time and its parts (like the relation between space and its parts) is essentially diﬀerent from the relation between a class and its members. I can think of a member of a class without thinking of the class, while I cannot think of a part of time otherwise than as a “limitation” of continuous, infinite time. “Idea temporis est singularis, non generalis. Tempus enim quadlibet non cogitatur, nisi tamquam pars unius eiusdem temporis immensi ... Omnia concipis actualia in tempore posita, non sub ipsius notione generali, tamquam nota communi, contenta” (De mundi sensibilis atque intelligibilis forma et principiis, (Kant 2002), paragraph 14). Time, in other words, is a continuum, not a discrete collection, like an extensional class. How does one verify the judgment of interpretation: “time is a continuum?” (“continuum” in the pre-Cantorian sense, i.e., defined solely by “density” or “compactness”). It certainly is not analytic. Then is it an inductive generalization? In that case one would have to admit the possibility that, if the process of dividing time were only continued far enough, we might encounter discrete parts of time, i.e., parts of time not separated from one another by other parts. Is one convicted of “psychologism” if one regards such a state of aﬀairs as inconceivable and hence the judgment “time is a continuum” as “synthetic a priori”? Surely, if one were to insist that “time is a continuum” is no more than an inductive generalization, one would have to admit that it is not the same kind of inductive generalization as those that apply to classes of natural objects. A judgment

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represents an inductive generalization if it presupposes as a necessary assumption the uniformity of Nature. The very concept of “uniformity,” however, is defined in terms of continuous time: the statement of a given functional relation (e.g., the law of gravitation) represents an inductive generalization if it is assumed to hold at all places and at all times; time, however, enters into the equations of physics (whether explicitly or implicitly, through the “time derivatives”) as a continuous variable. Hence, to say “time is a continuum” is an inductive generalization involves an obvious circularity. One can, of course, regard this judgment as “synthetic a priori” in the functional sense of Kant’s attribute: science assumes time to be continuous, just as it assumes space to be continuous, in order to be able to correlate it with the number-continuum, and thus to render a mathematical treatment of motion possible. This is true enough; but it is by no means conflicting to call a judgment both functionally and materially a priori. On the contrary, if the judgment under discussion is functionally a priori, i.e., adopted as a necessary presupposition of science, it is because it is materially a priori, i.e., intuited to be the case. But is not this appeal to “self-evidence,” “inconceivability” of the opposite, outmoded dogmatism? At one time, e.g., energy was assumed to be a continuous function. Nowadays, quantum-physics has produced experimental evidence that it is a discontinuous function. Could not the same happen with respect to time? No, it could not, because, as Kant has shown, time is not an object of experience at all; it is a “constitutive condition” of empirical objects. One can experiment with energy, and other empirical functions; one cannot experiment with time, one experiments in time. Before proceeding to a discussion of the functional a priori, let us emphasize that the material a priori has, by us, been defined negatively rather than positively: a materially a priori judgment is neither analytic nor inductive in the ordinary sense. Phenomenologists define the material a priori positively by self-evidence, disclosed in pure “Wesensanschauung” (eidetic insight): the evidence of the judgment is independent of multiplication of instances; as Goethe characterized Galileo’s discovery of the isochronism of the pendulum: “Ein Fall gilt ihm f¨ur Tausend.” But if one argues for the material a priori in terms of self-evidence, one throws oneself open to the accusation of confusing logic with genetic psychology. No doubt, Galileo did not discover the law of the proportionality of fallen height to t2 by the method of “simple enumeration.” But the question of how a law is discovered must not be confused with the question of the logical validation of that law. Being a statement about empirical objects, Galileo’s law is valid only insofar as it applies to a vast number of instances, unless it be deduced from other empirical laws; and insofar as the premisses from which it might be deduced must themselves be empirical, the reference to many instances will always be logically essential, although it may be genetically, in the order of discovery, inessential.

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The material a priori does not, indeed, belong to the province of formal logic. For formal logic the disjunction: either analytic or inductive, is exhaustive, because the formal logician deals only with “ready-made,” i.e., already defined, concepts, leaving the process of constituting concepts, the process of constructive definition, to the concern of “genetic psychology.” A judgment is analytic if it is true (directly or indirectly) by definition, and the raison d’ˆetre of the definition from which analytic truths derives is of no concern to the formal logician. Being aware of this limited scope of formal logic, Kant felt the need of supplementing it by a more comprehensive “transcendental logic” which investigates the process of noninductive synthesis which precedes and renders possible analysis. Accordingly, “transcendental logic” is, to use a characteristic term of Hermann Cohen’s, occupied with the “Urteil des Ursprungs.” If one dislikes the notion of “Bewußtsein u¨ berhaupt” because of its metaphysical “pathos of obscurity,” and holds that, in any proper sense of the term “Bewußtsein,” Bewußtsein is the subject-matter of psychology, not of logic, then, insofar as the subject-matter of transcendental logic is claimed to be Bewußtsein u¨ berhaupt, one may hold that transcendental logic reduces merely to genetic psychology. However, it is a historical fact that what is commonly called genetic psychology has not concerned itself with the problem of the origin of scientific concepts and their articulations in the form of definitions. Insofar it would be at least as improper usage to call this kind of investigation “genetic psychology” as to call it “logic.” Let me concretely illustrate the diﬀerence between formal and transcendental logic, and the place of the material a priori within the latter. I may refer back to the already discussed judgments “time is irreversible” and “time is a continuum.” The formal logician straightforwardly asks you for the definitions of subject and predicate, and once these definitions have been supplied, he can easily determine whether the judgments under analysis reduce to mere exemplifications of laws of logic and are thus logically true. The origin of the definitions themselves is irrelevant for the purpose of formal logic. Definitions, “primitive notions” and axioms (or rather, with respect to the postulational technique developed by Hilbert, axioms definitory of primitive notions) are the basic elements for formal logic. But there is no reason why these basic elements that are taken for granted, presupposed by formal logic, may not constitute a problem for “transcendental” logic. Let us analyze the following judgment: “If A is higher in pitch than B, and B is higher in pitch than C, then A is higher in pitch than C.” If the relation “being higher in pitch” is defined as a transitive relation, then, of course, this judgment is analytic. Or suppose what is defined as transitive is the more general relation “being higher than.” Then the above judgment may be analyzed into the analytic judgment “If A is higher than B, and B is higher than C, then A is higher than C,” and the synthetic judgment that this analytic judgment is applicable to tones as far as

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their pitches are concerned. Is this synthetic judgment, viz. that tones order themselves according to height of pitch, inductive in any ordinary sense? Is it conceivable that there should be found tones that do not order themselves according to height of pitch, or colors that do not order themselves according to lightness, the way it is conceivable that there should be found white crows, or masses that do not obey the law of gravitation? If it is not, thus a formal logician would presumably argue, it is because tones are defined in terms of that transitive relation, just as the integers are (ordinally) defined in terms of the transitive relation “being greater than.” This kind of reasoning befits a formal logician, because, as mentioned in the discussion of the formal a priori, for formal logic the essential can mean only the definitory. But unless one dogmatically dismisses all phenomenological analysis as “psychologism,” one must admit that it is not na¨ıve scholasticism to say “it is of the nature of tones to have that structural property, and therefore they are defined as having it”; the ground of that definition certainly is not inductive in the same sense in which the ground of the definition of whales as mammals is inductive. Also the definition of “being higher than” in terms of the structural property of transitivity may, from the standpoint of “transcendental logic,” be said to presuppose, genetically, the materially a priori judgment “it is of the nature of the relation ‘being higher than’ to be transitive.” Insight (“Wesensanschauung”) is required whenever a formal definition is applied to a specific case and the nature of the case is such that the validity of the application cannot be inductively verified (unless, indeed, the meaning of the term “induction” be stretched to such an extent that “thought experiment” is regarded as an inductive process). Inasmuch as such insights form an essential part of the cognitive process, it should be conceded to transcendental logic—which “transcends” the scopes of formal logic and inductive logic separately—to register judgments expressing such insights as belonging to a separate category, the category of the material a priori. III whether empirical or “eidetic,” may be made, by Any synthetic a conventional act, into an analytic, formally a priori sentence. But it is usually made formally a priori, in order to be taken as functionally a priori, i.e., a hypothetically necessary presupposition, a “procedural means,” as Dewey would say; or, as Kant called these functionally a priori principles, a “Grundsatz,” in contradistinction to an analytically demonstrable “Lehrsatz.” In the case of empirical systems, these sentences are, prior to their being adopted as functionally a priori, empirically synthetic (in part, at least); in the case of conceptual systems they are “eidetically” synthetic. Thus, the postulates sentence,6

6I

use here the term “sentence” instead of the term “proposition,” in order to obviate the objection that a proposition, interpreted as the “Sachverhalt” expressed by a sentence, ceases to be the same when the sentence that expresses it ceases to be synthetic and becomes analytic.

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which implicitly define the “primitive notions” of Peano’s system of arithmetic (e.g., the principle of mathematical induction: if x is an integer, then (∀P){P(0).(∀y)[P(y) ⊃ P(succ(y))] ⊃ P(x)}, or D. Hilbert’s postulates which implicitly define the primitive notions of pure geometry, are, prior to their use as postulates definitory of the primitive notions of the system, eidetically synthetic (materially a priori). In Carnap’s language of Carnap 1928, such postulates might be characterized as “constitutions” (“constitution” taken as a verbal noun, not as a past participle) of the “basic elements” of the respective system. Once the basic elements, or primitive notions, have thus been constituted, they can be reanalyzed in terms of these constitutive postulates, and the latter will then, of course, appear as analytic. But this analysis presupposes the synthetic process in which the primitive notions were first constituted. Insofar as there is no logical necessity in the choice of a specific set of notions as primitive (or postulationally defined), the basic postulates may be treated as conventions or functional necessities. However, it is especially with respect to the postulates of empirical science, that we must be on our guard against the identification of the conventional with the arbitrary. If the basic postulates of physics have methodological value and can be profitably taken as a priori in the functional sense, it is because they have fundamentum in re. Consider, for example, the principle of the conservation of mechanical energy: its formulation presupposes the definition of potential energy as the negative of the space-integral of force: s1 Fdξi −dV = Fdξi ; hence V = − s2

i

But in this definition the existential assumption is implicit that V exists in the sense of being operationally definable as a function of position only, in contradistinction to kinetic energy which is a function of both position and velocity. Now, that the conservation principle, as a consequence of what I like to call a “definition with existential import,” is not a priori in the sense of being analytic is evidenced by the fact that actually all real systems are nonconservative, since, owing to friction, some energy is always lost during the motion of a system. On the other hand, if the principle were a posteriori, it would have to be rejected as false in the face of this fact of non-conservation. But instead the physicist saves it by introducing a new form of energy, viz. heat energy. The mechanical energy that is dissipated by friction is conserved in the form of heat. This extension of the concept of energy—as expressed by the first law of thermodynamics—is, however, to use Poincar´e’s phrase, though conventional, not arbitrary. For its empirical basis is Mayer’s and Joule’s discovery of the quantitative equivalence of heat and mechanical work. If energy, in this extended form, should still fail to be conservative, the physicist would still search for a further quantitative equivalence between the lost amount of energy and some new form of energy. The conservation principle thus func-

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tions as a leading principle: it expresses the physicists’ faith that something is constant in Nature, and this faith progressively “enacts its own verification,” as James would put it. Again, Newton’s second law functions as an a priori principle; it is, in Dewey’s language, “operationally a priori with respect to further inquiry”; it tells the physicist how to measure force, thus prescribing a method. Methods cannot be directly refuted (in fact, the expression “to refute a method” does not even make syntactical sense); insofar methodological principles are a priori. But they may be indirectly refuted, i.e., they may be proven fruitless by the failure of the empirical laws that gave rise to them to be verified; insofar they have fundamentum in re and are open to revision by experience. The empirical fact that led to the methodological postulate expressed by Newton’s second law is that force manifests itself as change of velocity, while before Galileo force was supposed to be the cause of changes of position. As Mach describes the empirical character of Newton’s second law: “Before Galileo, force was known only as pressure. Now, nobody who has not experienced it can know that pressure does at all produce motion, and still less how pressure passes into motion, that pressure does not determine position nor velocity, but acceleration . . . . It is hence not at all obvious that the factors which determine motion (forces), determine accelerations” (Mach 1904, 142). Theoretically, any postulate, whether it be a formal axiom or an empirical hypothesis, can be adhered to, whatever deduction or induction may disclose. For the so-called experimentum crucis, based on the contrapositive mode of inference, is, if taken to apply to singular postulates, an illusion. One cannot deduce consequences from one singular postulate or hypothesis; there is, for the possibility of deduction, required a set of postulates or hypotheses. Hence, what the falsity (whether empirical or formal) of a deduced consequence entails is not the falsity of one definite postulate, but the inconsistency of a set of postulates. Let us symbolize the set of postulates (no matter whether these be formal postulates or empirical hypotheses) by the conjunction p1 . . . . . pn , and the deduced consequences by q1 . . . . . qn . Then ∼ qi entails ∼ (p1 . . . . . pn ). The negation of a conjunction, however, is equivalent to a disjunction of negations, i.e, ∼ (p1 . . . . . pn ) ≡ ∼ p1 ∨ . . . ∨ ∼ pn . We are thus left with the materially indeterminate conclusion ∼ pi , i.e., we are free to choose among alternative falsities; we are free to decide which one of our postulates we want to abandon and which ones we want to continue to adhere to. If, e.g., the planets should be observed to deviate considerably from the elliptic paths determined by Kepler’s laws, one would have to infer, by contraposition, the falsity of either the law of gravitation or the law of inertia or the law of the parallelogram of forces; provided, indeed, one is assured of a complete knowledge of the initial conditions: it may well be that the deviation from the expected path might be ascribed to the disturbing influence of a force exerted by a hitherto unknown

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planet. Since the law of gravitation is less general than either one of the two latter laws, upon which the very possibility of the geometrical construction of motion depends, the physicist would probably choose to abandon or revise it. It thus appears that whether a hypothesis functions as a priori or not depends on the degree of its generality. It is the most general laws that are under all circumstances adhered to as methodological postulates or leading principles, because the very possibility of science depends on their validity. The very possibility of statistical physics, e.g., or the applicability to physics of the calculus of probability, presupposes the validity of the two synthetic axioms that define a probability-aggregate: the axiom of randomness, which defines the non-causal nature of a series of events (in Schlick’s formulation, this axiom states, that a series of events is non-causal “if, in the case of a very long series of observations, each series to be formed out of the diﬀerent events by permutation (with repetition) has the same average frequency (whereby only the series would have to be small in comparison to the total series of observations)”; (cf. Die Kausalit¨at in der gegenw¨artigen Physik, in Schlick 1938, 71) and the axiom of the existence of limits to series of relative frequencies. These axioms are functionally a priori or, as Kant would say, “synthetic a priori principles of experience,” not because they express “apodiktische Wirklichkeitserkenntnis” (as Schlick wrongly interprets the meaning of Kant’s “synthetic a priori”), a priori insights into the nature of reality, but because they are universal “conditions of possible experience” (where “experience” means “science”). Being undoubtedly synthetic (although they are used to define the subject-matter of statistical physics), they are not formally necessary; their necessity is but functional. Kant always uses the terms “necessity” and “universality” as equivalents; and, indeed, one intuitively feels that there is an inner connection between these categories. We may now give a precise meaning to this equivalence, by showing that functional and formal necessity are, though distinct, continuous with each other in the Leibnizian sense of continuity: formal necessity may be interpreted as a limiting case of functional necessity. A formally necessary, or analytic, judgment, we said, is a judgment whose contradictory is inconsistent. But inconsistency is an irreflexive relation, i.e., no judgment can significantly be said to be inconsistent with itself, but only with other judgments. The diﬀerentia of formal inconsistency, in terms of which formal necessity is defined, as contrasted with material inconsistency, in terms of which functional necessity is defined, is, then, the fact that violation of the most general laws, viz., the laws of logic, is involved: pi , denoting a variable member of the set of postulates p1 , . . . , pn , is functionally necessary if it is so general that its negation would entail the negation of a vast number of empirical laws; ∼ pi would then be said to be to a high degree materially inconsistent. pi , now, is said to be formally necessary, if ∼ pi is formally inconsistent; this, however, means that it

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contradicts the laws of logic, and the latter are themselves functionally necessary in the highest degree, insofar as their rejection would force us to abandon all laws whatever. Formal necessity is thus seen to be equivalent to the highest universality of the laws by inconsistency with which functional necessity is defined. It is to be noted, though, that formal necessity is not thereby “reduced” to functional necessity. Inconsistency with the laws of logic remains sui generis, although it can be interpreted as, so to speak, the upper limit of material inconsistency. The laws of logic themselves cannot be said to diﬀer merely quantitatively from functionally necessary laws, such as to be defined by the property of having functional necessity in the highest degree. For functional necessity itself is defined in terms of the logical relation of inconsistency, which is itself defined in terms of the laws of logic. In summary, it should be noted that, although the formal a priori, the material a priori, and the functional a priori are, as categories or epistemological predicates, distinct, they are quite compatible in the sense of being predicable of one and the same sentence. As a matter of fact, the main intent of this analysis has been to mark out the diﬀerent types of epistemological status that accrue to statements in the process of scientific systematization. Synthetic statements, whether they be empirical or materially a priori, are made into analytic statements in order to be taken as “leading principles” or “conventions.” Hypostatization of the categorial distinction between synthetic truth and conventionalanalytic definition into existential separation, such as to think of the statements which are epistemologically qualified by these categories, as of mutually exclusive classes, gives rise to a radical misconception of science. If definitory or analytic statements are of any use in inquiry, if they have, in other words, existential import, and are not idle nominal definitions, it is because they are synthetic in origin. “Being conventionally definitory” and “being synthetically descriptive” (whether descriptive of “matters of fact” or of “eidetische Sachverhalte”) are no doubt distinct predicates; but this distinctness should not mislead us into throwing “conventions” into one basket, synthetic truths into another; for then we bring the puzzles of applicability upon us. And these puzzles should not be belittled, in the manner of Schlick, who dismisses applicability as a pseudo-problem due to lack of semantical analysis. For example, in his article “Gesetz und Wahrscheinlichkeit” (in Schlick 1938), he “solves” the puzzle that chance should be predictable, that there should be, paradoxically, “laws of chance,” by the simple semantical observation that “the rules of probability apply to chance events for the simple reason that what we call chance events are those events to which they apply.” This is like arguing that there is no problem of induction, since the word “nature” is defined as the totality of events insofar as they obey laws. If there were no uniformities in events, if mathematics, consequently, were not applicable to empirical data, we simply would not possess a concept of nature, hence it would not occur to us to set up

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such a definition. The same considerations apply to the possibility of statistical physics: if irregular or non-causal series of events did not, as a matter of fact, exhibit statistical regularity, in the sense that finite series of relative frequencies approximate to limits, we simply would not come to possess that well-defined concept of “chance-event” of which Schlick is talking; or, at least, it would be a purely formal concept without existential reference. Schlick’s quoted statement typically exemplifies lack of “transcendental logic,” i.e., awareness of the fact that concepts that have, in Kantian phrase, any “reference to objects,” do not exist “ready-made,” but are first constructed by synthetic judgment. Why should the principle of mathematical induction apply to integers? Because integers are defined in terms of that principle; integers that do not obey mathematical induction could not legitimately be called integers. But the question is: quid juris the concept of integer, or, which amounts to the same, quid juris that principle which defines the concept? Again: Why should Nature obey conservation-laws? Because Nature is defined in terms of conservation-laws; but the question is: quid juris such a concept of Nature? Why should there be conservation-laws? Why should that definition have existential import? Why should such a concept of Nature “refer to objects,” rather than being an idle fiction? If one forgets that analysis presupposes synthesis, one is led to perform the “ontological leap”: Why should God exist? Because God is defined as existing. But quid juris the concept of a being among whose definitory properties there is existence itself? Why should such a concept “refer to an object”? Why should such a definition be real?

Chapter 3 LOGIC AND THE SYNTHETIC A PRIORI (1949)

The distinguished American logician, C. H. Langford, recently published a paper (Langford 1949), as brief and alarming as what the title, “A Proof that Synthetic a priori Propositions Exist,” claims for it. Although this publication has, to my knowledge, had no noticeable repercussions in the literature of analytic philosophy, it deserves credit for reopening (for open minds, that is) an issue which according to the logical positivists has been decided once and for all. One of the merits of logical positivism which I would be the last one to deny is to have revealed a typical character of philosophical disagreements, viz., the fact that many (or most, or all?) philosophical controversies are rooted in diﬀerences of verbal usage. I am fairly sure that Langford’s paper constitutes, indeed, further confirmation of this positivistic thesis, for a positivist is not likely to deny the cogency of Langford’s proof of the existence of synthetic a priori propositions in Langford’s sense of “synthetic a priori.” He would rather criticize Langford for having suggested by his terminology an accomplishment which he cannot really claim. I hope, therefore, to shed some light on this issue by scrutinizing the Kantian concepts involved in terms of modern logic. Indeed, it seems to me just as futile to discuss the nature of logic without a clear understanding of the distinctions which Kant strove (though rather unsuccessfully) to clarify as to discuss those distinctions without regard (be it ignorance or oblivion) to modern logic. The line of attack against the Kantian theory of geometry most popular with modern analytic philosophers has been to call attention to the distinction between pure and physical geometry, and to show that synthetic a priori propositions disappear from geometry once this distinction is observed. The “axioms” of pure geometry (more aptly called “postulates”) are propositional functions, so are the derived theorems, and the concepts synthetic-analytic, empirical-a priori significantly apply only to propositions. The propositions of pure geometry really belong to logic (and are hence analytic), since they have the form “if the axioms are true, for a given interpretation of the predicate variables (the so-called primitive terms of the axiom set), then the theorems are true, for

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that interpretation.” On the other hand, once the axioms are interpreted, one obtains either analytic propositions or empirical propositions: if specifically an empirical interpretation is given, the interpreted deductive system refers to physical space (physical geometry) or to some other empirical subject matter. Now, Langford admits, in the cited paper, that if the postulates which have to be added to an “adequate” definition of “cube” in order to derive the theorem “all cubes have twelve edges” are propositional functions, then it cannot be supposed that this geometrical theorem expresses a proposition at all, and that if the postulates are interpreted in terms of physical space, the theorem is not (or, at least, may not be) a priori. Yet, he claims it to be a priori if an interpretation of the postulates in terms of visual space is assumed. But thus he must hold that, however mistaken Kant’s views about physical space may have been (specifically the view, suggested by the apparent finality of Newtonian physics, that physical space must necessarily conform to the axioms and theorems of Euclidian geometry), Kant was right in holding that there is such a thing as “pure intuition” which makes a priori knowledge of synthetic geometrical propositions possible. Langford emphasizes, indeed, that his proof “does not require that all theorems of Euclidean geometry should become true a priori with an appropriate interpretation.” But it seems to me evident that his proof, if valid at all, establishes that all theorems of geometry which require for their demonstration postulates (containing specifically geometrical terms), in addition to explicit definitions, are synthetic a priori propositions, provided only that we suppose them to refer to visual (= idealized?) space. Not that Langford is committed to the Kantian view that we are graced with a power of specifically spatial intuition which puts geometrical knowledge into a category by itself. Consider, for example, a proposition of phenomenological acoustics, like “if x is higher in pitch than y, and y is higher in pitch than z, then x is higher in pitch than z.” This proposition, which we are all inclined to regard as necessary on intuitive grounds (the contradictory is inconceivable) is certainly not derivable from logical principles with the help of definitions: the relational predicate involved admits only of ostensive definition; it defies analysis. And if so, then this proposition would be analytic only if the predicate “higher in pitch than” occurred inessentially in it, i.e. if the proposition “for every R, x, y, z, if xRy and yRz, then xRz” belonged to logic—which, of course, it does not. Some will no doubt say: “Granting, for the sake of the argument, that Langford has established the synthetic character of the proposition ‘all cubes have twelve edges’ (i.e., that it is not demonstrable with the help of explicit definitions, which do not beg the question, alone); has he given any argument at all for the claim that it is a priori (necessary)?” Indeed, Langford just takes it for granted that the proposition is not empirical. And I can hear those who hold all necessary propositions to be definitional truths argue: “What could one mean by saying ‘p is necessary’ if one at the same time admits that p is not demonstrable with the help of logical principles alone? Surely, the concept

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of necessity one has in mind must then be purely psychological, something like the inconceivability of the falsehood of p.” Now, for the sake of those who think, for some such reasons as I just outlined, that Langford’s proof stands and falls (or, rather falls) with his by and large discredited Kantian assumption of a faculty of pure intuition of visual space, I want to show as tersely as I can that it must be possible to know some propositions to be necessary before any can be known to be analytic;1 and that if the concept of synthetic entailment2 be held to be psychological, the concept of analytic truth (as applied to natural languages) will be no better oﬀ. First, what sort of a statement does a philosopher intend to make when he says “all necessary propositions are analytic (and, of course, conversely)?” An empirical statement, like “all dogs have the power to bark?” Obviously not. He intends this statement as an explication (to use Carnap’s term) or analysis (to use Moore’s term) of the concept of logical necessity. It therefore rather resembles such statements as “all cubes are regular solids bounded by square surfaces,” “all fathers are male parents.” I shall now attempt to show (a) that a definition of “logically necessary”3 in terms of “analytic” is untenable since it suﬀers from implicit circularity, and (b) that if a semantic system of logic is taken to include its meta-language (meta-logic), it must be held to contain synthetic propositions (which, of course, are not empirical). Suppose we define an analytic statement as one which is demonstrable with the help of adequate definitions and without the use of extra-logical premises (Langford’s “postulates”). The use of the word “demonstrable” in this definition should make it clear that the concept here defined is a semantic one (corresponding to Carnap’s L-truth), since demonstration, as commonly understood, involves the assertion of the premises from which a deduction has been made as true. An apparently equivalent definition is the one preferred by Quine: a true statement is analytic if either it contains only logical constants, or, provided it is written out in primitive notation, descriptive terms occur vacuously in it, i.e., the statement would remain true if the descriptive terms were arbitrarily replaced by others that are semantically admissible in the context. I want to show that if in these definitions “necessary” were substituted for “analytic,” the definitions would become circular on two accounts. It is important to realize, in the first place, that unless the classification of a statement as belonging to logic or an empirical science respectively is to be wholly arbitrary and devoid of philosophical interest, it will not do to define “analytic” as a predicate which accrues to statements relatively to arbitrary 1 See

also chapter 4. is here called “synthetic entailment” has nothing to do with the notion of causal entailment which is allegedly involved in subjunctive conditionals; for causal statements are at any rate empirical, while a statement expressing a synthetic entailment would be necessary. 3 There exists, to be sure, an established usage for the expression “logically necessary” according to which it is synonymous with “analytic” or “logically true”; and I am obviously departing from this usage here. But we obviously need a qualifying adjective in order to distinguish the sense of “necessary” under discussion from other, irrelevant, senses like “factual necessity,” “practical necessity,” and so forth. 2 What

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definitions of their constituent terms. We obviously want to say, for example, that while one could arbitrarily define “man” in such a way that “all men are mortal” would become a logical truth, the statement as commonly understood simply is empirical. But in that case the definitions from which analytic truth derives will have to be characterized as in some sense adequate. What, then, are the criteria of adequacy of definitions? Extensional equivalence is obviously an insuﬃcient criterion, otherwise it would be adequate to define “equilateral triangle” as meaning “equiangular triangle,” and, worse still, any proposition of physics that has the form of an equivalence (an “if and only if” proposition, in other words) could be made out as analytic and thus belonging to logic. This consideration suggests a further negative criterion of adequacy of definitions, to be added to extensional equivalence of definiendum and definiens: the definition should enable the logical demonstration only of such propositions as are not empirical. But to call a proposition nonempirical is the same as calling it necessary, hence the concept of adequate definition which was used to define analytic truth leads us back to the concept of a necessary proposition, and if “necessary,” here, were synonymous with “analytic,” the definition would be viciously circular.4 To illustrate: in constructing a definition of propositional truth, one will be guided by the criterion “to say of a proposition that it is true is equivalent to asserting that proposition” (if and only if “p” is true, then p), i.e., no definition of truth will be accepted as adequate unless it entails the mentioned proposition. Why not choose as a criterion of adequacy the proposition “if p is true, then, if p is asserted, p is believed by the speaker”? The obvious reason is that this proposition is not necessary, i.e., it is conceivable that people should assert true propositions which they fail to believe (say, because they are liars who, contrary to their knowledge, happen to disbelieve true propositions). Let us, now, illustrate the same point with regard to the definition of a logical constant, say, “not.” Why is the definition embodied in the conventional truth-table (if p is true, then not-p is false, and if p is false, then not-p is true) considered adequate? What does it mean to say, in answer to this question, that it “conforms to ordinary usage”? Suppose we wanted to decide which of the following two definitions of the function “I know that p” conforms to ordinary usage: (a) I believe that p, and p; (b) I believe that p, and p is highly probable on the available evidence. We would, or should, argue somewhat as follows:

4 It

may be noted in passing that C. I. Lewis’s definition of analytic truth, supplemented by his identification of analytic and a priori truth, suﬀers precisely from this circularity. (cf. Lewis 1946, Ch. V). Analytic statements are defined as statements derivable from principles of logic with the help of definitions which are not arbitrary terminological conventions but “explicative statements.” Explicative statements are said to be statements to the eﬀect that the intensions of two terms, “P” and “Q,” are identical. But then we are told that “P” and “Q” have the same intension if they are inter-deducible, i.e., if the formal equivalence “(∀x)[Px ≡ Qx]” is analytic!

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the proposition “if I know that p, then p is true” is clearly necessary, in other words, it would be self-contradictory to claim knowledge of false propositions; but according to definition (b) it would be logically possible that false propositions should be known, since a false proposition may be highly probable on the available evidence; hence definition (a), not (b) is correct.5 In order to conform to the ordinary usage of the defined term, a definition, then, must enable the demonstration of such sentences, and only such sentences, involving the defined term as are ordinarily held to express necessary propositions. Accordingly, the test of adequacy of the definition of “not” is that it enables the logical demonstration of certain fundamental necessary propositions involving the defined constant, such as the law of the excluded middle and the law of noncontradiction. And if we argued that what makes these principles necessary is the fact that they are demonstrable with the help of adequate definitions of the logical constants involved, our argument would evidently be circular. Since the definitions of such logical constants do not form part of the logical system as such but belong to the meta-language, it may be reasonable to demand that the criteria of adequacy themselves be formulated in the meta-language. Thus, one would properly distinguish the tautology of the propositional calculus “for every p, p or not-p” from the meta-linguistic statement “every proposition has either the truth-value ‘true’ or the truth-value ‘false”’ (here “true” and “false” are meta-linguistic terms and hence the “either-or” of this statement is diﬀerent from the “either-or” which is used, but not mentioned, in the calculus). But now we face the following situation: unless this meta-linguistic statement is accepted as necessary, no instruments, as it were, are provided for proving that the law of excluded middle of the object-language is analytic; and if the meta-language is not formalized in terms of a meta-meta-language, the meta-linguistic L.E.M. cannot be analytic; and since we cannot go on building meta-meta...meta-languages forever, some meta-language will have to contain an analogue of the object-linguistic L.E.M. which is at once synthetic and necessary. I anticipate the objection that my argument is completely worthless since it proceeds on the assumption that “analytic” is an absolute concept. The opposition might, indeed, demonstrate the absurdity of my argument by comparing it to the following: a body can be said to move only relatively to a referencebody; but unless we knew that the reference-body is absolutely at rest, we could

5 Incidentally,

it seems to me semantically inaccurate to distinguish, as philosophers frequently do, two kinds of knowledge: certain knowledge and probable knowledge. What could be meant by saying “I know with high probability that the sun will rise”? It could not mean that on the one hand I know, but on the other hand it is (or I am) not certain, for that surely sounds self-contradictory. I think what the intended distinction comes to is merely this: sometimes the proposition “I know that p” (which is always empirical regardless of whether p is empirical or not) is certain and sometimes it is only probable on the evidence. In the latter case one might appropriately say “I am not certain that I know p, but it is probable that I do.”

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not be sure that the first body really moves; hence, if we want to say that some body really moves, we shall have to assume that some other body is absolutely at rest, i.e., at rest regardless of what happens to any other body. The point of the analogy would presumably be that it is just as meaningless to call a statement of a given language synthetic relatively to no meta-language at all, as it is to speak, in old Newtonian fashion, of absolute rest (and the same analogy would, of course, hold for “analytic” and “in absolute motion”). To which I reply: If “analytic” is to be short for “analytic in L,” where L is formalized in terms of some meta-language which includes, among other rules, definitions of the defined terms of L, then “analytic” cannot be regarded as an explicatum of the common notion of logical truth, since by a suitable choice of definitions any statement could then be made out as a logical truth. Thus, to escape from this “conventionalism,” as Lewis calls it, one will have to make reference to a privileged meta-language (just as the earth is the privileged reference-frame tacitly referred to in common-sense statements of the form “x moves” or “x does not move”); and in order to mark out this privileged meta-language one will have to introduce precisely that notion of adequate definitions which leads up to necessary propositions defying formal demonstration. Now, to my second argument for the proposition that a definition of logical necessity in terms of analyticity would be circular. Langford defines “analytic,” as most logicians would, in terms of “logical principle”; and “logical principle” is defined, in the paper already referred to, as “principle involved in the extended function calculus.” But how are we to decide whether a given proposition is involved in the extended function calculus? Either we enumerate all such propositions, so that “logical principle” becomes an abbreviation for a finite disjunction of propositions, or else we shall have to state a common and distinctive property of such propositions by virtue of which they are classifiable as “logical principles.” The former method of definition is impossible since (a) the number of propositions belonging to a system of logic that can be fabricated is unlimited, owing to repeated applicability of the rule of substitution, (b) given such an enumeration of propositions which defines “logical principle,” it would be self-contradictory to suppose that one day a new logical principle should be discovered (or manufactured), since this would mean that an element both belonged and did not belong to the same collection. We are, therefore, faced with the necessity of supplying an explicit definition of “logical principle.”6 6 Some may think that this conclusion can be avoided since the general concept of tautology admits of recur-

sive definition, thus: one first enumerates a set of primitive propositions which one calls “tautologies,” and then extends the term “tautology” to any proposition which is derivable from these primitive propositions with the help of special rules of derivation. I think, however, it is perfectly evident that this amounts to a statement about tautologies and not to an explication of the meaning of “tautology.” The very choice of primitive propositions as well as of rules of derivation must be guided by a prior understanding of what a

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It seems that such a definition will have to make use of the notion of logical constant, e.g., “a logical principle is a true proposition containing only logical constants.” But what do we mean by “logical constant”? While I am unable to give a satisfactory explicit definition, I am sure that such a definition would involve the concept of validity (as predicated of deductive arguments), for the following reason: The rules of formal logic contain no descriptive terms (this is the reason why they are called formal rules), hence before they can be applied to the test of the validity of specific arguments, the latter must be formalized; which means that specific descriptive terms in the argument are replaced with variables until a more or less abstract schema is left over. Not all expressions can be replaced by variables, however, since otherwise it could not be said that the argument has one logical form rather than another: in order to have a specific form of argument, we need some constants. How, then, are we to tell which terms may be replaced by variables (of appropriate type) and which may not? Consider, for example, the syllogism “if Socrates is a man, then Socrates is mortal; Socrates is a man; therefore Socrates is mortal.” We see that if in this argument the proper name “Socrates” is replaced by any other name or description of an individual, the resulting argument would still be valid. Hence, instead of considering this specific argument, we consider any argument of the form “if x is a man, then x is mortal; x is a man; therefore x is mortal.” But then we also notice that the validity of the argument would not be destroyed if any other predicates of the first type were substituted for “man” and “mortal”; which leads us to consider any argument of the form “if Px, then Qx; Px; therefore Qx.” And one further degree of abstraction is seen to be possible: it does not matter what the forms of the constituent propositions are; hence we may introduce propositional variables and consider any argument of the form “if p, then q; p; therefore q.” But what prevents us from pushing this process of formalization one step further by introducing a variable whose values are binary connectives (like “if-then,” “and,” “or”), and studying the schema: pCq (where “C” is a binary-connective-variable); p; therefore q? The answer is obvious: we see at once that not all the argument-forms resulting from substitution of a binary connective for “C” are valid (as, e.g., p or q; p; therefore q). It is no doubt such observations that led logicians to stress the distinction between logical constants, as terms on whose specific meanings the validity of an argument depends, and descriptive constants which occur inessentially (to borrow Quine’s term) with respect to the validity of the argument—although they may occur essentially with respect to the factual truth of propositions en-

tautology is. We might want to say, for example, that not enough rules of derivation, or not enough primitive propositions had been laid down, since there are tautologies which could not be derived in the constructed system; but this would be a self-contradictory statement if “tautology” meant “proposition derivable in the constructed system.”

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tering into the argument. But to say that an argument is valid is to say that its conclusion necessarily follows from its premises; which is to say that the implication from premises to conclusion is a necessary proposition. Which completes the vicious circle I have been endeavoring to demonstrate. If the distinction between analytic and synthetic truths rests, as has been suggested, on the distinction between logical and descriptive constants, and if there should exist no sharp criterion by which the two types of expressions could be distinguished, then the distinction analytic-synthetic is less clear-cut than is commonly supposed. Superficially it looks as though the above description of the function of logical constants in formal logic suggested a perfectly simple explicit definition of “logical constant”: a logical constant is a term which cannot be replaced by a variable in the process of formally testing the validity of arguments which contain it. This definition, however, breaks down under the weight of three objections: (a) an expression which occurs essentially in one argument may occur inessentially in another argument. Take, for example, the identity sign. In the argument “x = y; therefore not-x y)” it occurs inessentially, since any argument of the form “xRy; therefore not-not-xRy” is valid (indeed the word “not” is the only expression in this argument which occurs essentially!). On the other hand, in the context “x = y; Px; therefore Py” the identity sign has an essential occurrence, since “xRy; Px; therefore Py” is not generally valid. (b) On the proposed definition, it will depend on the kind of variables that are available for formalizing arguments, whether an expression belongs to the vocabulary of logic or not. Suppose that we introduced symmetric-relation-variables, i.e., variables taking the names of symmetric relations as values, S , S , S , etc. In that case the argument “x = y; therefore y = x” might be regarded as a substitution-instance of the argument “xS y; therefore yS x,” and since the latter is generally valid, “=” would be classified as a descriptive (inessential) constant. But if the variables at our disposal are less variegated, and we can use only generic relation-variables R, R , etc., then the above argument will have to be considered as an instance of “xRy; therefore yRx,” and since this is not a valid argument-form, we could then with equal plausibility (or implausibility) conclude that “=” is a logical constant. This difficulty cannot be avoided either by stipulating that the formalization should be as abstract or generic as possible. For such a stipulation is presumably equivalent to the demand that the range of the variables used for formalization should correspond to the logical type of the values in question. But what is meant by saying that class C is the logical type to which entity x belongs, if not that “x is a member of C” is true, provided it is significant? Thus a criterion of significance would first be required, and, lest we end up with a circular definition of “logical truth,” we would have to formulate it without using the concepts “analytic” and “self-contradictory.” This, however, is a diﬃcult task which has hardly been tackled yet. In fact, the relation between nonsense and self-

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contradiction has not yet been suﬃciently analyzed. (c) Whether a term has an essential occurrence in an argument all depends on whether the argument is written out in primitive notation or contains definable terms. Thus the term “bounded by squares” occurs essentially in the argument “x is a cube; therefore x is bounded by squares,” but vacuously in the argument “x is a regular solid bounded by squares; therefore x is bounded by squares.” In order to decide with finality, then, whether a term belongs to the essential logical skeleton of arguments, we have to presuppose a completely analyzed language. Nowadays we regard, in the light of the impressive reduction accomplished in Principia, arithmetical terms as part of the logical skeleton of the language of the empirical sciences. But before this meta-mathematical analysis took place, it was natural to regard arithmetical terms as descriptive. In fact, in the sense of the word “descriptive” which most readily comes to one’s mind, such terms as “two,” “three,” clearly are descriptive: they designate observable features of collections; we perceive that a collection has three members just as we might perceive a common physical characteristic of its members. Can we say with finality, then, that, say, geometrical terms, like “straight line,” “circle,” cannot be incorporated into the vocabulary of logic, and that geometrical propositions like “any two points uniquely determine a straight line” could never be written out in primitive notation in such a way that the specifically geometrical terms “point,” “straight line,” “lying on,” drop out as inessential and the proposition becomes derivable from logic? For the reasons just stated, it would be a poor rejoinder to say, unlike arithmetical terms, geometrical terms are descriptive of observable features of the world. It is, moreover, somewhat naive to say flatly “arithmetic is reducible to logic,” or “geometry is not reducible to logic.”7 Apart from the consideration that the confines of the language of logic are largely determined by the intuitive acceptability of certain definitions—somebody might, for example, reject Russell’s contextual definition of descriptive phrases in terms of logical primitives as intuitively inadequate, and on that ground hold that statements about mathematical functions are not really reducible to logic—arguments by which such reducibility is commonly proved involve a subtle element of circularity. To say, for example, that “1 + 1 = 2” is really a truth of logic, is to say that it can be derived from the primitive propositions of Principia with the help of the logistic definitions of the specifically arithmetical terms “1,” “2,” “+.” But on what grounds are those definitions accepted as adequate? Either the grounds are intuitive, in which case one is left without a logical argument 7 What

I mean, here, by a reduction of a system of geometry to logic is a derivation of the postulates of such a system from postulates containing only logical terms. It is frequently said that pure geometry, whose propositions are formal implications in which specifically geometrical terms occur vacuously, is part of logic. This is, however, a rather trivial observation since the word “geometry” here has itself a vacuous occurrence: in the same sense pure mechanics, pure deductive sociology, etc., form part of logic.

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by which the reducibility of arithmetic to logic could be proved in the face of objections from the intuitive inadequacy of, say, the definition of the number one, or else one will have to argue: the definitions are adequate in the sense that they enable the demonstration of the theorems of arithmetic from logical premises alone. Considering, then, the vagueness of the expression “logical term,” I do not find the claim that arithmetic (or any other science which contains, prior to reduction to primitive notation, non-logical terms) is logic any more convincing than the contrary claim. The contrary claim might be supported by two reasons: (a) the whole reduction is circular if the underlying definitions can be defended only by showing that they lead to the desired result, (b) the definitions themselves express synthetic propositions, for a biconditional joining a proposition in purely logical notation with a proposition in partly arithmetical notation cannot be analytic unless arithmetic has already been reduced to logic; but if this reduction could be eﬀected independently of such definitions, there would be no need of setting the latter up as instruments of reduction. By way of clarifying this type of argument (which, it should be kept in mind, is not so much intended as a proof of the existence of synthetic a priori propositions than as a proof that, in default of a clear definition of “logical constant,” it is to a certain measure arbitrary whether a given a priori proposition be called “analytic” or “synthetic”), let us suppose that the confines of a system of logic are drawn, in the manner familiar to formal logicians, with the help of recursive, not explicit, definitions. We arbitrarily mark out some terms, say “not” and “or,” as logical constants, and then specify that any term exclusively definable with the help of “not” and “or” is also a logical constant. This kind of definition leaves us, of course, in the dark as to what “logical constant” means, yet it provides an eﬀective procedure for deciding questions of the form “is x a logical constant in the system L?” If contextual definitions are permitted, then such terms as “all,” “and,” “the,” “there is,” etc., would all be logical constants in this system.8 Yet, how would one decide whether a proposed definition of a derived constant in terms of the selected primitives was adequate? It seems to me that the proof of adequacy would rest on the intuitively accepted validity of certain forms of argument. How could we prove, for example, that “class A has exactly one member” is logically equivalent to “there is an x such that x is a member of A and for any y, if y is a member of A then y is identical with x?” All we can say in the end is that the two statements evidently entail each other. Since what is in question is just a definition of “one” with the help of which the extension of the term “logical constant,” and therewith of the term

8 In

order to reduce “the” to those primitive constants one needs, though, the symbol of identity which can be reduced to the primitive constants only if quantification over predicate-variables is permitted.

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“logical truth,” could be increased, it would be circular to attempt to show that this equivalence is analytic. But what guarantee, now, do we have that any new logical constant which we might discover in the process of analyzing arguments would be definable in terms of our logical primitives? What if what appears to belong to the logical skeleton of our language should defy definitional reduction to the selected primitives? If such a situation should arise, two alternatives would be open to us. We might say that the terms in question are logical constants since they evidently have an essential occurrence in some valid arguments, and that therefore statements containing these constants along with already accepted citizens of the vocabulary of logic are logical truths. Or, we might say that such terms are not logical constants, just because they are not definable in terms of our primitive logical constants, and that therefore statements which contain them essentially are synthetic. Take, for example, the intuitively valid argument: x and y are distinct points in a plane; therefore there exists just one straight line (in that plane) which contains both x and y (where “point” and “straight line” and “plane” are not predicate variables but have the customary geometrical meanings). To be sure, I might, using “point” as an undefined term, define a straight line (in a plane) as a class of points which is uniquely determined by any two members of itself, and relatively to this definition, of course, the argument would be formally valid. But such a definition would obviously be questionbegging in the present context of discussion, just as a definition of “cube” as “regular solid with twelve edges” would be question-begging if it were used as a refutation of Langford’s point. If we assume that “point” and “straight line” and “plane” belong to the undefined vocabulary (which assumption is justified in view of the fact that the meanings of these terms are commonly understood only by virtue of ostensive definition), we see at once that this argument is not formally valid: if “point” and “straight line” and “plane” and “x contains y” are replaced by variables, the resulting argument-form is invalid.9 Should we say,

9 It

might be objected that an explicit definition of “plane” could be constructed which would bring the analytic character of this axiom of Euclidean plane geometry into evidence. Suppose we wanted to verify whether a given surface was plane or curved, would we not endeavor to determine whether one and only one straight line could be drawn through any couple of points in it? And does this not suggest the definition of a plane surface as a surface such that any couple of points in it uniquely determines a straight line? The trouble with this attempt at explicit definition is that it lands us in a vicious circle. For one could similarly ask how one would distinguish a straight line from any other type of line? And with the same plausibility one might say that it is that type of line which, in a plane, is uniquely determined by a couple of points. It is, indeed, sometimes said that the geometrical primitives “reciprocally” define each other, but I have never been able to understand how such “reciprocal” definition diﬀers from viciously circular definition. We are then left, as far as I can see, with two and only two alternatives: either the primitives are predicate variables, in which case the axioms are no propositions at all; or else they have an empirical reference through ostensive definition. It should be noted, however, that the contingent character of such empirically interpreted axioms cannot be inferred from the fact that they refer to qualities of sense experience, no more than it follows that “2 + 2 = 4” is contingent from the fact that it is applicable to classes of empirical objects.

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then, that this is a case of synthetic, or material, entailment since the implication is not derivable from what has been defined as “logic,” but that the essential occurrence of the geometrical terms calls for an extension of our definition of “logical constant”? Or should we say that these geometrical terms could never be called “logical constants” at all, just because they are not definable in terms of the accepted primitive vocabulary of logic? Most logicians would seize the latter alternative, in line with the view that geometrical axioms, unlike the postulates of arithmetic, cannot be reduced to pure logic. But then they might also have refused to admit, say, “all” as a logical constant into a truth-functional logic, since it can be defined in terms of “and” only if infinity (expressed by “. . .”) is admitted as a logical concept—and would it not be arbitrary to do so? And arguments involving “all” essentially would, relatively to such a narrow definition of “logic,” have to be counted as synthetic entailments. The same point may well be illustrated further in terms of Langford’s second candidate to the honorable (or decadent?) title of “synthetic a priori truth,” viz. the proposition “whatever is red is colored.” Langford argues that “x is colored” cannot be formally deduced from “x is red,” since a man who is able to understand the meanings of the extra-logical terms in the premises of a formal argument should also be able to understand the meanings of the extra-logical terms in its conclusion (at least if the argument is written out in primitive notation, and does not have some such trivial form as “p, therefore p or q”); and he argues (convincingly, I think) that a man might well understand the meaning of “red” without understanding the meaning of “colored.” I shall oﬀer an independent, though perhaps similar, argument for the view that this proposition is synthetic. It is natural to suppose that the conditional “if x is red, then x is colored” has the form “if p, then p or q,” since to say “x is colored” is to say “x is blue or green or red or ...” But could such a disjunctive definition, as we may call it, ever be written out? Suppose we wrote it out, eliminating the convenient dots, by putting in as a disjunct each and every color that happens to have received a name. And suppose that thereafter we observed for the first time an object that had a color which, unknown as it was, failed to have a name; would we not want to say that this object is colored? We certainly would, yet if “colored” meant what we defined it to mean, we could not say that. I conclude that “colored,” like “red,” must be regarded as a term whose meaning is grasped only through ostensive definition and which therefore belongs to the primitive vocabulary. But in that case the argument “x is red, therefore x is colored” would be formally valid only if all arguments of the form “x is P, therefore x is Q” were formally valid. Have I shown that the true statement “whatever is red is colored” (and all similar species-genus statements like “whatever is round has shape,” “whatever is hot has temperature”) is synthetic (though presumably necessary)? Only if it can be assumed that “red” and “colored” are not logical constants. This

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assumption seems innocent enough: if these are not descriptive terms, what terms are? Yet, in the first place, the customary manner of distinguishing logical terms from non-logical terms as purely syntactic elements of language from denotative elements of language is, as pointed out already, not quite safe: numbers are, up to a certain point, observable properties of collections, still they are regarded as logical concepts; second, if all we can say by way of defining “logical constant” is that logical constants are those terms which occur essentially in some valid arguments,10 then “red” would have to be admitted as a logical constant. To be sure, this word occurs inessentially in many valid arguments, but the same is true, as was shown, of such full-fledged members of the vocabulary of logic as the identity sign. Perhaps the proper conclusion to be drawn from these observations is that it does not make sense to represent the distinction between logical and nonlogical expressions as absolute. Perhaps all that can be significantly said in answer to the question of what a logical expression is, is that an expression functions logically in the context of an argument in which it occurs essentially (in the already explained sense of Quine’s phrase “essential occurrence”). But in that case the whole problem of whether any necessary propositions could fail to be analytic, i.e., certifiable by reference to logical principles alone, is ill-defined. If, on the other hand, the meaning of “logical principle” should be, more or less arbitrarily, made precise by laying down a number of postulates and rules of derivation and defining “logical principle” as any proposition derivable in this system (including the postulates which are derivable from themselves!), then I think Langford’s proof is irrefutable, since any number of necessary propositions (necessary, that is, in terms of ordinary, pre-analytic usage of the term) could easily be produced which, like “all cubes have twelve edges,” cannot be demonstrated except with the help of extra-logical postulates.11 Of course, what inevitably happens is that such a logical system which, judged in the light of the small number of postulates and primitive terms, ap10 One

might think that much embarrassment could be spared by replacing “valid,” here, with “formally valid.” But such a replacement would make the definition grossly circular, since formal validity is defined in terms of a set of formal rules of deduction which cannot be formulated until after a choice of logical constants has been made for the given language. 11 Whether Langford is right or wrong may well depend on the precise extension assigned to the term “definition” in the characterization of analytic propositions as propositions demonstrable with the sole help of definitions. Specifically, if postulates (or, rather, sets of postulates) should be described as implicit definitions of the primitives they contain, then what Langford calls a “synthetic a priori” proposition is for those who use the term “definition” more liberally simply a species of analytic proposition. Once, however, one operates with the concept of implicit definition, the extension of the concept of logical truth is in danger of becoming paradoxically large, especially if implicit definitions are claimed, as by Schlick in Schlick 1925, to prevail not only in formal mathematical languages but also in the language of science. Thus one might hold that, like the primitives of a system of geometry, the primitives of a system of mechanics (“particle,” “mass,” “force,” etc.), are implicitly defined by the postulates of the science (see e.g., Margenau 1935); in fact, Poincar´e has impressively shown how attempts at explicit definition entangle one in logical circles. But the embarrassing consequence of this position is that the laws which appear as theorems in the

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pears rather meager, expands enormously with the help of definitions: just think of the way in which logic in Russell and Whitehead’s magnum opus swallows up the huge system of classical mathematics! But I see no escape from the conclusion, well worth repeating, that those definitions which syntactically function as rules of translation from one universe of discourse to another and thus enable incorporation of more and more material into logic, express (in a sense of “express” suﬃciently clear to go without analysis in this discussion) necessary or a priori propositions; and since the necessity of those propositions is the ground which makes those definitions cognitively acceptable, it would be circular to prove that they are analytic by reference to the very definitions which they are to support.

postulational development of the science would then have to be regarded either as propositional functions or as logical truths!

Chapter 4 ARE ALL NECESSARY PROPOSITIONS ANALYTIC? (1949)

The title question of this paper admits of two diﬀerent interpretations. It might be a question like “Are all swans white?” or it might be a question like “Are all statements of probability statistical statements?” “Are all causal statements, statements of regular sequence?” etc. If these two types of questions were contrasted with each other by calling the former “empirical” and the latter “philosophical,” little light would be shed on the distinction, since what is to be understood by a “philosophical” question is extremely controversial. Perhaps the following is a clearer way of describing the essential diﬀerence: the concept “swan” is on about the same level of clarity or exactness as the concept “white,” and one can easily decide whether the subject-concept is applicable in a given case independently of knowing whether the predicated concept applies. On the other hand, the second class of questions might be called questions of logical analysis, i.e., the predicated concept is supposed to clarify the subjectconcept. They can thus be interpreted as questions concerning the adequacy of a proposed analysis (frequency theory of probability, regularity theory of causation); and the very form of the question indicates that the suggested analysis will not be accepted as adequate unless it fits all uses of the analyzed concept. Now, when I ask, as several philosophers before me have asked, whether all necessary propositions are analytic, I mean to ask just this sort of a question. I assume that those who, with no hesitation at all, give an aﬃrmative answer to the question, consider their statement as a clarification of a somewhat inexact concept of traditional philosophy, viz., the concept of a necessary truth, by means of a clearer concept. I feel, however, that little will be gained by the substitution of the term “analytic” for the term “necessary,” unless the former is used more clearly and more consistently than it seems to me to be used in many contemporary discussions. And I shall attempt to show in this paper that once the concept “analytic” is used clearly and consistently, it will have to be admitted that there are propositions which no philosopher would hesitate to call “necessary” and which nevertheless we have no good grounds for classi-

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fying as analytic. Moreover, I shall show that even if the concepts “necessary” and “analytic” had the same extension, they would remain diﬀerent concepts. To prove this it will be suﬃcient to show that a proposition may be necessary and synthetic. Probably the most precise analysis of the concept of analytic truth is to be found in the logical writings of Carnap. In Carnap 1954a an analytic sentence is defined as a sentence which is a consequence of any sentence (§10). This definition makes the defined concept, of course, relative to a given language (i.e., “p is analytic” must be regarded as elliptical for “p is analytic in L”), since the syntactic concept “consequence” is defined in terms of the transformation rules for a given object-language. Now, it is clear that this definition is constructed with a view to syntactical investigations into the formal structure of artificial languages such as logical calculi and formalized arithmetic. It is therefore not very useful for philosophers who are interested in the analysis of natural languages which are obviously unprecise in the sense that their formal structure cannot be exhaustively described by stating complete sets of formation rules and transformation rules. Also, no philosopher who proposes “analytic” as the analysans for “necessary” could plausibly mean by “analytic” a syntactic concept, i.e., a concept defined for sentences of an uninterpreted language of which it cannot be said that they are either true or false (uninterpreted as they are) but at best—in case the system is complete, that is—that they are either derivable from the primitive sentences or refutable on the basis of the primitive sentences. For necessity of propositions has always been meant as a semantic concept: a necessary condition which any adequate analysis of “necessary” must satisfy is that the truth-value of a necessary proposition does not depend on any empirical facts. Carnap has since constructed a definition of “analytic” in semantic terms, which yields a concept corresponding to the earlier-defined syntactic concept in the sense that any sentence which is analytic in the syntactic sense (e.g., “p or not-p,” where the logical constants “or” and “not” are not defined by truth-tables but occur as undefined logical symbols in the primitive sentences) becomes analytic in the semantic sense once the language to which it belongs is semantically interpreted. A sentence of a semantic system (i.e., a language interpreted in terms of semantic rules) is said to be analytic or “L-true,” if it is true in every state-description.1 This definition is, of course, reminiscent of the old Leibnizian conception of “truths of reason” as those that hold in any possible world. But this semantics is analogously constructed with a view to the investigation of artificial, completely formalized languages. Specifically, the concept of a state-description is defined for a highly simplified molecular 1A

state-description is a class of atomic sentences of such a kind that the semantic rules of L suﬃce to determine whether any sentence of L is true in the world described by this class of sentences. Thus, if L were a miniature language containing two individual constants “a” and “b,” and two primitive predicates “P” and “Q,” the following would be an example of a state-description: Pa and Qa and Pb and not Qb.

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language containing only predicates of the first level, like “cold,” “blue,” etc. Also, the far-reaching assumption has to be made that the undefined descriptive predicates of the language designate absolutely simple properties and are hence logically independent. Otherwise, further analysis might reveal logical dependence, and what appeared before analysis as a “possible” state-description might turn out to be an inconsistent class of sentences. For such reasons, definitions of “analytic” that are fruitful from the point of view of the semantic analysis of natural languages (including scientific language), which is practiced by both the so-called left-wing positivists and the followers of G. E. Moore, have to be sought elsewhere. This does not mean, however, that we must altogether ignore what formal logicians say about the matter. The following definition by Quine, for example, is illuminating: An analytic statement, as ordinarily conceived, is a definitionally abbreviated substitution instance of a principle of logic. Thus, if the word “father” is introduced into the language as an abbreviation for “male parent,” then “All fathers are male” is synonymous with “All male parents are male,” and, assuming that the type “male” is univocal, this statement reduces to a substitution instance of the logical principle “For every x, P, Q; Px and Qx implies Px .” What are we to understand by a logical principle? Following Quine, a logical principle might be defined as a true statement in which only logical constants occur. This definition raises some problems, to be sure. To begin with, the statement, “something exists,” formalized in the familiar functional calculus by “There is an x and a P such that Px ,”2 would express a logical truth, which some philosophers would find diﬃcult to accept. But the paradox will be mitigated if one considers what would be entailed by the elimination of this statement from logic. According to the customary interpretation of the universal quantifier, “(∀x)Px ” is equivalent to “∼ (∃x) ∼ Px.” It can easily be seen to follow that two statements of the form “(∀x)Px ” and “(∀x) ∼ Px ” are incompatible only if something exists. And would it not be paradoxical if it depended on extralogical facts whether two given propositions are incompatible by their form? Second, it is not easy to give a general definition of “logical constant.” It would obviously be circular to define logical constants as those symbols from definitions of which the truth of logical principles follows. Perhaps we have to be satisfied with a definition by enumeration, just as we cannot define “color” by stating a common property of all colors but only by enumerating all the colors that happen to have names. Such a definition would be theoretically incomplete but practically complete enough. If, for example, we mentioned “or,” “not,” “all,” and any term definable in terms of these, we probably would not omit any logical constant that occurs in the familiar logical, scientific, and 2 The

use of a predicate variable here cannot be circumvented since there are compelling reasons for not admitting the various forms of “to exist” as logical predicates.

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conversational languages (I assume, of course, the reducibility of arithmetic to logic). According to some uses, the statement “All fathers are male” would be called analytic, and the statement “All fathers are fathers” would be called an explicit tautology. But it is clear that those who roughly identify analytic truth with truth certifiable by formal logic alone would include explicit tautologies as a subclass of analytic statements. It appears, therefore, convenient to widen the above definition as follows: A statement is analytic if it is a substitution instance of a logical principle or, in case defined terms occur in it, a definitionally abbreviated substitution instance of a logical principle. This definition commits us to acceptance of an interesting consequence, whether we like it or not: if a statement, like “No part of any surface is both blue and red at the same time,” contains undefined predicates (“blue,” “red”), we cannot know it to be analytic unless replacement of all descriptive terms by appropriate variables leaves us with a principle of logic.3 This point will prove to be important in the subsequent discussion. The question might be raised whether logical principles themselves could be called “analytic” on the basis of the proposed definition. Certainly an adequate definition of analytic truth should allow an aﬃrmative answer to this question. What makes me know that it will rain, if it will rain, is the same as what makes me know the law of identity, “if p, then p,” viz., acquaintance with the meaning of “implies” or “if, then.” It sounds admittedly awkward to say of a statement that it is a substitution instance of itself —but perhaps such language is no more uncommon than, say, the use of implication as a reflexive relation. Thus, stretching language somewhat to suit our purposes, as is quite common in logic and mathematics, logical principles like “if p, then p or q” will be said to be their own substitution instances. And when definitional abbreviations are spoken of, not only definitions of descriptive constants, like “father,” are referred to, but also definitions of logical constants, like “if, then.” This convention enables us to say that “not (p and not p)” is a definitional expansion of “if p, then p,” for example. The fact that a principle of logic is analytic leaves it, of course, an open possibility that it might also be necessary in a sense in which synthetic propositions likewise may be necessary. It will be emphasized, in the sequel, that “p is necessary” does not entail “p is analytic,” although the converse entailment undeniably holds. I pointed out that Carnap’s definitions of “analytic” (or “L-true”) are constructed with reference to (syntactically or semantically) formalized languages 3 There

are certain technical details concerned with a fully satisfactory definition of “logical principle,” such as whether a logical principle may contain free variables or whether all variables must be bound. But these questions are unimportant in this context. Thus I shall call “Px or not-Px ” a logical principle, although customarily variables are used to express indeterminateness rather than universality, and such an expression is, therefore, regarded as a function, not as a statement.

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and have therefore a limited utility. But I should not be misunderstood to imply that reference to a given language ought to be, or can be, avoided in the construction of such a definition, if “analytic” is treated as a predicate of sentences at all.4 The same relativity characterizes the definition proposed above, since “analytic” is defined in terms of the systematically ambiguous term “true.” The so-called semantical antinomies (like the classical antinomy of the liar) are well known to arise from the treatment of truth as an absolute concept, i.e., a property meaningfully predicable of any sentence, no matter on which level of the hierarchy of meta-languages the sentence be formulated. What is to be taken as defined, then, is “analytic in the object-language L,” although a schema is provided by the definition for constructing analogous definitions for each level of language. Another appropriate comment on the proposed definition of “analytic” should be made. It is well known that if our language refers to an infinite domain of individuals, there is no general decision procedure with respect to quantified formulas, i.e., no automatic procedure by which it can be decided, in a finite number of steps, whether such a formula is tautologous, indeterminate, or contradictory. For this reason, it might not be possible to decide in a given case whether the formula which results from a statement suspected as analytic when the descriptive constants are replaced by variables is a logical truth. This is again an admitted theoretical defect of the proposed definition, but not a defect that might really prove fatal to the practice of linguistic analysis. For such undecidable formulas (like Fermat’s theorem, for example) are usually complicated to a degree which the formulas resulting from the formalization of controversial “necessary” statements never are. Thus, the formula corresponding to “No space-time region is both wholly red and wholly blue” would be “∼ (∃x)(∃t)[Pxt.Qxt]” (where we might consider surfaces as constituting the range of “x”), and this formula is certainly not logically true, since we can easily find predicates which, when substituted for “P” and “Q,” would yield a false statement. If “analytic” is thus defined as a semantic predicate of sentences of a language of fixed level, the proposed substitution of “analytic” for “necessary” at once raises the question: Does it make sense to speak of analytic propositions? If it does not, then our new concept cannot replace the old concept of necessity, since obviously necessity is intended as an attribute of propositions. If Leibniz, for example, were asked whether “All fathers are male” and “Alle V¨ater sind M¨anner” are diﬀerent truths of reasons, he would undoubtedly deny it. These two sentences express the same proposition, and it is the proposition which is

4 Carnap, indeed, speaks in his semantical writings at times of analytic or L-true propositions. But he would regard this merely as a convenient mode of speaking: “The proposition that . . . is L-true” is short for “The sentence ‘. . . ’ and any sentence that is L-equivalent to ‘. . . ’ is L-true.”

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said to be necessary. Also, a sentence may obviously be analytic at one time and synthetic at another time, viz., in case the relevant semantic rules undergo a change. But nobody who believes that there are necessary propositions at all would admit that a proposition which is now contingent may become necessary, or vice versa. If we adopt the semantic rule “Nothing is to be called ‘bread’ unless it has nourishing power,” then the proposition expressed by the sentence “Bread has nourishing power” is necessary; and this proposition was necessary also before this semantic rule was adopted, although the sentence by which it is now expressed may at that time have been synthetic. However, the method of logical construction shows a way toward construing reference to analytic propositions as an admissible short cut for talking about classes of sentences that are related in a certain way. To say, “The proposition that all fathers are male is analytic,” might be construed as synonymous with saying “Any sentence which should ever be used, in any language at all, to express what is now meant by saying ‘all fathers are male’ would be analytic.” Those who hold, with C. I. Lewis, that analytic truth is grounded in certain immutable relations of “objective meanings,” not aﬀected by accidental changes of linguistic rules,5 could therefore consistently accept a definition of “analytic” which makes this term primarily predicable of sentences. It will, indeed, be my main point against the linguistic theory of logical necessity, to be discussed shortly, that the necessity of a proposition, whether the proposition be analytic or synthetic, is a fact altogether independent of linguistic conventions. On the other hand, I do find C. I. Lewis and those who share his views concerning the nature of analytic or a priori truth (where “analytic” and “a priori” are regarded as synonyms) guilty of a diﬀerent inconsistency. Analytic truth, they say, is certifiable by logic alone; and I have attempted to clarify what this means by defining “analytic” as above. They also say that what makes a statement analytic is a certain relationship of the meanings of its constituent terms. But it seems to escape their notice that these assertions are by no means equivalent; the first implies, perhaps, the second, but the second, I contend, does not imply the first. Consider a simple statement like “If A precedes B, then B does not precede A.” I assume that few would regard this statement as factual, i.e., such that it might be conceivably disconfirmed by observations.6 And if it is not factual, then it must be true on the basis of its meaning. But it seems that just because all analytic statements are true by the meanings of their terms, it has been somewhat rashly taken for granted that whatever statement is true by what it means is also analytic. To see that the above statement is

5 Cf.

Lewis 1946, chapter 5. hope nobody will make the irrelevant comment that it is meaningless to speak of absolute temporal relations, and that it is empirically possible to disconfirm the statement by a shift of reference-frame. Obviously, I assume that the verb “to precede” is used univocally.

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not analytic, in the sense defined, we only need to formalize it, and we obtain “(∀x)(∀y)[xRy ⊃ ∼yRx],” which is certainly no principle of logic. This statement, then, is not deducible from logic; hence if we want to call it necessary (nonfactual), we have to admit that there are necessary propositions which are not analytic. Some will no doubt reply: “If we knew the analysis of the predicate of this sentence, we would have a definition with the help of which we could demonstrate its analyticity.” But then one would at least have to admit that the statement is not known to be analytic. And since we know it to be true on noninductive evidence, it follows that there is a priori knowledge which is not derived from our (implicit or explicit) knowledge of logic. I anticipate the objection that if the above statement is not factual, then at least I cannot be certain that it is synthetic; after all, I have no ground for asserting that the relation of temporal succession is unanalyzable. But I fully admit that, as Carnap has recently emphasized, we cannot be certain that a given true statement is not analytic unless we assume that our analysis has reached ultimately simple concepts. At least this would seem to be correct as far as nonfactual statements are concerned. What I maintain is only that, if by a necessary proposition we mean a proposition that is true independently of empirical facts (or that is not disconfirmable by observations), then a necessary proposition may be synthetic, and that therefore “analytic” will not do as an analysans for “necessary.” I propose to show, now, that there is a temptation to beg the question at issue in trying to prove that a necessary proposition like the above follows from logic after all. Obviously, to oﬀer such a proof would amount to the construction of a definition of “x precedes y” with the help of which the asymmetry of this temporal relation could be formally deduced. But one could not significantly ask whether a proposed definition is adequate unless one first agreed on certain criteria of adequacy, i.e., propositions which must be deducible from any adequate definition. Thus, most philosophers would agree that no definition of “xPy” (to be used as an abbreviation for “x precedes y”) could be adequate unless it entailed the asymmetry of P. If it should leave this open as a question of fact, it would be discarded as failing to explicate that concept we have in mind. Now, by enumerating all the formal properties which P is to have, one could not construct a definition suﬃciently specific to distinguish P from all formally similar relations with which it might be confused. If I define P as asymmetrical, irreflexive, and transitive, the relation expressed by “x is greater than y,” as holding between real numbers, would also satisfy the definition. But there is a simple device by which uniqueness can be achieved. I only have to add the condition, “The field of P consists of events.” It is easily seen that with the help of this definition our necessary proposition reduces to a substitution instance of the logical truth, “xPy . [(xPy ⊃ ∼ yPx).Q] ⊃ ∼ yPx” (where “Q” represents the remaining defining conditions for the use of “P”). But is

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it not obvious that acceptance of the definition from which the asymmetry of temporal succession has thus been deduced presupposes acceptance of the very proposition “Temporal succession is asymmetrical” as self-evident? This way of proving that the debated proposition is, in spite of superficial appearances, analytic, is therefore grossly circular. I should say, then, that such propositions as “The relation of temporal succession is asymmetrical, transitive, and irreflexive,” “No space-time region is both wholly blue and wholly red,” are necessary, but that nobody has any good ground for saying they are analytic in any formal sense.7 In general, this seems to me to be true of two classes of necessary propositions, of which the first asserts the impossibility for diﬀerent codeterminates (i.e., determinate qualities under a common determinable quality) to characterize the same space-time region, and the second the necessity for certain determinables to accompany each other. A classical representative of each group will be selected for discussion, viz., “Nothing can be simultaneously blue and red all over,” from the first, and “Whatever is colored, is extended,” from the second. I have already insisted that the statement “∼ (∃x)(∃t)(blue xt .red xt )” is not deducible from logic. Indeed, if “blue” and “red” designate unanalyzable qualities, it is diﬃcult to see how analysis could ever reveal that this statement is a substitution instance of a logical truth. Perhaps, however, the above statement (S ) could be formally demonstrated as follows. The sort of entities to which colors may be significantly attributed are surfaces, no matter whether they may be located in physical or perceptual space. But, so the argument runs, if I say “x is blue at t” and also “y is red at t” (where the values of the variables “x” and “y” are names of surfaces), I have already implicitly asserted “x y”; in the same way as simultaneous occupancy of diﬀerent places is tacitly regarded as a criterion of the presence of diﬀerent things at those places. If the proposition here asserted is formalized, we obtain: (∀x)(∀y)(∀t)(if blue xt .red xt , then x y) (T ). Obviously, S follows from T , hence we may say that “if T , then S ” is analytic. But thus we would first have to prove that T is analytic before we could assert that S is. As I follow the rule, “Any proposition is to be held synthetic unless it is derivable from logic alone,” I hold T to be synthetic until such time as conclusive proof of the contrary is produced. And the same applies, of course, to S . Such left-wing positivists as practice an “informal” (or “non-formal”?) method of linguistic analysis will probably disown this kind of discussion as too formal. As I promised, an examination of their linguistic theory of a priori truth is to follow. At the moment I only want to point out that I would not find it enlightening to be told: “S is obviously analytic, since in calling a given 7I

postpone examination of the familiar argument of the “verbalists” that such propositions are analytic in the sense that anybody who denied them would be violating certain linguistic rules.

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part of a surface ‘red’ we already implicitly deny its being blue. ‘Non-blue’, that is, forms part of the meaning of ‘red’.” I do not find this easy argument in the least cogent, since the only meaning I can attach to the statement “ ‘Non-blue’ is part of the meaning of ‘red’ ” is just “ ‘x is red at t’ entails ‘x is not blue at t’,” and the question at issue is just whether such an entailment may be regarded as analytic (or formal). Surely, “non-blue” could not be an element of the concept “red” in the sense in which “male” is an element of the concept “father.” Otherwise it would be diﬃcult to understand why any intelligent philosophers should ever have held it possible that there should be unanalyzable qualities, and specifically that color qualities should be such. Next, let us consider the statement, “If x is colored, then x is extended,” which may be classified together with such necessary propositions as, “if x has a pitch, then x has a degree of loudness,” “if x has size (i.e., length, area, or volume), then x has shape.” Which determinate forms of these determinables are conjoined in a given case is contingent, but that some determinate form of the second should accompany any given determinate form of the first is generally held to be necessary. Here again, the reason for our inability to deduce these propositions from pure logic would seem to be the fact that the involved predicates cannot be analyzed in such a way as to transform the propositions into tautologies; they can only be ostensively defined. I shall examine two counterarguments with which I am familiar. To what else, it is asked, could colors be significantly attributed except surfaces?8 If to nothing else, then only names or descriptions of surfaces are admissible values of “x” in the debated universal statement. But then each substitution instance is analytic, since it has the form, “If this surface is colored, then it is a surface,” which reduces to, “If this is a surface and colored, then it is a surface.” And therefore the universal statement itself, which may be interpreted as the logical product of all its substitution instances, must be analytic. Notice that a similar argument also would prove, if it were valid, the analytic nature of the other necessary propositions of the same category. Pitch can be meaningfully attributed only to tones; it is not false so much as meaningless to say of a smell or feeling that it has a given pitch. In fact, we mean by a tone an event characterized by pitch, loudness, and whatever further determinables be considered “dimensions” of tones. And to say, “If a tone has a certain pitch, then it has a certain loudness,” is, then, surely analytic. But such arguments beg the question. The statement, “Only surfaces can significantly be said to have a color,” diﬀers in an important respect from such statements as “Only animals (including human beings) can significantly be

8 There

may be some who wish to defend the possibility of colored points. Whether such a concept is meaningful is, however, a question of minor importance in this context, since we can easily stretch the usage of “surface” in such a way that points become limiting cases of surfaces.

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called fathers,” or “Only integers can significantly be called odd or even.” For the latter statements involve analyzable predicates and may well be replaced by the statements, “Fathers are defined as a subclass of human beings,” “Oddness and evenness are defined as properties of integers,” while we cannot assume, without begging the very question at issue, that “x is colored” entails by definition “x is a surface.” Similarly, if “x is a tone” is short for “x has a pitch and x has a degree of loudness and . . . ,” then to say that pitch is significantly predicable only of tones is to say that pitch is significantly predicable only of events of which loudness is also predicable. But this semantic statement cannot be replaced by the syntactic statement “‘x has pitch’ entails by definition ‘x has loudness,”’ unless the question at issue is to be begged.9 The second argument in support of the thesis that a necessary proposition like “If x is colored, then x is extended” is analytic can be stated very briefly. If we had the analysis of “x is colored,” we could deduce the consequent from the antecedent and would therefore see that the connection is analytic. Here my reply is twofold. (1) Nothing has been proved that I wish to deny. I contend only that there are necessary propositions, i.e., propositions which are known to be true independently of empirical observations, which are not known to be analytic. One might, indeed, insist that no proposition can be known to be necessary before it is known to be analytic. But I propose to show shortly that this view is untenable. (2) If it is stipulated in advance that an analysis of the antecedent will be correct only if it enables deduction of the consequent, it is not surprising that any correct analysis of the concept in question will reveal the analytic nature of the statement. Consider the following parallel. Everybody would agree that the proposition expressed by the sentence, “if x = y, then y = x,” is necessary; quite independently of our knowledge of logic, one feels that it would be self-contradictory to deny any substitution instance thereof. But as long as the relation of identity of individuals remains unanalyzed, there is no way of deducing it from logic. “If xRy, then yRx” is not true by its form. Now, one will be perfectly safe in claiming that this proposition will turn out to be analytic once the involved relation is correctly analyzed. For formal deducibility of this proposition from logical truths will be one of the criteria of a correct analysis of identity. Indeed, if Leibniz’s definition of identity,

9 It

would be irrelevant to point out that pitch is physically defined in terms of frequency, which is by definition a property of waves with definite amplitude; and that the physical definition of loudness is just amplitude. In the first place, it is causal laws that are here improperly called “definitions”: pitch may be produced by air vibrations of definite frequency, but nobody means to talk about air vibrations when referring to pitch. Let it not be replied that if I am not talking about such inferred physical processes I must be discussing an empirical law of psychology concerning correlations of sensations of pitch with sensations of loudness. I use the word “pitch” as it is used in such sentences as “The pitch of the fire siren periodically rises and falls”: “pitch,” here, refers to a power of producing certain auditory sensations—if the phenomenalist analysis of material object sentences is correct—and such a power would exist even if nobody actually had any auditory sensations.

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(∀P)(Px ≡ Py), is used, the symmetry of identity becomes deducible from the symmetry of equivalence, which is in turn deducible from the commutative law for conjunction. Is it my contention, then, that even a “formal” statement, as it would commonly be called, like, “for any x and y, if x = y, then y = x,” is a synthetic a priori truth? This would, indeed, amount to going more Kantian than Kant himself; for, on the same principle, it could be argued that all logical truths, which Kant at least conceded to be analytic, are synthetic. Take, for example, the commutative law for logical conjunction, just mentioned. Obviously, I cannot prove that “(p and q) ≡ (q and p)” is tautologous, unless I first construct an adequate truth-table defining the use of “and.” But surely one of the criteria of adequacy for such a truth-table definition consists in the possibility of deriving the commutative law as a tautology. If, for example, a “T” were associated with “p and q” when the combination “F T” holds, and an “F” when the combination “T F” holds, the resulting definition would be rejected as inadequate just because it would entail that the commutative law is not a tautology. Indeed, I should belabor the obvious if I were to insist that the laws of logic are not known to be necessary in consequence of the application of the truth-table test, but that the truth-table definitions of the logical connectives are constructed with the purpose of rendering the necessity of the laws of logic (or at least of the simpler ones, like the traditional “laws of thought”) formally demonstrable.10 But my point can be made far more clearly if the term “synthetic a priori” is not used, since it is used neither clearly nor consistently in Kant’s writings. Philosophers who regard “analytic” as the only clear analysans of “necessary” are inclined to hold that we have no good ground for calling a given proposition necessary unless we can formally deduce it from logic. This, however, amounts to putting the cart before the horse. In most cases it is impossible to deduce a proposition from logic unless one or more of the constituent concepts are analyzed—as I have already illustrated more than once. But we accept such an analysis as adequate only if it enables the deduction of all necessary propositions that involve the analyzed concept. We therefore must accept some propositions as necessary before we can even begin a formal deduction.11 One more illustration may be helpful to clarify my thesis. Several logicians are at the present time engaged in the construction of a definition of the central concepts of inductive logic, viz., confirmation and degree of confirmation. But their analytic activities would be altogether aimless if they did not lay

10 I

shall nonetheless belabor this point at some greater length in the sequel. here use “formal deduction” in the sense of “deduction from logical truths alone,” not in the sense of “deduction by (with the help of) logical rules.” In the latter sense, empirical propositions are, of course, likewise capable of formal deduction. 11 I

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down beforehand certain criteria of adequacy, such as the following: the degree of confirmation of a proposition relatively to specified evidence does not vary with the language in which the proposition is formulated; hence, if “degree of confirmation” is treated as a syntactic predicate of sentences, logically equivalent sentences should have the same degree of confirmation relatively to the same evidence. Similarly, if evidence E confirms hypothesis H, and H is logically equivalent to H , then E must also confirm H . Unless these propositions are accepted as intuitively necessary—or, if you prefer, “true by the ordinary meaning of ‘to confirm”’—by all competent inductive logicians, the latter will never agree as to what definition of the concept is adequate. It might be suggested that all we could mean by calling such propositions “necessary” is that if we had suitable definitions we could formally deduce them. But to say of a definition that it is “suitable” is to speak elliptically: suitable for what? Evidently they must be suitable for deducing those very propositions. The proposed analysis of “necessary,” then, reduces to the following: p is necessary if and only if with the help of definitions that enable the deduction of p, it is possible to deduce p. It hardly needs to be explicitly concluded that on the basis of this analysis any proposition would be necessary. Carnap and his followers will undoubtedly protest against this analysis of what they are doing. Those criteria of adequacy which I interpret as preanalytically necessary propositions they would simply call conventions in accordance with which a definition is to be constructed. I should not, of course, deny that a logical analyst may specify such criteria of adequacy without committing himself to any assertion of their necessary truth. Just as a theoretical physicist who is more interested in elegant mathematical deductions than in the discovery of experimental truth may work on the problem of constructing a theory from which some arbitrarily assumed numerical laws would follow, so the logical analyst may formulate his problem merely as the construction of a definition which will satisfy some arbitrarily stipulated conditions. But, to spin this analogy a little further, just as nobody would regard that physicist’s work as being in any way relevant to physics, so the logical analyst’s constructions will have no relevance to the problems of analytic philosophy if the criteria to which they have to conform have no cognitive significance. It might be replied that such conventions are not held to be arbitrary; that, on the contrary, their choice is limited by the dictates of intuitive evidence. But in that case I suggest that what may properly be called a “convention” is the act of selecting some necessary propositions involving the concept to be analyzed as criteria, and not the object selected; the latter is a proposition, and there is no more literal sense in calling a proposition a “convention” than there is in calling a color a sensation or in calling a murdered bird a “good shot.” The results so far obtained may also throw some light on the status of socalled explicative propositions, which occupy a prominent place in analytic

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philosophy. Thinking of the literal meaning of the word “analytic” (dividing, separating), it is, of course, natural to suppose that explicative propositions, like “A father is a male parent,” are analytic. But are they analytic in the sense of being deducible from logic? I want to call attention to the consequences of the triviality that unless certain definitions are supplied, “A father is a male parent” is no more deducible from logic than, say, “A father is a mature person with a keen sense of responsibility.” Relatively to the definition “father =d f male parent,” our explicative statement becomes obviously deducible from the law of identity. But “father” could arbitrarily be defined in such a way that the explicative statement which we regard as necessary would become synthetic, and other statements involving the subject “father,” normally interpreted as empirical statements, would become analytic. Only, such definitions would be rejected as inadequate (in traditional terminology, as merely nominal, not real). We have to admit, then, that by an adequate definition of “father” we understand one with the help of which necessary statements that involve the word “father,” and only such statements, become formally deducible or analytic. It follows that to say of a statement that it is necessary is diﬀerent from saying that, relatively to such and such transformation rules, it is analytic. It is now time to face the objections I expect from the camp of the Wittgensteinian “verbalists.” “Your point is trivial,” they will say. “Nobody has ever maintained that necessary propositions formulated in natural, non-formalized languages are analytic in the sense of being logically demonstrable on the basis of explicitly formulated semantic rules. When we assert that a necessary statement is the same as an analytic statement we use the word ‘analytic’ in the broader sense of ‘true by virtue of explicit or implicit rules of language.”’ The question I now want to raise is: What precisely is meant by an “implicit rule of language”? It is not any sort of insight or intuition, according to the verbalists, which makes a man know the proposition, “If A precedes B, then B cannot precede A” (p), but merely an implicit rule governing the usage of the verb “to precede.” Presumably this means that people familiar with the English language follow the habit of refusing to say “B precedes A” once they have asserted “A precedes B”—provided, of course, that they are serious and mean what they say. Now, it seems to me very obvious (as it may have seemed to such anti-verbalists as Ewing) that this is a grossly incorrect account of what makes a proposition like p necessary. Just suppose that the linguistic habits of English-speaking people changed in such a way that the verb “to precede” came to be used the way the verb “to occur at the same time as” is now used. People would then be disposed to say, on the contrary, “If A precedes B, then B must precede A.” If the verbalist theory were correct, the proposition expressed by the sentence “if xPy , then not-yPx ” would not be necessary, but in fact self-contradictory, in such a changed sociological world. To be sure, the proposition which was formerly expressed by this sentence would remain nec-

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essary, as the verbalist is certain to point out. But if the modality of p is thus invariant with respect to changes in its sentential expression, in what sense can it be said that the modality of a proposition “depends upon” linguistic rules? Notice that I am not misinterpreting the verbalist thesis, as some may have been guilty of doing, to assert that necessary propositions are propositions about linguistic habits and hence a species of empirical propositions. Of course, nobody would maintain (I hope) that in asserting p one makes an assertion about implicit linguistic rules or linguistic habits. What I take the verbalists (like Malcolm, for example) to claim is that the existence of certain linguistic habits relevant to the use of a sentence S is a necessary and suﬃcient condition for the necessity of the proposition meant by S . As I think such a fact is neither a suﬃcient nor a necessary condition for the necessity of a proposition, I reject the verbalist analysis of what a necessary proposition is. It is tempting to regard the existence of a certain linguistic habit relevant to some constituent expressions of S as a suﬃcient condition for the necessary truth of what S means, through some such reasoning as this: If there exists a verbal habit of applying the word “yard” to distances of three feet and only to such distances, then the proposition expressed by “Every yard contains three feet” is identical with the proposition that every yard is a yard; hence, given that linguistic convention and no further facts at all, the truth of the proposition follows. To detect the flaw in this argument we only need to ask, “follows from what?” What is tacitly assumed is that the law of identity, of which the proposition “every yard is a yard” is a substitution instance, is a necessary truth. If it were not, no amount of linguistic conventions would suﬃce to make any proposition necessary. The verbalist may reply that the law of identity (if p, then p) itself derives its necessity from a certain linguistic habit as to the usage of the expression “if, then.” And I would similarly maintain that the existence of such a habit is at best a suﬃcient ground for saying that “the proposition expressed by ‘if p, then p’ is identical with the proposition expressed by ‘not (p and not-p)’, and hence the first proposition is necessary if the second is.” And how could it be maintained that the existence of a certain linguistic habit is a necessary condition for the necessity of a given proposition? If linguistic habits were to change in such a way that, say, a length of two feet came to be called a “yard,” then, of course, the proposition now expressed by the sentence “Every yard contains three feet” is false, and hence not necessary. But surely the proposition which was formerly expressed by that sentence remains necessary? That proposition is eternally necessary, if you wish, in the sense that any sentence which happened to express it would be true independently of empirical facts, including the sociological facts which the verbalists call “implicit rules.” If the rules by which a given sentence now expresses a proposition p were to change in such a way that the same sentence came to express a diﬀerent proposition, p would still be necessary if it ever was.

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Notice that “If not A, then the proposition expressed by S is not necessary” is not synonymous with “If not A, then S does not express a necessary proposition.” If “A” refers to the existence of certain linguistic habits by which the meaning of the sentence S is determined, then the latter statement may be true. But the first statement would be false, since it does not depend on the contingent verbal expression of a proposition whether the latter is contingent or necessary. This would remain the case even if “The proposition p is necessary” were a mere mode of speech, short for “Any sentence which meant p would be necessary.” If linguistic rules change in a certain way, a given sentence may cease to mean p; but it will still be true that it would be necessary if it did mean p. I may as well take the opportunity to call attention to a neat paradox in which the verbalist thesis entangles itself. It asserts the synonymity of the following two statements: (A) it is necessary that every yard contains three feet; (B) “every yard contains three feet” (S ) follows from the rules governing usage of the constituent terms. But rules, especially implicit rules (= linguistic habits), are not propositions from which any proposition could follow. Hence B should be modified as follows: S follows from the proposition asserting the existence of those rules (abbreviate this existential proposition by “S ”). Now, S is empirical, and whatever follows from an empirical proposition, in the ordinary sense of “to follow from,” is itself an empirical proposition;12 which contradicts the original assumption. On the other hand if the statement which A asserts to be necessary is necessary, then it either does not follow from any empirical proposition (viz., if “to follow” is used in the sense in which it is nonsense to make a statement like “‘If it is hot, then it is hot’ follows from ‘It is cold now”’), or else it vacuously follows from any empirical statement. But in the latter case it will be true independently of what linguistic habits happen to exist, if true at all. And therefore B might as well be changed into “‘Every yard contains three feet’ follows from the rules which governed usage of ‘yard’ 50,000 years ago.” In case this paradox should be held to apply only to an unfortunate formulation of verbalism, I proceed to advance a more serious argument against verbalism. To say that it is an implicit semantic rule to apply “B” to anything to which “A” is applicable is presumably equivalent to saying “People who are acquainted with the language never refuse application of ‘B’ to anything to which ‘A’ is applicable.” But how could it be maintained that observation of such a habit is suﬃcient ground for holding that the proposition “If anything 12 It

is, of course, one thing to argue that, on the verbalist theory, necessary propositions are really a species of empirical propositions, and another thing to argue that modal statements of the form “p is necessary” are empirical, if the verbalist theory is correct. I am not sure whether the latter would amount to a pertinent criticism of the verbalist theory, since I am not convinced that “It is necessary that p” entails “It is necessary that it is necessary that p.”

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is A, then it is B” is necessary? Is it not easily conceivable that people use language that way because they firmly believe that whatever has in fact the property A also has in fact the property B? How, then, can observation of such habits be a reliable method for distinguishing necessary propositions from empirical propositions? I notice, for example, that people apply the word “hard” to certain things. Although I have never troubled to carry out the experiment, I am quite sure that if I asked anybody who calls a thing “hard” whether that thing is weightless, he would say “of course not.” But I would not hence infer, and I doubt whether any verbalist would, that the proposition “Nothing that is hard is weightless” is necessary. The verbalist may reply, “You have oversimplified my thesis. To make the sort of observations you describe is not enough. In order to be sure that ‘if A, then B’ is a necessary proposition, you must moreover get a negative reply to the question: “Can you conceive of an object to which you would apply ‘A’ and would refuse to apply ‘B’?”’ This rejoinder, however, amounts to an unconditional surrender of verbalism. If the final test of necessity is the inconceivability of the contradictory of p, then what linguistic rules happen to be followed by people is irrelevant to the question whether p is necessary.13 A knowledge of linguistic rules is necessary only for knowing what proposition it is that a given sentence is used to express. Once this is determined, we all discover the necessity of the proposition in an intuitive manner, viz., by trying to conceive of its being false, and failing in the attempt. Before I embarked on a critique of the linguistic theory of logical necessity, I endeavored to show that “p is necessary” (in the sense of “p is true independently of empirical facts” or, in the Leibnizian language revived in Carnap’s pure semantics, “p is true in any possible world”) cannot be synonymous with “p is analytic,” since the analytic nature of a proposition is presumably knowable by formal demonstration, while an infinite regress would ensue if formal demonstration were the only available method of knowing the necessity of a proposition. In conclusion I shall apply this thesis to the propositions of logic, i.e., true statements containing no descriptive terms (Quine). It is tempting to suppose that with the help of the truth-table method such fundamental proposi13 In

an unpublished paper by a Wittgensteinian friend of mine I have seen the following analysis of the verbalist theory that all necessary propositions are verbal: “Whatever the sentence or combination of signs may be which expresses a given necessary proposition, it is always possible to ascertain the truth of the proposition by ascertaining the syntactic and part of the semantic rules which govern the constituents of the combination.” This statement seems to me to be equivalent to the statement, “If S expresses a necessary proposition, then, in order to know that S is true, it is suﬃcient to know what proposition it expresses.” If this is what the verbalist thesis amounts to, I have no quarrel with it at all; but I should say that “verbalism” is in that case merely a redundant, and moreover misleading, name. Actually, however, I think verbalists want to assert more than this; they want to assert that the necessity of a proposition is somehow produced by linguistic conventions, and this I hold to be a fallacy. That S expresses a necessary proposition is, of course, a consequence of linguistic conventions, simply because it is a consequence of linguistic conventions that it expresses the proposition which it does express. But the verbalists slip in their inference that the necessity of the proposition expressed by S is a result of linguistic conventions.

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tions of logic as the law of the excluded middle or the law of non-contradiction could be shown to be true in any possible world (or, using more formal language, true no matter what the truth-values of their propositional components may be) in purely mechanical fashion, without any appeal to intuitive evidence. Although this view may have the weight of authority behind it, I consider it gravely mistaken. The lights of intuitive evidence can be turned oﬀ only if the T’s and F’s of the truth-tables are handled as arbitrary symbols with no meaning at all. But in that case one obviously does not establish, for example, that “p or not-p” expresses a proposition true in all possible words; one only establishes the far less interesting syntactical theorem that it is a T-formula—which is a result of the same order as, say, that “x2 = 4” is a quadratic equation. In order to establish the semantic theorem first mentioned, I have to interpret “T” and “F” as meaning “true” and “false” respectively. Once this is done, the primitive truth-tables for the primitive connectives “not,” “or” are really semantic rules.14 What, now, is the principle of selection from all the formally possible semantic rules or truth-table definitions? It would seem to be the following: a semantic rule is adequate if it enables the demonstration of the T-character of those basic propositions, like the laws of non-contradiction, excluded middle, etc., which we already know to be necessary in the sense of being true no matter what the empirical facts may be. A keen student of logic should laugh in his teacher’s face if he were told that with the help of the truthtables the “laws of thought” which we always take for granted can be formally demonstrated as necessary propositions. For he should quickly apprehend that in deciding to assign to each elementary proposition at least and at most one of the two truth-values “true” and “false,” one has already assumed the law of the excluded middle and the law of non-contradiction. I do not, of course, deny that the law of the excluded middle, or any of the similarly simple laws into which it is transformable, can be formally demonstrated as a T-formula (or better, tautology, to use the semantic term) without circularity, if only the distinction between object-language and meta-language is observed. What I claim is that its necessity is not known in consequence of such a formal test; that, on the contrary, the semantic rules which render it demonstrable are chosen in such a way that those and only those formulas will turn out as T-formulas which express propositions that are materially known to be necessary truths or follow from such propositions in an axiomatically developed logic. Similar comments would apply to state-description tests of necessity, if such should be proposed. The definition of a necessary proposition as one that holds in any state-description is, of course, formally unobjec-

14 Carnap

interprets them, in Carnap 1942, as truth-rules; but since he accepts Wittgenstein’s principle that to know what a sentence means is to know the truth-conditions of the sentence, he would undoubtedly agree with the above interpretation.

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tionable. But unlike such definitions as “A square is an equilateral rectangle” it does not indicate a method of verifying that the definiendum applies in a given case. The formation rules defining “state-description” are deliberately constructed in such a way that no state-description can be incompatible with such recognized necessary propositions as the law of non-contradiction. One only needs to refer to the “stipulation” that a state-description is to contain any atomic sentence that can be formulated in the given language or its negation, but not both!

Chapter 5 NECESSARY PROPOSITIONS AND LINGUISTIC RULES (1955)

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Are there Necessary Propositions?

Logical empiricism, a powerful and in many ways sanitary philosophical movement which was started by the “Vienna Circle,” propagated in England mainly through Wittgenstein, and in the United States mainly through Carnap, has always been committed to some kind of “linguistic” or “conventionalist” theory of necessary propositions, though it would be diﬃcult to pin down a party line as regards the precise form of such a theory. Such jargons as “the laws of logic are rules of the transformation of symbols,” “all a priori knowledge consists in decisions concerning the use of symbols,” are well known to students of logical empiricism. “Logic formulates rules of language—that is why logic is analytic and empty,” writes a famous logical empiricist. In a sense this theory denies that there is such a thing as a priori knowledge, knowledge of necessary propositions. If, as Schlick wrote, “7 + 5 = 12” is just a rule of symbolic transformation, telling us that we may interchange “12” and “7 + 5” in any context, and a proposition is something that is true or false and that may be believed or disbelieved, then this equation does not express a proposition. To say “I know (indeed, I know for certain) that 7 + 5 = 12” would be either to misuse the word “know” or to use it in a diﬀerent sense from what might be called “propositional knowledge”: it would be a case of knowing a rule, like knowing a rule of chess, and therefore the object of knowledge in this sense of the word would not be a specially exalted kind of truth, traditionally called “necessary” or “a priori” truth. In this sense, it would not be incorrect to ascribe to the logical empiricism of the Vienna Circle and its descendants the view that all knowledge in the sense of propositional knowledge—in contrast to knowledge in the sense of acquaintance as well as knowledge in the sense of knowing how to do something (“he knows how to play the piano,” e.g.)— is empirical knowledge. As a matter of fact, this conclusion was in perfect harmony with another basic tenet of logical empiricism, the famous verifiability principle of cognitive meaning. At first glance, the latter may seem to

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be neutral with respect to the question whether a priori knowledge is possible. There are many diﬀerent methods of verification, of establishing the truth of a proposition; why not recognize purely intellectual operations, the sort of operations by which mathematicians and logicians establish their theorems without recourse to empirical data, as one such method? And are not, then, such sentences as “(a + b)2 = a2 + b2 + 2ab” verifiable and therefore, by the verifiability principle, cognitively meaningful, i.e. linguistic expressions of true-or-false propositions? However, “S is verifiable” was meant in the sense of “it is logically possible to verify S .” Clearly, it is not logically possible to verify a self-contradictory sentence1 , like “there are round squares”: if it is self-contradictory to suppose that there is a round square, then it is self-contradictory to suppose that anyone perceives one. Self-contradictory sentences, therefore, must be ruled out as cognitively meaningless. But if a sentence is cognitively meaningless, so is its negation. Therefore analytic sentences must likewise forego cognitive significance: in saying “no squares are round” we express our knowledge of how words are conventionally used, but there is no such thing as knowing that no squares are round—there just is no such proposition waiting to be known. Since the disjunction “either analytic or self-contradictory or empirical” was regarded as exhaustive by the logical empiricists, their verifiability principle of meaning committed them indeed to identification of genuine propositions with empirical propositions, i.e. propositions to be validated by experience and in principle refutable by experience2 . However, most logical empiricists, I think, shy away from the flat doctrine that there are no necessary propositions. After all, they hold that there are no necessary propositions that are not analytic—though some of them have become aware of grave diﬃculties besetting the explication of “analytic”—and they do not mean to assert this as a trivial consequence of there not being any necessary propositions; they do not mean to assert it, that is, in the sense in which one might truly assert that there are no unicorns that do not live in the Bronx zoo. Nevertheless, they hold that the concept of a necessary proposition, which they identify with the concept of an analytic proposition in some suitable sense of “analytic,” is analyzable in terms of concepts referring to language and linguistic rules. Confusedly, but strongly just the same, it is felt that the necessity of a proposition is somehow “rooted” in linguistic rules—the 1 It

is here assumed that “to verify” means “to establish as true.” Schlick must have had this meaning in mind when he drew from his verifiability principle the consequence—criticized later by Carnap, in Carnap 1937—that self-contradictory statements are meaningless. The consequence could easily have been avoided by interpreting “to verify” in the sense of “to establish as true or false,” for if S is self-contradictory it is logically possible to establish the falsehood of S (viz. by analysis and-or formal deduction), and hence it is logically possible to establish the truth of S or the falsehood of S . 2 Both Carnap and Ayer seem to have attempted an escape from this consequence by restricting the domain of application of the verifiability principle (or its liberalized form, the confirmability principle) to synthetic, or “putatively factual” sentences. But such a restriction seems to make the meaning criterion itself redundant: before we could apply it to a sentence, we would have to make sure that the latter is synthetic, but this involves making sure that it expresses a proposition: hence one would have to decide the question of cognitive significance before application, and therefore independently, of the verifiability principle.

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problem being how to get beyond such metaphorical suggestions as “rooted.” I wish to prove, now, that though the linguistic theory of necessary propositions claims to do no more than analyze a concept which admittedly has instances, it entails paradoxically that the analyzed concept has no instances. The view that necessary statements owe their truth to linguistic conventions alone, such that they could be deprived of necessity by arbitrarily changing linguistic conventions, has some surface plausibility if one thinks of such statements as “there are no married bachelors” (S ) and of linguistic conventions in the form of explicit definitions, like: “bachelor” is synonymous with “unmarried man.” But the air of plausibility will vanish once we take a look below the surface. The cited definition, or statement of synonymy, entails that S must be true if “there are no married men who are unmarried” (T ) is true. As Quine has put it, in “Truth by Convention” (Quine 1936), explicit definitions transmit truth from one statement to another, they cannot generate truth. Further, if the derivability of S from T is to prove that S is necessary, T itself must be necessary. If the descriptive terms occurring in T , viz. “married” and “men,” are primitives, no further explicit definitions can be invoked as the source of the necessary truth of T ; and if they are not primitive and nevertheless T is necessary, the process of reduction to primitive notation will eventually lead to a statement whose necessary truth is independent of explicit definitions. Now, one might refer to a logical principle of which T is a substitutioninstance as the ground of T ’s necessity: ∼ (∃x)(∃P)(∃Q)(Px.Qx. ∼ Qx). How could definitions in the sense of rules of interchangeability of synonyms be relevant to the assumed necessity of the logical principle? To be sure, if “(p.q) ⊃ p” seems more evident than “∼ (p.q. ∼ p),” we can invoke the definition “p ⊃ q ≡d f ∼ (p. ∼ q)” in order to drive the process of validation still “deeper,” and if we attach the highest degree of self-evidence to logical principles in the disjunctive normal form, we may eventually appeal to “ ∼p ∨∼q ∨ p”—provided “∼” and “∨” are the primitives of our logical system. But clearly we must sooner or later reach necessary statements in primitive notation, whose necessity therefore cannot depend on the kind of linguistic rules we have considered. “But such statements are simply implicit definitions of the primitives occurring in them; the linguistic rules which fix the meanings of the primitives take the form of postulates,” the conventionalist will reply. Although a separate section will be devoted to this notion of “implicit definition,” let us note immediately that the conventionalist who takes this line has already reduced the concept of “necessary truth” to absurdity. For if such allegedly necessary statements as S derive their truth from postulates which are assignments of meanings to expressions, then they derive their truth from sentences to which truth cannot significantly be ascribed—which is to say that they themselves cannot significantly be called “true.” If a sentence like “any two points determine a straight line” is interpreted as a postulate which, though

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incompletely only, specifies what “point” and “straight line” mean, then it says in eﬀect: let “point” mean any element x and “straight line” any class of elements A such that any two x’s determine one and only one A. And such a sentence clearly does not express a proposition—it rather expresses a proposal regarding the use of symbols. Suppose, now, that the conventional definitions which are alleged to be the source of necessary truth are statements about the conventional usage of certain expressions of a natural language (“natural language” is contrasted with “language system”), such as “people who understand English never apply ‘father’ to an object x unless they are willing to apply ‘male’ and ‘parent’ to x and vice versa.” Alternatively, this proposition of descriptive semantics could be expressed as follows: People who understand English use ‘father’ and ‘male parent’ synonymously. It is essential to the argument in process of construction that such statements about linguistic usage be admitted to be empirical. One might deny this on the ground that a language is defined by a set of rules, so that anybody who does not know some of these rules eo ipso does not really understand the language. But this argument confuses natural languages with language-systems. A natural language, unlike a language-system, can without contradiction be said to undergo changes. If any of the rules (rules of sentence formation, semantic rules, rules of deduction) which collectively define a given language-system are changed, a new language-system has been constructed. But the very fact that we are making a contingent historical statement about a given natural language L when we describe, say, changes of meaning undergone by a certain expression of L, indicates that a natural language is not similarly definable by a set of static rules. Accordingly, if “understanding English” were defined as “knowing all the rules (at least those in operation during a certain epoch) of usage of English expressions,” this strong definition itself would be in conflict with the conventional rule governing the usage of “understanding English.” And therefore the claim that a statement like “people who understand English use ‘father’ and ‘male parent’ synonymously” is, if true, simply analytic of what is meant by “understanding English,” is untenable. The point is of suﬃcient importance to deserve a little more elaboration. Suppose the use of a language L were compared to playing a game of chess in the following respect: to say that an expression E is used as an expression of L implies that its use is governed by rules which constitute L. If E were used in accordance with a diﬀerent rule, it would not be an expression of L (example: if the graphic sign “hut” is used to refer to hats, then it belongs to German language; if it is used to refer, not to an article of clothing, but a primitive kind of cottage, then it belongs to the English language). Similarly, moves made with chess figures constitute a game of chess only if they are governed by the rules of chess. Consider now the statement “anybody who plays chess moves the bishop diagonally only, not vertically.” Clearly, this

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is not an empirical generalization about what chess players do; it is a statement analytic of “playing chess.” Those philosophers of language who hope to gain insight into the nature of language by looking at linguistic behavior as the same sort of rule-governed behavior as playing a game—while recognizing, of course, diﬀerences in the motivating purposes—might use the analogy to support the view that statements like “anybody who understands English applies the expression ‘father’ only to males” are analytic. “To understand English” analytically entails “applying ‘father’ to males only,” just as “to know how to play chess” analytically entails “moving the bishop diagonally only,” they might argue. However, this is to overwork an analogy which is helpful up to a certain point. It is in fact to ascribe to natural languages features which only language-systems share with games. The rules of usage of, say, English expressions must be discovered empirically by observing how English speaking people commonly use them; they are not norms which are first stipulated, and then conformed to. Rather linguistic habits develop before the grammarian catalogues them, and the norms are taken notice of ex post facto, if at all, by those who learn the language “naturally.” Of course, in order to determine whether a given person speaks English and understands English, and whether his linguistic behavior, accordingly, is evidence relevant to a generalization about the usage of an English expression, one must test his acquaintance with some rules of usage of English expressions. If, for example, a technique of interrogation is used by the descriptive semanticist engaged in a “field trip,” evidence should first be obtained that the subject who is interviewed understands English to the extent of being able to interpret the interview questions correctly. The point to be emphasized is that once this evidence has been obtained, the question is still logically open how the subject will respond to the interrogation and, therefore, which hypothesis about the rules governing the English expression in question his responses will confirm. To illustrate, suppose we wanted to test the semantic hypothesis that as English speaking people use the verb “to see” in such contexts as “I see a snake,” “A sees an x at time t and at place P” entails “an x exists at t and at P.” Our questionnaire might contain a question like “is it possible (conceivable) that somebody sees a snake at t and at P if there is no snake at t and at P?” Clearly, we shall have to make sure that our subject understands the meaning of “snake,” of “there is,” of “conceivable,” and that he understands the meaning of “to see” to the extent of being able to distinguish between “to see” and “to imagine,” to recognize a sentence like “he saw a sweet smell” as nonsensical, etc. But this test of eligibility as experimental subject need not include a response to the very question through which we want to test alternative hypotheses about the meaning of “to see.” To deny the possibility of such non-circular experimental tests would be as gratuitous as to deny the possibility of a non-circular test of the physical hypothesis that all metals are electrical conductors on the ground that we could not be sure

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whether the material at hand is a metal until it had been tested for electrical conductivity. But once such statements about linguistic usage are admitted to be empirical, the question arises how the fact they state can be a reason for the necessity of any statement, such as “there are no fathers who are not male.” Surely, if by pointing to an empirical fact F we oﬀer a reason for accepting a statement p, then p itself must be an empirical, and so a contingent, statement. It is not that a necessary statement could not be entailed by a contingent statement. Indeed, if “p entails q” is synonymous with “it is impossible that p and not-q”3 , then demonstrably any contingent statement entails any necessary statement. But it is precisely because the entailment from a contingent statement p to a necessary statement q is thus “vacuous,” that p could not be a reason for acceptance of q. Nobody would say “any father must be a parent because there are nine planets,” although undoubtedly “there are nine planets” entails “any father is a parent.” Would it be any less absurd to say “any father must be a parent because, at least according to present linguistic conventions, it is incorrect to apply ‘father’ to an object to which ‘parent’ is inapplicable”? At first sight such a statement may, indeed, seem reasonable. After all, would we still say “any father must be a parent” if, say, “father” were used in the sense in which “brother” is used at present? This superficial impression, however, is due to confusing the admittedly contingent statement of the metalanguage “the sentence ‘any father is a parent’ expresses at the present time a necessary proposition” and the necessary statement whose necessity is to be explained, viz. “any father is a parent.” If the indicated change of linguistic rules occurred, the same sentence would henceforth express a non-necessary proposition, but the proposition previously expressed by it would remain necessary if it ever was. Notice also that, say, a German advocating the same linguistic theory of necessary truth would give a completely diﬀerent reason for holding the same proposition to be necessary, for he would refer to the usage of the German expressions “Vater” and “Elter,” not to the usage of the English expressions “father” and “parent.” This indicates that something has gone wrong somewhere. Some philosophers may be quick to point out that what has gone wrong is that the statement of the object-language “any father is a parent,” whose necessity was to be accounted for, has been confused with the modal statement of the meta-language “‘any father is a parent’ is necessary.” The latter statement, they might say, is not necessary, and therefore it can without absurdity be validated by reference to such empirical facts as linguistic habits. And from the fact that

3 Some

have rejected this definition precisely because it has the “paradoxical” consequences that an impossible proposition entails any proposition and that a necessary proposition is entailed by any proposition. But I think their arguments are untenable. See chapter 11.

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“‘p’ is necessary” is contingent it does not follow, they would say, that “p” is contingent. Therefore the linguistic theory of necessity is compatible with there being necessary propositions. It is important to show carefully the error of this position, for it is held by some analytic philosophers of eminence. Thus G. E. Moore wrote in his essay on “External and Internal Relations” (Moore 1922) that “p entails q” is not itself a necessary proposition, but that “p ⊃ q” is a necessary proposition if “p entails q” is true. Since “p entails q” means “‘p ⊃ q’ is necessary,” Moore’s statement is equivalent to a denial of what I shall call the NN thesis: that if a statement is necessary, then it is necessary that it is necessary (in other words, necessary statements are necessarily, not contingently, necessary). While Moore made this claim en passant, without supporting it by any argument, more recently Strawson has argued in favor of the view that entailmentstatements are contingent statements about the usage of expressions, and that it is therefore a mistake to construct a modal logic with modal operators in the object-language. Also Reichenbach maintains, in Elements of Symbolic Logic (Reichenbach 1947), that the meta-linguistic statement to the eﬀect that a given formula of the object-language is a tautology is, even if true, not itself a tautology. From Quine’s writings on analyticity one gathers that he too regards the NN thesis as at least problematic. For, using “analytic” and “necessary” interchangeably, he holds that the concept of necessity is obscure to the extent that it is wider than the concept of logical truth, because the concept of “synonymy” enters into its definition; and the latter concept he finds obscure because he knows of no operational criterion for deciding questions of synonymy. For example, “all bachelors are unmarried” is not logically true, for the descriptive terms “bachelor” and “unmarried” occur essentially in it, i.e. not all statements of the same form, “all A are B,” are true. If nevertheless it is claimed to be a necessary statement, the reason is that “bachelor” seems to be synonymous with “unmarried man,” and by substitution of “unmarried man” for “bachelor” a logically true statement is obtained. But Quine finds the meaning of such an assertion of synonymy obscure. Now, leaving aside doubts about a sharp criterion of distinction between true statements that are logically true and those that are not, Quine, it seems, would be satisfied with the definition of a necessary statement as one which is either logically true or transformable into such a statement by substitution of synonyms for synonyms, if an operational definition of “synonymy” were at hand. But to ask for an operational definition, in behavioristic terms, of “synonymy” is to presuppose that statements of the form “T is synonymous with T ”4 are empirical. And since the meta-statement

4 Strictly

speaking, such statements are of course incomplete. There should be at least a reference to a group of sign-users and the context of usage. If I omit these variables, it is not because I overlook them, but just for the sake of expository simplicity.

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“ ‘all bachelors are unmarried’ is necessary” would then have to be established empirically, by investigating whether “bachelor” is synonymous for English speaking people with “unmarried man,” it is not itself necessary5 . Now, my argument in support of the NN thesis is that the primary object of a priori knowledge is always a modal proposition. If the latter has the form “N(p),” we pass from our a priori knowledge of the modal proposition to a priori knowledge of the proposition asserted to be necessary via the necessary proposition “if N(p), then p.” Therefore, if “N(p)” means “p can be known a priori, i.e. by just thinking about p, without ‘looking at the world’,” no proposition at all would be necessary if propositions of the form “N(p)” were not necessary. Consider again our simple example of a necessary proposition: there are no fathers that are not male. If we are a priori certain of it, i.e. in advance6 of having observed all fathers, past, present, and future, it is surely because we see that fatherhood entails malehood. This is to say that we see a priori the truth of the modal proposition “father(x) male(x)”—alternatively expressed as an assertion of impossibility “∼ 3(father(x) . ∼ male(x))”—and hence derive the corollary (∀x)(father(x) ⊃ male(x))—alternatively expressed as an assertion of non-existence “(∃x)(father(x).male(x)).” The point is that since it is our knowledge of the entailment which is the ground of our certainty with regard to the universal proposition, the universal proposition would itself have only the status of inductive generalization if the entailment were not known a priori. The argument “there are at no time fathers who are not male, because fatherhood entails malehood” has the form “p, because N(p)”: for “p q” is definable as “N(p ⊃ q).” Therefore our acceptance of p would be based on empirical evidence if our acceptance of N(p) were based on empirical evidence. Therefore one cannot consistently hold that p is necessary and N(p) contingent.

5 Incidentally,

Quine’s definition of “analytic,” as a wider concept than “logically true,” in terms of “synonymy” would be inadequate even if the latter concept could be clarified to Quine’s satisfaction. For concepts of natural kinds which are formed by abstraction usually have a certain amount of intensional vagueness which makes it impossible to give a strict explicit definition of the name of the kind, and yet the intensions are definite enough to permit a number of analytic entailments about the kind to be stated. For example, “all swans are animals” is surely analytic: it is self-contradictory to assert the existence of swans which are not animals. But to complete the definition by stating diﬀerentiating characteristics P1 , . . . , Pn of such a kind that it would be self-contradictory to apply “swan” to an animal which lacked any of them, is very diﬃcult, since we might be strongly inclined to classify an animal lacking Pi as a swan on the ground of its strong resemblance in other respects to animals normally called “swans” (For more details on “intensional vagueness,” see chapter 19 and Hospers 1953, 42-45.) 6 I deliberately exploit the etymological meaning of “a priori” in this context. In spite of Kant’s warning not to confuse the temporal origin of knowledge with its ground of validity, there is an obvious connection between the temporal and the epistemological meaning of “a priori”: to know a universal proposition a priori is to know it in advance of testing its substitution-instances—though observation of some instances of its constituent concepts may be causally necessary for “having” the concepts.

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It is true that no parallel argument can be constructed with respect to modal propositions asserting possibility, specifically compatibility, i.e. possibility of a conjunction of propositions, since no non-modal proposition about the world of particulars is deducible from such a proposition. But once it is granted that propositions of the form “p is necessary” are not contingent, it ought also to be granted that those of the form “p is possible” are not contingent, since “p is necessary” is equivalent to the negation of “not-p is possible.” For from the necessity of “p is necessary” follows the impossibility, and hence noncontingency, of “not-p is possible,” and by contraposition, from the contingency “p is possible” follows the contingency of its negation “not-p is necessary”7 . The case for the NN thesis is strong also if we consider that special class of necessary propositions which are instances of tautologous truth-functions, i.e. functions of propositional variables which are true for all values of the variables. Take, e.g., a proposition of the form: (p ⊃ q) ⊃ (∼ q ⊃ ∼ p). If we knew it to be true merely on the basis of having empirically ascertained the truth-values of the atomic propositions and then computed the truth-value of the molecular proposition with the help of the definitions, given in the form of “truth-tables,” of the connectives, we would not know it to be necessarily true. But to say that we know it to be necessarily true is to say that we know that every proposition of the same form is true, i.e. that it is an instance of a tautologous truth-function. And surely this knowledge is a priori: by reflecting on the meanings of the logical constants “not” and “if, then” we discover that it is an instance of a tautologous truth-function, hence we know a priori that it is necessarily true. Therefore here again our a priori knowledge of the truth of p is based on our a priori knowledge of the necessity of p.

2.

The Confusion of Sentence and Proposition

In my opinion the view just criticized, that modal propositions are contingent propositions about linguistic usage, arises from confusion of sentence and proposition, a confusion which is a special case of a confusion between using and mentioning an expression. If “it is necessary that p” were meant in the sense of “the sentence ‘p’ is used to express a necessary proposition,” then the NN thesis would be false indeed: propositions about what given expressions mean as used in natural language, are surely contingent. But the very appearance of the expression “necessary proposition” in the meta-linguistic interpretation of “it is necessary that p” indicates that the alleged interpretation is no interpretation at all: after all, “S expresses, in L, a necessary proposition”

7 That a contingent proposition has a contingent negation follows by simple commutation from the definition

of contingency: p is contingent ≡d f p is possible and not-p is possible.

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means:(∃p)(Des(S , p, L).N(p))8 , and thus “N(p)” has not been interpreted at all. Indeed, anybody who can grasp the diﬀerence between direct and indirect quotation, ought likewise to see that when a proposition is asserted to be necessary, nothing at all is asserted about the sentence which happens to be used to mention the proposition. In asserting “Aristotle said that the good is that which all men desire” one is not asserting “Aristotle said ‘the good is that which all men desire”’: indeed, the latter proposition is certainly false, since Aristotle did not speak English. In Frege’s terminology, the proposition that p is the sense of the sentence “p”; in Carnap’s terminology, it is the latter’s intension. To be sure, this explanation leaves the diﬃcult question open what exactly is to be understood by “proposition.” To answer this question is to propose a satisfactory explication of the concept of synonymy, i.e. an exact statement of the conditions under which two sentences express the same proposition. But any such explication would have to be guided by what might be called a “pre-analytic” understanding of the sentence-proposition distinction. Such pre-analytic understanding is, for example, manifested by the statement that “Aristotle said that the good is that which all men desire” reports an assertion of a proposition but says nothing about what sentence was used to assert the proposition. Now, the relation between “he said that p” and “he said ‘p”’ is precisely like the relation between “it is necessary that p” and “‘p’ is used to express a necessary proposition” in the respect noted. There is a simple device, recently used by Alonzo Church (Church 1950) to criticize Carnap’s attempt to translate statements of beliefs and assertions (indirect quotations) into a meta-language mentioning sentences but no propositions, for making it plain that statements of the form “it is necessary that p” are not about the sentences replacing the variable “p.” We simply translate both the interpreted sentence of the object-language and its proposed metalinguistic interpretation into another language. The interpretation is correct if and only if these translations are themselves synonymous. For example, “it is necessary that there are no fathers who are not male” is correctly translated into German by the sentence “es ist notwendig, daß es keine V¨ater gibt die nicht m¨annlichen Geslechtes sind” (S 1 ) but “‘there are no fathers who are not male’ expresses a necessary proposition” translates into “ ‘there are no fathers who are not male’ ist Ausdruck eines notwendigen Urteils” (S 2 ), and clearly S 1 and S 2 are not synonymous. Let us return for a moment to the contrast between direct and indirect quotation. The statements “he said that the earth is round” and “he said ‘the earth is round”’ seem to be alike in violating the postulate of extensionality: the extension of a compound expression remains unchanged if a component ex-

8 “Des(S ,

p, L)” reads: sentence S designates proposition p in language L.

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pression is replaced by a diﬀerent expression having the same extension. Thus substitution for the true sentence “the earth is round” of the true sentence “the moon revolves around the earth” would change the truth-value of both statements. Even if we replace the singular term “the earth” by another singular term having the same extension, i.e. denoting the same object, leaving all else unchanged, we might change the truth-value of the indirect quotation just as we—necessarily—change the truth-value of the direct quotation. For example, it might well be the case that the earth is the only planet which is inhabited by philosophers, among other animals. But although on this assumption “the earth is round” is materially equivalent to “the only planet which is inhabited by philosophers, among other animals, is round,” the indirect quotation obtained by making this substitution would be bound to have the opposite truthvalue from “he said that the earth is round,” since the substituted singular term has a diﬀerent sense. But if this apparent similarity leads one to construe indirect quotation as meta-linguistic statements, then one has been misled by a linguistic accident. It is a linguistic accident that we form a name of an expression, in written discourse, by putting the named expression itself within a pair of quotes. If we used instead an entirely diﬀerent expression as name of the expression, it would be obvious that neither the singular term “the earth” nor the sentence “the earth is round” is a component of the direct quotation at all; the latter therefore satisfies the postulate of extensionality trivially9 , and accordingly a careful comparison of the direct and indirect quotations with respect to extensionality reveals a significant dissimilarity rather than similarity. Mutatis mutandis, the same holds for “it is necessary that p” in comparison to “‘p’ expresses a necessary proposition”10 . Logicians who admit the diﬀerences noted but nevertheless adhere, for certain pragmatic reasons, to the postulate of extensionality, may still attempt to interpret modal statements and indirect quotations (as well as other types of apparently non-extensional statements, above all statements about beliefs) in an extensional meta-language. Thus they might be inclined to interpret “it is necessary that p” in a semantic meta-language11 as follows: for any language 9 It

is precisely for this reason that Carnap, at the time firmly committed to the thesis of extensionality, proposed in Carnap 1954a to translate “A believes that p,” respectively “A says that p,” into formal mode of speech simply by “A believes ‘p’,” respectively ‘A says ‘p’.” 10 In Quine 1953b, Quine maintains that “necessity as statement operator,” i.e. statements of the form “it is necessary that p,” is capable of being reconstrued in terms of “necessity as a semantical predicate,” i.e. statements of the form “‘p’ is necessary. But I have not discovered any better reason, in his paper, for this claim than the superficial similarity between “it is necessary that p” and “‘p’ is necessary” consisting in their non-extensionality with respect to “p.” This similarity, to repeat, is superficial because it is entirely accidental that the named sentence “p” is a component of its name “‘p’.” If it were not for considerations of convenience, we might use meta-linguistic names which do not contain their nominata any more than the name “Rome” contains the city of Rome. 11 The semantic meta-language employed by Carnap in Carnap 1942 is not extensional, since it contains the triadic predicate “Des(S , p, L)” which is not extensional with respect to “p”: if “Des(S , p1 , L)” is true, and

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L and sentence S , if S is synonymous with “p” in L, then S is L-true in L, i.e. true by virtue of the semantic rules of L. While postponing a closer examination of this interpretation of modal statements to the next section, let us just note that the argument from translation into another language retains its force even against more complicated meta-linguistic interpretations such as this one. For if a sentence contains a name N of an expression E, then any correct translation of the sentence will contain a synonym of N, not of E; E will be mentioned by it. But no correct translation of an English sentence “it is necessary that p” into another language will mention the English sentence “p”; rather it will mention the sense (intension) of “p.” It was noted above that the contingent meta-linguistic statement “‘p’ expresses a necessary proposition” in no way sheds light on the meaning of “it is necessary that p,” since in fact its intelligibility presupposes an understanding of that meaning; after all, “‘p’ expresses a necessary proposition” asserts “there is a proposition p which ‘p’ expresses and which is necessary.” But many philosophers of language, foremost those influenced by the teachings of Wittgenstein, will object to the analysis of semantic statements in terms of a designation-relation, holding between expressions and designated entities. Thus Carnap, who speaks of propositions as being designated by sentences, and properties as being designated by predicates, just as he speaks of things as being designated by proper names, has been accused (quite unjustly, I think) of confusing meaning with naming.12 Naming, so the argument runs, is a relation of a symbol to an independently existing entity, but to assume that for every meaningful symbol, even those that are not “names” in any usual sense, there exists an entity which is its meaning, is to be guilty of Platonic reification. More formally, the argument may be put as follows: a statement to the eﬀect that an expression which is not a name in any ordinary sense (e.g. a predicate, a sentence) means such and such does not have the logical form “xRy,” though it may share this grammatical form with a statement identifying the designatum of a name, like “‘Peter Shannon’ is the name of the person I had lunch with today.” As Wittgenstein and Russell have emphasized, grammatical similarities between sentences of natural language mislead philosophers to wrong analyses of their meanings: they take grammatical form naively as a clue to “logical form” (where “logical form” may perhaps be defined as the grammatical form

“p1 ” is just materially, not logically, equivalent to “p2 ,” then “Des(S , p2 , L)” is not true. But in (Carnap 1956a) Carnap attempts to extensionalize the semantic meta-language by taking both statements of designation, the one about the proposition p1 and the one about the proposition p2 , as true, yet distinguishing in the meta-meta-language between statements of designation which are conventions or provable on the basis of conventions, and such as are not, or are “F-true.” 12 See e.g. Ryle 1949b.

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of the translation of the sentence into an ideal language)13 . Consider the semantic statements a) “blue” means the color of the sky, b) “die Erde ist rund” means the proposition that the earth is round14 . Since the phrases following the verb are nouns, or noun-clauses, the naive philosophers of language construe them as names of entities constituting the range of “y” in the sentential function “x means y.” But this amounts to no smaller sin, in the eyes of the opponents of what is sometimes called the “representative theory” of language (in contrast to a “functional theory” of language, whatever it may be), than treating all meaningful expressions as names. And to speak of necessary propositions as of entities designated by those sentences which cannot be denied without violating the conventional rules of usage, is to commit just that sin. The detailed reply to this criticism is deferred to the next section where it will be argued 1) that the construal of meaning as a relation whose converse domain consists of such “abstract” entities as properties and propositions does not commit the crime of Platonic “reification,” 2) that the alternative theory of propositions as “logical constructions” out of sentences is untenable. But in the meantime it should be understood that the validity of the NN thesis does not depend on a sentence-proposition distinction which the mentioned philosophers of language find objectionable. For it holds even if for propositions in the sense of (possible or actual) states of aﬀairs designated by meaningful declarative sentences we substitute statements in the sense of sentences as meaning such and such. Thus, instead of saying that the sentences “A is larger than B” and “B is smaller than A” are ordinarily used to designate the same proposition, let us say that they are ordinarily used the make the same statement; for short, we might say that while being diﬀerent sentences they are the same statement. Now, that the sentence “A is larger than B if and only if B is smaller than A” is used to assert a necessary statement, is indeed a contingent fact about the English language. But that the statement which anybody familiar with the rules of contemporary English would make if he uttered this sentence is necessary, is not a contingent fact. If it were, then it would be conceivable that the same statement might not be necessary. But what is conceivable is only that the same sentence might be used to make a diﬀerent statement which is not necessary. Since the NN thesis thus stands independently of any commitment to “propositions,” it follows that a linguistic theory which interprets modal statements as contingent statements about linguistic rules is committed to the denial of the existence of necessary propositions—or necessary statements. 13 For a discussion of problems raised by this definition of “logical form”—a term abounding in the writings

of Wittgenstein and Russell, and crying out for clarification—see Kalish 1952. avoid the formulation “ ‘the earth is round’ means the proposition that the earth is round,” which occurs in Carnap’s semantical writings, because the semantic statement is not informative if the meta-language is the same natural language as the object-language it talks about, unless the semantic statement is a meaninganalysis within the same language—which a statement of the form “‘p’ means that p” is not.

14 I

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3.

Are Propositions “Logical Constructions”?

Are Propositions “Logical Constructions”?

Although some proponents of a linguistic theory of necessary truth must be accused of simple carelessness in not observing the distinction between “‘p’ is used to a express a necessary proposition” and “it is necessary that p,” it would be unjust to say generally that logical empiricists who combat “rationalism” overlook the sentence-proposition distinction. Nor can they justly be accused of such crude expressions as “it is nonsense to speak of propositions, for they are by definition unobservable entities.” No, the distinction is usually admitted, but an explication of “proposition” is proposed which somehow conforms to the empiricist criterion of cognitive significance: that statements ostensibly about unobservables be given a cognitive meaning by reduction to statements no matter how complicated, about observables. Just as statements about physical objects are reducible, according to phenomenalism, to statements about sense-data—which is what a phenomenalist means by saying that physical objects are “logical constructions” out of sense-data—so the view is sometimes taken that statements about propositions are shorthand, as it were, for statements about sentences, specifically about classes of synonymous sentences. Thus Ayer writes: “Regarding classes as a species of logical constructions15 , we may define a proposition as a class of sentences which have the same intensional significance for anyone who understands them” (Ayer 1952b, 88). And he clarifies the meaning of this definition in the introduction to the second edition of his “manifesto” of logical positivism as follows): Thus, if I assert, for example, that the proposition p is entailed by the proposition q I am indeed claiming implicitly that the English sentence s which expresses p can be validly derived from the English sentence r which expresses q, but this is not the whole of my claim. For, if I am right, it will also follow that any sentence, whether of the English or any other language, that is equivalent to s can be validly derived, in the language in question, from any sentence that is equivalent to r; and it is this that my use of the word ‘proposition’ indicates. (Ayer 1952b, 6)

This relatively clear statement of the logical-construction-theory of propositions will now be used as a platform for criticizing the theory. I shall ignore the minor defect of the theory in Ayer’s formulation, that logical equivalence instead of a stronger relation of synonymy is used for the logical construction of propositions16 . For whatever the diﬃculties involved in a satisfactory 15 Presumably Ayer has in mind the theory of Principia Mathematica that classes

are “incomplete symbols,” i.e. that statements ostensibly about classes are translatable into statements which do not mention classes (but propositional functions instead). 16 If all logically equivalent sentences express the same proposition, then there is but one analytic proposition and but one self-contradictory proposition. But even a definition of “proposition” from which it only follows that any two synthetic sentences which are logically equivalent express the same proposition has counterintuitive consequences: “Mt. Everest is the highest mountain of Asia” is logically equivalent to “anybody

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explication of “synonymy,” the theory could easily be restated in terms of this stronger relation without thereby becoming immune to the criticisms to be presented. However, one serious confusion should be noted. What exactly does Ayer mean by saying that the assertion “p is entailed by q” implicitly claims that the English sentence s which expresses p can be validly derived from the English sentence r which expresses q? I take him to mean “implicit claim” in the sense in which one might say, e.g., the assertion “I have an older brother” implicitly claims that the speaker’s parents have more than one child, i.e. in the sense of entailment. Ayer, then, claims that “q entails p,” where “p” and “q” are propositional constants (left undetermined), entails “if s expresses p and r expresses q, then s is validly derivable from r.” But here the propositions p and q are mentioned again, in the semantic contexts “s expresses p” and “r expresses q,” which contradicts the aim of the translation. If, on the other hand, the antecedent is omitted, and “the proposition that there are husbands entails the proposition that there are wives,” e.g., is claimed to entail “‘there are wives’ is validly derivable from ‘there are husbands’,” the claim is obviously false: for if the sentences “there are wives” and “there are husbands” happened to express logically independent propositions, which is logically possible, neither would be derivable from the other, and such a contingency, of course, could not alter the necessity of the entailment between the propositions. Now, in the second part of the translation Ayer shifts from “any sentence expressing p” to “any sentence (logically) equivalent to s.” But he is silent about the criterion of logical equivalence. If to say that two sentences are logically equivalent is just to say that they express the same proposition, the translation is implicitly circular again. And it would not help to define “s is logically equivalent to t” syntactically as “the same sentences are derivable from s and t.” Suppose, e.g., that s = “I have two brothers” and t = “I have two male siblings.” From t “I have two siblings” is syntactically derivable, i.e. by applying rules of deduction which refer only to the forms of sentences. Is this sentence syntactically derivable from s? Surely not, unless the definition “brother = male sibling” is used as a rule of substitution. But if with the help of this definition I propose to prove that s and t are logically equivalent, I must first prove that it itself expresses a logical equivalence; otherwise it would be easy to prove, by the debated syntactical criterion of logical equivalence, that “I have two brothers” is, say, logically equivalent to “I have two wives,” by using an arbitrary definition of “brother” as, say, “attractive wife.” And thus we either return to “s and t are logically equivalent because they express the same proposition,” or

who has seen the highest mountain of Asia has seen Mt. Everest,” yet since the proposition expressed by the latter sentence contains the concept of seeing, while the proposition expressed by the former sentence does not, we can hardly say the propositions are the same.

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are faced with an infinite regress in attempting to establish logical equivalence in a purely syntactic way. To maintain that propositions are logical constructions out of sentences is to claim that contextual definitions of statements containing names of propositions or propositional variables can be set up whose definientia mention sentences, either by way of names of sentences or by way of sentential variables (not excluding both), but do not mention propositions by way of names of propositions or by way of propositional variables. Of course, it need not be claimed that this problem of reduction could be solved by a single contextual definition that is applicable to all possible contexts of “propositional talk.” One may have to set up separate contextual definitions for diﬀerent types of statements about propositions, e.g. “it is true that p,” “the proposition that p entails the proposition that q,” “so-and-so believes that p,” “it is logically necessary that p,” “any proposition entailed by a true proposition is true.” Let us examine such a contextual definition for “it is true that p.” Since this is just about the simplest kind of “propositional talk,” one would expect the attempted reduction to succeed in this instance if it is possible at all. But I wish to show that even here it fails. Suppose the objection were raised against the “reduction” of propositions to classes of synonymous sentences that 1) a class is just as abstract an entity as a proposition, so that the reduction, even if it were successful, would not be an application of Occam’s razor at all, 2) this identity could not be meant literally, since there are predicates which are significantly applicable to propositions but not to classes: e.g. it does not make sense to say of a class that it is true, or that it entails another class. The proper reply to this objection would be that what is meant is that statements about propositions are translatable into statements containing universal quantifiers that refer to any sentences satisfying certain conditions, but which do not mention classes at all17 . Let us, then, state formally such a reduction of “it is true that p,” which happens to be identical with the reduction of the “absolute” concept of truth (truth as predicable of propositions) to the “semantic” concept of truth (truth as predicable of sentences of a language-system) in Carnap 1942: for any sentence S and language L, if S designates p in L, then S is true in L. We notice at once that we are still left with a propositional variable, in the semantic context “S designates p in L.” To get rid of the propositional variable—or of constant substituends for the propositional variable, if we consider translation of a specific statement of the form “it is true that p”—in this context is a far more diﬃcult task than it may seem superficially. The obvious suggestion is that we replace “S designates p in L” by “S

17 We

may overlook in this context the fact that a sentence in the sense of a repeatable type of expression, in contrast to tokens exemplifying the type, is itself a class; for the same reply would mutatis mutandis be appropriate to this objection.

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is synonymous with ‘p’ in L.” But apart from the criticisms already advanced in connection with Ayer’s reduction, the argument from translation is applicable again: a translation of “it is true that the earth is round” which mentions the English sentence “the earth is round” cannot be adequate, since if it were, the translated English sentence would not be synonymous with, say, the German sentence “es ist wahr, daß die Erde rund ist.” Further, as G. E. Moore has pointed out (Moore 1946), to say that two expressions have the same meaning is diﬀerent from telling what they mean (just as to say that two classes have the same number is not to say what that number is). However, let us ignore these diﬃculties, and assume for the sake of the argument that “S designates that p” can in some way be translated into a purely extensional meta-language (cf. footnote 11), so that he who cannot countenance propositions as “real entities” may in good conscience employ this mere “fac¸on de parler.” Let us see, then, whether the following equivalence is necessary: It is true that p, if and only if, for every S and every L, if S designates p in L, then S is true in L.

(D)

The answer depends on the sense of “if, then” in the definiens. Is the definiens a generalized material implication (what Russell called “formal implication”)? In that case the definiens will be true of a given proposition, provided there are no sentences designating that proposition in some language. But surely this is a logical possibility. Thus, consider the statement “there was a time when nobody believed or disbelieved or supposed or in any other mode thought of the proposition that there are 9 planets.” Is it not highly probable that it is true? But it entails that there is a proposition p and a time t such that p is not verbally expressed at t. “Yes, but this does not prove that it is logically possible for a proposition to be verbally unexpressed at all times. After all, the very act of saying ‘it is logically possible that nobody should ever have thought of p1 ’ constitutes a verbal expression of p1 , hence the supposition that there is a proposition which is unexpressed at all times cannot be consistently made, just as one cannot consistently suppose that he is never thinking.” This objection, however, rests on the confusion between pragmatic and logical contradiction. A proposition is pragmatically contradictory if its falsehood follows from its assertion; more precisely, “p” is pragmatically contradictory if “p is asserted” entails “p is false.” In this sense the proposition that no proposition is ever asserted is obviously a pragmatic contradiction; but which moderately clear-headed person would seriously doubt its logical possibility? Similarly, the supposition that the proposition that p1 is never at all verbally expressed is pragmatically contradictory, since one cannot assert (or even suppose) that p1 is never verbally expressed without producing, physically or mentally, a sentence which expresses p1 . But this is entirely compat-

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ible with the logical possibility that p1 is never verbally expressed. Whether or not there is such a thing as completely nonverbal thought is an interesting problem of psychology but completely irrelevant to this issue. One cannot think of a proposition without thinking of it: very true. Hence to infer that one cannot consistently think of a proposition as not being thought of (i.e. think of the proposition that a given proposition is not being thought of) is to commit the same howler which Russell attributes to Berkeley: “‘It is impossible for a nephew to exist without an uncle; now Mr. A is a nephew; therefore, it is logically necessary for Mr. A to have an uncle’. It is, of course, logically necessary given that Mr. A is a nephew, but not from anything to be discovered by analysis of Mr. A. So, if something is an object of the senses, some mind is concerned with it; but it does not follow that the same thing could not have existed without being an object of the senses” (Russell 1945, 652). You cannot discover by analysis of a proposition that some mind is thinking of it, whether verbally or not. Now, if p is verbally unexpressed, then by D (whose “if, then” we suppose to be an extensional connective) p is true, but also not-p is true, since there can be no sentence designating not-p if there is no sentence designating p. But that unexpressed propositions are both true and false is an absurd consequence of D which speaks decisively against this rule of translation18 . If the “if-then” in D is interpreted as a strict implication (logical deducibility of consequent from antecedent), the translation does not become more plausible. If a strict implication is true, then, in accordance with the NN thesis, it is necessarily true. But then the translation would entail that only necessary propositions are true, in other words, that there are no true contingent propositions. The same conclusion can be derived from a diﬀerent consideration. We just need to ask ourselves how a conjunction of the form “Des(S , p, L) and T rue(S , L)” could be self-contradictory. Surely, if p is contingent, then it is logically possible that a sentence designating p in some language should be false in that language. To assert the logical impossibility

18 Carnap

might challenge this argument on the ground that the propositional variables of a semantic system range only over propositions for which there are constructible sentences of the system that designate them. But the question whether there are propositions that are not designatable by any sentence of a given semantic system is surely a significant one. Indeed, suppose the language-system to be a “continuous” one, i.e. containing real-number-variables. Consider the proposition, expressible in the system, that a given object has a certain determinate form of a property assumed to be continuously variable, say length. This proposition is one out of a non-denumerably infinite set of similar propositions, yet the set of sentences constructible (“sub specie aeternitatis”) in any language-system would seem to be denumerably infinite. Are we not forced to the conclusion, then, that not all propositions are expressible in language-systems though any given proposition is? Notice that even if the reduction of “absolute truth” to “semantic truth” were satisfactory for statements “it is true that p” within a semantic system, we would be left with the question whether statements outside and about a semantic system, such as “half of the propositions that are not expressible in system L are true,” are thus reducible.

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(self-contradictoriness) of such a conjunction, therefore, is equivalent to asserting the necessity of p. There remains the interpretation of the conditional as a subjunctive conditional which does not hold by logical necessity but is an inductive generalization. At least, one might suggest, the universal conditional of the semantic meta-language has inductive character if the proposition of which truth is predicated is contingent. But, assuming we have found empirically that a sentence S 1 is true if its constituent terms have such and such meanings, are we really drawing an inductive inference in saying “for any S , if S were synonymous with S 1 , S would also be true,” the way it is an inductive inference to infer from the proposition that a given sample of iron melted at temperature t, that any other sample of iron would also melt at temperature t? Clearly not, for it is a contradiction to suppose that S 1 is true, that S 1 is synonymous with S and yet that S is not true. In other words: if a criterion of adequacy to be satisfied by an explication of the diﬀerence between “‘p’ is true” and “it is true that p” is that the latter statement should say more than the former (see the quotation from Ayer, p. 122 above), then the explication in terms of a universal subjunctive conditional referring to any (actual or possible) sentences synonymous with “p” is demonstrably inadequate. If “p” logically entails “q1 . . . . . qn ” then the conjunction “p.q1 . . . . . qn ” has no more logical content than “p” alone. These criticisms apply mutatis mutandis to applications of the logical-construction-theory to other kinds of propositional talk, such as statements of entailments, statements of belief, etc. If one wishes to construe propositions as “pseudo-entities,” “logical constructions,” in the precise sense in which classes are logical constructions in Principia Mathematica, then one will have to show that propositional names, i.e. such expressions as “that Mary is baking pies now” in the context “Clarence believes that Mary is baking pies now”19 are eliminable through contextual definitions. In recent discussions of possible translations of propositional talk into a language acceptable to “nominalists,” attention has been paid primarily to statements of belief. One strong argument against such translatability is Church’s “translation argument,” already discussed above. I would like to add here a few criticisms: In order to translate “A believes that p” into an extensional meta-language, a behavioristic disposition concept must be used, something like Carnap’s “be-

19 It

should be noted that one who countenances propositions as “real entities”—in the formal mode: who employs without compunction a language containing, in the primitive notation, variables ranging over propositions—is not committed to construal of true-or-false statements as names of entities. True-or-false statements intend propositions, but that does not turn them into names any more than predicates are names. In fact the analogy here is fairly close: predicates have properties as intensions, which latter may be named by abstract nouns (“roundness” is a name of the intension of the predicate “round”); similarly names of propositions can be derived from the statements intending them, by using the gerundive (“Mary baking pies now” names the intension of the statement “Mary is baking pies now”).

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ing disposed to an aﬃrmative response” to some kind of sentence. Thus Carnap proposed in Carnap 1956a “A is disposed to an aﬃrmative response to some sentence which is synonymous with ‘p”’ as a tentative analysis, where synonymy is a stronger relation than logical equivalence (explicated by Carnap as “intensional isomorphism”; see Carnap 1956a, para. 13, 14). He was led to the distinction between L-equivalence and synonymy in this context by the following reasoning: if “p” is L-equivalent to “q,” then “A is disposed to an aﬃrmative response to some sentence which is L-equivalent to ‘p”’ entails “A is disposed to an aﬃrmative response to some sentence which is L-equivalent to ‘q”’ (since L-equivalence is transitive). Therefore the analysis of belief in terms of L-equivalence entails that “A believes that p” entails “A believes that q”; yet, since A may not be aware of the L-equivalence of “p” and “q,” it is possible that he should believe that p and yet not believe that q (assume, e.g., that p = (p.(p ⊃ q)), and q = (p.q), and that A does not know the propositional calculus)20 . However, the analysis of belief in terms of synonymy does not overcome this kind of diﬃculty at all; it is open to a perfectly analogous objection. For, suppose that “p” and “q” are not only L-equivalent, but even synonymous. Could not A fail to know that they are synonymous, and hence respond aﬃrmatively to the question “Do you believe that p” but not to the question “Do you believe that q”? But further, “A believes that p” is not a necessary consequence of “A is disposed to an aﬃrmative response to some sentence in some language which is synonymous21 with ‘p’,” since A may interpret the sentences towards which he is thus disposed to express a proposition other than p. To try to obviate this diﬃculty by qualifying A as a subject who understands the questions to which he responds aﬃrmatively or negatively, would be to readmit the banished propositions through the backdoor: for to 20 Since for Carnap L-equivalent sentences express the same proposition, Carnap here asserts that it is possible for a person to believe and yet not to believe the same proposition at the same time! This is not just a question of psychology, nor can such an assertion be defended by pointing to the sad fact that many people hold inconsistent beliefs most of the time; for “A believes at t that p, and not-(A believes at t that p)” is as good an instance of a logical contradiction as any. Carnap, it seems to me, should have drawn the consequence that his weak identity-condition for propositions is not only in conflict with ordinary usage of the word “proposition,” but with the obvious fact that people may fail to be aware of some L-equivalences holding in a language which they understand as well as anybody can be expected to understand any language. 21 An especially serious diﬃculty connected with Carnap’s analysis is that it introduces an interlinguistic synonymy-relation, whereas Carnap’s explications are usually relative to a given language. This is like the diﬀerence between simultaneity defined for events in a given system and simultaneity of events occurring respectively in systems that move relatively to each other. Even if one could plausibly define “A, as expression of L, is L-equivalent to B, as expression of L”’ as “it follows from the combined semantic rules of L and L’ that A and B have the same extension,” the criterion of intensional isomorphism could not be used for interlinguistic synonymy, since two languages may be structurally so diﬀerent that it is impossible to give a syntactic criterion for “correspondence” of constituent signs. Take, e.g., the Latin sentence “ignorabimus” and the English sentence “we shall never know”: since the former is not a compound designator and the latter is, these sentences are not intensionally isomorphic, yet they are obviously synonymous (if Carnap does not agree that this is, preanalytically, an instance of synonymy, then I do not know what his explicandum is).

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say that A understands the question “do you believe that p” is to say that A interprets the sentence p as standing for the same proposition as the interrogator intended by it. Finally, such a behavioristic analysis of belief cannot be adequate for the simple reason that the subject whose beliefs we investigate by interrogation may be a liar. If we take this possibility into account by including the proviso that A is honest (A believes that p = if A were asked, with respect to each and every sentence S which is synonymous with ‘p’, “do you believe that S ?,” or a synonymous question, then, provided A is honest, A would respond aﬃrmatively at least once), the translation becomes grossly circular since “A is honest” means nothing else than “on the evidence that A says that p it is highly probable (or even certain) that A believes that p.” There is a close parallel between the phenomenalist attempt to reduce statements about material objects to statements about sense-data, as well as statements about postulated physical entities, like electrons and force-fields, to statements about homely observables, and the kind of semantic reductionism that is under discussion. Perhaps it will come to be realized more and more widely that semantics needs the postulation of such “abstract entities” as propositions, just as science needs to operate with “constructs” that are not just shorthand devices for formulating highly complex propositions about the phenomenally given. However, while it is unreasonable, and a betrayal of a mere prejudice in favor of a certain kind of language, to dismiss “proposition” as an obscure word unless its meaning can be explained by means of contextual translation into an extensional meta-language, it is a legitimate request that an identity-condition for propositions be formulated. Now, to formulate an identity-condition for a type of entity is to specify a context in which two names of entities of the type in question are interchangeable salva veritate if and only if they designate the same entity of that type. Thus, to mention the least problematic identity-condition, “A” and “B” designate the same class if and only if it does not aﬀect the truth-value of sentences of the form “x ∈ . . .” whether “A” or “B” occupies the dotted place. Similarly, we need a criterion of the form “p and q are the same proposition (or semantically: “p” and “q” designate the same proposition) if and only if “. . . p . . . ≡ . . . q . . .” But what kind of context is symbolized by the dots? To require universal interchangeability would be to lay down a criterion at once ineﬀective and pregnant with paradox. It would be ineﬀective because one of the possible contexts of “p” is the very identity-statement “the proposition that p = the proposition that q” whose truth-value is at issue. And the paradox it is pregnant with is the paradox of analysis: if the synonymy of “p” and “q” entails that nobody can know that p = p unless he knows that p = q, then everybody must know the analysis of every proposition, hence analysis of propositions must be a trivial enterprise

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that does not enlarge anybody’s knowledge22 . Notice that the requirement of universal interchangeability of names of identical entities is equally unreasonable for other kinds of entities that are generally considered less “problematic” than propositions. Thus, if we define identity of individuals x and y, following Leibniz, as “(∀ f )( f x ≡ f y),” where “ f ” ranges over all properties that are meaningfully predicable of individuals, then 1) decision of “x = y” would presuppose itself since “x = y” is one of the values of “ f x,” 2) everybody would have to know every true identity-statement about every individual, since “everybody knows that x = y” would follow from “x = y” and “everybody knows that x = x.”23 Now, I propose as the kind of context in terms of which identity of propositions is to be defined belief-contexts and similar “intentional” contexts—with the exclusion however of the context “A believes (respectively “intends” in some other mode) that p = q,” in order to avoid the paradox of analysis. This definition seems to me to be suggested quite naturally by the reasons for which Carnap’s identity-condition, viz. L-equivalence, is unsatisfactory. “A believes that p but does not believe that q” seems to be perfectly compatible, as Carnap noted, with “‘p’ is L-equivalent to ‘q”’—especially if we consider that any two analytic statements are L-equivalent. But if “‘p’ is L-equivalent to ‘q”’ entailed “the proposition that p = the proposition that q,” there would be incompatibility, since “A believes that p and does not believe that p (at the same time)” is self -incompatible and hence incompatible with any other statement (if “p” is self-contradictory, then “p . q” is self-contradictory). The objection to be faced by his proposal is that such a definition is just as ineﬀective as the one in terms of universal interchangeability, since one cannot decide whether it is logically possible that A believes that p yet not believes that q, independently of deciding whether p = q. But the objection is surely invalid. In fact, the method of proving non-identity of propositions p and q by inviting attention to the evident possibility that someone knows that p yet does not know that q (or mutatis mutandis for other modes of intentionality) is a powerful and perfectly sound method of philosophical criticism24 . One important fruit of

22 For

a detailed discussion of this “paradox of analysis,” see Pap 1955h and Pap 1955a, chapter VI D. be sure, the strong definition of identity would not have this unwelcome consequence if the ramified theory of types were adopted, and the range of “ f ” were accordingly restricted to first-order properties. But the ramified theory of types, according to which it is meaningless to speak of “all the properties” of an entity, turned out to be so restrictive that the axiom of reducibility had to be invoked to mitigate the blow. As it follows from this axiom that if x and y agree in all first-order properties, they also agree in all second-order properties, the above paradox would be reinstated. 24 I deliberately avoid the semantic formulation “it is possible to prove the non-synonymy of ‘p’ and ‘q’ by showing that it is possible for someone to know that ‘p’ is true without knowing that ‘q’ is true.” For, if the latter possibility entailed the non-synonymy of ‘p’ and ‘q’, then no two statements could be synonymous: it is always conceivable that someone fails to understand one of two statements and hence knows one to be true without knowing the truth-value of the other. If, to take account of this objection, the person who allegedly 23 To

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the method is the revelation of the inadequacy of many behavioristic or physicalistic analyses of psychological concepts. Thus, since it is possible for A to know immediately, without induction, that A is in a state of seeing blue, yet not possible for anybody to know without induction that anybody has a certain behavioral disposition, the proposition that A sees blue at a given time cannot be identical with a proposition about a behavioral disposition25 . The choice of intentional contexts for the definition of propositional identity is perfectly natural because propositions are abstracted objects of intentional acts (vide Russell’s suggestive term “propositional attitudes” for what the Brentano school called “intentional acts”). This accords with the traditional definition of a proposition as anything which may be believed or disbelieved or doubted or supposed or asserted, etc. “But, if propositions are thus abstractions from certain kinds of mental states, is not your Platonic realism, according to which propositions may exist without being apprehended by any subject, a gross hypostasis?” Not so. For in asserting the possibility that p1 is not apprehended in any mode by anybody at any time, I merely assert the possibility that nobody apprehend in any mode at any time the proposition that p1 , which is to say that the occurrence of a mental state objectively characterized by p1 is a contingent event. The diﬀerence is merely linguistic: in the one case the propositional noun is grammatical subject and the passive form of “to apprehend” is used, in the other case the propositional noun occurs as the grammatical subject and “to apprehend” in active form. Reification (or “hypostasis”) is the crime committed by those who suﬀer from the compulsion to think of every significant noun as referring to a spatio-temporal particular, not by those “realists” who recognize diﬀerent types of entities. If a philosopher who smells an infantile belief in ghosts and fairies, which is incompatible with what Russell has called a “robust sense of reality,” whenever he hears a fellow philosopher mention entities that are not spatio-temporal, asks me, with an aﬀected air of puzzlement “but where are those propositions? Where is Plato’s heaven?,” I have nothing to reply except to invite him to familiarize himself with type distinctions, and so with the diﬀerence between meaningful and meaningless questions. And if a sophisticated philosopher of language, reared at Oxford, who professes to be well aware of the “category mistake,” claims that propositional nouns are merely incomplete symbols, that they do

might know one to be true without knowing the other to be true is qualified as one who understands both statements, then Goodman’s criticism (see Goodman 1952a) seems inescapable: one who understands both statements must already know whether or not they are synonymous, hence the test in terms of knowledgepossibilities becomes redundant. However, my formulation of the method is invulnerable to Goodman’s criticism, since knowledge of the truth of the proposition that p does not presuppose understanding of the sentence “p” (cf. Pap 1955h). 25 An excellent critique, along this line, of dispositional analysis of psychological concepts may be found in A. C. Ewing’s article (Ewing 1949).

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not refer to anything, he should be invited to make good his claim by answering the presented critique of the logical-construction-theory of propositions.

4.

Necessary Truth and Semantic Systems

It will be recalled that my argument in support of the claim that there are no necessary propositions at all if the linguistic theory of necessary propositions is correct, rested on two premises: 1) the NN thesis, 2) modal statements are, according to the linguistic theory, contingent statements about expressions—if they have a truth-value at all, and are not prescriptive rules of usage misleadingly formulated as declarative sentences. The second premise, however, is meant to refer to natural languages. If modal statements are meant as relative to a language-system, specifically those interpreted language-systems which Carnap calls “semantic systems,” then we get a diﬀerent picture. Thus it can easily be shown that the NN thesis holds if “N(p)” is interpreted as “‘p’ is L-true in L” (where L is a semantic system). Hence one might argue that acceptance of the NN thesis does not compel the conception of necessity as an intrinsic property of extra-linguistic propositions, but is perfectly compatible with the view that “necessary” is to be construed as a relational predicate whose first argument is a sentence and whose second argument is a system of linguistic rules that determines the meaning of the sentence. Before evaluating this species of linguistic theory, whose chief exponent is Carnap, let us clarify the issue. What does it mean to say that necessity is an “intrinsic” property of a proposition? The explanation “it means that a necessary proposition would not be the same proposition if it were not necessary” is not suﬃciently clear. It might give rise to the complaint that it does not at all diﬀerentiate intrinsic from non-intrinsic properties, that indeed any property of any entity is “intrinsic” to the entity in that sense. For is not, quite generally, “x has P and y does not have P” incompatible with “x = y”? And this objection is perfectly valid. It seems to make sense to say “but couldn’t the same proposition which happens to be believed by A not be believed by A, whereas it is a contradiction to suppose that the same proposition which is necessary might not be necessary?,” but the appearance of sense is due to an incomplete formulation. To say of p that it is believed by A is not to ascribe a property, relational or not, to p at all. Since nearly every non-angelic rational being believes some proposition at one time and not at another time, an enormous number of propositions would have incompatible properties if “being believed by A” expressed a property, and the law of non-contradiction would, to the satisfaction of the “dialectical” philosophers, break down not only for the dynamic world of changing particulars but even for the static world of abstractions. Of course, all this excitement stems just from a simple semantic mistake; only “being believed by A at time t” expresses a property, and then our proposition turns out to have the perfectly

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compatible properties “being believed by A at time t1 ” and “not being believed by A at time t2 .” However, that the same proposition should be believed by A at t1 and not be believed by A at time t1 , is just as impossible as that the same proposition should be both necessary and not necessary. On the other hand, the distinction which we vaguely apprehend in feeling that there is, after all, something “in” the intrinsic-extrinsic distinction is simply the distinction between properties expressed by one-place predicates and properties expressed by relational predicates. “p is believed” is an incomplete statement, neither true nor false; it requires expansion into “p is believed by A at time t”; and to say that necessity is an intrinsic property of a proposition is simply to say that statements of the form “p is necessary” (where the values of “p” are propositions, not sentences) are complete. In fact, the expansion “p is necessary in L” yields a meaningless statement, just as would the expansion “p is necessary in New York,” if p is an extra-linguistic proposition. It should be noted that the criterion of whether a predicate expresses a property (can occur as predicate of complete statements) is not syntactic, but rather whether or not predications of the predicate satisfy the law of noncontradiction. “A is to the left of B” is syntactically a complete sentence, unlike “A is between B.” But “being to the left of B” does not express a property any more than “being between B,” since if it did “A is to the left of B” would be both true and false: if A is to the left of B relatively to an observer north of A and B, then A is to the right of B relatively to an observer south of A and B. Now, if “necessary” is construed as a semantic predicate, i.e. a predicate applicable to sentences of an interpreted language (in contrast to sentences of a purely syntactic calculus, which can be characterized as provable, refutable, or undecidable, but not as true, and hence not as necessary), then statements of the form “S is necessary” (here “S ” is a sentential variable) are indeed incomplete: S 1 may be necessary in L1 , and non-necessary in L2 . The basic question at issue, then, can be formulated as follows: is “p is necessary” irreducibly complete, or can it be translated into a complete statement containing a corresponding, semantic and dyadic, predicate “L-true in L”? Carnap defines “S is L-true in L” as “S holds in all the state-descriptions of L,” where “state-description of L” is defined as a conjunction containing for every atomic sentence of L either it or its negation but not both. Thus “Pa. ∼ Qa.Pb.Qb” is a state-description of a semantic system containing just the two individual constants “a” and “b” and the two primitive (one-place) predicates “P” and “Q.” That an S holds in all state-descriptions of L can be discovered without running through all the state-descriptions of L individually. Indeed, if this were not the case, then L-truth relatively to an infinite language, i.e. a language containing quantifiers of unlimited range, would be undecidable. Whether L be finite or not, we can give a finite proof for the Ltrue character of an L-true sentence of L by looking at the semantic rules of

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L, especially those fixing the meanings of the logical constants (the semantic rules for the logical connectives in particular are usually given in the form of truth-tables). To illustrate by a simple example: “Pa ∨ ∼ Pa” holds in all statedescriptions of any L containing “a,” “P” and the logical constants “∨” and “∼ ,” because “Pa” holds in all those state-descriptions of which it is a conjunctive component and “∼ Pa” holds in all those state-descriptions of which it is a conjunctive component, and by the above definition of “state-description,” every state-description of the specified L contains either “Pa” or “∼ Pa” as a conjunctive component. It is clear, then, that “S is L-true in L” is decidable by inspecting the semantic rules of L, just as “S is true in L” is thus decidable if S is L-true in L26 . That the NN thesis holds if “N” is interpreted the way Carnap interprets it is particularly obvious in the case of those necessary statements, like “there are no married bachelors,” whose necessity depends on synonymy-relations. For whether the statement “‘bachelor’ is synonymous-in-L with ‘unmarried man”’ is true can be decided by just looking up the semantic rules of L, hence it is Ltrue (in the meta-language of L) if it is true at all. If one should object that the “looking up” of semantic rules involves after all sense-perception of signs, and that therefore the statement of synonymy rests on empirical evidence after all, he would consistently have to dismiss the very notion of necessary statement, whether explicated in Carnap’s way or in some other way, as absurd. Of course one cannot determine the logical status of a statement without perceiving signs, or imagining signs of a kind some instances of which one has perceived at some time. But the NN thesis, being a conditional proposition, is perfectly compatible with the proposition, be it absurd or not, that there are no necessary statements. Now, whether “‘p’ is L-true in L” can serve as an explication for “p is necessary” surely depends on what L is picked as reference-frame, so to speak. It is necessary that there are no married bachelors, it will surely be agreed. But how can we tell whether the sentence “there are no married bachelors” is Ltrue in L? It all depends on what L is meant. If L contains the usual semantic rules for the logical constants and besides the semantic rule “‘bachelor’ means the property being an unmarried man,” then I am confident that the quoted sentence is L-true in L. My point is that it is nonsense to propose “‘p’ is Ltrue in L” as an interpretation of the complete statement “it is necessary that p” if “L” is a variable. Carnap proposes “to take as the explicatum for logical necessity that property of propositions which corresponds to the L-truth of sentences,” and accordingly lays down “the following convention for ‘N’: For any sentence ‘. . .’, ‘N(. . .)’ is true if and only if ‘. . .’ is L-true” (Carnap 1956a,

26 Cf.

my arguments in support of the NN thesis on pp. 116-117, also Carnap 1956a, 174.

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174). But “ ‘p’ is L-true” is elliptical for “ ‘p’ is L-true in L,” and similarly “ ‘N(. . .)’ is true” is elliptical for “ ‘N(. . .)’ is true in L.” Carnap’s convention, therefore, must have been intended by him in the following sense: For any semantic system L and for any sentence ‘. . .’, ‘N(. . .)’ is true in L if and only if ‘. . .’ is L-true in L. But then I don’t see how it is relevant to the explication of, not “ ‘N(. . .)’ is true in L,” but “N(. . .).” If an analogous convention were laid down for “N(. . .),” it would read either: (a) for any sentence ‘. . .’, and for any L, N(. . .) if and only if ‘. . .’ is L-true in L, or: (b) for any sentence ‘. . .’, N(. . .) if and only if ‘. . .’ is L-true in L. Now, that (a) is false can easily be seen as follows. Consider the substitution-instance with “there are no married bachelors” substituted for the dots in the context “N(. . .),” “ ‘there are no married bachelors’ ” substituted for “ ‘. . .’,” and a language containing the semantic rule “‘bachelor’ means the property being a married man” as value of “L,” which we shall denote by “L1 ”: it is necessary that there are no married bachelors if and only if “there are no married bachelors” is L-true in L1 . But the quoted sentence is not L-true in L1 ; and if the meta-meta-language in which convention (a) is formulated contains the usual semantic rule for “bachelor,” then “it is necessary that there are no married bachelors” is true in it; hence the biconditional is false27 . On the other hand, convention (b) does not even allow us to derive a substitution-instance which is a true-or-false statement, since it contains “L” as a free variable. One cannot answer the question whether it is necessary that there are no married bachelors if and only if “there are no married bachelors” is L-true in L, until L is specified. At best, “it is necessary that p” could be interpreted as “ ‘p’ is L-true in every L which is logically adequate,” where a logically adequate L is a semantic system whose state-descriptions are, according to the customary meanings of the terms, descriptions of possible worlds. Since a possible world is any world that conforms to all necessary propositions, this interpretation slips, none too subtly, the concept of “necessary propositions,” which was to be replaced by a language-relative concept of L-truth, in again through the backdoor. Indeed, that Carnap is guided by an apprehension of entailments and incompatibilities which are “absolute” in the sense of not being relative to a language, in his very choice of the linguistic reference-frame for the construction of his semantic “explicatum” L-truth, is obvious. Thus he noticed that no more than one member of a family of codeterminate predicates, like the family of color predicates, can be admitted as primitive into a descriptive semantic system. For, suppose that both “blue” and “red” were primitives of L. Then conjunctions like “blue(a).red(a)” would occur in some state-descriptions of L, and conse-

27 Perhaps the reader will follow the argument more easily if its form is made explicit: “(∀x)(p ≡ f x)” entails “p ≡ f a”; but “p” is true and “ f a” false; hence the universal premise is false.

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quently (∀x)(blue(x) ⊃∼ red(x))” would not be L-true in L28 . But the explication of “N(p)” as “‘p’ is L-true in L” would then be inadequate, since some sentences which according to their customary interpretation express necessary propositions would fail to be L-true in L. Again, it was noticed that relational descriptive predicates would cause the same sort of trouble if they were admitted as primitives. For, it is surely necessary that the relation Warmer, e.g., is asymmetrical. But some state-descriptions would contain conjunctions like “warmer(a, b).warmer(b, a),” hence “(∀x)(∀y)(warmer(x, y) ⊃∼ warmer(y, x))” would not be L-true in L. After considering various possible solutions of these diﬃculties, Carnap has indicated his preference for the method of “meaning-postulates” and a corresponding redefinition of L-truth. The primitive predicates are introduced into the language-system in the context of postulates which might be regarded as partial explications of their meanings. Thus, if various codeterminate predicates (determinate predicates falling under the same determinable) P1 , . . . , Pn are introduced as primitives by incompatibility postulates of the form “(∀x) (Pi (x) ⊃ ∼ Pr (x))” (where i is not r), we see right away that they are codeterminate, though the question is left open which specific qualities they designate. And if a relational predicate “R” is introduced by postulates expressing asymmetry and transitivity, then we know that it designates an asymmetric and transitive relation, though we don’t know whether it is the relation Warmer, the relation Earlier, or the relation Louder etc. The original definition of “S is L-true in L” as “S holds in all state-descriptions of L” is now replaced by “the conditional with the conjunction of the meaning-postulates of L as antecedent and S as consequent holds in all state-descriptions of L” (Carnap 1952). But what determines the choice of meaning-postulates? Is it not one’s insight into the necessary character of the propositions ordinarily expressed by such and such sentences? Is not Carnap here again constructing the languagesystem in such a way that the extension of the systematic concept (the “explicatum”) “sentence which is L-true in L” should coincide as closely as possible with the extension of the presystematic concept (the “explicandum”) “sentence which, according to the customary meanings of its terms, expresses a necessary proposition”? Carnap knows that the proposition, say, that the relation Warmer is asymmetrical (here “Warmer” is used as a name of the relation which is usually meant by English speaking people when they use the relational predicate “warmer”), is necessary, and this a priori knowledge which is not knowledge of, or about, linguistic rules, but an apprehension of incompatibility (of the propositional functions that x is warmer than y at time t and that y is warmer than x at time t), motivates his adoption of the corresponding

28 Cf.

Carnap 1950b, §18c.

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meaning-postulate. Carnap, however, does not admit that such “conventions” are motivated by any sort of a priori knowledge, and it is pretty obvious to me that he is dodging the issue by committing just the confusion between the empirical proposition that a given sentence is ordinarily used to express a necessary proposition, and the non-empirical proposition that the expressed proposition is necessary, which I have already called attention to. This seems to me clearly brought out by the following passage: Suppose that the author of a system wishes the predicates ‘B’ and ‘M’ to designate the properties Bachelor and Married, respectively. How does he know that the properties are incompatible and that therefore he has to lay down postulate P1 ?29 This is not a matter of knowledge but of decision. His knowledge or belief that English words ‘bachelor’ and ‘married’ are always or usually understood in such a way that they are incompatible may influence his decision if he has the intention to reflect in his system some of the meaning relations of English words. (Carnap 1952)

Carnap seems to say that there is no such thing as knowledge that the properties B and M are incompatible, but only the empirical knowledge that the words “B” and “M” are “always or usually understood in such a way that they are incompatible.” But he overlooks that the knowledge that two predicates are commonly so meant as to be incompatible is compounded of an empirical and an a priori component. For, to say that “P” and “Q” are so meant as to be incompatible is to say: (∃P)(∃Q)(Des(‘P’, P).Des(‘Q’, Q).Inc(P, Q))—where “Des” symbolizes the designation-relation of descriptive semantics which is really at least a triadic relation, since reference must be made to a specified group of sign-users or sign-interpreters, but which I here simplify as a dyadic relation on the assumption that the “pragmatic” variable is kept constant. Suppose that P1 and Q1 are the properties for which this existential statement holds. Then “Des(‘P’, P1 )” and “Des(‘Q’, Q1 )” express empirical propositions, but the proposition expressed by “Inc(P1 , Q1 )” is, if true, necessary.

5.

Implicit Definitions

The reply which Carnap and other “conventionalists” with respect to a priori knowledge may be expected to make is that my argument is based on the mistaken presupposition that the designata of predicates are somehow given independently of the meaning-postulates that express their logical relations or properties. But if, to illustrate, observations of linguistic behavior led me to the semantic hypotheses that Frenchmen mean by the words “bleu” and “rouge” the properties Blue and Red, and I later noticed that they frequently applied these predicates to the same object (“cet objet est bleu et dur et chaud et rouge”), 29 P 1

= (∀x)(Bx ⊃∼ Mx).

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would I not revise my semantic hypotheses? Does not the sentence asserting the incompatibility of these properties function as an implicit definition of the predicates, then? Similarly, if a visiting anthropologist who wants to discover synonymies between words of the tribe language and English words with the assistance of an interpreter who is able to question the natives, is in particular looking for a synonym of the English word “warmer,” and arrives, after some questioning done by the interpreter, at the hypothesis that, say, “krut” is the synonym, he would surely abandon that hypothesis if the natives subsequently gave an aﬃrmative answer to the question whether krutness is a symmetrical relation30 . Is not the sentence “for any x and y, if x is warmer than y, then y is not warmer than x” an implicit definition of “warmer,” then? But let us ask what exactly is meant by saying of a sentence that it is an implicit definition. “Being an implicit definition” surely refers to a functional, not a structural property of a sentence, unlike such predicates as “containing three words,” “disjunctive,” “conditional,” etc. That is, a sentence can significantly be said to be used as implicit definition in certain contexts, not to be an implicit definition intrinsically. Now, I propose as analysis of “S is used as implicit definition in context C”: S does not have the form of an explicit definition, but is used in C as a means for explaining the meaning of one or more constituent terms. Thus, if somebody with a mediocre knowledge of English were to say “I met a happily married bachelor yesterday,” I might say to him “look here, no bachelors are married” in order to get him to see that he was misusing the word “bachelor.” Again, it is conceivable that I might say “if a is earlier than b, then b is later than a, but b can’t be earlier than a if a is earlier than b” to a foreigner who frequently used the word “earlier” where he should have used the word “later,” or vice versa, just in order to teach him the diﬀerence in meaning between “earlier” and “later.” Undoubtedly, the philosophers who, like Wittgenstein certainly and Carnap probably, feel that to speak of necessity as an intrinsic property of propositions is to speak a kind of metaphysics which arises from a fundamental misunderstanding of language, hold that “the proposition that p is necessary,” as a sentence of a natural language, has either no cognitive meaning at all or else means just “the sentence ‘p’, as well as its synonyms, is used, not to assert a fact, but as an implicit definition,” where “being used as an implicit definition” means more or less what it has been analyzed to mean. Yet, is it not obvious that sentences which are ordinarily held to express contingent propositions can exercise just the same function? Thus I might complete my explanation of the diﬀerence in meaning between “earlier” and “later” by asserting the contingent propositions a) event a happened earlier than event b, b) event b happened later than event a, using the words “earlier” 30 Naturally, in the absence of evidence that the natives have studied symbolic logic, the interpreter will have

to ask this question in a roundabout way.

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and “later” to express them. The method of teaching the meanings of words by implicit definition instead of by explicit definition is comparable to challenging the instructed person to solve a crossword puzzle. It amounts to saying that the word “X” means something of which such and such statements containing “X” are true, and it does not matter in principle whether these statements express contingent or necessary propositions. If the use, for the purpose of semantic instruction, of the latter kind of statements is preferable it is because the probability that the instructed person shares the instructor’s belief that p is greater if p is a necessary proposition than if p is a contingent proposition. Suppose, e.g., that B has seen crows and by abstraction formed a concept of crows, but does not understand the meaning of the English word “crow” and furthermore does not believe that all crows are black. If A then tells him “crows are a species of birds all of whom are black,” this assertion will not lead to the goal of communicating to B at least part of the meaning of “crow.” It may be replied that, indeed, sentences expressing contingent propositions may occasionally be used as implicit definitions, but that they are at any rate sometimes, or even most of the time, used to assert facts, whereas the distinctive characteristic of so-called “necessary” statements is that they are always used as implicit definitions and are never used to assert facts. Now, if “fact” here is simply a synonym for “empirical” fact, it is of course undeniable that necessary statements are never used to assert facts, since if they were in given contexts so used, they would, by definition of “necessary,” not be necessary statements in those contexts. But suppose that “fact” is used in the sense in which “it is a fact that p” is cognitively synonymous with “p.” We use the expression in this sense, as an “assertion sign,” when we make statements like “the fact that p, entails that q.” But it is used in just the same sense, whether the premised proposition p be contingent or necessary. If, e.g., it is proper to say “the fact that p, entails that q,” where “p” and “q” stand for contingent propositions, why should it not be proper to say “the fact that p entails r, entails that not-r entails not-p”? It may be countered that it is naive to infer from the grammatical propriety of the expression “the fact that p” that “p” is used to assert a proposition; the utterance “it is an undeniable fact that no bachelors are married” may just be an emphatic way of urging the person addressed to observe a conventional way of speaking. But remembering that propositions are abstractions from intentional states of mind (or dispositions manifesting themselves in such states), the relevant question to ask here is simply whether it is significant to make a statement of the form “I believe (resp. disbelieve, doubt, etc.) that p” where p is a necessary proposition. This will have to be granted if it is granted that “I believe that (p entails q)” is significant, since “p entails q” expresses, if true, a necessary proposition. And of course this must be granted. I remember a time when I did not believe that “∼ q” is entailed by

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“p ≡ q. ∼ p”; finding that an authoritative logician made this claim, I applied the methods of the propositional calculus, and now I believe (and probably know) that the entailment holds. The answer given by the linguistic theory to the perennial question why some universal statements, like the laws of logic and of arithmetic, but also statements like “the relation Warmer is asymmetrical” which cannot plausibly be classified as logical truths since a descriptive term occurs essentially in them, are absolutely undeniable although it is impossible to observe all the cases to which they apply, is that to deny them is equivalent to changing their meanings, to violate conventional linguistic rules. But this answer, popular as it is with empiricists who want to get rid of the “inner eye of Reason,” cries out for clarification. Suppose the assertion made by these philosophers is the following: to say that S expresses, according to the customary interpretation of its terms, a necessary proposition is equivalent to saying that if anyone were to deny S then he would take S to express a diﬀerent proposition, so that his disagreement with those who would stake their lives on the truth of S is merely verbal. Let us waive the objection that one might, after all, correctly interpret a sentence S to express a proposition p which is objectively necessary but disbelieve it just the same, just the way I personally once disbelieved the necessary proposition expressed by the sentence “(∀p)(∀q)[(p ≡ q . ∼ p) ⊃ ∼q].” For the “conventionalist” might with good reason say that this could not happen if the necessary propositions in question are of the “self-evident” variety, like the laws of non-contradiction and of the excluded middle. But let us ask whether “if, then” in the conditional “if anyone were to deny S then he would take S to express a diﬀerent proposition” stands for necessary implication or for factual implication. If the former, then the claim that is made by the conventionalist is that it is self-contradictory to suppose that a “self-evident” necessary proposition be disbelieved. Now, in the first place, it is not self-evident that such a supposition is self-contradictory. At any rate, if anyone maintains that he can derive a contradiction from the statement “A believes that there are two particular events (not kinds of events!) x and y, such that x happens before y and y happens before x,” and from similar statements, the burden of proof rests on him. But second, reference to linguistic rules would be conspicuously absent from such an analysis, hence it would be obscure in just what sense our theory has given a “linguistic” explanation of necessity. The other horn of the dilemma is that the conditional is meant to express a factual implication only: from the fact that A says, for example, “there are (or may be) events x and y such that x is earlier than y and y also earlier than x” it can be inferred with high probability that A uses some constituent term, especially the term “earlier,” in an unconventional sense. But since this condition would obviously be satisfied if the sentence in question expressed conventionally a contingent proposition

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which is probably believed by A, our theory gives no criterion of necessity at all on this alternative. Suppose we found a man who, having convinced himself antecedently of the truth of the empirical proposition “all metals conduct electricity” and “this is a metallic object,” discovered that the object in question does not conduct electricity, and thereupon concluded that the principle of deductive inference “if all A are B and x is an A, then x is a B” is invalid. Undoubtedly we would say that, if he is not just perpetrating a joke, he must be misinterpreting at least one of the logical constants, perhaps “if, then,” which enters into the formulation of the principle. But to say that our conclusion that the man must have misunderstood this abstract sentence is inevitable because the latter implicitly defines the logical constants, is not illuminating since it amounts to a mere repetition, in obscurer language, of the inference to be justified. How, after all, is one to prove that it is impossible for a man to believe the conjunctive proposition that all A are B and x is an A and x is not a B (where A and B are classes and x is an individual)? If C, who utters sentence S in order to assert proposition p, is cocksure that D, who verbally contradicts him, has misunderstood him, i.e. is not really contradicting p, this is because C is cocksure that D believed the proposition that p just as firmly as he himself. The view that those necessary statements in particular which are called “logical truths”—the exact boundary between logical truths and necessary truths that do not belong to logic being a controversial subject which I cannot go into here—are implicit definitions of logical constants, and hence terminological proposals or announcements and hence do not express “truths” at all, seems to be inspired by a dangerously misleading analogy between the axiomatization of bodies of descriptive statements containing extra-logical terms, like geometry or mechanics, and the axiomatization of logic itself. The axioms of a system of pure geometry, for example, implicitly define the geometrical primitives in the sense of specifying the structural meanings they have within the deductive system—not to be confused with their material meanings in empirical applications of the system. The truths of, say, pure Euclidean geometry, are, not the axioms themselves, nor the theorems themselves, but logical truths of the following form: for all values of x, y, z if F(x, y, z), then G(x, y, z), where “x,” “y,” “z” represent the primitives of the system that admit of various interpretations, “F(x, y, z)” the conjunction of the axioms, and “G(x, y, z)” a theorem. The axiomatic system is, as it were, embedded into a logical calculus whose axioms and theorems are used as rules for deducing the theorems of the axiomatic system from its axioms. Now it is tempting to construe the axioms of the logical calculus in turn as implicit definitions of the primitives of the logical calculus, like “or,” “not,” “all,” “if, then.” Suppose, for example, that a propositional calculus contained as sole primitives “if, then,” “not” and “and,” occurring in the following axioms: 1) if (if p, then q) and (if q, then r), then (if p, then r); 2)

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if (if p, then q) and not-q, then not-p; 3) if p and q, then p. We might take the conventionalist line, now, that these axioms are not truths at all, but simply explanations of the meanings of “if, then,” “and” and “not” in the calculus. If we replace these connectives by corresponding operation variables, two of them binary and one of them singularly, writing: 1) R(S (R(p, q), R(q, r)), R(p, r)); 2) R(S (R(p, q).Nq), N p); 3) R(S (p, q), p), we would indeed be left with formulae which are syntactically meaningful insofar as, given certain formal rules of derivation, certain other formulae are derivable from them, but their assertion as true would at once be recognized as nonsensical. Yet, the logical truths would simply reappear in the meta-language in which the rules of derivation are formulated in the non-symbolic, “natural” idiom. Thus, no formal proof at all can be conducted without the ponendo ponens rule: if “p” is an axiom or theorem of the calculus, and “R(p, q)” is an axiom or theorem of the calculus, then “q” is a theorem of the calculus. To say that this rule again is an “implicit definition” of “if, then” and the other connectives that occur in it, in the same sense in which this was asserted of 1), 2) and 3), is to start on an infinite regress of formalization. Some twin of our principle of inference will always remain alive on the highest level of the meta-linguistic hierarchy, which cannot be said to be an implicit definition in the same sense. Now, what is the justification for adoption of the above rule? Is this like asking for the justification of the adoption of a particular rule of a game? On the risk of acquiring the reputation of an impenitent reactionary, I would side with “logical realists” in firmly denying that deductive logic is analogous to a game of chess in this respect. Is it not obvious that this rule is laid down with a view to interpretation of the calculus? Specifically, the formalizing logician thinks of the “if p, then q” which was formalized as “R(p, q)” in the first place. And he knows that the stated rule will invariably lead from true axioms to true theorems if the calculus interpretation includes the interpretation of “R(p, q)” in the sense of “if p, then q.” But how can he know this without proving it as a “meta-theorem,” which presupposes formalization of the meta-language and application of an analogous rule of inference formulated in the meta-meta-language? Now, the usual way of proving that the ponendo ponens rule has the “truthpreserving” character which any acceptable rule of deduction must have, is to prove that the corresponding calculus formula “if p and (if p, then q), then q” is a tautology. Since a tautology is a truth-function of propositional variables which is true for all values of the variables, such a proof cannot get started until “if, then” is interpreted as a truth-functional connective, specifically in the sense of the symbol of Principia Mathematica “⊃” (material implication). Once this has been done, the proof is elementary: by definition, “p ⊃ q” is false if and only if “p” is true and “q” is false; hence “(p.(p ⊃ q)) ⊃ q” could be false only if “q” were false and “p.(p ⊃ q)” true. But the latter is impossible since in order for “p.(p ⊃ q)” to be true, “p” must be true, and the joint truth of

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“p” and falsehood of “q” is incompatible with the truth of “p ⊃ q” and hence with the truth of “p.(p ⊃ q).” But what is the justification for the truth-functional interpretation of “if, then”? None other than that it preserves that common core of meaning in the various uses of “if, then” (to assert causal connections, logical deducibility, resolution to act in a certain way if such and such conditions are realized, etc.) which enables us to justify those and only those methods of deduction which we intuitively accept as valid, i.e. as corresponding to entailments31 . Thus the interpretation of “if, then” in the sense of material implication makes it easy to prove that arguments of the forms “p; if p, then q; therefore q” and “if p, then q; not-q; therefore not-p” are always truth-preserving (i.e. leading from true premises to true conclusions) whereas arguments of the forms “q; if p, then q; therefore p” and “if p, then q; not-p; therefore not-q” are not. In that case, however the proof of the truth-preserving character of the ponendo ponens rule is, if not formally circular, based on our intuitive knowledge that any proposition expressed by “p and (if p, then q)” entails the corresponding proposition expressed by “q.” Such intuitive apprehension of entailments may then motivate our “adoption” of the criterion of adequacy: no analysis of “if, then” is adequate unless it enables a formal demonstration of the ponendo ponens entailment. The apprehension of logical necessity is thus prior to the adoption of linguistic conventions, such as the definition of “if p, then q” as “not both p and not-q,” that render formal proofs possible. Such apprehension alone can explain why we accept just this analysis of the logical constant “if, then” as adequate, and not, say, the analysis “not both q and not-p.” To explain, therefore, the necessity of the customary principles of deductive inference in terms of “linguistic conventions” is to put the cart before the horse.

31 Cf.

the following statement by I. Copi: “although most conditional statements assert more than a merely material implication between antecedent and consequent, we now propose to symbolize any occurrence of ‘if-then’ by the truth-functional connective ‘⊃’. It must be admitted that such symbolization abstracts from or ignores part of the meaning of most conditional statements. But the proposal can be justified on the ground that the validity of all valid arguments involving conditionals is preserved when the conditionals are regarded as asserting material implications only . . . ” (Copi 1954, 18).

III

SEMANTIC ANALYSIS: TRUTH, PROPOSITIONS, AND REALISM

Chapter 6 NOTE ON THE “SEMANTIC” AND THE “ABSOLUTE” CONCEPTS OF TRUTH (1952)

In Carnap 1942, Carnap tells us, following Tarski, that a criterion of adequacy to be satisfied by an acceptable definition of truth is that to assert that a sentence is true means the same as to assert the sentence itself ; e.g., the two statements “the sentence ‘The moon is round’ is true” and “The moon is round” are merely two diﬀerent formulations of the same assertion. (Carnap 1942, 26)

If we are permitted an interlinguistic use of “if and only if” (a usage involved in Carnap’s own formulation of truth-rules for atomic sentences, such as “‘P(a)’ is true if and only if a has property P”), we may formalize this informally stated criterion of adequacy as the following necessary truth:1 (∀S )(∀p)[if S designates p, then (S is true if and only if p)], where the variable “S ” takes names of sentences as substituends (or the values of S are sentences) and the variable “p” takes sentences as substituends (or the values of p are propositions, designata of sentences). Obviously, in committing himself to the above criterion of adequacy, Carnap (or, for that matter, Tarski) is committed to the view that the following proposition is self-contradictory: not-(‘the moon is round’ is true) and (the moon is round). (For reference purposes, this conjunction will be abbreviated by “not-S 1 and p1 ,” where it is understood, of course, that the proposition p1 is the designatum of the sentence S 1 .) Later on in the book, however, Carnap discusses a “corresponding” absolute concept of truth which is not semantical at all since it is predicable of 1 Since

by a “criterion of adequacy” is meant a proposition which must be logically demonstrable on the basis of an adequate definition of the concept in question, it follows indeed that in laying down “p” as a criterion of adequacy one asserts its necessity, though preanalytically.

147

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The “Semantic” and “Absolute” Concepts of Truth (1952)

propositions, not sentences, such that no reference to a definite language is required in predications of this absolute concept. Thus sentences of the form “it is true that p” are complete as they stand; being themselves object-linguistic they do not refer to any language at all. On the other hand, sentences of the form “S is true” are elliptical; in order to become unambiguous they must be expanded into “S is true in language L.” Or, as it might be put, it is logically possible that the sentences “S 1 is true” and “S 1 is not true” should both be true, since the former sentence may be short for “S 1 is true in L1 ” and the latter sentence short for “S 1 is not true in L2 .” But a statement of the form “it is true that p and it is not true that p,” being logically equivalent to “p and not-p,” would necessarily be self-contradictory. It would appear, then, that Carnap asserts the self-contradictoriness of both of the following conjunctions: not-(S 1 is true in L1 ) and p1

(1)

(where it is understood, and signalized by the use of identical subscripts, that p1 is the proposition designated by S 1 in L1 ), and it is not true that p1 , and p1 .

(2)

What I wish to prove in this note is that (1) is not self-contradictory, though (2) is self-contradictory, and that for this reason predications of the absolute concept of truth cannot be regarded as merely diﬀerent formulations of the corresponding semantic statements. I shall further argue that the theory of propositions as possible states of aﬀairs, which was subsequently sketched by Carnap in (Carnap 1956a), brings him close to a wholly nonsemantical conception of truth which is defensible, unlike the semantic conception, by reference to ordinary usage, and moreover involves no commitment to Platonism. My argument is exceedingly simple: “S 1 is true” could not be true unless the sentence named by “S 1 ” existed, while “p1 ,” the object-linguistic sentence, does not entail the existence of any sentence at all; hence “S 1 is true” and “p1 ” cannot be logically equivalent. That “S 1 is true” does entail the existence of at least one sentence follows from the following general logical principle: if “F(x)” is any sentential function at all (here “F” is not meant as a predicate variable but as an undetermined constant predicate), and xi an admissible value of the variable x, then “F(xi ),” the singular statement, entails “(∃x)F(x).” If the sentential function belongs to the object-language, then a familiar illustration of the principle would be as follows: if John is tall, then there are tall individuals and, a fortiori, there are individuals. If, now, “F” happens to be the semantic predicate “true,” and the variable x ranges over object-linguistic sentences as its values, it follows that “S 1 is true,” which is an instance of the form “F(xi ),” entails that there are true sentences and, a fortiori, that there are sentences.

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The same result can be obtained via the theory of descriptions in case the grammatical subject of a semantic predication of truth should be, not a name of a sentence such as the usual quotes, but a definite description of a sentence, as in “The paradoxical sentence on p. 200 is true.” For this is analyzable into “there is one and only one paradoxical sentence on p. 200, and that sentence is true.” It was with a view to the theory of descriptions that the negation in (1) was given the inelegant form “not-(S 1 is true in L1 )” instead of the more colloquial form “S 1 is not true in L1 .” For, suppose that “S 1 ” is an abbreviation of a description of a sentence. In that case the latter negation would naturally be interpreted to mean that the described sentence exists and is not true, while the former negation, being equivalent to the disjunction “either S 1 does not exist, or S 1 exists and is not true,” would be true in case S 1 did not exist. The conjunction “S 1 is not true in L1 , and p1 ” is, indeed, self-contradictory; but “S 1 is not true” is not the contradictory of “S 1 is true,” just as “the king of Switzerland is not jolly” is not the contradictory of “the king of Switzerland is jolly.” To illustrate the point in terms of Carnap’s sample sentence “the moon is round” (which I prefer to Tarski’s “snow is white” only because it is less worn in the vast literature on the semantic concept of truth): A universe which is just like ours except that it does not contain language, and thus contains no sentences, is surely logically possible. In such a universe it would still be the case that the moon is round, but nothing could be the case in such a universe which logically presupposed the existence of sentences, hence it would not be the case that the sentence “the moon is round” is true. I have already indicated one possible source of the erroneous belief that (1) is self-contradictory, viz. confusion of “S 1 is not true” (the contrary of “S 1 is true”) with “not-(S 1 is true)” (the proper contradictory). Another possible source is the confusion between formal and pragmatic contradiction. A sentence is pragmatically self-contradictory if it is necessarily falsified by the occurrence of a token of itself but not by what it asserts; in other words, if S is pragmatically self-contradictory, then a contradiction is deducible, not from S alone, but only from the conjunction “S , and there exists a token of S .” In this sense “no proposition is ever asserted” is pragmatically contradictory but not formally so (unlike “no proposition is true”). Now, if anyone should feel that from “the moon is round” it follows that the sentence “the moon is round” exists, this feeling would evidently be due to an unnoticed shift from the object-linguistic premise (which is a statement of the physical language) to the statement that the premise has been asserted (which is a statement belonging to the pragmatic meta-language). Now, I propose to show that Carnap, in addition to being wrong in maintaining the self-contradictoriness of (1), is moreover inconsistent in holding both

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(1) and (2) to be self-contradictory. As has been shown, in holding (1) to be self-contradictory one is committed to the view that “p1 ” (e.g., “the moon is round”) entails the existence of at least one sentence; for if it does not entail this consequence (and it surely does not), then it might be true even though “S 1 is true” is false on account of the nonexistence of S 1 . Now, in Carnap 1942 Carnap implicitly asserts the logical equivalence of “it is true that p1 ” and “p1 ” by laying down definition 17-1: p is true ≡d f p.2 He also holds, as we have already seen, “p1 ” to be logically equivalent to “S 1 is true.” If he were consistent, he would accordingly have to hold that “S 1 is true” is logically equivalent to “it is true that p1 .” But as a criterion of adequacy to be satisfied by an acceptable definition of the absolute concept of truth, i.e., of such sentences as “it is true that p1 ,” he lays down the following logical equivalence: (a proposition) p is true if and only if for every language L and for every sentence S , if S designates p in L, then S is true in L. (slightly altered version of T 17-A in Carnap 1942)

Hence consistency would require him to hold that “S 1 is true” is logically equivalent to: (∀S )(∀L)(if S designates p1 in L, then S is true in L).

(3)

Now, if “p1 ” entails the existence of at least one sentence, and “p1 ” is logically equivalent to “it is true that p1 ,” then “it is true that p1 ” cannot be logically equivalent to (3), for (3) is a nonexistential universal implication: you cannot deduce from (3) that there exist any values of S and L which satisfy the antecedent! After what has been said already, it hardly needs to be added that the proper way to resolve the inconsistency is to recognize that “p1 ” does not entail the existence of sentences, not to reject T 17-A. As a matter of fact, if the latter criterion of adequacy were employed as the definition of “it is true that p” (Carnap leaves it undecided whether any other definitions, likewise satisfying T 17-A, would be preferable), we would have a precise formulation of the logical-construction theory of propositions according to which propositions are classes of sentences. It has not been suﬃciently realized perhaps that, even if one holds statements about propositions to be reducible to statements about sentences in the sense of T 17-A, one can consistently hold that propositions may exist without being verbally expressed. But T 17-A clearly shows this to be the case, since by this theorem (or definition) “(∃p)(p is true)” is logically equivalent to 2 I am replacing Carnap’s “p is true” by “it is true that p,” because one cannot derive from “p is true” by substitution grammatical sentences if, as is customary, statements are substituted for “p.”

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(∃p)(∀S )(∀L)(if S designates p in L, then S is true in L),

151 (4)

which still does not assert the existence of sentences. It may be objected, justifiably, that T 17-A can hardly be regarded as a formulation of the logical-construction theory of propositions, for this reason: it shows, indeed, how predications of truth upon propositions may be dispensed with in favor of predications of truth upon classes of sentences, but it does not show how any reference to propositions as entities is eliminable since the propositional variable remains with us in the context of the function “S designates p.” But the sketched logical construction of propositions out of sentences can be purified by replacing the relation of designation, which calls for a propositional variable, with the relation of synonymy. The expression “for any sentence S , if S designates that the moon is round, . . . ” may simply be replaced with the expression “for any sentence S , if S is synonymous with ‘the moon is round’, . . . ´’ (and if it is objected, in keeping with current fashion, that the relation of synonymy is not clear, it is easy to reply that it is at least as clear as the relation of designation). My purpose is not to argue in favor of the logical-construction theory of propositions, with its implication that the absolute use of “true” (predicating it upon propositions) is merely a fa¸con de parler which is reducible to the semantic use of “true” (predicating it upon sentences). Rather I wish to show that even if one endorses the thesis of the reducibility of the language of abstract entities to the nominalistic language, it still makes good sense to speak of the existence of propositions which are not verbally expressed. In accordance with the above proposal of replacing “designation” with “synonymy,”3 we obtain the following translation of “p1 is true”: (∀S )(∀L)(if S is synonymous with ‘p1 ’ in L, then S is true in L).

(5)

But this universal implication does not entail the existence of sentences any more than the corresponding implication containing “S designates p.” If the existence of the sentence “p1 ” were deducible from it, as a superficial glance might make one think, then “(∃S )(S is synonymous with ‘p1 ’)” would be deducible from it (for “p1 ” itself is such an S ). But the latter existential sentence 3I

would like to remind the reader that I am here only concerned with the implications of the logicalconstruction theory of propositions, not with the question of its tenability. I have, as a matter of fact, serious doubts whether propositional variables are eliminable in the indicated fashion, since the synonymy of “S 1 designates that p1 ” and “S 1 is synonymous with S 1 ” is questionable for this reason: given the assumption that the language containing S 1 is univocal, the latter statement is a tautology; but the former statement is clearly contingent since its translation into another language leaves us with an informative statement of descriptive semantics. For example, “‘the moon is round’ designates, in English, that the moon is round” translates into “‘the moon is round’ heißt, auf Englisch, daß der Mond rund ist.”

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is not deducible, by the principle that no universal implication entails the existence of values satisfying its antecedent. Now to the second thesis I promised to argue, viz. that the absolute use of “true,” involved in statements of the form “it is true that p,” is far more common than the semantic use, and can be logically reconstructed in terms of the theory of propositions as possibilities which Carnap sketches (perhaps all too briefly) in Carnap 1956a. For one thing, if a sentence like “it is true that X voted against the motion” were synonymous with the semantic statement “‘X voted against the motion’ is true,” then its correct translation into, say, German, would be: “‘X voted against the motion’ ist wahr”; for the sentence thus construed would contain a name of the sentence asserted as true which belongs equally to all languages containing the quoting device. But while a translation of “he is the author of a book entitled ‘Meaning and Truth”’ into “er is Verfasser eines Buches betitelt ‘Meaning and Truth”’ is undoubtedly correct, I doubt whether anybody would accept the above as a translation of the English statement of the form “it is true that p.”4 A more important point against the semantic conception of truth, however, is that the interpretation of “true” as a systematically ambiguous predicate goes against ordinary usage. I think ordinary usage of “true” would justify a criterion of adequacy, to be satisfied by an acceptable explication of the concept of truth, according to which “true” in the meta-linguistic statement “it is true that language L contains some atomic statements” means the same as “true” in the object-linguistic statement “it is true that some things are blue.” One might reply that one is surely justified in departing, in the logical reconstruction of a term T , from the ordinary usage of T in case the latter is inconsistent in the sense of giving rise to contradictions; and that the reconstruction of “true” as a systematically ambiguous term is demanded precisely by the well-known antinomy of the liar which can be constructed in natural languages. But while splitting “true” into “true relatively to object-linguistic sentences” and “true relatively to meta-linguistic sentences” is one way of solving the antinomy, it is not the only way. It can, of course, equally be solved by forbidding self-referential statements, i.e., splitting “statement” into “object-linguistic statement” and “meta-linguistic statement.”5 Whether the 4 The

same point is made by P. F. Strawson, in his subtle article “Truth” (Strawson 1949), at 84. does not seem to be any inconsistency between the use of “true” as an absolute predicate, i.e., a predicate primarily applicable to propositions, not sentences, and Tarski’s method of solving the antinomy by the prohibition of “semantically closed” languages. The stratification of sentences yields indirectly a stratification of propositions, where “object-linguistic proposition” would mean “proposition expressed by an object-linguistic sentence,” and similarly for the higher levels. Even though a given meta-linguistic proposition might entail that some propositions are verbally expressed (e.g., the meta-linguistic proposition that “the moon is round” designates that the moon is round), it could never entail that it itself is verbally expressed, and thus the concept of verbally unexpressed propositions is equally applicable to meta-linguistic propositions.

5 There

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price which this method of solving the antinomy costs in terms of violence done to ordinary usage is just as high as the price to be paid for adoption of the former method is at least an open question. I am not denying that “S 1 is true” is ambiguous until it is expanded into “S 1 is true in L1 ,” just as “x is in motion” is ambiguous until it is expanded into “x is in motion relatively to y.” But this fact lends no support to the view that “true” is ambiguous in the sense that it might mean “true in L1 ” or “true in L2 ”— just as, say, “sentence” is ambiguous in that it might mean “sentence relatively to formation-rules F1 ” or “sentence relatively to formation-rules F2 .” For the ambiguity of “S 1 is true” is easily accounted for in terms of the ambiguity of the sentence of which “true” is predicated. The expansion “. . . in L1 ” (or, more specifically, “relatively to semantic rules S R1 ”) simply amounts to an indication of what proposition, expressed by S 1 , it is that truth is predicated of. To argue that “true” is systematically ambiguous because the meaning of “it is true that p1 ” varies with the semantic rules determining the meaning of “p1 ,” is no more plausible than to argue that, say, “believed” is systematically ambiguous because the meaning of “it is believed that p1 ” varies with semantic rules in just the same way. In Carnap 1956a, specifically in the section on “Extensions and Intensions of Sentences,” Carnap moves pretty close to an absolute theory of truth, i.e., a theory according to which “true” is primarily predicable of propositions and only derivatively of sentences: a true sentence is a sentence designating a true proposition. He speaks of true propositions as of possibilities exemplified by a fact and of false propositions as of unexemplified possibilities. An absolute theory of truth as such a relation of exemplification holding between propositions and facts, by analogy to exemplification as a relation between properties (universals) and individuals, has been worked out by Baylis, in Baylis 1948, and I cannot go into an examination of this theory in this brief note. I wish to confine myself here to a reply to the frequent objection against absolute theories of truth that they necessarily involve a foggy Platonic metaphysics of abstract entities. What, it is asked, could be meant by the “existence of unrealized possibilities”? Is not Carnap’s talk of “intensions” or “designata” a revival, in technical jargon, of Meinong’s “Gegenstandstheorie” which Russell put in its place long ago? It seems to me that Carnap could easily silence these accusations by applying once again his fruitful maxim (if I may put it in Russellian paraphrase): wherever possible, translate from the material to the formal mode of speech. Consider the sentence “the possibility that Mary should be baking pies now exists, and exists no matter whether it is exemplified or not.” If this is deemed obscure, let us translate this piece of “ontology” into the semantic meta-language, as follows: Let “ S 1” represent the sentence “it is possible that Mary should be baking pies now,” and “S 2 ” the sentence “Mary is baking pies now.” Then

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the truth of S 1 is compatible with the falsehood of S 2 ; in other words, S 1 does not entail S 2 . Thus the assertion of the existence of possibilities reduces to the assertion of the truth of certain modal propositions. Surely, we cannot accuse a man of talking obscure metaphysics because he freely uses “possible” as a primitive term?

Appendix: Rejoinder to Mrs. Robbins (1953) [Editors’ note: Pap here replies to B. L. Robbins’ criticisms of chapter 6 in Robbins 1953.] There is, indeed, one serious misrepresentation of my argument in Mrs. Robbins’ “Some Remarks on Semantic Systems” (Robbins 1953). It may be described as a confusion of material and logical equivalence. She claims that I reject: (1) S 1 is true in L1 if and only if p1 . But I don’t. What I reject is the claim that this biconditional is necessarily true, not that it is true. After all, in order to show that the biconditional is false I would have to show that either “p1 ” is true and “S 1 is true” is false or that the latter statement is true and the former statement false! What I claimed was that it is logically possible that “p1 ” is true while “S 1 is true” is false. It follows that Mrs. Robbins’ reductio ad absurdum of my argument on Robbins 1953, 26, is fired at a strawman. For while the negation of “p ≡ q” is indeed equivalent to “p ≡∼ q,” the negation of “‘p ≡ q’ is necessary” is obviously not equivalent to “p ≡∼ q”! The same oversight leads her to impute to me the claim that (1) is inconsistent with (2): it is true that p1 if and only if p1 . What I actually claimed, on page 4 of my article (January 1952), is that it is inconsistent to hold both (1) and (2) to be necessary equivalences. I did not deny that (1) is provable in the semantic systems of Tarski and Carnap on the basis of their definitions of truth. My point is that insofar as their definitions turn (1) into a necessary proposition they do not accord with the ordinary meaning of “true”; for (1), taken as a statement of natural language, does not express a necessary truth. This non-accordance with the ordinary meaning of “true” is the “error in the construction of the definition of truth” which Mrs. Robbins challenges me to detect. To take an analogy, suppose I constructed a definition of “significant” which makes the following statement provable in my system: There is a statement S such that S is significant but not-S is not significant. She would presumably reject my definition just because it entails a statement which she finds, on the basis of the ordinary meaning of the defined term, unacceptable. To ask for an independent argument against the definition would be unfair; the only way you can refute a theory, be it scientific or philosophical, is by showing that it entails unacceptable consequences. As for the “gratuitousness” of my eﬀort “to prove that predications of the absolute concept of truth cannot be regarded as merely diﬀerent formulations of the corresponding semantic statements,” I would like to say this: “it is true that p” diﬀers from ordinary truth-functional compounds (“if p, then q,” “p or q,” etc.) in that it contains the name of an intension, “that p1 ” names the intension of the sentence “S 1 .” Since Carnap still maintains the thesis of extensionality, one would expect him to hold that names of intensions are eliminable. I wanted to show that “that p” is not eliminable from the context “it is true that p” via translation into “S is true (in L).” Mrs. Robbins may say that it is easily eliminable since “it is true that p” is synonymous with “p.” But when Carnap says in Carnap 1956a that a (factual) proposition is something which may or may not be exemplified, one gathers that truth is conceived as a genuine property of a proposition (its being exemplified) just as nonemptiness is a genuine property of a property (something is predicated of a property when it is said to be nonempty, though this can be said without using the second-level predicate).

Chapter 7 PROPOSITIONS, SENTENCES, AND THE SEMANTIC DEFINITION OF TRUTH (1954)

Those philosophers who conduct analyses of concepts in a formal way generally lay down formal and material conditions of adequacy for the definition which is to be constructed. The formal conditions of adequacy concern the formal features of the language in which the definition is to be constructed; an example would be the rule that any variable which occurs free (unbound by quantifiers) in the definiendum must likewise occur free in the definiens. The material conditions of adequacy, however, are philosophically more interesting: they are sentences which must be provable on the basis of an adequate definition, and which are intended to guarantee that the defined term A designates on the basis of the constructed definition the same concept it designates in its ordinary usage in the “natural” language (whether conversational or scientific)—or at least a closely similar concept.1 The material conditions of adequacy, then, are sentences containing A which must themselves satisfy the following requirements: R1 they are necessary statements (for they are supposed to be provable on the basis of a definition, without the help of empirical premises) R2 they contain A in its usual sense (otherwise one would not be justified in calling them conditions of adequacy, for an “adequate” definition of A is a definition formulating the usual sense of A).2 The separation of these requirements is, indeed, somewhat artificial; one might simply state the requirement that the material conditions of adequacy 1 That

the “explicandum” and the “explicatum” are diﬀerent concepts is shown by the fact that some necessary statements involving the explicandum cease to be necessary when the explicatum is substituted for the explicandum. Thus material implication is diﬀerent from the concept usually meant by “implication,” because in the usual sense of “implication” it is necessary that “for any p there is a q such that p does not imply q nor not-q,” while this is false for material implication. 2 Frequently, what is called the condition of adequacy is, not the statement which is to be proved on the basis of the definition, but that the statement be provable on the basis of the definition. This, however, is an inconsequential diﬀerence of terminology.

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be sentences which, in the usual sense of the definiendum, express necessary propositions. But for expository purposes this artificial separation may be salutary. By way of illustration, suppose the concept of truth were so defined that the statement “whatever is believed by everybody is true” is a logical consequence of the definition. This statement is, however, ineligible as a material condition of adequacy for the definition of truth, for, since it is not self-contradictory to suppose that at a given time a given false proposition is believed by everybody, R1 is violated. Should one object that the statement is necessary for the simple reason that it is a logical consequence of a definition, then R2 would be violated, since the statement would be necessary only because the definition gives “true” an unusual sense. As is well known, Tarski selects as material conditions of adequacy for the definition of truth sentences of the form: p is true if and only if p

(T)

where “p” is a functor that turns into the usual kind of sentence-name, viz. the sentence itself put within quotes, when a sentence is substituted for its argument “p.” (Notice, however, that if by the “values” of a variable we understand the entities designated by the substitutable constants, then the values of “p” are propositions, and the values of “p” are sentences.) What I wish to show in the following is that statements of the form (T) satisfy neither R1 nor R2. I begin by showing that R1 is not fulfilled, i.e. that statements of the form (T) are contingent, not necessary. Admittedly, Tarski has not explicitly claimed that such biconditionals are, according to the ordinary meaning of “true,” necessarily true, but only that they are true. Yet the former claim is implicit in his selecting them as sentences that should be provable by means of the semantic definition of truth. The thesis to be established, that biconditionals of the form (T) are contingent, may be made intuitively plausible by the following reasoning. Let us consider the example “‘the moon is round’ is true if and only if the moon is round.” Now, it is consistently thinkable that, while the moon is indeed round, the sentence “the moon is round” is not true for the simple reason that it does not express the proposition that the moon is round, but instead some false proposition. From the proposition that the moon is round we can logically deduce such propositions as that the earth’s satellite is round (assuming the identity “the moon = the earth’s satellite” to be analytic), or that there exists at least one round celestial body, not however the proposition that the sentence “the moon is round” expresses the proposition that the moon is round. In other words, the truth-value of the semantic proposition that “the moon is round” is true depends on what proposition is expressed by this sentence, while the truthvalue of a proposition of astronomy hardly depends upon semantic facts. To be sure, nobody would ever say “the moon is round, but ‘the moon is round’ is not true.” Similarly, nobody would ever say “John is healthy, but the individual I am talking about is not named ‘John”’: the speaker’s belief that the subject of

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the proposition he asserts (viz. the proposition that John is healthy) is named “John” causes him to express the proposition by the words “John is healthy,” hence the belief indicated by the statement “but ... is not named ‘John”’ cancels, as it were, the belief which led him to say “John is healthy.” It would be disastrous, however, to confuse this kind of “contradiction,” which might be called pragmatic, with a logical contradiction. It certainly is no logical contradiction to suppose that John is healthy but is not named “John.” In order to put this argument more formally, let us introduce a descriptive function: ( p)(Des(“p”, p)), where “p” is a propositional variable, “ ‘p’ ” a sentential variable, and “Des” stands for the semantic relation of designation. If the domain of this relation consists of sentences of a semantic system, then a third term-variable is required, ranging over semantic systems, since the propositions designated by the sentences of a semantic system are determined by the semantic rules of the semantic system. Thus, if in such a system L we have the semantic rules “‘John’ designates John,” and “‘intelligent’ designates the property intelligent,” we can write:3 That John is intelligent = ( p)(Des(“John is intelligent”, p, L)). However, since we are at present concerned with sentences of a natural language (attempting to show that sentences of the form (T) do not, as sentences of a natural language, express necessary propositions), we need not complete the above descriptive function in this way. Instead, we have to complete it by adding a time-argument, since the same sentence may at one time be used to express one proposition and at another time to express a diﬀerent proposition.4 We thus get the three-termed descriptive function: ( p)(Des(“p”, p, t)). It appears, now, that semantic sentences of the form “‘p’ is true” are to a considerable degree incomplete, requiring completion to ( p)Des(“p”, p, t) is true. 3 Propositions are related to sentences the way properties are related to predicates, i.e. they are meanings, something which synonymous sentences have in common. But no expressions, except perhaps the highly problematical “logically proper names,” name their own meanings. A predicate is not a name of a property, and analogously sentences are not names of propositions. However, names are sometimes constructed in order to talk about the meanings of expressions. Thus “blueness” is a name of the meaning of “blue,” and analogously “that p,” in such contexts as “X believes that p,” “it is true that p,” is a name of the meaning of “p.” 4 Strictly speaking, several more variables would have to be taken into account, e.g. the language-user, the situation of utterance of a token of the sentence etc. But since the time variable is of special relevance to a major question that will be discussed in the sequel, viz. whether “true” is a time-independent predicate, it is here made explicit in preference to other suppressed variables.

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If the constant “p1 ” abbreviates the sentence “the moon is round” (or designates the proposition that the moon is round), and the meta-linguistic constant “ ‘p1 ’ ” abbreviates the name of the sentence “the moon is round,” and t0 is a particular time during which we may suppose the relevant linguistic conventions to be invariant, then the proposition whose necessity is here disputed may be formulated thus: ( p)Des(“p1 ”, p, t0 ) is true if and only if p1 . It is easy to see that this proposition can be negated without self-contradiction. For, consider the conjunction, incompatible with it: p1 and ∼ (( p)Des(“p1 ”, p, t0 ) is true). If the contingent statement: Des(“p1 ”, p, t0 ), were true, this conjunction would indeed be contradictory, since “p1 and ∼ (p1 is true)” is, on account of the logical equivalence “p1 is true if and only if p1 ”—here truth, being attributed to a proposition, is not a semantic concept at all!—equivalent to “p1 and ∼ p1 .” But just because the above statement of designation is contingent, the conjunction could be true; and it would be true if “p1 ” designated at t0 a false proposition while the proposition p1 is true. It must be concluded that the conjunction “p1 and ∼(‘p1 ’ is true)” expresses a possible state of aﬀairs. It would be a grave error to suppose that this possibility is ruled out by the convention that instances of (T) are to be constructed by substituting for “ ‘p’ ” the names of sentences substituted for “p,” such that “‘p1 ’ is true if and only if p1 ” could not fail for the reason that “ ‘p1 ’ ” is not the name of the sentence “p1 .” For the relation of designation which is involved in the conjunction that was argued to be logically possible goes from sentences to propositions, not from names of sentences to sentences. A defender of (T) as a material condition of adequacy for the systematic definition of truth may reply that my argument is irrelevant since it is based on the absolute concept of truth (in the terminology employed by Carnap in Carnap 1942, henceforth to be referred to by “ta ”), i.e. truth as predicable of propositions, whereas Tarski explicitly uses “true” as a predicate applicable to sentences (henceforth to be referred to by “t s ”), and repudiates propositions as obscure entities. However, if “ ‘p’ is true” is not interpreted in the sense of “the proposition designated by ‘p’ is true,” then it is, as a sentence of a natural language, simply incomplete; in which case we have no means of assessing the truth-values of such sentences, and therefore no means of evaluating the claim that the instances of (T) are according to ordinary usage of “true” true—let alone necessarily true. In the first place, natural languages allow the construction of ambiguous sentences. If we regard sentences of the form “‘p’ is true” as complete and at the same time take sentences of a natural language as the values of “ ‘p’,” we get into conflict with the law of non-contradiction, since the occurrence of an ambiguous sentence “p” could lead to both “‘p’ is true” and “ ‘∼ p’ is true.” And the natural way of saving the law of non-contradiction is then to expand “‘p’ is true” into “the proposition p1 , designated by ‘p’ in

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some usages, is true,” and “ ‘∼ p’ is true” into “the proposition p2 , designated by ‘∼ p’ in some usages, is true,” where p1 and p2 are compatible propositions. That predications of “true s ,” as a monadic predicate, upon sentences are incomplete statements is further evident from the time-dependence of “true s .” When Carnap argues, in Carnap 1949, on the contrary that “true,” unlike “confirmed,” is a time-independent predicate, in the sense that it is a complete statement to say of a sentence that it is true and the question “at what time is it true?” is meaningless (just as it would be meaningless to ask at what time an entailment holds between two given propositions), he must be taken to refer to “truea ,” otherwise his claim is indefensible. For, it clearly makes sense to suppose that a sentence is true at one time and false at another time because at diﬀerent times it designates diﬀerent propositions. This is particularly evident if we consider sentences containing the indicator term “now,” which express a diﬀerent proposition each time they are uttered or written. Further, it is not unnatural to say that if an object has a given property P at one time and does not have it at another time, then the sentence “x has P” is true at one time and false at another time. The reply which probably would be made by the defenders of the theory of the time-independence of truth is that the incompleteness of “‘p’ is true” is, in the alleged cases, of course due to the incompleteness of “p.” For example, to say “John is shaved” is to make an incomplete, ambiguous statement; if truth, however, is predicated of the complete statement “John is shaved at 9 a.m. April 4 1954,” then it is nonsense to go on asking at what time the statement is true. But thus the theory is saved by restricting it to truth-predications upon complete sentences, specifically sentences complete with respect to time. Yet, since “S designates the proposition that p” is always incomplete with respect to time if S belongs to a natural language, it is always significant to ask at what time S is true, provided that “S is true” means “S designates a true proposition.” In order to vindicate Carnap’s claim, therefore, the assumption that truth is predicated of sentences unambiguous as to time is not suﬃcient. We must moreover interpret “‘p’ is true” in the sense of “the proposition that p is truea .”5 For example, it is perfectly meaningful to say “the sentence ‘John is shaved at 9 a.m. April 4 1954’ is true at 9 a.m. April 4 1954, but will be false at any time thereafter,” because it is conceivable that after the mentioned date any of the constituent expressions of that sentence change their meanings in such a way that the same sentence henceforth expresses a false proposition. However, what would be meaningless is the statement (about a proposition) “it is true at 9 a.m. April 4 1954 that John

5 I deliberately write “the proposition that p ...” instead of “the proposition designated by ‘p’ ...” because those who accept the theory of descriptions may say that a statement about a proposition still makes an assertion about contingent usage if the proposition is referred to by a semantic description, such as “the proposition designated by ‘p’.”

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is shaved at 9 a.m. April 4 1954.” For, “it is truea that p,” unlike “‘p’ is true s ,” expresses a proposition whose truth-value is unaﬀected by contingencies of linguistic conventions (unless, of course, p is about linguistic conventions), and therefore may be regarded as cognitively synonymous with “p.”6 But it would be even ungrammatical to say “John is shaved at 9 a.m. April 4 1954, at 9 a.m. April 4 1954.” Thus a close analysis reveals that, even if a completely unambiguous language is presupposed, the claim of the time-independent character of truth holds only for the absolute concept of truth, i.e. truth as predicable of propositions. Now, the argument to the eﬀect that “true s ” is time-dependent even if the sentences of which it is predicated are unambiguous as to time could be countered by construing this predicate as relational, as in fact it is construed in Carnap 1942: S is true s in language L, where in fixing L we fix the semantic rules that determine the meaning of S . For it could be significant to suppose that S , as meaning p (where the coordination of the proposition p to S is just what the semantic rules of L eﬀect), changed its truth-value, only if it were significant to suppose that p changed its truth-value; but, as we have just seen, “truea ” is time-independent. Notice that if predications of “true s ” are completed in this way, then the argument, presented above, to the contingency of the instances of (T) from the contingency of linguistic conventions ceases to be valid. For while the sentence “‘the moon is round’ is true” could change its truth-value owing to a change of meaning of the sentence “the moon is round” without the moon itself changing its shape, this fate could not befall the sentence “‘the moon is round’, as meaning that the moon is round, is true.” My argument was based on the assumption that “truea ” is needed for the completion of “S is true s ” into “the proposition designated by S at t is truea .” And since S could at diﬀerent times designate propositions of diﬀerent truth-values, it followed that S could change its truth-value. But if S designates diﬀerent propositions at diﬀerent times then L has changed; and the possibility of S being true s in L1 and false s in L2 is, of course, consistent with the time-independence of “true s (respectively false s ) in L.” However, there remain two arguments against the claim that (T) is a condition of adequacy satisfying R1 and R2. In the first place, (T) violates R2 because the semantic use of “true” just is not its ordinary use at all. If in re-

6 Carnap

obliterates this important distinction when he says, “We use the term (‘true’) here in such a sense that to assert that a sentence is true means the same as to assert the sentence itself ; e.g. the two statements “The moon is round” and “The sentence ‘The moon is round’ is true” are merely two diﬀerent formulations of the same assertion” (Carnap 1942, 26). Notice that “to assert” is inconsistently used by Carnap in that in “to assert that a sentence is true” the object of the assertion is a proposition (albeit a proposition about a sentence), whereas in “to assert the sentence itself” a sentence is the object of assertion. I should say it is a proper usage of “assert” to say that a sentence is used to assert something, but not to say that a sentence is asserted.

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sponse to a statement one says “that’s true,” “that” refers to the proposition he takes the speaker to have asserted by means of the uttered sentence, not to the uttered sentence.7 If the speaker repeated his assertion by means of a slightly diﬀerent, but clearly synonymous, sentence, and then asked “is that true?,” it would be perfectly proper to reply “I have already admitted it.” Further, “it is true that the moon is round,” or “the proposition that the moon is round is true”—which expressions are surely more usual in ordinary language than “‘the moon is round’ is true”—cannot be adequately translated into a metalanguage, whether the proposed translation by which the propositional name “that the moon is round” (cf. note 3, p. 157) is to be removed be “‘the moon is round’ is true” or “all sentences synonymous with ‘the moon is round’ is true” or “some sentence synonymous with ‘the moon is round’ is true.” Such a translation must be inadequate because of the principle that if S 2 is a correct translation of S 1 , then any sentence which is a correct translation of S 2 must be a correct translation of S 1 . For, any of those meta-linguistic translations contain a name of the English sentence “the moon is round,” hence any correct translation of them into another meta-language (e.g. “ ‘the moon is round’ ist wahr,” if the meta-language be German) would have to contain a name of an English sentence; yet no correct translation of “it is true that the moon is round” into another language could mention the English sentence “the moon is round.” A still more important argument against such translations, however, which at the same time proves that (T) violates R1 as well, can be constructed on the premise that no statement of the form “ f (a),” where “a” is a proper name or definite description, can be true unless the entity denoted by “a” exists.8 For, a contingent consequence which follows from “‘the moon is round’ is true” by virtue of this premise is the existence of a sentence, whereas “it is true that the moon is round” has no such consequence (which is a further reason why it may be considered cognitively synonymous with “the moon is round”). This argument against the logical necessity of the instances of (T), unlike the argument from the contingency of linguistic conventions, survives the expansion of “S is true s ” to “S is true s in L,” for the latter sort of statement entails the existence of a sentence no less than the incomplete statement which it completes. That predications of truth upon propositions do not entail the existence of sentences is implicitly acknowledged by Carnap, when he proposes as a translation of “p is true” into the semantic meta-language “for any sentence S and semantic system L, if S designates p in L, then S is true in L”: for the truth of

7 Cf.

D. R. Cousin’s illuminating article (Cousin 1950). significantly diﬀerent variant of this premise is obtainable by substituting “can be significant” for “can be true.” But while this diﬀerence is, indeed, significant (cf. Pap 1953c), a similar argument leading to the same conclusion could nonetheless be constructed.

8A

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this universal conditional is compatible with there not existing any sentences that designate the proposition of which truth is predicated. However, for this very reason the attempted reduction of “truea ” to “true s ” is demonstrably inadequate. For, let us ask how the “if-then” of the semantic meta-language is to be interpreted. Suppose that material implication is meant, such that the translation may be formalized: (∀S )(∀L)(Des(S , p, L) ⊃ true s (S , L)). Then it would follow that any verbally unexpressed proposition must be true.9 If, on the other hand, the translation is meant as an entailment, a no less paradoxical consequence follows. For, let us ask what kind of p is such that from the fact that a given S designates p in a given L it follows that S is true in L? The obvious answer is: a necessary proposition. With regard to a contingent proposition it cannot be logically impossible that a given sentence expresses it in a given language and yet that sentence is false; in fact, if p is false, then any sentence designating it (in L) will be false (in L). In order, then, for the falsehood (in L) of S to be logically incompatible with the fact that S designates (in L) p, it must be logically impossible for p to be false. The translation thus entails that only necessary propositions are true! In order to escape from this dilemma one may be tempted to try as translation a contingent, but existential conditional: (∀S )(∀L)(Des(S , p, L) ⊃ true s (S , L)).(∃S )(∃L)Des(S , p, L). It would then follow that only expressed propositions can be true. And since it would be an absurd asymmetry to hold that propositions cannot be true unless they are expressed yet can be false without being expressed, the view to be examined is really that it is logically impossible that there should exist unexpressed propositions. I propose to prove, however, that if we make the sentence-proposition distinction at all (and I don’t see how one can think clearly, above all about semantic questions, without making this distinction), then we must grant that it is possible that there should be unexpressed propositions, true or false. Let t be the time which was the first time when a given proposition p was, graphically or orally, expressed, such that it is true to say with respect to p “before t nobody said or wrote that p.” In fact, if only t is some time before the origin of language, this will be true with respect to any proposition. Obviously, such a statement, involving what is commonly 9 This

counterintuitive consequence is reminiscent of Carnap’s reductio ad absurdum of explicit definitions (in an extensional language) of disposition predicates, in “Testability and Meaning” (Carnap 1937). Indeed, consistency would require him to say that “truea ” is reducible to “true s ,” yet not explicitly definable in terms of it—unless the ideal of an extensional language be abandoned and “if-then” be construed as the subjunctive connective which so far has resisted all attempts to define it in an extensional language.

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called indirect quotation, is not about the sentence “p,” but about its meaning, the proposition that p. For example, it may be true to say “before t nobody said or wrote ‘the earth is round’,” yet false to say “before t nobody said or wrote that the earth is round” because before t some Greek uttered a Greek sentence expressing the proposition that the earth is round. Now, by the rule of existential generalization, “before t nobody said or wrote that p” entails “(∃t)(∃p)(Unexpressed(p, t)).” And this entails, by a simple shift of quantifiers, “(∃p)(∃t)(Unexpressed(p, t))]”: there is a proposition which is at-sometime-unexpressed. But that there should come a time when such a, at-sometime-unexpressed, proposition comes to be expressed, is clearly a contingent event. Therefore the statement “there is a proposition which is at-all-timesunexpressed” ((∃p)[(∀t)(Unexpressed(p, t))]) must be conceded to describe a logical possibility. Conceivably it may even be provable in some way analogous to the way in which mathematicians prove that there are unnamed numbers (though it would, of course, be contradictory to mention a particular member of the class thus proved to have members). If all this is correct, then “truea ” is not reducible to “true s ,” at least not by the method shown by Carnap in Carnap 1942. Further, since predications of “truea ,” i.e. statements of the form “it is true that p,” which, as I have argued, cannot be translated into meta-statements containing “true s ,” are perfectly clear provided that “p” is clear, I do not see why the “Platonistic” commitment, which according to some critics of Carnap’s semantics is involved in such uses of absolute concepts, should worry us. Existential generalization is a mode of deductive inference which makes explicit a part of what is asserted by its singular premise; thus “(∃x) f x” expresses part of the information expressed by “ f (a),” regardless of what the type of “x” may be, and accordingly it is impossible that one who understands “ f (a)” and who understands the logical constant “there is,” should fail to understand “(∃x) f x.” But then anybody who understands, for example, “that the earth is round, is true, yet was not believed by anybody 3000 years ago,” ought to understand “there is something which is true yet was not believed by anybody 3000 years ago.” And if it is further explained that the word “proposition” refers to the sort of things (entities) that can significantly be said to be true, to be believed, to be entailed, to be meant by declarative sentences etc., no obscurity should surround statements beginning with “there are propositions which ....” To be sure, in the spirit of Occam’s razor logicians and semanticists may hail the reduction to the indispensable minimum of the types of entities to which scientific discourse is committed. In this spirit, one may hope that quantification over propositional variables is dispensable in favor of quantification over sentential variables. It will be impossible, however, to carry this reduction through, unless singular statements involving names of propositions, like “it is true that p”—here “that p” names a proposition, not “p,” which is not a name at all (cf. footnote 3)—“it is be-

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lieved by x that p,” can be translated into statements devoid of such names. Once such a translation has been achieved, and not before, one may justifiably look upon propositional names as “incomplete” symbols in the sense in which class-names are incomplete symbols in Principia Mathematica, i.e. existential generalization from “ f (p1 )” to “(∃p) f (p)” will then be illegitimate. To suppose that propositional names are eliminable without diﬃculty at least from truth-predications, since “it is true that p” is cognitively synonymous with “p,” would be to overlook that the subjects of truth-predications are more frequently descriptions, than names, of propositions. Thus, consider “the first proposition he asserted in his speech is true.” Even if the proposition satisfying the description could be named, say, “the proposition that he would attempt to prove two propositions,” the statement containing the description is not logically equivalent to the statement containing the name, since it is a contingent fact that the named proposition satisfies the description. Therefore we cannot move by mere substitution of synonyms from “ p which satisfies φ is true,” via “it is true that p1 ,” to “p1 ”: for, the identity “p1 = ( p)φp” is usually factual. And if the description be eliminated from the truth-predication by means of the contextual definition of descriptions known as the “theory of descriptions,” we are left with a statement in which “truea ” occurs in the context of propositional variables. But he who dreads propositional names will dread propositional variables at least as much,10 and therefore an elimination of names or descriptions of propositions serves no philosophical purpose if it necessitates the employment of propositional variables instead.

10 Thus

Quine says repeatedly that the ontological commitments of a theory or a piece of discourse are more reliably learnt from the latter’s variables of quantification than from its names, since names may be eliminable shorthand devices, like e.g. class-names in Principia Mathematica.

Chapter 8 BELIEF AND PROPOSITIONS (1957)

The repudiation of propositions as “obscure entities,” which is prevalent among logicians and philosophers of “nominalistic” persuasion, is frequently justified by pointing out that no agreement seems ever to have been reached about the identity-condition of propositions. And if we cannot specify, so they argue, under what conditions two sentences express the same proposition, then we use the word “proposition” without any clear meaning. Quine, for example, feels far less uneasy about quantification over class-variables than about quantification over attribute-variables and propositional variables, because there is a clear criterion for deciding whether we are dealing with two classes or with only one class referred to by diﬀerent predicates: two classes are identical if they have the same membership. And such a criterion of identity is alleged to be lacking for intensions. The problem is a serious one and cannot be disposed of by applying a generalized Leibnizian principle of identity of indiscernibles: two propositions are identical if they agree in all properties. For this would mean that two sentences express the same proposition if one can be substituted for the other in any context without change of truth-value. But either this criterion is applied to extensional contexts only, or it is also applied to modal and intentional1 contexts. If the former, it leads to the untenable conclusion that all sentences of the same truth-value express the same proposition (“proposition,” then, becomes a redundant term; to ask what proposition is expressed by a given sentence would simply be asking whether the sentence is true or false). If the latter, then the criterion seems to be ineﬀective for several reasons. First, a non-extensional language permits the construction of such non-extensional sentences as “the proposition that p = the proposition that q” (such sentences are non-extensional with respect to “p” and “q” because obviously diﬀerent propositions may have identical truth-values). And how is one to decide whether “p” is substitutable for “q” on the right-hand side of this identity without changing the truth-value 1 By

an intentional context of “p” I mean a statement like “A believes that p,” “A doubts that p” etc.

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of the identity-statement, unless one already knows whether or not “p” and “q” express the same proposition? Second, consider a modal context like “it is necessary that (A is a father if and only if A is a male parent).” This modal statement is true if and only if “A is a father” is L-equivalent to “A is a male parent,” but although L-equivalence is not, as I shall explain presently, in general a suﬃcient condition for identity of propositions, I doubt whether the Lequivalence of these two sentences could be established independently of the assumption of their synonymy2 —which is the question to be decided by the substitution test. It may be replied that Leibniz’s principle leads to similar diﬃculties when applied to individuals, and that the intensionalist, therefore, is really no worse oﬀ than the nominalist who uses the concept of “individual” without philosophical qualms. Indeed, there is something in this objection. If the predicatevariable in “(∀P)(Px ≡ Py),” the Leibnizian definiens for “x = y,” is completely unrestricted except as to type, then peculiar consequences follow: in the first place, “x = y” would be a value of “Px,” hence the attempt to decide an identity-statement on the basis of such a definition would be circular. Second, if an intensional function “A knows that x = y” is admitted as value of “Px,” it would even follow that no identity-statement about individuals can be both informative and true. For, since undoubtedly everyone knows that x = x, whatever individual x may be, “x = y” could not then be true unless everyone knew that x = y. This is a generalization of the paradox of analysis and of what Carnap has called the “antinomy of the name-relation,”3 anticipated by Russell’s observation that King George wished to know whether Scott was the author of Waverley, but not whether Scott was Scott. Quine has shown that unrestricted substitutivity of identity also leads to paradox if modal functions are admitted: it used to be thought that the morning star is identical with the evening star, but this was to overlook that the morning star is necessarily identical with itself whereas it is not necessarily identical with the evening star.4 Nevertheless, the nominalists are right in feeling that identity of individuals is less problematic than identity of intensions, such as propositions. For, whether or not a consistent calculus of modal functions—to be distinguished from modal operators, prefixed to names of propositions—be possible, none of the mentioned paradoxes would arise if the Leibnizian definition were restricted to first-order functions in the sense of the ramified theory of types. This does not mean that the ramified theory of types, which is widely held to 2 Given

the synonymy of “father” and “male parent,” the above modal statement is of course derivable from the truth of modal logic “it is necessary that A is a father if and only if A is a father.” 3 See Carnap 1956a, §31. 4 Incidentally, some logicians would solve Quine’s paradox of necessary identity by pointing out that it cannot arise in the primitive notation in which the identity sign does not occur between descriptions but between logically proper names: there the identity-statements have either the form “a = a” or the form “a = b,” and hence are L-determinate since diﬀerent individual constants are defined as names of diﬀerent individuals. But as the informative identity-statements of natural language always involve descriptions, I doubt whether this solution would satisfy anyone who doubted on philosophical grounds whether an identity-statement like “the morning star is identical with the evening star” could be true.

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be dead, must be resuscitated in order to rescue the concept of identity of individuals. Expressions like “all the properties (of a given type, but of any order) of x” may be admitted as perfectly meaningful, yet a definition of “x = y” as “x and y share all extensional first-order properties, i.e. extensional properties not defined in terms of a totality of properties” is perfectly adequate. Nobody, for example, would seriously doubt the identity of Scott and the author of Waverley just because a king some time ago was doubtful of this identity while being perfectly certain of Scott’s self-identity. It is perfectly satisfactory, then, to define identity of individuals as agreement with respect to all extensional first-order properties, whereas a similar restriction of substitutivity of identity does not, as we have seen, work in the case of intensions. According to Carnap two designators have the same intension if and only if they are L-equivalent. A special case of this criterion is that two declarative sentences express the same proposition if and only if they are L-equivalent. But this will not do either. For, an obvious criterion of adequacy which an explication of “proposition” (via the explication of synonymy of declarative sentences) should satisfy is that “A believes that p, and p ≡ q” should entail “A believes that q.” Yet, any two logically necessary statements are L-equivalent, but it could hardly be maintained that, where “p” and “q” are logically necessary, “A believes that p” entails “A believes that q.” For example, anybody with a rudimentary knowledge of the propositional calculus will believe a simple tautology like “((p ⊃ q). ∼ q) ⊃∼ p,” yet there are tautologies with respect to which he could profess neither belief nor disbelief because he does not recognize them as tautologies. In general, it is surely possible that, being familiar with the semantic and syntactic rules for the symbols of a logical system, one understands the sentences of the system, i.e. knows what propositions they express, yet does not know whether the propositions expressed are logically necessary. We all understand the sentence “for n greater than 2, there are no solutions for the equation: xn + yn = zn ,” i.e. know what proposition it expresses, but according to my information it is not yet known whether the proposition is logically necessary. But according to the L-equivalence criterion of propositional identity, we already believe this proposition if it is logically necessary! In order to bring the L-equivalence criterion into accord with the concept of (propositional) belief, one would in fact have to stipulate, as a postulate partially defining “belief”: ([A believes that p] and p entails q) entails (A believes that q). But this postulate is obviously not satisfied by the concept ordinarily meant by “belief.” It is only “A believes that p” together with “A believes that p entails q” that could be said to entail “A believes that q.” The L-equivalence criterion does not seem to be satisfied by contingent propositions either. This can again be shown in terms of the evident requirement that “A believes that p, and p ≡ q” should entail “A believes that q.” It will surely be granted that it is impossible to have a propositional attitude,

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whether belief or disbelief or doubt or any other, towards a proposition some of whose constituent concepts one does not “have.” A man who does not understand the meanings of color predicates, e.g., cannot have a propositional attitude towards a proposition containing a color concept. Now, take any true contingent proposition p; it is L-equivalent to the proposition “if anyone asserted that p, he would make a true assertion.” It seems to me to be logically possible that a man who “has” the descriptive and logical concepts that are constituent of p should fail to have the concept of asserting, and therefore believe p without having a propositional attitude towards this conditional proposition which is L-equivalent to it. I am not, however, convinced by this argument for the thesis that even in the case of contingent propositions L-equivalence is not a suﬃcient condition for identity. For its validity depends on the notion of “concept constituent of a proposition,” which requires to be clarified. Is the concept expressed by “red,” for example, a constituent of the proposition expressed by “a is round and red, or round and not red”? Now, it is not, according to a plausible definition of “constituent concept”: the concept expressed by a constituent expression of a sentence is a constituent of the proposition expressed by the sentence, if and only if the expression occurs essentially in the sentence. Clearly, “red” does not occur essentially in the above disjunction of conjunctions. But it might be argued that “assert” likewise fails to occur essentially in the sentence “if anyone asserted that p, he would make a true assertion.” For, the latter can be transformed into “if anyone asserted that p, he would assert a true proposition.” Now, a verb occurs inessentially in a sentence if any consistent and grammatically admissible substitution of a diﬀerent verb for it leaves the truth-value of the sentence unchanged. But the only verbs that can be inserted for the dots in “so-and-so . . . that p” without producing nonsense are intensional verbs, i.e. verbs like “assert,” “believe,” “disbelieve,” “presuppose” etc., and it is clear that any such substitution into “if anyone . . . that p, he would . . . a true proposition” is truth-preserving. Nevertheless, I do not think that by reference to inessential occurrence of terms the failure of logically equivalent contingent propositions to be identical can always be shown to be merely apparent. “x is orange” is logically equivalent to “x is intermediate-in-color between red and yellow,” but all the descriptive terms occur essentially, hence we can argue that a man having a propositional attitude towards the proposition expressed by the first sentence might fail to have a propositional attitude towards the proposition expressed by the second sentence, since he might, say, have a concept of the color orange without having a concept of the color red: for the latter concept is a genuine constituent of the proposition expressed by the second sentence. Let us return to our adequacy criterion: A believes that p, and p ≡ q, entails that A believes that q. Would it be satisfied if we strengthened the identitycondition by requiring, instead of merely L-equivalence, intensional isomor-

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phism of “p” and “q”? It has been argued recently that even this extremely strong identity-condition—I shall argue presently that it is much too strong— does not necessarily satisfy it. Thus Benson Mates (Mates 1952) seems to think that, though whoever believes that all Greeks are Greeks believes that all Greeks are Greeks, it is not necessarily the case that whoever believes that all Greeks are Greeks believes that all Greeks are Hellenes; yet, on the assumption that “Greek” and “Hellene” are L-equivalent, the sentence “whoever believes that all Greeks are Greeks, believes that all Greeks are Greeks” (D) is intensionally isomorphic with the sentence “whoever believes that all Greeks are Greeks, believes that all Greeks are Hellenes” (D ). If we substitute D and D for “p” and “q,” and agree with Mates that it is possible to doubt that D but not possible to doubt that D, then I suppose we must agree with him that no explication at all of synonymy which allows diﬀerent sentences to be synonymous could satisfy our adequacy criterion. However, while it is of course conceivable that a person respond aﬃrmatively to the question “do you believe that everybody believes that all Greeks are Greeks” yet negatively to the question “do you believe that everybody believes that all Greeks are Hellenes,” this does not establish Mates’ contention. For if the subject were asked to support his doubt whether everybody believes that all Greeks are Hellenes, he could only do so by pointing out that somebody may have an imperfect knowledge of the English language so as to fail to know that “Greek” and “Hellene” are synonyms, and therefore fail to know that the proposition expressed by “all Greeks are Greeks” is the same as the proposition expressed by “all Greeks are Hellenes.”5 But then Mates’ argument is simply based on the confusion between “A believes that p” and “A believes that ‘p’ expresses a true proposition” (as I have argued in Pap 1955b). Mates’ argument does not prove that no explication of synonymy that is compatible with applicability of “synonymous” to pairs of distinct sentences must fail to satisfy our adequacy criterion (substitutivity in belief sentences). If it proves anything, it proves the inadequacy of a superficial behavioristic analysis of “belief” according to which a subject’s response to questions about his beliefs is conclusive evidence for or against hypotheses about his beliefs.6 It would indeed be strange if an experimental linguist reported that some people do not accept the law of identity, his evidence being that some people who professed belief when they were asked whether all Greeks are Greeks, expressed doubt when they were asked whether all Greeks are Hellenes although the two sentences express (to him, the interrogating linguist!) the same proposition. Such a response would normally be 5 It

is but for the sake of the argument that I assume here that a trivial tautology of the form “all A are A” expresses a proposition at all. 6 It is interesting to notice, though, that on Carnap’s behavioristic analysis of “belief” viz. “A believes that p = A is disposed to an aﬃrmative response to some sentence that is intensionally isomorphic to ‘p’,” it is logically impossible that A should believe that D, yet fail to believe that D .

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taken as evidence, rather, that “Greek” and “Hellene” do not mean the same to the subject. On the other hand, intensional isomorphism in Carnap’s sense7 is too strong an explicatum for “synonymy,” because a simple designator cannot be intensionally isomorphic with a compound designator. I believe that “father” is synonymous with “male parent,” “ignoramus” with “we do not know,” “x3 ” with “x · x · x”; further, “p and q” with “q and p,” “p or q” with “q or p,” “some A are not B” with “some A are non-B.” And since none of these pairs is intensionally isomorphic, I conclude that either Carnap’s explication is incorrect or else his explicandum is not a concept of synonymy that is of interest for philosophical analysis. Carnap’s reply to a similar objection by Linsky (Linsky 1949) was that there is a family of stronger and weaker synonymy-concepts and that it is unfair to criticize an explication of a stronger synonymy-concept on the ground that it does not fit a weaker synonymy-concept. But this reply is insuﬃcient, since the explicandum which analytic philosophers are interested in is a semantic relation of which the pair “father, male parent,” e.g., is an instance; and second, since in a language in which “x3 ” is explicitly introduced as abbreviation for “x · x · x,” “33 ” is surely synonymous with “3 · 3 · 3” in precisely the same sense in which, say, “3 · 3 · 3” is synonymous with “3 × 3 × 3.” The time is ripe for suggestion of a new approach to the problem of propositional identity. To begin with, it is utopian to look for an absolute criterion. Instead, propositional identity should be relativized to specified kinds of substitution-contexts. The weakest concept of propositional identity is propositional identity relative to extensional contexts: relative to an extensional language, “p ≡ q” is a suﬃcient condition for “ f (p) ≡ f (q).” Hence there is here no need to distinguish propositions from truth-values: the range of the sentential variables in the propositional calculus comprises only two values, the true and the false. The idea that a law of the propositional calculus is a truthfunctional tautology could then be expressed as follows: if “ f (p)” is a tautology, then “ f (T). f (F)” holds, if “ f (p, q)” is a tautology, then “ f (T, T). f (F, T). f (T, F). f (F, F)” holds, etc. The next stronger concept of propositional identity is propositional identity relative to modal contexts: relative to modal logic, strict equivalence, not material equivalence, is the suﬃcient condition for truthpreserving substitutivity of statements. “If p is L-equivalent to q, then f (p) ≡ f (q)” expresses this stronger identity-condition. Accordingly, there are distinct propositions of the same truth-value in modal logic. However, relative to modal logic there is but one necessary proposition and but one impossible proposition, since all necessary statements are strictly equivalent and all impossible statements are strictly equivalent. Now, we have seen that it is the

7 See

Carnap 1956a, §§14, 15.

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adequacy condition concerning belief which forces us to distinguish one necessary proposition from another. This is to say that L-equivalent statements are not necessarily interchangeable in intensional contexts. The strongest identitycondition for propositions accordingly reads: if “x believes that p”8 is strictly equivalent to “x believes that q”—which entails that “p” is strictly equivalent to “q”—then f (p) ≡ f (q). Or alternatively, f (p) ≡ f (q) if p is strictly equivalent to q and moreover B(x, p) ≡ B(x, q). Relative to an extensional language, propositional identity means material equivalence; relative to a modal language it means strict equivalence; relative to an intensional language it means strict equivalence of the corresponding statements of belief. The objection is likely to be raised that no operational (“eﬀective”) criterion of synonymy has been provided at all. For the only possible strict proof of the strict equivalence of “x believes that p” and “x believes that q” would rest on the proof of “p = q,” and hence the whole procedure would be circular. Now, in the first place, the same circularity arises already in connection with the weaker criterion of propositional identity which is in eﬀect Carnap’s criterion: in order to prove, e.g., that “father” is synonymous with “male parent” one would have to prove that the biconditional joining them is just as necessary as the trivial biconditional “all and only fathers are fathers,” which proof could be conducted only by substitution of “male parent” for one occurrence of “father” in the trivial biconditional, which substitution could be justified only by the assertion of synonymy which is in question. The lesson to be learnt from this, it seems to me, is that the clarification achieved by analyzing “proposition” (via “same proposition”) in terms of modal or intentional concepts, in a sense of “analysis” which requires that the analysans be clearer than the analysandum, is illusory. In the following, I shall propose an alternative method of, as it were, simultaneous clarification of the category-term “proposition” and the non-extensional operators, both modal and intentional, through axiomatic definition. In the meantime, it might be pointed out that at least negative conclusions about synonymy can be arrived at without apparent circularity through the test of substitutivity in intensional contexts. I think the following type of argument is perfectly respectable: “p” and “q” do not express the same proposition, because it is possible to know that p without knowing that q. It is by this type of argument that I try to convince students who are accustomed to a loose use of the word “definition,” that so-called definitions of color predicates in terms of wavelengths do not express the ordinary meanings of the color predicates, and that physicalistic definitions of mentalistic terms likewise do not express the meanings of the latter: it is possible to know that a thing is blue, or that one is feeling sad, without knowing which is the wave-

8 This

statement-form will henceforth be abbreviated: B(x, p).

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length of the light the thing is disposed to reflect, and without knowing anything about one’s physiological condition or behavior at a time one is feeling sad. And just to forestall an irrelevant debate about the criterion of “knowledge,” I hastily add that the argument remains unaﬀected by a substitution of “belief” for “knowledge”: if “p” and “q” are synonymous, then the statement “it is possible to believe that p without believing that q” is self-contradictory. Hence, if the latter statement is not self-contradictory, the sentences are not synonymous. Nelson Goodman in Goodman 1952a has decried this negative test of synonymy as a pseudo-test, on the following ground: with respect to any two statements at all, it is possible that one should know one to be true without knowing the truth-value of the other, since one might not understand the other. And if the test is applied only to persons who understand both statements, then it becomes redundant since one could not be said to understand both of two statements unless one knows whether or not they are synonymous. However, Goodman’s objection does not apply to my formulation of the test, for the latter involves statements of the form “A knows that p,” which are object-linguistic and non-semantic, not meta-linguistic statements of the form “A knows that ‘p’ is true.” The first part of Goodman’s argument presupposes that “A knows that ‘p’ is true” entails “A knows what proposition is expressed by ‘p’.” But such semantic knowledge is obviously not presupposed by A’s knowledge of the proposition that p. Now, it must be admitted that to say two sentences express the same proposition if they are interchangeable, not only in extensional contexts, but also in modal contexts, is not illuminating unless the meanings of the modal operators, “necessary,” “possible”9 etc. are understood. But the relevant meaning of “possible,” e.g., can, apart from illustrations, be indicated only by giving a set of axioms which are satisfied by the intended meaning: if p, then possibly p; if necessarily p, then possibly p; if possibly (p and q), then possibly p and possibly q, but not conversely (for “not-p” is substitutable for “q”) etc. Such an axiomatic definition10 at the same time defines “proposition”: the propositions to which modal logic is “committed” are the values of the sentential

9 Notice

that only one primitive modality is required, e.g. “possible.” is sometimes objected to axiomatic (or “implicit”) definitions that they are not unique, since there is more than one model for any consistent (and nontrivial) set of axioms. But neither is there any guarantee that an explicit definition is unique in a sense in which an implicit definition is not. For either the primitives occurring in the definiens are implicitly defined by the axioms of the deductive theory in which the defined term occurs, and then their ambiguity is communicated to the defined term (e.g., since “straight line” as a primitive of a geometrical system admits of several interpretations, so does “triangle” which is explicitly defined in terms of “straight line”). Or else they are interpreted by means of ostensive definition. But there is no guarantee that an intended meaning is really communicated by means of ostensive definition, since the latter limits it only to a property shared by all the instances pointed to (and absent from all the negative instances pointed to, if such are used). At any rate, ostensive definition is applicable only to descriptive terms, not to logical constants, hence only the first alternative is relevant in the present context.

10 It

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variables in the axioms of a system of modal logic. If, indeed, the variables are given completely unrestricted ranges, then the terms which would otherwise be used in the meta-language to characterize the subject-matter referred to by the axioms simply appear as additional primitives in the axioms. For example, one could diminish the number of primitives in the axioms of formal Euclidean geometry, by using special variables ranging over points, special variables ranging over straight lines, and special variables ranging over planes. This would be analogous to using the term “proposition” in the meta-language to define the ranges of the bound variables of modal propositional logic. But we could alternatively put it as a primitive into the axioms, just the way one normally handles “point,” “straight line,” “plane” as primitives along with the relational predicates that are used to make assertions about these entities; and then it would be obvious that “proposition” is defined simultaneously with the modal operators. Following this approach, let us formulate the stricter requirements of propositional identity that are imposed by belief-contexts by laying down axioms of a logic of belief, and saying that propositions are the values of the sentential variables in those axioms. The latter may be conceived as implicitly defining at once “belief” and “proposition.” I proceed to formulate five axioms of such a logic of belief which in terms of the intended meaning of “belief” are selfevident—indeed this is to assert a tautology since the axioms serve the purpose of explicating the meaning of this new primitive.11 A1 A2 A3 A4 A5

(B(x, p) B(x, q)) (p q) (B(x, p).B(x, (p q)) B(x, q) B(x, ∼ p) ∼ (B(x, p)) B(x, (p.q)) B(x, p).B(x, q) ∼ (B(x, p) p)

The sense of A1 is that we cannot analytically infer from the fact that a person believes p that he believes q, if p does not entail q—though “p entails q” is compatible with “somebody believes p but does not believe q.” Of course, if a person believes p without believing q although the latter proposition is entailed by the proposition he believes, this must be because he is not aware of the entailment. Hence A2 says, not that “p entails q” warrants the inference of “A believes q” from “A believes p,” but that “A believes that (p entails q)” does. A3 is likely to be disputed by those who insist that people frequently 11 Strictly

speaking, “believe” requires a time variable, so that a triadic, not dyadic, predicate should be used to express the concept of belief. The axioms are of course meant as universally quantified with respect to the (omitted) time variable. A5 is added in order to diﬀerentiate belief from knowledge, for A1 -A4 are satisfied by the relation of knowledge as well as by the relation of belief. It should be noticed that the possibilities of interpretation of the axioms are severely limited by the meta-linguistic explanation that “x” ranges over persons and “p” and “q” take declarative sentences as substituends.

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believe contradictory propositions; they may say that what is here implicitly defined is rational belief, not actual belief. I would justify the axiom, however, by the consideration that if a man believes incompatible propositions, this is because he is not aware of the incompatibility. As a matter of fact, from A3 together with A2 we can deduce: B(x, p).B(x, p ∼ q) ∼ (B(x, q)

(T 1 )

which means that it is impossible to believe propositions which one believes to be incompatible. This theorem may seem to be much weaker than ∼ 3(B(x, p. ∼ p))

(T 2 )

deducible from A4 and A3 ,12 since substitution of “∼ p” for “q” in T 1 yields the apparently more cautious entailment: B(x, p).B(x, p ∼ ∼ p) ∼ (B(x, ∼ p)). But the caution is unnecessary since a man who grasps the ordinary meaning of “not” cannot fail to see that the propositions expressed by “p” and “not-p” are contradictory. It would be futile to try to prove empirically that a man may (at the same time) believe explicitly contradictory propositions by obtaining aﬃrmative responses from the same person to two sentences which, as interpreted by the interrogator, are contradictory. I should think one would inevitably infer from such responses that the subject has misinterpreted at least one sentence. If I found a man insisting with great earnestness that the same part of a surface can at the same time be both green and red, I would conclude that he is either an excellent actor or else does not mean by “green” or “red” what one usually means. However, it is possible to believe propositions which are incompatible if their incompatibility is not self-evident. Therefore the inference from an affirmative response to several sentences which in their intended interpretation are contradictory to the subject’s misinterpretation of at least one sentence, is not always warranted. This is the reason why the second conjunct of the antecedent of A2 is “B(x, (p q))” instead of “p q.” A definition of “understanding ‘p”’ from which it follows that one does not understand “p” unless one knows all that is entailed by “p,” is both arbitrary and ineﬀective—though it cannot be denied that the test of whether one has understood “p” includes a test for awareness of some consequences of “p.” The specified axioms serve as adequacy criteria for behavioristic interpretations of “belief,” in much the same way as interpretations of “probability” will 12 Indirect proof: Suppose B(A, (p. ∼ p)). By A , this entails B(A, p).B(A, ∼ p). By simplification, B(A, ∼ p). 4 By A2 this entails ∼ (B(A, p)). Since the hypothesis thus entails the contradiction B(A, p). ∼ (B(A, p)), it is self-contradictory.

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usually be accepted as adequate only if they satisfy the axioms of the calculus of probability. By a behavioristic interpretation is meant an interpretation of “belief” as a disposition manifested in responses to specified stimuli. For example, “A believes that p = A is disposed to respond aﬃrmatively to some sentence synonymous with ‘p’,” as suggested by Carnap in Carnap 1956a. When combined with Carnap’s L-equivalence criterion of propositional identity, this interpretation leads to the contradictory consequence that a person may believe and also not believe (at the same time) one and the same proposition, since he may respond aﬃrmatively to some sentence synonymous in L with “p” yet fail to respond aﬃrmatively to some sentence synonymous in L with “q” though “p” and “q” are L-equivalent in L. This would of course be due to his not recognizing the L-equivalence in question, but as already pointed out it would be gratuitous hence to infer that he misinterprets at least one of the two sentences. This contradiction can be avoided by strengthening the requirements for propositional identity, so as to make “p ≡ q” entail “A believes that p if and only if A believes that q.” For if p ≡ q in the sense that “p” and “q” are interchangeable even in belief-contexts, then obviously it must be the case that a person believes that p if and only if he believes that q. And furthermore, on this assumption of strong synonymy of “p” and “q,” it is logically necessary that A is disposed to respond aﬃrmatively to some sentence synonymous with “p” if and only if he is so disposed towards some sentence synonymous with “q.” But having defined synonymy in terms of belief and belief implicitly in terms of a set of axioms, we face the question whether the behavioristic definition of belief satisfies those axioms. Now, it obviously does not satisfy A3 , for example, since it is quite possible that A respond aﬃrmatively to some sentence synonymous with “∼ p” and also to some sentence synonymous with “p,” because he does not interpret these sentences as contradictory. This indicates that the behavioristic definition is inadequate unless an important “mentalistic” condition is added to the linguistic stimulus: at best we may infer “A believes that p” from “A was asked to respond aﬃrmatively or negatively to the sentence S , and A interpreted S to mean the proposition p, and A responded aﬃrmatively to S .” I say “at best” because even this better grounded inference is not necessary: A may assert the proposition that p without believing it; he may be lying, in other words. This consideration shows that an interpretation of “belief” in behavioristic terms can hardly be adequate if it takes the form of an explicit definition in terms of a causal implication, unless the antecedent of the latter contains a “ceteris paribus” clause which covers our ignorance of relevant “intervening variables” (such as correct interpretation). The same objection would apply to introduction of “belief” by reduction sentences conceived as analytic, like

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Carnap’s “bilateral” reduction sentences.13 Carnap, therefore, is surely on the right track in advocating, more recently,14 that “belief” be treated as a theoretical construct which is implicitly and incompletely defined by the postulates of a psychological theory—though whatever theoretical postulates Carnap may have in mind must be supplemented by reduction sentences which tie the term to the observation-language in the loose way of indefinite probability implications. The important point in the present context is that “proposition,” being defined correlatively with “belief,” will inevitably share the latter’s openness of meaning.15 Notice that the terms “belief” and “intension”—a proposition being a special kind of intension—are almost inseparable in reduction sentences connecting them with behavioristic terms. As we saw, no reliable inference can be drawn either from a belief-hypothesis to a verbal response or conversely unless an assumption about an act of interpretation is warranted. But likewise, a hypothesis about a habit of interpretation—e.g., A is in the habit of interpreting “dog” to refer to a quadruped of kind K—is not highly confirmable by bare observation of linguistic behavior, for it is highly relevant to the question what a person means by an expression to know what beliefs motivate him to apply the expression to such and such objects. This interconnection of the constructs “belief” and “intension” might be expressed by the following postulate: if predicate “P” has property P as its intension for A, then, ceteris paribus, A applies “P” to an object x if and only if A believes that x has P. The “ceteris paribus” proviso covers such, not exhaustively known, conditions as suitable stimuli to a verbal utterance, a desire to express the belief and not to mislead, absence of relevant inhibitions, etc. The postulates for other kinds of intensions, like propositions conceived as intensions of declarative sentences, would be similar. Such postulates are perhaps less “analytic” than the axioms stated above, in the sense that observations of human responses would be more likely to lead to their revision than to a revision of the more “logical” axioms. But a strict analytic-synthetic distinction must be abandoned if a postulational method of meaning-specification is adopted.16 It remains to contrast the outlined approach to the definition of “intensional” concepts—which in its recognition of semantical “constructs” that are not definable in the nominalistic vocabulary is akin to the liberalization of strictly positivistic meaning criteria with respect to physics and psychology—with the approach of the logical-constructionists. According to the theory of Principia Mathematica, classes are logical constructions in the sense that names

13 See

Carnap 1937, §8. Carnap 1954b. 15 For a detailed discussion of “openness of meaning” see chapter 19, Pap 1963b, and Hempel 1952, section II. 16 This is lucidly argued by Hempel, in Hempel 1954. 14 See

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of classes as well as class-variables are contextually eliminable. It is in this sense that according to old-fashioned phenomenalism material objects are logical constructions out of sense-data: expressions referring to material objects were supposed to be in principle eliminable by translation of material-objectstatements into sense-data-language. In just the same sense Ayer, for example, regards propositions as logical constructions out of sentences—a convenient fa¸con de parler that can in principle be dispensed with. Since I have advanced detailed arguments against this kind of “reductionism” elsewhere,17 I shall confine myself here to one fundamental criticism. The logical-constructionists hold that statements ostensively referring to such pseudoentities as propositions are shorthand for meta-linguistic statements about classes of synonymous sentences. And this implies that they are translatable into a meta-language containing sentential variables and names of sentences (formed by means of the familiar quotes), but no propositional variables nor names of propositions. But surely the statement “there are propositions which are not expressed” is not so analyzable. Therefore the logical-constructionists could make good their claim only if they could show that, unlike statements about expressed propositions, this statement is meaningless. Now, the latter is incomplete in two respects: it does not specify a time, nor a specific language. Let us take it in the strongest form: there are propositions which are not expressed in any language at any time. It is not my purpose to argue for the truth, but only for the meaningfulness, of this statement. For this purpose we may begin with the innocent and undoubtedly true statement: there was a time when nobody believed, or even thought of, the proposition that the earth is round, and when no language at all existed. The proposition that the earth is round, therefore, was not expressed at that time. Therefore there is a proposition which was not expressed at that time. But that a proposition comes to be expressed in a language, is a contingency. It is therefore possible that this proposition should have remained unexpressed forever—though we could not have mentioned this possibility had it been actualized. And if “it is possible that p is at all times unexpressed” is true, then “p is at all times unexpressed” must be a meaningful statement. That it is a pragmatically self-refuting statement, in the sense that its falsehood follows from its assertion, is irrelevant. In the same sense “I am not asserting anything now” is pragmatically self-refuting, yet it is logically possible that the person denoted by “I” should not be asserting anything at the time denoted by “now.” The enemies of propositions will be quick to point to the fallacy in the foregoing argument: the step from “the proposition that p1 is not expressed at t” to “there is a p such that p is not expressed at t” begs the question. It presupposes that “that p1 ” is a genuine name, which it can be only if there are

17 See

chapter 5, especially section 3.

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propositions. It is like the inference from “roundness is a property shared by all nickels” to “there is a property which is shared by all nickels.” But since the apparent name “roundness” is contextually eliminable by translation of the above sentence into “all nickels are round,” the apparent basis for existential generalization disappears.18 Indeed, if the claim of the contextual eliminability of propositional names and variables can be made good, then the nominalist will be justified in saying that there are no propositions—or in a more tolerant vein, that we need not assume that there are. Since I have expressed serious doubts about the possibility of such contextual elimination, some may label (or libel?) me as one holding a metaphysical belief in propositions as “real entities.” I would like to conclude, therefore, by inquiring what it means to believe that “there are propositions.” Quine has oﬀered a much debated criterion of a man’s “commitment” to a special kind of ontology: Look at the variables of quantification in his language; if they belong to the primitive notation, then the user of that language is committed to the belief that the entities over which they range exist. But he has not, to my knowledge, discussed the question what it means to say that such entities exist. Now, within a realistic language we encounter only qualified existence assertions, not what might be called categorial existence assertions: “there are attributes (or propositions) satisfying the function f (φ) (or f (p))”—e.g., “there are propositions which nobody believes,” “there are attributes which are possessed by only one individual”—not “there are attributes (or propositions).” In Carnap’s terminology, categorial existence assertions are external to a given language.19 We may not like Carnap’s claim that the corresponding questions like “are there propositions” are devoid of cognitive meaning (unlike the questions concerning qualified existence which are formulated within a given language), but whosoever claims the contrary should oﬀer an interpretation of such questions. I suggest that we take Quine’s criterion of ontological commitment as the very definition of ontological commitment and thereby assign a cognitive meaning to categorial existence assertions. That is, to say “there are propositions” is to say that the propositional names and variables which we employ in order to say what we want to say are not contextually eliminable, that they belong to the “ultimate furniture” of our language. Thus nebulous questions about the ultimate furniture of the (extra-linguistic) universe are reduced to less nebulous questions about the ultimate furniture of cognitive language. The question whether propositions and other abstract entities exist is not, indeed, decidable empirically, not even in the indirect empirical way in which scientists decide whether atoms and electrons exist, but it is nonetheless a cognitive question: it is decidable the way questions of 18 See 19 See

Quine 1949. Carnap 1950a.

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semantic analysis are decidable, by examining whether proposed translations into a language with a specified primitive vocabulary preserve the meanings of the translated statements. Of course, in what precise sense of “meaning” such philosophical translations into ideal languages are required to preserve meanings, is another question—fortunately beyond the scope of this paper.

Chapter 9 SEMANTIC EXAMINATION OF REALISM (1947)

1.

Universals in Re and the Resemblance Theory

It is by no means beyond dispute what precisely the terms “realism” and “nominalism,” in the age-long controversy about the status of universals, have stood for. Without any regard to historical complexities and shifts of meaning, I shall, in this paper, define “realism” and “nominalism” as follows: According to realism, universals exist, to employ the scholastic phrase, in re, i.e., one and the same property (in the wide sense in which both qualities and relations are properties) is often simultaneously exemplified by several particulars. “Property” and “universal” are here used as synonyms. A property, in this usage, is the intension (or logical connotation) of any predicate, of whichever degree (relations are thus the intensions of predicates of degree 2 or any higher degree). According to the nominalists, on the other hand, there are no “ontological” universals. In the impressive language of metaphysicians, “only particulars have ontological status,” according to nominalism. There are, indeed, general words; but it is a mistake to suppose that, like proper names and definite descriptions, general words stand for, or refer to, an entity. Predicates (which are general words) are, indeed, applicable to several particulars that resemble each other in certain respects. But if the word has a unique referent, the latter is not a universal whose identical presence constitutes the resemblance, but at best a class of similar particulars. It is not always clear whether nominalists deny the existence of universals, substituting instead the existence of classes that are constituted by resemblance, or whether they maintain that universals are classes. The latter statement means presumably that the terms “universal” and “class” are synonyms. However, such a view cannot be taken seriously. Any material equivalence of the form “for every x, if and only if x has the property F, then x has the property G” expresses the fact that the class of particulars having the property F is identical with the class of particulars having the property G, where F and G are distinct properties since they do not mutually entail their presence. Hence, if “universal” is used in the sense of “property,” “universal” and “class” cannot

181

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be synonyms. A universal is an entity capable of having instances; a class has members, but no instances. To speak of an instance of a class makes no more sense than to speak of a member of a property or universal. No doubt many of the controversies between nominalists and realists are due to the fact that the term “universal” is used in diﬀerent senses by the disputants. Besides the confusion mentioned above, the tendency to speak of universals as though they were mental entities may be cited. Berkeley’s attack on abstract ideas, for example, has nothing to do with the issue whether there are universals in re or whether there are only classes constituted by resemblance and predicates with multiple applicability. Ideas, whether they be abstract thoughts or concrete memory images, are themselves particulars, since they are dated events (even though it might be questioned whether, like physical particulars, they can be spatially located). Hence, in identifying universals with ideas, one could not be using the word “universal” in the sense in which universals are contrastable with particulars. Nominalism, in the sense in which I have defined this traditional doctrine, may assume a stronger or a weaker form. By the stronger variety of nominalism I mean the flat assertion that it is never the case that the same property is instanced in more than one particular at the same time.1 In its weaker variety, nominalism merely asserts that it may always be doubted whether one and the same property is instanced in more than one particular. Now, it seems to me that the only way in which the nominalist could attempt to defend the stronger of the two assertions is by refuting any apparent contradictory instance of his universal negative proposition. But how else could this be achieved than by showing that in each proposed counter-instance it may be doubted whether the same property is instanced more than once? Hence it will suﬃce, for the refutation of nominalism,2 to show that such multiple exemplification of the same universal cannot always be doubted. Nominalism has historically gone hand in hand with empiricism, while realism has been fought as a metaphysical school. Thus the nominalist is prone to point out that classes of similar particulars are all that is empirically given, whereas universals identically present in diﬀerent particulars are purely metaphysical, non-verifiable entities. It is my endeavor to show that realism, in the sense defined, can be established on purely semantic grounds and is hence in no way opposed to empiricism. 1 Particulars

constitute a genus, of which things (Broad’s “continuants”), events (Broad’s “occurrents”), and processes are species. Traditionally, in speaking of particulars most attention seems to have been paid to things with qualities lasting over a finite duration. Strictly speaking, of course, the strong variety of nominalists would have to contend not only that spatially separate things sharing an identical property do not exist, but even that a thing is not ever in a definite state for more than an instant of time. For if the latter were not contended, it would be admitted that several successive events (which, like things, are particulars) may be instances of the same universal. 2 The reader should keep in mind that when I speak of the “refutation of nominalism, ” I mean by “nominalism” only what I said I mean. There may be senses of the word in which nominalism is quite unobjectionable. For example, if by “nominalism” be meant a semantically-founded criticism of Platonic reifications, I have no quarrels with it at all.

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First, let us see what precisely is involved in the claim that only similarities, not identical properties, are empirically given. The favorite examples used by the defenders of the resemblance theory are color shades. Can I be sure that two juxtaposed color patches are instances of precisely the same specific shade of red? Suppose that this may, indeed, be doubted. Can it be doubted, then, that we have here, at least, two instances of the determinable “redness”? Suppose this is also considered doubtful, since, after all, color terms are vague, and we might be confronted with a borderline case where one of the compared patches still fell within the class “red” while the other just fell outside it, even though extraordinary discrimination might be required to detect this fact. Surely, all doubt must vanish if we push our abstractive procedure one step further and consider the determinable “being colored.” When we compare patches or surfaces with respect to color and we make comparative judgments (a resembles b more than c in hue, but it resembles c more than b in color saturation, e.g.), we certainly imply that the compared surfaces all have the self-same property of being colored. In fact, the respect in which they are said to be similar or dissimilar is precisely a property (usually a determinable) identically present in all of the compared particulars; otherwise there would be no “ground of comparison.” To take another example: we may be in doubt as to whether there are two hats in the universe of precisely the same size; but how could we even raise the question whether they have the same determinate size, unless the hats shared the determinable property of having size (i.e., some size or other)? It could not be replied that determinables are universals manufactured by the mind and not really perceived in things. For if determinates are perceived, determinables must be perceived also. When I perceive a red patch I cannot fail to perceive a colored patch; and in fact I may perceive a patch as having some color or other without being able to tell which determinate color it is. My first point, then, is that resemblance is always resemblance in a certain respect, and if we only choose as our “respect” a suﬃciently abstract property, we can eliminate all doubt as to the presence of an identical universal in the compared instances. And since the more abstract properties that constitute a ground of resemblance are usually related to the more concrete (specific) grounds of resemblance as determinable to determinate, it cannot be argued that while the latter are perceivable, the former are mental constructions only. At least I strongly doubt whether a philosopher could convince an ordinary mortal that he never saw any human being even though he saw plenty of women and plenty of men.3

3 It

might be added that it is hard to be sure in any given case whether a given determinate is really an “ultimate” determinate in the sense of not being itself a determinable with respect to still more specific determinates; hence the boundary line between perceivable and merely conceivable determinables would be fairly arbitrary. Thus, if coloredness is held to be not strictly perceivable since whatever we perceive must

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Now to my second point against the resemblance theory. Resemblance is itself a property or universal, even though a relational one. Let “R1 ” stand for the relation of resemblance between particulars; “R1 ” is thus a polyadic predicate of type 1. Suppose we have two perceptual judgments: aR1 b and bR1 c. To be consistent with their resemblance theory, the nominalists could not assert “R1 = R1 ” (where the first occurrence of the predicate refers to the context a, b, and the second occurrence to the context b, c). Instead they have to introduce a relation of resemblance holding between first-order resemblances: R1 R2 R1 , where “R2 ” is of type 2. The infinite regress is obvious. Not that the infinite regress by itself proves the falsity of the resemblance theory. However, let us apply our abstractive procedure again. In which respects does relation R1 in the context a, b resemble relation R1 in the context b, c? One obvious respect in which this second-order resemblance holds is the respect of being a relation. If a and b are related in a certain way, and b and c are related in a certain way, it may, indeed, be doubted whether they are related in quite the same way. But how could it be doubted that they are both related? Now, being related is a universal, even though an extremely generic or abstract one; and being related by an n-adic relation (where n is a constant greater than 1) is a universal,4 a little more specific. I confess to being absolutely convinced that this universal is instanced in the pair a, b as well as in the pair b, c. I proceed now to the specifically semantic argument I promised. We saw that with regard to the atomic sentences “aR1 b” and “bR1 c” the nominalist, to be consistent, would have to deny that R1 = R1 (where “=” is the identity sign). But what could such a denial amount to, if not the assertion that the symbol “R1 ” in the first application (“aR1 b”) has a diﬀerent meaning from the symbol “R1 ” in its second application (“bR1 c”)? To take a concrete example: suppose I pointed to an American Indian and said “he is red,” and then pointed to a communist, saying likewise “he is red.” These sentences have the syntactic form of the atomic sentences, in an artificial symbolic language, “Pa” and “Pb” (where “P” is the symbolic substitute for “red”). Suppose I added to my symbolic object-language the sentence “P P.” How should we interpret it? I think it could only be regarded as a misleading formulation of what is properly expressed by the following sentence from the semantic part of the meta-language: the predicate “P” has diﬀerent meanings, in the sense of conhave a specific color, why not say that redness likewise cannot be perceived, since whatever is red must have a specific shade of red? 4 “Being a relation” and “being related,” as well as “being an n-adic relation” and “being related by an n-adic relation,” must, of course, be distinguished as predicates of diﬀerent types and diﬀerent degrees. A couple of related particulars constitute an instance of the relation (of type 1) “being related by a dyadic relation,” but not of the property (of type 2) “being a dyadic relation.” In this connection it might also be noticed that the relation “being an instance of” is diﬀerent from the relation of determinate to determinables. Determinates are always themselves universals, while instances may be particulars; and the former relation is analogous to class-membership while the latter relation, being transitive, is analogous to class-inclusion.

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noting diﬀerent properties, in diﬀerent applications (such as the application of “red” to both the American Indian and the communist). Thus the doubt as to whether an identical property can ever be instanced more than once is in no way distinguishable from the doubt as to whether any predicate may be univocally applied to more than one instance. When we say of a given predicate that it is non-univocal or ambiguous, we mean that in diﬀerent classes of usages it refers to diﬀerent properties. The predicate “red” is ambiguous in this sense, as shown by the above illustration. But the mere fact that a predicate is applicable to more than one instance does not make it ambiguous. Yet if we can never be sure that the same property is present in diﬀerent instances, then we cannot be sure, from an examination of actual usage of predicates, that there are any univocal predicates at all. For how else can we establish the univocality of a predicate than by observing the instances to which it is applied by competent users of the language, and noting the recurrence of a common property in those instances?5 This lack of certainty with regard to univocality of predicates, which seems to be entailed by the resemblance theory, does not mean merely that natural languages are inevitably beset with ambiguities and that an ideal language in which all predicates are univocal is—just an ideal. For, if multiple applicability constitutes ambiguity, then any predicate is ambiguous in virtue of being a predicate, and the only univocal names would be proper names which are literally “proper”—unlike proper names in the grammarian’s sense of “proper name”—to one and only one individual. The question “Are a and b instances of the same property P?” is translatable from the “material mode of speech” into the “formal mode of speech”: “Is the predicate ‘P’ applicable to both a and b?” In whichever mode, material or formal, the question be stated, it is undeniably a factual one (assuming, of course, that “P” is a non-logical, descriptive predicate). But is the question “Is ‘P’ applicable to more than one instance?” likewise factual? In some cases it certainly is. Thus it is a fact that the property “being dictator of Germany between 1935 and 1940” has only one instance. However, one will find that as a rule such unit classes are specified in terms of predicates which are compounded out of simpler predicates, and that the latter are applicable to more than one instance. In our example, this is true of “being dictator between 1935 and 1940” (Mussolini is another instance) and still more of the still simpler predicate “being dictator.” Now, the meaning of complex descriptive predicates is a function of the meanings of the constituent predicates. But since no descriptive predicate in actual use is infinitely complex, we must begin with simple predicates

5 Naturally, the instances to which a univocal predicate is applied will always have more than one property in

common and the instances to which an ambiguous predicate is applied will also share common properties, however glaring the ambiguity may be. Hence, in any such semantic investigation a prior judgment as to which properties could possibly be meant by the predicate is indispensable.

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whose meaning can only be denotatively shown. With regard to such simple predicates it is not a factual question at all whether they apply to more than one instance. For they can acquire a meaning (an intension) for the language user only by being applied to several instances. Hence it follows from the very fact that they are meaningful that they specify classes of several members. It is an elementary truth about the process of learning one’s native language that it is impossible to give an ostensive definition of a predicate by pointing to one and no more than one particular. For the particular that is denoted has a variety of properties; and how is the instructed person to tell which of these properties is being pointed out to him? Thus, if I point to a white billiard ball and tell a child “that is white,” the child might say “white” the next time he sees a red billiard ball. To prevent such misunderstandings, I have to point to other white things which share, besides their color, as small a number of properties as possible with the first thing and with each other. Suppose that no particular except that memorable billiard ball possessed the property of whiteness. In that case, “white” would not function as a predicate at all, but it would be indistinguishable in function from a proper name. For “a is white” (where “a” refers to the particular which we suppose to have, besides whiteness, several properties which are exhibited in other instances as well) would be a conventional baptismal, as it were, involving no judgment of similarity. Of course, an act of recognition would be involved if the same term “white” were applied to a on several occasions. But the same sort of recognition is involved in saying “this is John Smith,” if one has met John Smith before, and we would not say that “John Smith” is for that reason a predicate and not a proper name.6 The alleged diﬃculties besetting the realistic theory of multiple exemplification of an identical universal are all of them purely linguistic—or rather there are no genuine diﬃculties but only apparent diﬃculties arising through the tricks which language plays upon some of us. One speaks of universals as of entities named by predicates just as particulars are entities named by proper names. This reification of universals is facilitated by the grammatical accident that not only adjectives but also nouns may function as predicates. Adjectives like “cubical” or “human” explicitly indicate properties; nouns like “cube” or “man” make us think of abstract substances. Substances may resemble each

6 Theoretically

it is, indeed, possible to render an ostensive definition unambiguous by a purely eliminative process, involving no multiplication of instances of the ostensively defined quality. I might point to a second billiard ball which resembles the first in all respects except the color, and utter the words “not white.” This eliminative method of removing ambiguities as to which property of the particular pointed at is being pointed out is indispensable when it is predicates designating determinates that are ostensively defined. If I pointed only to white things and not to non-white things, the child might come to think that “white” means what “color” means. However, it remains true that a simple predicate must refer to a universal that has more than one instance. For, suppose, indeed, the white surface pointed at for purposes of definition had no duplicates at other places, and moreover disappeared as soon as it was perceived. At least it is divisible into spatial parts, and each of its parts is an instance of whiteness just as much as the whole surface.

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other; hence such relations must hold between universals and particulars likewise, especially between a particular and a universal of which the former is an instance. Thus we get Aristotle’s “third man,” as the respect in which a particular man and the universal man resemble each other. The confusion, of course, is one of logical types: the domain and the converse domain of the relation of resemblance must contain entities of the same logical type; the difference of logical type between the property chosen as ground of resemblance and each of the compared entities must be the same, viz., one. Again, there is the old paradox, discussed in Plato’s Parmenides, of multiple location. How can one and the same entity be simultaneously located at diﬀerent places?7 But this, again, is paradoxical only because “same” is surreptitiously given the meaning of numerical identity, which is a property that can be significantly attributed only to particulars, not to universals. If I say, for example, “sphericity has the dispositional property of being simultaneously exemplifiable at several places,” my ontological language is likely to raise puzzles; if universals have dispositions, are they not analogous to particulars, like pieces of sugar, which have the disposition to dissolve in water? But if they have dispositions, then these dispositions are likely to become actualized some time. Like particulars, then, universals have a history; in Whitehead’s idiom, they “ingress into actual entities.” Which is their locus before ingression takes place? The divine mind, a Platonic heaven of subsistence, human minds? There is no end to pseudo-questions of this sort. But the sentence which generates them is of the kind aptly termed by Carnap “pseudo-object-sentences.” Once we replace it by the corresponding meta-linguistic sentence “‘spherical’ is a predicate which is in principle applicable to more than one instance,” the metaphysical mystery vanishes. For what do I add to the information conveyed by “‘spherical’ is a predicate,” if I continue “which is in principle applicable to more than one instance”? I merely make explicit what characterizes a predicate as such, i.e., in contradistinction to a proper name.

2.

Platonism and the Existence of Universals

So far, in discussing “realism,” I have neglected the historical distinction between Platonic and Aristotelian realism. I confess that it is far from clear to me what the phrases by which the distinction is commonly expressed mean. According to the Platonists, so it is said, universals constitute an autonomous 7 It

is noteworthy that while many philosophers found simultaneous multiple location of universals in space paradoxical, multiple location of a single universal in time did not give rise to any puzzles. But just as multiple location in space is paradoxical if it is associated with continuants (common sense “things”), so multiple location in time is paradoxical if it is associated with another kind of particulars, viz., events; while the latter kind of multiple location is perfectly consistent with the nature of continuants. This suggests that those philosophers thought (or think) of universals not so much as another class of particulars, but more specifically as another class of continuants.

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realm; they are “independent of particulars.” The Aristotelian, on the other hand, is supposedly more earth-bound; and while he admits that universals as well as particulars exist (presumably in opposition to the nominalists), he insists that universals have no “independent being,” but exist only in particulars as attributes.8 Now the question is what literal meaning we can attach to the phrase “universals have a being independent of particulars,” as well as to the phrase “the being of universals depends upon the being of particulars.” If the former, the Platonic, phrase means “no universals are exemplified by particulars,” and the latter, Aristotelian phrase means “all universals are exemplified by particulars,” then both Platonism and Aristotelianism are plainly false views. The simple truth (or truism) is that some universals are exemplified and some are not. Suppose, then, we analyze the Platonic phrase as follows: “universals would exist, even if they had no instances.” And suppose that this is what the Aristotelian realist means to deny. The controversy, couched in these terms, looks seductively like a controversy concerning the existence or non-existence of a certain relation of causal dependence. Would noises exist if no auditory nerves existed? Of course not, since a noise is a sensation which arises when, under otherwise normal conditions, certain physical stimuli hit the auditory nerves. Waiving all that is inessential in this analogy, it may be presumed that the Aristotelian conceives universals to be as intimately dependent upon particulars as noises are dependent upon auditory nerves. The Platonist thinks otherwise. But unless it is specified in which sense one uses the word “existence” when one speaks of the existence of universals, it is absurd to take sides in the controversy. What could the Platonist mean in contending that whiteness, unadulterated, pure, and absolute, would still exist even if no white particulars existed? And what would the Aristotelian be denying if he denied this? Now, if we examine ordinary uses of the verb “to exist,” we find that existence assertions have the form “the so-and-so exists” or “x’s of such and such a kind exist.” In the former case we assert that a certain description applies to an individual, in the latter case we assert that a certain class has members or, in intensional language, that a certain property has instances. In this ordinary usage, the verb “to exist” has, like all verbs, a tense. We can significantly ask “do bears still exist in Switzerland?” or “will there ever exist a dictator in the United States?” or “how long did the Athens of Pericles exist” etc. A moment’s reflection shows that no such questions could be significantly raised about pure universals. It makes sense to ask “when did the human race (i.e., human beings) come into existence”; but it would be nonsense to ask “when

8 The

apprehensive reader will naturally wonder what the Aristotelian realist would have to say about (a) unexemplified universals, such as “golden mountain” or “mermaid”; (b) universals that cannot possibly characterize particulars, since they are properties of properties, e.g., “universal” or “color.”

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did manhood come into existence?” Indeed, the lovers of subsistence will probably say “You have simply put your finger upon the decisive diﬀerence between universals and particulars. Particulars are in time, but universals are timeless and unchanging.” This reply, however, is misleading for the following reason. “Unchanging” and “timeless” are predicates which designate, as they are commonly used, a logically possible, though rarely actualized, state of particulars (specifically, of continuants or processes). Thus the Sphinx may be said to be unchanging and to be timeless in the sense of persisting unaltered through time; the same may be said of the planetary revolutions and, in general, of any periodic process. To say of a thing that it changes is to say that at diﬀerent successive instants it has diﬀerent properties, i.e., exemplifies diﬀerent universals. This being the way the terms “changing” and “unchanging” are ordinarily used, the only sensible interpretation of which the statement “universals are unchanging” appears to be susceptible is: the predicate “changing” cannot be significantly applied to universals (indeed, such application would violate the theory of types). Suppose a man said “smells have a certain brightness of their own, colors are not unique in that respect.” You then point out to him that such brightness is very peculiar, since the brightness of colors varies with conditions of illumination, while presumably such optical changes could not aﬀect the brightness of smells. And he replies “of course, that’s because smells are smells and colors are colors; this is just the decisive diﬀerence.” I think this reply is on par with the Platonist’s statement “the reason why existence, as predicated of universals, is tenseless, is just that it is of the nature of universals to be timeless.” In the above analogy, brightness corresponds to existence, the smells correspond to universals, and the changes in optical conditions correspond to the flow of time which cannot aﬀect the existence of universals. The Platonists have failed to give a meaning to the verb “to exist” in its application to universals, just as, in the above analogy, no meaning is provided for the word “bright” as applied to smells. To be sure, most Platonists would prefer to say universals subsist; they more or less dimly recognize the misuse of language involved in the statement “universals exist.” But how does this new word help? It is as though, to refer again to the analogy, one exchanged the word “brightness” for the word “grightness,” saying “colors alone have brightness, but smells have grightness; that’s what makes them smells rather than colors.” That statements like “whiteness exists” have no sense even though they have the same grammatical form as statements which do make sense seems to be furthermore evident from the following consideration. “Whiteness” is undoubtedly synonymous with the corresponding adjective “white” in the sense that both of these designative expressions have the same intension. But “white exists” is unquestionably meaningless, since the quoted expression is not even a sentence conforming to grammatical syntax. Hence, if “whiteness exists”

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were admitted as a meaningful sentence, one would have to accept the extraordinary consequence that a sentence may be synonymous with an expression which is not a sentence.9 Indeed, it would be a worthwhile objective to see whether among people whose language does not admit the construction of abstract nouns corresponding to adjectives or verbs anything similar to the Western Platonic belief in the existence of universals may be found. Perhaps, however, Platonism cannot be disposed of so easily. Historically, Platonism has grown up concomitantly with mathematics, and maybe Platonists can be found as frequently among mathematicians as among metaphysicians. Something ought to be said, therefore, about the use of the verb “to exist” in connection with mathematical entities, specifically numbers. Let us note at the outset that numbers are universals in precisely the same sense as qualities, relations, and processes (i.e., kinds of processes, designated by verbs) are universals: they are properties, and as such belong to the converse domain of the relation “being an instance of.” As meta-mathematical analysis by Frege and Russell has revealed, the natural numbers are properties of properties or, in Russell’s extensional language, classes of classes.10 For example, fiveness is a property of the property “being a finger on my right hand” as well as of the property “being a toe on my left foot.” To say “the fingers on my right hand are five” is to say “the property ‘finger on my right hand’ has five instances.” Again, to say “God is one” is to say (assuming “one” is used as a numerical predicate, and not as a metaphysical predicate whose meaning is ineﬀable) “there is one and only one God,” i.e., the property “divine” has one and only one instance. It is in the sense here illustrated that numbers are said to be properties of properties.11 What, now, could be meant by the assertion that the natural number n exists? According to Russell’s analysis of number it means that there are properties which have just n instances. The latter existen-

9 One

might object that there are cases where a single word is synonymous with a whole sentence, as with “alas” and “I feel terrible.” However, “synonymous,” here, could not be used in the sense in which to say of two expressions that they are synonymous entails that they necessarily have the same truth-value; for there is no sense in attributing truth or falsehood to a mere exclamation. If “alas” is held to be synonymous with “I feel terrible” or some such introspective report, this could only mean that both kinds of expressions literally “express” the same kind of mental state of the speaker, and would hence be interpreted as causal signs of the same kind of state. 10 Frege and Russell’s analyses may conveniently be combined by saying that the intension of a numeral (which latter is a symbol of type 2) is a property of a property, while its extension is a class of similar classes. 11 The main consideration that led Russell to his extensional analysis of numbers as classes of similar classes is this: suppose one defined the number 5, e.g., as the common property of all properties that have 5 instances (the definition is only apparently circular, since with the help of symbolic logic one can express the fact that a property has n instances without using the concept of the number n). How does one know that there is only one such common property? To insure the fulfilment of this uniqueness condition, Russell substitutes for the common property of all properties having n instances the class of all classes similar (in the sense of one-one correspondence) to a given class of n members. This definition of number evidently does not militate against the characterization of a number as a property of a property.

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tial statement is itself translatable into “there are classes (of individuals) which have just n members.” Thus, ultimately, if n individuals (of a certain kind) exist, the number n exists. And in order to insure the existence of all the natural numbers,12 Russell had to assume the existence of infinitely many individuals (axiom of infinity). Many philosophers feel uncomfortable about this consequence of Russell’s analysis, viz., that the existence of numbers should depend upon the existence of individuals, and in fact that only a finite number of numbers could be assumed to exist if the universe happened to contain only a finite number of individuals. But one thing clearly speaks in favor of Russell’s approach: it involves a clear analysis of the notion of existence. The existence of numbers “depends upon” the existence of individuals simply in the sense that numbers are properties, and to say of a property that it exists means that it has instances. Since numbers are properties of type 2, their existence definitionally reduces to the fact that they are properties of properties which have instances. In Russell’s logic, the statement “fiveness exists” is analyzable in one and only one way, “there is an x, such that x is five”13 (where the values which x, consistently with the theory of types, may assume, are properties of individuals), or, in the class calculus “the class of quintets has members.” Any idiomatic statement which contains “exist” or “exists” as grammatical predicate can be formalized in such a way that it exhibits the structure of an assertion of classmembership. Examples: “zebras exist” = “the class of zebras has members”; the author of Tom Jones exists = “the class of authors of Tom Jones has one and only one member”;14 “the even prime exists” = “the class of even primes has one and only one member.” Russell himself occasionally reduced existential statements to statements about sentential functions, thus: “zebras exist” = “the sentential function ‘x is a zebra’ is true for some values of ‘x’.” However, this is not an analysis in the ordinary sense, since here the analysans is expressed in the meta-language, while usually both analysandum and analysans are expressed in the same object-language (this is, of course, consistent with the fact that the statement expressing the analysis is meta-linguistic).

12 Once

the existence of the natural numbers is certain, the existence of all the other kinds of numbers (rational, irrational, real, etc.) follows, since the latter are all “constructable” out of the former. 13 For the sake of illustration, I assume a symbolic language which contains number concepts as primitives. In the system of Whitehead and Russell, of course, there are no such primitives, since numbers are defined in terms of purely logical concepts. 14 Sentences having a proper name as grammatical subject and existence as grammatical predicate are either meaningless or disguised assertions of class-membership. Thus “Hitler exists” might mean “Hitler is now alive.” But if “exists” is used in a tenseless sense, the statement is meaningless, since there cannot be proper names of non-existent individuals. In the grammarian’s sense of “proper name,” “Apollo” or “Jehovah” or “Saint Nicholas” are, indeed, proper names. But they are not proper names in the sense in which the referent of such a name must be, directly or indirectly, given through denotation; for, obviously, these mythological names are merely abbreviations for definite descriptions, which is not the case with strictly proper names.

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Platonism and the Existence of Universals

We have already seen that those Platonists who maintain that universals exist could hardly be using “exist” in the sense here analyzed. For then they would either mean that all universals have instances or that some have; but they could not mean the former, since that is so patently false that it could not escape their notice that it is false, and since it is moreover typical of Platonists to assert the existence of unexemplified essences (such as perfect beauty and perfect sphericity); and they could not mean the latter, since that is too trivially true to deserve emphasis. But as long as no alternative analysis is forthcoming (and to my knowledge none has yet come forth), the much debated statement that universals exist is meaningless. The analysis in terms of the notion of classmembership certainly fits the mathematician’s usage of the verb “to exist.” To ask, for example, whether the square root of 2 exists is precisely synonymous with the question “is there an x such that x2 = 2” or “does the class of integers whose square is equal to 2 have a member?” The psychological root of the puzzle about the existence of universals such as numbers may well be this: most ordinary uses of “exist” are—prima facie in contradiction to the class-membership analysis—not tenseless.15 Since existence is usually predicated of particulars that come into being and perish, a sentence of the form “x’s exist” or “the so-and-so exists” automatically evokes the question “where, and when?” Such questions can be meaningfully addressed to properties of type 1: when and where does this property have instances? (or, in the equivalent language of classes, “. . . this class have members?”). Therefore, when we speak of the existence of properties of higher type, the same quest for spatio-temporal specification obtrudes itself, and our inability to satisfy it leaves us in a mystery. But all that is rational in our feeling that numbers and other properties of non-elementary type exist and still do not exist in the way particulars exist (i.e., in space and time) is that these properties are not properties of individuals with spatio-temporal position. Of course, it may be presumed that hardly any Platonist would be satisfied with “P has instances, or the class determined by P has members” as the analysis of “P, the universal, exists.” For he feels that the universal would still exist even if it were not exemplified. After all, before being exemplified, the universal must already be, must it not? But this question arises from a confusion between logical intensions to which temporal predicates (such as “existing before exemplification”) cannot be significantly applied and ideas which are mental events or mental dispositions.16 If “universals exist, even if they 15 Actually,

however, there is no contradiction. For a statement involving a temporal use of the verb “to exist” may be transformed into an assertion of class-membership by introducing temporal predicates. Thus the statement “there will be an atomic war” may be transformed into “the class of future atomic wars has a member.” 16 In a statement like “I have no idea of complex numbers,” “idea” obviously refers to a disposition, not to an event, since I do not intend to assert merely that I am not apprehending the nature of complex numbers at the

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have no instances” means “we may have ideas of kinds of things that do not exist,” then, of course, the statement is both meaningful and true. We can all form ideas of golden mountains and dragons. Who would deny it? Conceptual apprehension of universals is possible, whether the apprehended universal have instances or not. “But, surely, unless unexemplified universals had some sort of being, they could not be apprehended?” Well, if “x has being” means “x is a possible object of apprehension,” it must be admitted that any universal which we happen to apprehend has “being,” no matter whether it be exemplified in the actual universe or not. For, as the scholastics said, “de esse ad posse valet consequentia”; in plain English, whatever is the case, must be possible. This, however, is entirely too trivial to be insisted upon. It may well be that the Platonists, with their talk of universals as independent “essences,” are led by the nose by grammatical forms. The statement “I think of a universal” has the same grammatical form as the statement, to pick one at random, “I hear of the death of Roosevelt”: the death of Roosevelt would have occurred even if I had never heard of it, and unless it had occurred in the first place it is unlikely that I would have heard of it. In this case, the grammatical object of the verb corresponds to an object that exists independently of, and in a sense prior to, the activity expressed by the verb. This leads one to suppose that it is the same with the object of the verb “to think of” or “to think about.” But perhaps thinking of universals is like dancing dances or singing songs or smelling smells.

present moment. Even though, however, the word may more often refer to a disposition than to an actual mental event or process, ideas in this latter sense are logically prior to ideas in the former sense; for, in psychology as well as in physics, dispositions are defined in terms of events or states evoked by appropriate stimuli.

IV

PHILOSOPHY OF LOGIC AND MATHEMATICS

Chapter 10 LOGIC AND THE CONCEPT OF ENTAILMENT (1950)

The main thesis of this paper will be best approached by raising a question of the philosophy of logic (or “meta-logic”) which most practicing logicians neglect to raise, presumably for the same reason that most mathematicians neglect to raise philosophical questions about mathematics: what is a logical constant? The problem of defining what is meant by a “logical constant” (logical term, logical sign) is crucial for a satisfactory theory of logical truth, since it seems impossible to analyze the latter concept without using the concept of a logical constant. Definitions of logical truth which do not use this concept are easily shown to be unsatisfactory. If, for example, we define a logical truth as a statement which is true by the very meanings of its terms, we are either defining a concept of psychology, not of logic, or else the definition is implicitly circular. The former is the case if we interpret the definition to say that anybody who understands what the constituent terms of the statement mean (who understands, in other words, what proposition the sentence is used to expressed) will assent to it; and the definition is circular if it tells us that the statement will turn out to be derivable from logic alone once the definitions of its terms are supplied. Again, it is implicitly circular to define a logically true statement as one that cannot be denied without self-contradiction. For, surely, we want to say that p is logically true if a contradiction is derivable from not-p with the help of logic alone, without the use of factual premises. A definition which, prima facie, is free from the vice of circularity is the following one: a logical truth is a true statement which either contains only logical constants (besides variables) or is derivable from such a statement by substitution (this is essentially the definition preferred by Quine).1 It remains to be seen, however, whether this appearance will stand the test of analysis. The crucial question is obviously whether we could construct an independent definition of “logical constant.” 1 See

Quine 1947b.

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The customary procedure of logicians who define their meta-logical concepts with respect to a specified postulational system is to define the logical constants simply by enumeration. But while such definitions serve the function of criteria of application, they clearly cannot be regarded as analyses of intended meanings. To give an analogy, suppose we defined “colored” by enumerating n known colors, i.e., colored = C1 or C2 ... or Cn . And suppose we subsequently became acquainted with a new color which we name “Cn+1 .” On the basis of our definition it would be self-contradictory to say that Cn+1 is a color, or at any rate we could not say that it is a color in the same sense as the initially enumerated ones. Thus, so-called definitions by enumeration do not tell us anything about the meaning of the defined predicate, and the same is true of many recursive definitions. In fact, recursive definitions of “logical constant” given by logicians usually reduce themselves to an enumeration of logical signs with the addition that any sign definable in terms of these alone is also a logical sign. The problem of defining this basic meta-logical concept explicitly, however, cannot be said to have been solved.2 Frequently the explanation is given that logical signs are purely formal (or syntactic) constituents of sentences, as though it were perfectly clear what was meant by that. But in classifying a sentence as having such and such a form we presumably point out what it has in common with other sentences. Why, then, could not, for example, the sentences “the sky is blue” and “all exam-booklets are blue” be said to be formally similar on account of sharing the constituent “blue,” if it is all right to call the sentences “the sky is blue” and “the weather is nasty” formally similar because they contain the “is” of predication as a common element? If all we can reply is that the latter is a formal sign while “blue” is a descriptive sign, the above explanation leaves us no wiser than we were before. Logical terms are, as usually understood, contrasted with descriptive terms, and if, therefore, an independent definition of “descriptive term” were at hand, a logical term could simply be defined as a non-descriptive term. It will appear, however, that such an independent definition presents grave diﬃculties. To begin with, it would not be clarifying to define a descriptive term as one that refers to an observable feature of the world as long as we have no clear criterion of observability. Are numbers, for example, observable features of the world? This could not plausibly be denied since numbers are observable properties of collections (counting is surely a mode of observation), although of a higher type than the properties which could be ascribed to the elements of the collection singly; and yet number-predicates would by most logicians be classified as logical terms, in view of the logistic reduction of arithmetic. Again, it would not be illuminating to define descriptive terms as those terms that may function as values of variables, where variables are divided, say, into individual, predicate, and propositional variables. For, if the logician were asked why, 2A

critical comment on the proposed solution by Professor Reichenbach, in Reichenbach 1947, will be found later in this chapter.

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for example, no connective-variables, i.e., variables taking connectives as values, occur in his system, he would presumably reply that connectives are not descriptive terms. Perhaps we shall fare better if, instead of looking for an explicit definition, we try a contextual definition, like the following: T occurs as a descriptive term in argument A, if A would remain valid, respectively invalid, when any other syntactically admissible term is substituted for T in all its occurrences. But this definition is open to objections from two angles: (1) if it is our aim to define “logical term” negatively, as “non-descriptive term,” then this definition makes it impossible to define a valid argument as one such that the implication from premises to conclusion is true by virtue of the meanings of the logical constants involved. At least this is an objection from the point of view of those who are not satisfied with accepting “valid” as a meta-logical primitive. (2) While the mentioned condition is no doubt necessary, in line with the idea that the logical validity of an argument does not depend upon the subject-matter which the argument is about (briefly, the idea that logic is a formal science), it is not suﬃcient. For example, the argument “x = y, therefore not-not-(x = y)” would remain valid no matter what relation were substituted for identity, and still one would not call “identical” a descriptive term. The point is that a term may occur inessentially in an argument without being descriptive. It should be mentioned, though, that this objection would lose its force if the distinction between logical and descriptive terms were altogether functional, i.e., if “T is descriptive” should be regarded as elliptical for “T is descriptive in A.” But while we may in the end have to accept this possibility—which would lead to the, perhaps surprising, consequence that no general definition of logical truth could be given, let us first see whether we have better luck in beginning with a positive definition of “logical constant.” The process by which terms used in deductive arguments are actually identified as logical and as determining the logical form of the argument, is to replace constants with variables until only those constants are left over on whose meanings the validity of the argument depends. But the definition, thus suggested, in terms of essential occurrence in deductive arguments, has already been seen to be unsatisfactory, since one and the same term may occur essentially in one argument and inessentially in another. This is particularly obvious if our arguments contain defined terms: in “x is a triangle, therefore x has three sides,” “triangle” occurs essentially, but in “x is a triangle, all triangles have property P, therefore x has P,” “triangle” occurs vacuously. The explicit definition of “logical term” recently proposed by Reichenbach (Reichenbach 1947, §55) seems to me, indeed, to break down because of this circumstance, viz., that the concepts “logical term” and “term occurring essentially in every necessary implication in which it occurs” do not have the same extension. Reichenbach attempts to clarify the distinction between logical and descriptive terms by means of the distinction between expressive and denotative terms.

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Denotative terms are values of individual, predicate, or propositional variables, and expressive terms are those which do not denote. A logical term is then defined as an indispensable expressive term, or one definable in terms of such. This definition leads, however, to such embarrassing consequences as that the connectives “or,” “and,” etc., are not logical terms. For while “or,” for example, may be used, and mostly is used, as an expressive term, it could be used as value of a two-term predicate variable. Instead of writing “p or not-p” one would then write “or (p, not-p).” To be sure, as such a denotative term it is definable by means of the corresponding expressive term, but ipso facto the latter is not indispensable: definitional eliminability works both ways. If I understand Reichenbach correctly, he hopes to overcome this diﬃculty by defining “logical term” with respect to a language in which the following notational convention is observed: in a tautologous formula only such denotative terms may occur as have a vacuous occurrence. Thus we should not express universality by a second-level predicate “Un” and write, for example, Un (p or not-p), since “Un” here would occur essentially, i.e., the function obtainable from this proposition by substituting an appropriate variable for the secondlevel predicate would not be universally assertible. But this convention will not do the job of saving the initial definition since it presupposes that the concepts “logical term” and “term having an essential occurrence in tautologies” have the same extension, which they do not. To add an illustration to the one already oﬀered, in the tautology “p ∨ q ⊃ p ∨ q,” the logical term “∨” occurs inessentially. And any tautology that contains defined predicates illustrates the possibility of essential occurrence, in tautologies, of non-logical terms.3 But even if these anomalous cases of vacuous occurrence of logical constants could be discounted for some reason, the general definition of “logical constant” in terms of “essential occurrence in deductive arguments” would be open to further objections. In the first place, since terms that would not normally be classified as “logical” may occur essentially in arguments containing defined terms, one would have to specify either that the arguments referred to are to be formally valid or invalid, or that they are to contain no definable pred3 In personal correspondence, Professor Reichenbach has answered my objection to his definition of “logical

term” by pointing out that while logical terms may, indeed, occur vacuously in some tautologies they do not occur vacuously in all tautologies and that this is the reason why they cannot be treated as values of variables. The suggested definition of “logical term” in terms of “term occurring essentially in some tautologies” leads, however, into the following dilemma. Either “tautology” is so used that only sentences in primitive notation could be tautologies, or more broadly (and more in accordance with ordinary usage) so that sentences containing defined terms could also be tautologies. In the former case, there could be no defined logical terms at all, since obviously a defined logical term cannot occur essentially in sentences which contain no defined terms. In the second case, however, terms that are ordinarily regarded as descriptive, like “bachelor,” would turn out to be logical, since they occur essentially in such tautologies as “all bachelors are unmarried men.” If, to avoid the latter consequence, the definition of “logical term” be restricted to non-descriptive languages, it becomes circular again, since a non-descriptive language is presumably a language containing only logical constants.

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icates. This, however, is a real dilemma. Relatively to the first specification, the definition is circular, since an argument is said to be formally valid if it is valid by virtue of the meanings of the logical terms alone; and relatively to the second specification we get a concept which is applicable only to fictitious completely analyzed languages. Instead of hunting any longer after a satisfactory general definition of “logical constant,” let us now focus attention upon an important consequence of our mainly negative results. It has been seen that the discrimination of logical terms from descriptive terms amounts to identification of those terms in a given deductive argument on whose meanings the validity of the arguments depends.4 But since any formal test of validity presupposes identification of logical constants (how could I know to what degree a given argument under scrutiny is to be formalized, unless I knew which terms may not be replaced by variables?), validity could not here, on pain of circularity, be established by a formal test. This suggests that no adequate epistemological account of the construction of semantic systems of logic can be given without countenancing the concept, held in disrepute by many logicians and philosophers, of material (= non-formal) entailment. It is widely held that entailment is essentially a formal relation, i.e., “ ‘p’ entails ‘q’ ” is held to be equivalent to “ ‘if p, then q’ is true by virtue of its logical form.” But the latter statement is presumably equivalent to the statement “‘if p, then q’ is true by virtue of the meanings of the logical terms involved, all other ingredients occurring vacuously.” But I have shown that judgments of entailment are presupposed by the very process which leads to the definition of the meta-logical concept logical form. For the only way in which this concept can be defined (if it be proper to call such a procedure “definition” at all) is to exhibit logical forms by the use of logical constants. What, indeed, is the attitude of logicians when they are faced with an evidently non-empirical conditional statement which, though expressing an a priori truth, does not seem to be demonstrable with the sole help of the definitions of specified logical terms? Their endeavor may be described as the analysis of the non-logical concepts that seem to occur essentially by means of logical

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frequently debated thesis that logic is a purely formal science, in the sense that questions of logical validity can be answered without any appeal to meanings, is ambiguous. If the contention is that, provided no defined descriptive terms occur, the meanings of the descriptive terms are irrelevant to questions of logical validity, that is true enough—although, as I have suggested, possibly truistic. But if the claim is that no semantic rules at all need be consulted in such investigations, then “logic” is defined as a science that is competent to answer only questions of purely syntactic derivability, which therefore does not concern itself with relations of truth-values, and is therefore inapplicable to problems of logical validity as they arise in natural languages.

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concepts, so that what appeared as a material entailment5 reduces to a formal entailment once the proposition is fully analyzed. The obvious illustration of this procedure that comes to one’s mind is the reduction of arithmetic to logic. To take a very simple example, since the relational predicate of arithmetic, “being greater than,” does not belong to the vocabulary of logic, the sentence “if G(x, y), then not-G(y, x)” could not, oﬀhand, be said to express a formal entailment. This postulate of arithmetic, however, can be reduced to pure logic by defining numbers as properties of classes and defining the mentioned arithmetical relation in terms of the logical concepts “similarity” (= one-one correspondence) and “proper subclass.” But the definition which enables such a reduction is obviously not an arbitrary stipulation; rather it expresses an analysis of a primitive concept of arithmetic. And I would like to know what else I judge, in judging this analysis to be correct, but that the proposition “the number of A is greater than the number of B” entails and is entailed by the proposition “B is similar to a proper subclass of A (where A and B are finite classes).” But since the non-logical term “greater” occurs essentially in the corresponding conditional statement6 this is definitely not a formal entailment. We have succeeded in formalizing the entailment from “G(x, y)” to “not-G(y, x)” only by accepting the material entailment just mentioned. I anticipate the objection that analysis of a concept belonging to an interpreted language must not be confused with interpretation of the primitives of a postulational system; that there is no sense in speaking here of an entailment holding between a proposition of arithmetic and a proposition of logic, since a postulate of uninterpreted arithmetic is no proposition. I submit, however, that if the logistic thesis is to be redeemed from the charge of triviality, it must be taken to assert that the terms of interpreted arithmetic are reducible to logic. For the thesis that uninterpreted arithmetic, as erected upon Peano’s postulates, is logic could only mean one or the other of equally trivial propositions: (a) assertions of the form “if the postulates are true, for a given interpretation, then the theorems are true, for that interpretation” belong to logic; (b) Peano’s postulates are satisfied by a logical interpretation. (a) is trivial since in this sense any uninterpreted postulational system is logic, (b) is trivial, since the logical interpretation is by no means the only one which satisfies Peano’s postulates. Any class of objects which is the field of an asymmetrical one-one relation, which contains an object which does not stand in the converse relation to any member of the field, and whose members are characterized by properties hereditary with respect to

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only reason I use the qualifying adjective “material” rather than “synthetic” is that “material” is the natural term to use in contrast to “formal.” My use of the word “material” in this context has, of course, nothing to do with its use in the phrase “material implication.” 6 By the conditional statement corresponding to the entailment-statement “‘p’ entails ‘q”’ I mean the statement “if p, then q,” which occurs in the object-language and does not contain names of propositions.

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the ordering relation, would be a model for Peano’s postulates. If the model is sociological, for example, one could then with equal plausibility say that arithmetic is reducible to sociology. If I were to permit myself the use of inexact metaphorical language in the midst of an austerely exact discussion, I would say that as logic expands by naturalizing more and more alien concepts in its economic household, one material entailment is used to kill oﬀ another, and ipso facto material entailments are there to stay. Furthermore, that we can reduce an apparently non-formal entailment to a formal entailment by supplying a correct analysis for the troublesome non-logical terms is no surprise. For is not such reducibility a tacit criterion of a correct analysis? Would Russell and Whitehead, for example, have proposed those definitions of the primitives of arithmetic in terms of logical concepts which they did propose, if they had not enabled them to reduce, say, Peano’s postulates to pure logic? Or, to take a simpler example, how could one controvert the claim that the admittedly necessary connection between the attributes “colored” and “extended” is at bottom a formal one, which would no doubt become evident if we knew the correct analysis of the attribute “colored”? Why, no such analysis would, of course, be judged correct unless it enabled the purely formal deduction of the attribute “extended.” I have deliberately postponed the dropping of the bomb to the concluding section of my paper, lest I prejudice from the outset my chances of getting an impartial hearing. Here is the bomb: if by a synthetic proposition you mean a proposition not deducible from logic alone, and by an a priori proposition you mean one that is not empirical, and if you define logic by means of an enumeration of a set of concepts called “logical constants”—to which there is no alternative in the absence of a satisfactory general definition of “logical constant”—then you have to accept the conclusion that synthetic a priori propositions are acknowledged whenever the territory of logic expands. And it appears, then, that in “reducing” the non-geometrical parts of mathematics to logic, the logisticians have not eliminated the synthetic a priori7 from mathematics; they have merely dislocated it to those regions where mathematical and logical concepts make definitional contact. It may be clarifying to refer to an analogous logical situation in the reduction of one empirical science to another, say thermodynamics to mechanics. Once the temperature of a gas is defined as

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do not intend to resuscitate the ghost of Kantian epistemology by using this Kantian expression, and would be sorry if my terminology had this eﬀect. I would justify my usage by pointing out that the term “synthetic” as well as the term “a priori” is in good standing with philosophical analysts who are fully emancipated from metaphysics; so why should the term “synthetic a priori” be disreputable? If, indeed, the term “analytic” is used so broadly, as, e.g., in C. I. Lewis, that any statement which can be established by reflecting upon meanings is analytic, then “analytic” becomes synonymous with “a priori” or “nonempirical” and the thesis of the analytic character of all a priori truth becomes irrefutable on account of triviality (see, on this point, chapters 3 and 4).

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average molecular kinetic energy, thermodynamic laws containing the variable “temperature” are translatable into mechanical language. But in terms of the meaning attached to “temperature” in the language of thermodynamics before such a reduction took place, the proposition “the temperature of a gas is proportional to the average kinetic energy of its molecules” was obviously synthetic. As far as I can see, the only ground on which the suggested analogy between reduction in the empirical sciences and reduction in the formal sciences could be questioned would be the following: one might hold that if a term of formal science S is analyzed with the help of terms of formal science S , the term of S has no independent meaning; that it only receives a meaning by virtue of its definitional reduction to the terms of S . In order to refute this view it will suffice to point out that it is equivalent to the view that such analyses are arbitrary stipulations. For to see the untenability of such a position—which is motivated by what I regard as an irrational dread of intuitionism—we only need to observe that according to it the question whether S is reducible to S would be nonsensical; all one could sensibly ask would be whether it is convenient to reduce S to S . In conclusion, I shall briefly reply to the comment, which I may well anticipate, that I have here argued in terms of an absolute concept of entailment which is an intuitionistic superstition as far behind the times as, say, the Newtonian superstition of absolute space, time, and motion. The thesis that “‘p’ entails ‘q”’ is, if meaningful at all, to be construed as elliptical for “‘q’ is derivable from ‘p’ in terms of the transformation-rules (including definitions) of a specified language-system” seems to me to involve a fatal paradox. The paradox is simply that through the construction of suitable definitions statements which would normally be held as logically independent could be made to entail one another. And if it be replied that those definitions, of course, must be adequate, the paradox is merely dodged, not solved. For, in order to determine whether a definition of A in terms of B is adequate, philosophers normally do not resort to statistical investigations of linguistic habits— if they did, how could the results of philosophical analysis obtained in one language-community be of any philosophical interest to philosophers in other language-communities?—but instead ask themselves whether it would be selfcontradictory to predicate at once A and non-B. But this is to ask whether a predication of A entails a predication of B. And it should be obvious that a consistent adherence to the criticized theory of entailment makes this procedure either circular or infinitely regressive.

Chapter 11 STRICT IMPLICATION, ENTAILMENT, AND MODAL ITERATION (1955)

Ever since C. I. Lewis oﬀered the concept of “strict implication,” defined explicitly in terms of logical possibility (p q ≡d f ∼ 3(p . ∼ q)) and implicitly by the axioms of his system of strict implication, as corresponding to what is ordinarily meant by “deducibility” or “entailment,” there have been analytic philosophers who denied this correspondence. They denied it specifically because of the paradoxes of strict implication: that a necessary proposition is strictly implied by any proposition and an impossible proposition strictly implies any proposition. These theorems, it is maintained, do not hold for the logical relation ordinarily associated, both in science and in conversational language, with the word “entailment.” It is my aim in this paper to show that it is extremely diﬃcult, if not downright hopeless, to maintain this distinction. I shall refer specifically to a subtle paper by C. Lewy (Lewy 1950), which deals with the intriguing problem of modal iteration, and which emphatically endorses the distinction here to be scrutinized. Let me begin by presenting a brief argument against the distinction which seems to me conclusive, though I do not intend to rest my case on it. It is simply that anybody who wishes to maintain the distinction must abandon one or the other of two propositions which seem equally unquestionable: (a) p is necessary if and only if ∼ p is impossible, (b) p entails q if and only if it is necessary that (if p, then q)—where “if p, then q” is a material implication. For, from the conjunction of (a) and (b) we can deduce: p entails q if and only if (p and ∼ q) is impossible. Since the impossibility of the latter conjunction follows from the impossibility of p, we have already the conclusion that an impossible proposition entails any proposition—which is what those who insist on the diﬀerence between entailment and strict implication deny. Notice that this argument does not presuppose that either (a) or (b) are definitions of “necessity” and “entailment” respectively. The conclusion follows even if the equivalences asserted by (a) and (b) are just material. In the above mentioned essay, Lewy confesses that he cannot completely de-

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fine “entailment,” but thinks that he is nevertheless justified in distinguishing entailment from strict implication because he can state two conditions which are necessary for p to entail q, over and above the condition that p strictly imply q, and which are not necessary for p to strictly imply q: p entails q only if (1) “R counts in favor of p” strictly implies “R counts in favor of q” and (2) “R counts against q” strictly implies “R counts against p.” Lewy uses “counting in favor of” as a primitive concept; the illustrations he gives show that it is meant as a very broad concept which covers both “confirming evidence” in the sense of inductive logic and deductive entailment as special cases. Thus he would say presumably that a sample of black ravens counts in favor of “all ravens are black,” but also that the proposition “all birds are black” would, if it were true, count in favor of “all ravens are black.” He gives the following examples of strict implications which are not entailments because they either do not satisfy (1) or do not satisfy (2): “the proposition that there is nobody who is a brother and is not male is necessary” strictly implies “there is nobody who is a sister and is not female,” because the implied proposition is necessary and a necessary proposition is strictly implied by any proposition. But it is no entailment, he says, because (1) is not satisfied. (1) is not satisfied because it is logically possible that there should be an R1 which counts in favor of the first proposition but is irrelevant to the truth of the second proposition. Lewy does not produce an example of such an R, but he might have produced the following: the concept being a brother is identical with the concept being a male sibling. Perhaps he would say that this proposition—the classical example of a “correct analysis” in Moore’s sense—entails, and therefore counts in favor of, the modal proposition “it is necessary that there are no brothers that are not male”; and surely we could agree that this proposition is irrelevant to the strictly implied proposition, since the latter does not contain the concept being a brother at all. His example of a strict implication which fails to satisfy condition (2) is, however, more convincing, since it involves nothing more problematic than that empirical evidence is, favorably or unfavorably, relevant only to contingent propositions: the false contingent proposition “Cambridge is larger than London” is strictly implied by the impossible proposition “there is somebody who is a brother and is not male,” but while there is empirical evidence counting against the implicate, there can be no empirical evidence that counts against the logically impossible implicans. If it were significant to say “since so far no brother has been found anywhere that was not male, it is unlikely that there is somebody who is a brother and is not male,” then the sentence “there is somebody who is a brother and is not male” would presumably express a contingent proposition. Lewy, then, assumes the following principle: if p entails q, then, for any R, “R counts in favor of p” strictly implies “R counts in favor of q”; and he be1I

take Lewy’s “R” to be a propositional variable, such that “counting in favor of” designates, like “strict implication” and “entailment” in his usage, a relation between propositions.

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lieves that this principle (together with the analogous principle corresponding to condition (2) above) serves to diﬀerentiate entailment from strict implication, because it is false if, for “entails,” “strictly implies” is substituted. I wish to show, however, that Lewy’s principle is false, its falsehood being a consequence of another principle nicely established by Lewy himself in the very same essay, viz., that a contingent proposition may entail, not just strictly imply, a necessary proposition. For, let R1 be empirical evidence which counts in favor of a contingent proposition p, and let q be a necessary proposition entailed by p. Since by the very meaning of “necessary proposition,” no empirical evidence can be relevant to a necessary proposition (and, a fortiori, cannot count in favor of it), R1 cannot count in favor of q; but this, by Lewy’s principle, contradicts the assumption that p entails q.2 In order to gain a clear insight into this subtle matter, it is necessary to consider the kind of entailment from a contingent proposition to a necessary proposition adduced by Lewy. The contingent proposition (J) “There is somebody who is French and is not under fifty years of age,” he says, entails the necessary modal proposition (H) “The proposition that not-J is not necessary.” While Lewy just takes it to be intuitively evident that J entails H, we can postpone, if not dispense with, the appeal to intuitive evidence by deriving this entailment from the formal entailments: (1) 2p entails p, (2) p entails ∼ ∼p.3 It seems, therefore, that Lewy is right in holding that J entails H. But consider, now, the singular proposition (A) “Pierre is French and is not under fifty years of age.” Inasmuch as A entails J, it counts in favor of J. Yet, if a necessary proposition is, by definition, such that no empirical evidence can be relevant to it, and the modal proposition H is, as maintained by Lewy, necessary, must we not conclude that A does not count in favor of H? Suppose, however, that Lewy replied: A does count in favor of H, because, entailment being transitive, “A entails H” follows from “A entails J” and “J entails H”; and if A entails H, then A counts in favor of H. To this I would make a threefold rejoinder: (1) If one holds that empirical evidence may count in favor of a necessary proposition, it may prove very diﬃcult, if not impossible, to explain the distinction between necessary and contingent propositions. (2) Lewy’s principle is presumably oﬀered as a partial analysis 2 One

might think that Lewy’s principle could be saved by considering strict implication itself as a case of “counting in favor of,” such that “p counts in favor of q” would be equivalent to “p is confirming evidence for q, or p entails q, or p strictly implies q.” But if “counting in favor of” were, in gratuitous violation of the ordinary usage of the expression, construed in this way, then Lewy’s attempt to diﬀerentiate entailment from strict implication would come to naught anyway: since strict implication satisfies the syllogism principle “if (if p, then q), then, if (if r, then p) then (if r, then q),” the proposition “for any p and q, if p strictly implies q, then ‘R counts in favor of p’ strictly implies ‘R counts in favor of q’ ” would not fail for the “paradoxical” case of a necessary q being strictly implied by any p. For, for any R, it will be impossible that R counts in favor of p but does not count in favor of q, simply because it will be impossible that R does not strictly imply q. 3 From (1): ∼ p entails ∼ 2p (3); from (3): ∼ ∼ p entails ∼2(∼ p) (4); from (2) and (4): p entails ∼2(∼ p) (5); from (5): J entails ∼ 2(∼ J).

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of the relation of entailment, which describes a method for deciding whether a given strict implication is also an entailment. Thus we ought to be able to decide whether the strict implication from J to H is also an entailment, by deciding whether “R counts in favor of J” strictly implies “R counts in favor of H.” But then the question is begged if we infer “‘A counts in favor of J’ entails ‘A counts in favor of H,”’ and hence “‘A counts in favor of J’ strictly implies ‘A counts in favor of H,”’ from “J entails H.” Of course, this strict implication follows, by virtue of Lewy’s principle, from “J entails H,” but he ought to show that it holds, independently of the assumption of the entailment. (3) The appeal to the transitivity of entailment becomes irrelevant to the question at issue, viz., whether the entailment from J to H satisfies Lewy’s principle, if instead of A we consider some empirical proposition which confers a high probability upon A, and therewith upon J, e.g., “Pierre is French and his birth certificate—which probably has not been falsified—indicates that he is not under fifty.” Such a proposition will not entail H, as Lewy surely would admit. If, then, it is held to “count in favor of” H, this must be in the sense of “making H highly probable.” Yet, is it not nonsense to say, with respect to a necessary proposition p, “The available empirical evidence makes it highly probable that p is true”? Now, Lewy’s attempt to diﬀerentiate entailment from strict implication might still be successful if a flaw could be found in Lewy’s demonstration of an entailment from a contingent to a necessary proposition. Perhaps modal propositions, like H, are contingent? If only we were permitted to use Lewy’s principle, we could easily prove that propositions to the eﬀect that some other proposition is necessary (propositions of the form “It is necessary that p”) are not contingent—and one would expect that the same holds for any kind of modal proposition, and thus for H in particular. For suppose that 2p is contingent while p is necessary. If 2p is contingent, then there is an empirical R which, if it were true, would count in favor of 2p. But 2p entails p. Hence, by Lewy’s principle, R would count in favor of p. But this contradicts the hypothesis that p is necessary (compare the first rejoinder above). Therefore 2p cannot be contingent. However, it would be poor strategy to use this argument in the present context. For, we set out to prove the noncontingency of modal propositions in order to defend the claim, made and substantiated by Lewy, that a contingent proposition may entail a necessary proposition (i.e., that there are pairs of propositions (p, q) such that p is contingent and q is necessary and p entails q), and this claim we wanted to defend in order to be able to refute Lewy’s principle. Yet, the above argument presupposed Lewy’s principle, hence we would assume the validity of the latter in proving its invalidity. But actually nothing more elaborate is required to see that modal propositions are noncontingent than reflection on the meaning of the words “necessary” and “contingent.” When we call a proposition “necessary” we are saying

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that it can be known to be true without empirical investigation by reflecting upon meanings and, in some cases, applying logical principles. Now, it is clear that we do not make any sort of empirical investigation in order to answer a question like “Is it necessary that anything which is a brother is also male?” Should one object that it can be answered only by discovering the rules governing the use of the words “brother” and “male,” then one would be guilty of confusing the proposition about verbal usage “The sentence ‘anything which is a brother is also male’ expresses, in ordinary usage, a necessary proposition”—which is admittedly empirical—with the proposition in question, which is not about a sentence at all, but rather about a proposition. The distinction is analogous to the distinction between “The property being a round square is L-empty” and “The expression ‘being a round square’ designates an L-empty property”; it is of course a contingent fact that a given expression is used to designate the property it designates, but this has no tendency to prove that the designated property has whatever properties it has contingently, not necessarily. Some philosophers no doubt will say that these distinctions are intelligible only if one conceives of propositions and properties as of entities named by expressions (what has been satirized as the “‘Fido’ means Fido” pattern of semantic analysis). Meaning, they would say, has been fallaciously assimilated to naming, as though “meaning” were a transitive verb that can occur in sentences of the form xRy, where the values of x are linguistic expressions and the values of y—meant meanings! But I think the question at issue can be settled without any prior commitment for or against Platonistic semantics. For while a philosopher may have misgivings about propositions as extra-linguistic entities designatable by sentences, he will surely accept the distinction between sentences and statements, where a sentence is a special kind of sequence of physical marks or noises, and a statement is a sentence as meaning such and such, “meaning” here being used as an intransitive verb, a verb without object. With this terminology the above distinction can be reproduced as follows: that the sentence “If anything is a brother, then it is male” is used to make a necessary statement, is indeed a contingent fact if it is a fact at all; but that the statement which this sentence is ordinarily used to make is necessary, is not a contingent fact. There is no possible world in which this sentence, as meaning what it usually means, is false; a world in which it would be false is a world in which it would be used with a diﬀerent meaning. But this we find out by reflecting on the meaning the sentence is commonly used to convey, not by ascertaining empirically what meaning it is that the sentence is commonly used to convey. That statements to the eﬀect that a given statement is necessary are themselves necessary has been denied by Strawson (Strawson 1948) on the ground that modal statements are contingent meta-statements about the uses of expressions. Thus, “It is necessary that there are no brothers that are not male” turns

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on this theory presumably into an empirical statement about the usage of the expressions “brother” and “male” (the linguistic theory of logical necessity advocated by Strawson and others who shun the “inner eye of reason” would also lay stress on the prescriptive function of such modal statements, their “recording our determination” to adhere to a certain usage, as Ayer put it, but this aspect of the theory is not under discussion now). Strawson faces the obvious objection to this theory, viz., that the modal statement talks about the concept of being a brother, or, if you wish, the meaning of the expression “brother,” not about the expression “brother,” in the usual manner: a concept is a class of synonymous expressions, hence it can be conceded that a statement is made about the concept of being a brother without surrendering the claim that a statement is made about the expression “brother.” The modal statement, if correctly interpreted, is a meta-statement which mentions the expression “brother” in order to define a class of expressions, viz., the class of expressions synonymous with “brother,” and it makes an assertion at one stroke about each member of this class. Thus it says not only that “brother” is inapplicable to x unless “male” is also applicable to x, but further that if “Bruder” is synonymous with “brother” and “m¨annlich” synonymous with “male,” then “Bruder” is inapplicable to x unless “m¨annlich” is also applicable to x, etc. According to this theory, then, the statement about the concept being a brother, viz., “being a brother entails being male,” is a statement of the same sort as the statement about the expression “brother,” only it says much more of the same sort. This additional content can be expressed as a conjunction of conditionals of the form “If E is synonymous with E1 and D is synonymous with E2 , then E is inapplicable to x unless D is applicable to x.” Yet, conditionals of this form are surely analytic: it is surely self-contradictory to suppose that there existed two expressions E and D which are respectively synonymous with “brother” and “male,” but are such that E is correctly applicable to something to which D is not correctly applicable. But a conjunction of a contingent statement and an analytic statement asserts no more than the contingent statement alone.4 Therefore Strawson has utterly failed to analyze the diﬀerence between the statement about “brother” and the statement about the concept being a brother. He might reply that on his theory the concept of synonymy is not used in the meta-linguistic translation of “being a brother entails being male”; that the latter is to be translated into a conjunction of categorical statements about the uses of synonyms of “brother” and “male,” such statements as “ ‘Bruder’ is not correctly applicable to x unless ‘m¨annlich’ is correctly applicable to x.” But this analysis is even

4 Perhaps

it would be more accurate to say that the conditionals in question are, not analytic in themselves, but analytically entailed by the statement about the English expressions “E1 is not correctly applicable to x unless E2 is applicable to x.” But the same conclusion would follow: if p analytically entails q, then (p and q) has no more factual content than p alone.

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less plausible, since it entails that only a person who knows something about all languages could know an entailment-statement to be true. That necessary propositions are necessarily necessary is particularly evident in the case of tautologies of propositional logic. Consider, e.g., “[(A ⊃ B). ∼ B] ⊃ ∼ A” (F), where “A” and “B” abbreviate definite statements substitutable for the variables “p” and “q” of the propositional calculus. The same method of analysis of logical ranges, the “truth-table” method, which assures us of the truth of this statement though we may be ignorant of the truth-values of the component statements, also assures us of its necessary truth, its truth in all “possible worlds.” For the unbroken column of T’s under the major operator signifies not only that this statement is true but also that any statement of the same form is true. Hence, if the possibility of establishing the truth of a statement without appeal to empirical evidence marks it as necessary, then not only F but likewise “It is necessary that F” must be held to be necessary. If this attempt to show that modal statements like H are, as maintained by Lewy, not contingent, has been successful, then we must conclude that Lewy’s principle is invalid and therefore does not enable us to distinguish the relations of entailment and strict implication. Perhaps, however, the distinction can be drawn in some other way. Let us try the following suggestion: an entailment between p and q is decidable, if it is decidable at all, without knowledge of the modalities of p and q. Notice that this condition may be satisfied even though one or both of the propositions standing in the entailment-relation are modal propositions: just as (p.q) entails p regardless of whether (p.q) is a contingent, necessary, or impossible proposition, so 2(p) entails p regardless of whether 2(p) is a contingent, necessary, or impossible proposition; similarly, we can say with certainty that p entails that p is possible (where “possible,” of course, is so used that “p is possible” is compatible with “p is true,” “p is false,” and “p is necessary”) prior to having settled the question whether or not “p is possible” is contingent. Now, “p strictly implies q” is defined as “It is impossible that p and ∼ q.” From this definition it follows that (p and ∼ p) strictly implies q, whatever propositions p and q may be, because (p and ∼ p) is impossible, and the conjunction of an impossible proposition with any other proposition is of course likewise impossible. It seems, therefore, that the condition above stipulated for entailment is not satisfied by the “paradoxical” strict implications: we know that q is entailed by (p and ∼ p) because we know that (p and ∼ p) is impossible; and we know that [∼ (p and ∼ p)] is entailed by q, because we know that [∼ (p and ∼ p)] is necessary. Yet, this attempt to diﬀerentiate entailment from strict implication overlooks that the “paradoxes” of strict implication do not depend on the stated explicit

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definition of strict implication5 at all, but can be derived within a system of strict implication in which this concept is but implicitly defined, by a set of axiomatic strict implications, just as well. As was shown by Lewis himself, the following strict implications (supplemented by the rule of substitution and the ponendo ponens rule) suﬃce for the derivation: (p.q) strictly implies p; p strictly implies (p∨q); [(p∨q). ∼ p] strictly implies q. These strict implications, however, satisfy the condition imposed upon entailment: that no assumptions about the modalities of the terms of the entailment are required to see that the entailment holds. I conclude that it is obscure what the alleged distinction between strict implication and entailment is. Perhaps, however, the feeling that there is a distinction may be traced to the following origin: those strict implications which are “paradoxical” are inferentially useless. If the antecedent of a strict implication is impossible, we cannot use the implication as a basis for proving the consequence, since the antecedent is not assertible; and if the consequent is necessary, the implication is useless as a rule of inference simply because no premise is required for the assertion of the consequent (it is “unconditionally” assertible). But if it is only by reference to inferential utility that strict implication is distinguishable from entailment, then there is no basis for saying that these are distinct logical relations; for the concept of inference, and any concept defined in terms of it, is of course no logical concept at all.

5 Alternatively, “p strictly implies q” could be defined as “2(p ⊃ q)”; if the definition of “2(p)” as “∼ 3 ∼ p” is added, the above definition turns into a theorem about strict implication.

Chapter 12 MATHEMATICS, ABSTRACT ENTITIES, AND MODERN SEMANTICS (1957)

A science can be in a highly advanced state, even though its logical foundations are far from being clarified. Mechanics, for example, reached a stage of astounding perfection by the end of the 18th century, mainly through the genius of Galileo and Newton, although much room was left for controversy about the meanings of its fundamental concepts: length, simultaneity, mass, and force. Even the meaning of the simple law of inertia remained controversial right up to Einstein’s “unification” of inertia and gravitation. Similarly, mathematics was not prevented from reaching breathtaking heights of perfection by the Grundlagenstreit (dispute about foundations) which began in the 19th century and still continues. The fundamental question of the philosophy of mathematics concerns the very nature of its subject matter. Traditionally, mathematics was defined as the science of quantity, but this definition was decisively criticized by Bertrand Russell in terms of a conception which reduces pure, abstract mathematics to pure, abstract logic, thereby lifting any restriction to a special subject matter. But though pure mathematics in this conception is, in contrast to the special empirical sciences, unrestricted in subject matter, its propositions are analyzed as referring to classes and attributes. In probing into the foundations of mathematics, therefore, one cannot avoid facing, sooner or later, the old metaphysical problem of the status of abstract entities. The latter has been tied up with the problems of linguistic meaning and reference ever since Plato but never before as closely as at the present time. Modern semantics, as cultivated by analytic philosophers in the United States and England, grew up in close contact with symbolic logic. Accordingly, the problem to be discussed in this article should be of equal concern to mathematicians who reflect on the foundations of their science, to semanticists, and to symbolic logicians.

1.

Traditional Problem of Universals

When a high-school teacher demonstrates that the sum of the interior angles of a triangle equals 180◦ , he usually draws a triangle on the blackboard, then

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draws a straight line through the top vertex parallel to the base, in order to be able to reduce the proof of the Euclidean theorem to the proposition that “corresponding” angles are equal. How would he satisfy a “philosophic” pupil who protested that he had not established that all triangles whatsoever have the property in question but only that the particular triangle on the blackboard had it? Obviously, the proper reply would be this: We can be perfectly sure that every Euclidean triangle has the property, without illustrating the proof over and over again by diﬀerent triangles, because any properties of the particular triangle that distinguish it from other triangles—such as its being equilateral, or its comparatively small size, or its being bounded by white lines—were disregarded in the proof. In the language of Plato: What the mathematician is thinking about when he discovers the geometric proof is not a particular triangle but triangularity as such—in other words, that which all particular triangles have in common and by virtue of which they are triangles. It is this sort of thing that various philosophers, past and present, under Plato’s influence call a “universal,” or an “intelligible form,” or an “essence.” These entities were supposed by both Plato and Aristotle to be the objects of scientific thought, as contrasted with “particulars” that are given in senseexperience with a unique location in space and time. Of course, these universals did not have to be geometric forms. In the case of every predicate whatsoever, the things to which the predicate is applicable can be distinguished from the property the possession of which by a particular thing is the criterion of the predicate’s applicability: redness is to be distinguished from red things, hardness from hard things, humanity from particular human beings, and so on. Again, numbers seemed to be a kind of universals, for, to ask a question which unphilosophic people never ask at all: What is the number 2? It surely is not the symbol “2,” for we can use diﬀerent symbols to talk about the same number (for example, “II,” “two,” “zwei”). It must, then, be something that the symbol stands for. But it cannot be identified with a particular pair of objects, say a pair of gloves, or a pair of apples, or a pair consisting of wife and husband. It seems, rather, to be something that all particular pairs have in common (“twoness”), an object of abstract thought, not a sensible object. Once universals are conceived as objects of thought, the realm of universals will be found to be populated not only with those that are somehow “exemplified” in the particulars we sense but also with unexemplified universals: no physical sphere corresponds exactly to the mathematical concept of a sphere; hence, the universal sphericity which the mathematician thinks about is not, strictly speaking, exemplified in the physical world. And even if one never physically constructed a regular polygon with 1000 sides, this mathematical object could be reasoned about, and in order that it may be reasoned about it must in some sense “exist,” say the Platonists. In the heaven of Platonic forms, then, we find not only perfect mathematical objects but also unexemplified forms that move the imagination of less intellectual people: mermaidhood, centaurhood, unicornhood, and so on. The Platonists’ argument seems to be

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that, in order that something may be thought of, it must in some sense be; we can obviously think of mermaids, though there are no physical mermaids in space and time, but we could not have such thoughts unless there existed the universal mermaidhood. Since these universals do not exist in space and time, some Platonists try to avoid confusion by calling the sort of “being” that is to be ascribed to them “subsistence.”1 It is important to understand that universals as traditionally conceived, especially by Platonists, are not mental in the sense in which a thought or an act of imagination or a feeling is mental. They are, rather, postulated objects of thought whose subsistence is alleged to be independent of their being mentally apprehended in any way. It is for this reason that Platonism is usually called a kind of “realism.” Universals are held to be real in the sense that there would be such entities even if no consciousness existed, just as a physical realist holds that physical objects are there, whether or not any conscious organism or disembodied mind is aware of them. The most important traditional arguments, then, for there being universals over and above the world of sensible particulars in space and time are the following two. (i) With the exception of proper names, all meaningful words are general. To say that they are meaningful is to say that they stand for something. But, unlike proper names, words like horse, round, and red do not stand for particular things. Therefore they stand for universals. And these universals would be “there” somehow, even if they were never referred to by means of words. (ii) Every thought has an object; to think is essentially to think of something, and the object of one’s thought is clearly distinct from the thinking of it. When two mathematicians, for example, converse with each other about the properties of a certain number, they assume without hesitation that they are thinking about the same object and that they are merely trying to discover the properties of that object, in the same sense that a chemist is trying to discover the properties of a compound he is experimenting with, though by essentially diﬀerent methods. The square root of two would have been irrational even if no mind had ever thought about it, let alone discovered its irrationality. This second argument is much more tempting than the first. As we shall see presently, the first argument is rather easily disposed of by exposing the underlying confusion between meaningful words and names. But the second argument is a natural outgrowth of a psychological situation with which even non-mathematicians are familiar. Imagine a child who has just learned the meaning of “square number” by means of the examples 4 = 2×2, 9 = 3×3, and curiously investigates how many square numbers there are between 1 and 100 and whether their frequency increases or decreases as the numbers increase.

1 See,

for example, Montague 1958, chapter 4; Russell 1912, chapter 9.

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He seems to explore a domain of objects that have invariable properties in no way dependent on human thoughts, and yet these objects are not physical objects with spatial location. The child may at first suppose himself to be talking about marks on paper when he says “9 is a square number but 10 is not,” but you can easily convince him that these are merely symbols we use for referring to invisible numbers: the sentences “9 is a square number” and “nine is a square number” refer to the same number, but the symbols “9” and “nine” are obviously diﬀerent.

2.

Modern Semantics and the Traditional Dispute

Many traditional philosophers, especially Occam and his followers in medieval philosophy and the British empiricists since Hobbes, have tried to discredit Platonism as a belief that has no better rational foundation than a belief in ghosts and fairies. Whatever exists, said the nominalists, is a particular in space and time, and there is no need to postulate any “subsistent” abstract entities. A general word does not stand for a universal but is just an economic device for referring to several particular things that are in certain respects similar. According to the realists, a meaningful general word, like man, stands for a universal, and when we judge that a particular object is correctly called “man,” we compare it, as it were, to our idea of the universal. But the nominalists denied that we can form ideas of universals—“abstract ideas”—at all. To quote the subtlest of them, George Berkeley: it is thought that every name has, or ought to have, one only precise and settled signification, which inclines men to think there are certain abstract, determinate ideas that constitute the true and only immediate signification of each general name; and that it is by the mediation of these abstract ideas that a general name comes to signify any particular thing. Whereas, in truth, there is no such thing as one precise and definite signification annexed to any general name, they all signifying indiﬀerently a great number of particular ideas. (Berkeley 1710, sec. 18)

This is a striking anticipation of modern semantic analysis. Berkeley is criticizing the scholastic belief, ultimately derived from Plato, that the generality of a word consists in its representing a determinate entity which is not a particular and of which we have an “abstract” idea. Rather, he held, the generality of a word is its capacity to evoke any one of a set of similar particular ideas. If I apply the general word triangle to a particular figure, I do not do so as a result of finding the particular figure to correspond to an abstract idea of triangularity—there is no such thing—but simply because I recognize it as similar in a certain respect to objects with which I have been conditioned to associate the word triangle. Berkeley might have said, as some contemporary analytic philosophers have said explicitly, that the meaningfulness of a word does not presuppose the existence of an entity which is what the word means.

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That the word triangle is unambiguous in the context of geometric discourse does not entail that it functions in that context as a name of a unique entity, call it a “universal.” One chief source of Platonism, according to this point of view, is the naive and mainly unconscious assumption that every meaningful word is a name of something. But the verb “to mean” is, in the relevant respect, more like the verbs “to dream” and “to wish” than like the verbs “to name” and “to eat”; there must be something that can be named in order that an act of naming may take place, and there must be something that can be eaten in order that a process of eating may occur. But would it not be grotesque if I argued to the existence of green dogs that can sing Verdi arias from the fact that I dreamed of such an animal, or the existence of a woman who is my wife and combines more virtues than any other woman from the fact that I wish for such a wife? In grammatical terminology, this criticism of Platonism accuses it of construing “to mean” as a transitive verb, whereas this verb is obviously intransitive. The chief philosophic motive behind Bertrand Russell’s theory of descriptions.2 was to expose this confusion and thereby to inhibit the tendency to postulate entities that do not exist in the physical universe. To borrow Russell’s own example, consider the sentence “the present king of France is bald.” On a crude level of analysis one might say that, since the sentence is meaningful (all constituent words are meaningful and are arranged in a syntactically correct way), it must be about something. And what is it about if not the present king of France? But there is no present king of France! Is it, then, about a Platonic essence or Idea that happens to be unexemplified in the physical world? Russell’s answer was that no such metaphysical postulations are required to account for the significance of the sentence. We have only to distinguish between contextual meaning and denotation: A definite description, like “the present king of France,” has meaning in context but does not denote anything at all. It has meaning in the context of sentences that contain it, and this meaning can be explained by means of a synonymous sentence in which no definite descriptions occur: “there is one and only one individual which is present king of France, and that individual is bald.”3 Once meaning and denotation are thus distinguished—a symbol may be meaningful (in context) without denoting anything—the argument for the existence of universals from the meaningfulness of predicates collapses, for a predicate can be meaningful in the context of entire sentences without being a name of anything. This, in fact, is a fundamental tenet of contemporary semantic nominalism (I call this philosophic school “semantic” nominalism to distinguish it from metaphysical nominalism, because its members tend to avoid such metaphys-

2 It

was first formulated in Russell 1905. See also Russell 1950, chapter 16. course, “France” is, in turn, an abbreviation for some definite description, but Russell believed that by repeated application of his rule of translation, any definite description could eventually be eliminated.

3 Of

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ical locutions as, “only particular things or events in space and time are real, universals are abstractions that have no reality”): to this school, predicates are contextually meaningful, and to explain the meaning of a predicate is to formulate a rule for applying the predicate to particular objects, but predicates are not names of anything. The predicate “blue” is not related to its meaning in the way that the name “John” is related to John. Indeed, it is already misleading, the semantic nominalists urge, to speak of the meaning of “blue,” since this expression suggests an elusive entity. It is preferable to speak of the rule governing the use of “blue,” and to give this rule is to formulate the conditions under which it is correct to apply “blue” to a thing. In the case of “blue,” the rule of usage cannot be formulated without pointing at particulars. In this respect the rule of usage for, say, “mermaid” is, of course, of a diﬀerent sort, since we can be said to understand what “mermaid” means, although we have never encountered any mermaids. But, the semantic nominalist insists, to grasp this meaning is not to be face to face, as it were, with a universal, call it mermaidhood, but just to know under what conditions it would be correct to apply the predicate “is a mermaid” to an object.4 An important application of the theory of contextual meaning to mathematics arises if one reflects on the meaning of infinitesimals, expressed by “dx,” “dy,” and so on. Mathematicians used infinitesimals successfully long before they were clear about their meaning. At first glance there is an air of hocus-pocus about infinitesimals: how can a genuine quantity be neither zero—insofar as an infinite sum of infinitesimals equals a finite quantity—nor finite—insofar as a finite sum of infinitesimals does not equal a finite quantity? And how can the ratio of dy (where y is a function of x) to dx equal a finite number if neither dx nor dy are finite numbers? The diﬀerential quotients themselves seemed perfectly legitimate, since they admitted of geometric interpretation, namely, as measures of the slopes of tangents. But what troubled both mathematicians, who used the device of infinitesimals without being able to justify it “philosophically” (as contrasted with “pragmatic” justification: it works!), and philosophers, who were critical of the calculus (Berkeley, for example), was that the components of the diﬀerential quotient had no intelligible meaning. The theory of contextual meaning, however, justifies the use of infinitesimals as “incomplete symbols,” in Russell’s phrase—that is, as symbols which do not denote any funny kinds of numbers but have meaning in context. dy = c” means that the limit approached by a seAn equation of the form “ dx

4 A programmatic formulation of this semantic theory, which derives from L. Wittgenstein and M. Schlick, is

as follows: the meaning of an expression consists in the rules for its use. Wittgenstein, a dominant member of the “Vienna Circle” (the founders of logical positivism see Kraft 1953), exerted a powerful influence on English analytic philosophy. For a lucid application of Wittgensteinian semantics to the problem of universals, see Lazerowitz 1946.

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∆y quence of diﬀerence quotients ∆x , as ∆x becomes smaller and smaller, equals c, and the meaning of “limit” can be explained without postulating any numbers other than finite ones. When one has explained the meanings of equations that involve diﬀerential quotients, both such as equate them to a constant and such as express them as a function of a variable, one has explained the contextual meaning of diﬀerential quotients and of their components.

3.

Classes, Attributes, and the Logical Analysis of Mathematics

It was stated in the beginning of this article that it is diﬃcult to avoid Platonism if one becomes philosophic at all and asks what a number, as distinct from a numeral, is. It seems that a number is either a universal—that is, a common property of classes of discrete objects that can be set into one-one correspondence—or else a class of such “similar” classes. The number two in particular, then, is revealed as either the universal “twoness” or the class of all pairs. Although classes are, for good or for bad reasons, somehow more acceptable to nominalists than are universals (or attributes, as we shall henceforth call them), they are likewise abstract entities. By “the class of all men,” for example, a logician does not mean a group of men that could possibly be seen but the conceptually apprehended totality of men past, present, and future.5 Can these Platonic temptations be overcome by means of the theory of contextual meaning? That is, can it be maintained that numerals have only contextual meaning and do not denote any entities at all? Now, contextual definitions of the natural numbers in terms of logical constants and non-numerical variables can be constructed, as was shown by Frege and Russell6 independently (although Frege has historical priority and influenced Russell’s thinking on the foundations of arithmetic); and these contextual definitions clarify, in a very important way, the language of applied arithmetic. Let us begin with the first natural number, zero. What is meant by a statement like “the number of French judges in the Supreme Court is zero”? Obviously, it means that there are no such persons, or that the class of such persons

5 There

is, incidentally, a neat proof that classes are diﬀerent from wholes, whether the parts of the whole be spatially or temporally contiguous or discrete. If x is part of a whole y and y is part of a larger whole z, then x is part of z. But if x is a member of class y and y is a member of class z, then x is not a member of z. For example, John is a member of the class of men, the latter is a member of the class of classes of living organisms, but John is not a member of the latter class, since he is not a class. Furthermore, the same whole may be conceptualized as diﬀerent classes: we can consider an organic body, for example, as a composite of cells but also as a composite of molecules. We then have a composite of cells that is identical with a composite of molecules; but a class of molecules cannot be identical with a class of cells, for, since cells are diﬀerent from molecules, nothing can be a member of both classes. 6 For an elementary exposition of “logicism” (as the reduction of mathematics to logic, attempted by Frege and Russell, is called), see Hempel 1983.

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is empty. Here, “no,” “there are,” and whatever symbol is used to denote membership in a class are logical constants. Note that zero, just like the larger numbers, is predicable of classes, not of individuals; that is, statements of the form “there is nothing of the kind K” are meaningful, but statements of the form “this thing is not” or “this thing is nothing” are not. Next, consider the number one. Again, we cannot predicate unity in the arithmetical sense of an individual: “the sun is one” makes no sense; “there is one (and only one) sun” makes sense. That is, we might say that the class of solar objects has exactly one member, and this means, according to the FregeRussell analysis, that there is an x such that x is a member of the class of solar objects, and for any y, if y is a member of the class of solar objects, then y is identical with x. Similarly, “there are two things of kind K” means that there is an x and a y, y distinct from x, such that x and y are of kind K, and for any z, if z is of kind K, then z is identical with x or with y, and so on for the larger numbers. It should be noted that the contextual definition of the arithmetical predicate of classes “one” is not circular, because “there is an x” does not have to be read as “there is at least one x” but may be read as “it is not the case that there is no x.” Are these definitions correct? Do they formulate what we mean by arithmetical symbols in the context of applied arithmetic? This question cannot be answered unless criteria are specified which must be satisfied by correct definitions. Now, an important criterion of correctness which these contextual definitions can be shown to satisfy is that the equations of arithmetic which are actually used in deducing numerical statements about empirical subject matters from other numerical statements should be logically demonstrable. For example, we would expect from adequate contextual definitions of “2,” “3,” and “5” that they make possible a formal proof (that is, a proof based on nothing but laws of logic) of the proposition “if a person has 2 daughters and 3 sons, then he (or she) has 5 children.” It may be thought that this proposition of applied arithmetic is just a special case of the general equation “2+3=5” and that there is no further problem, since the equation itself is simply “true by definition.” But insofar as the equation is just a provable formula in an uninterpreted system of arithmetic, its constituent arithmetical symbols do not have the sort of meaning that is presupposed by empirical applicability. In such a system (a “formal calculus” in the terminology of the logicians) 1 is defined as the successor of 0, 2, as the successor of 1, 3, as the successor of 2, and so on, where 0 and successor are primitive terms. These primitives also enter into the recursive definition of +: x + y = (x + y) ; x + 0 = x (here, “. . . ” is short for “the successor of . . . ”). In terms of these definitions, and of the rule that equals are interchangeable, it is indeed possible to transform 2+3, step by step, into 5, but as long as the primitives remain uninterpreted, the defined symbols likewise are without interpretation. The definition 2=1 , which occurs within the

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formal calculus, does not enable one to decide whether a given class has two or more or fewer members; indeed, there is nothing within the formal calculus itself to suggest even that numerals designate properties of classes that can be ascertained by counting. We require, then, an interpretation of the arithmetical primitives in terms of an understood vocabulary before we can even significantly raise the question of whether the equations are true. And if we want to justify the use of 2+3=5 as a rule for deducing “x has 5 children” from “x has 2 sons and 3 daughters” (with the additional help of the definition or theorem “x is a child of y = x is a son or daughter of y”), we have to assign to the numerals just those meanings which they have in such empirical contexts as “x has two sons.” This is precisely what the Russellian contextual definitions in terms of logical constants accomplish; they at once clarify the meanings of empirical statements in which numbers are applied and justify the belief that “x has 5 children” logically follows from “x has 2 sons and 3 daughters.” However, as both Frege and Russell clearly saw, such contextual definitions are insuﬃcient for the elucidation of numerical symbols. In eﬀect, what has been defined so far is only the statement form “class A has n members.” But we still do not know what 2, for example, means in the context “2 is a prime number.” Here, 2 does not occur as predicate but as subject. This is the kind of consideration that led Frege and Russell to ask what the number 2 itself is. Formally speaking, in the context “2 is a prime number,” the symbol “2” is a name, not an incomplete symbol. What is it the name of? Two alternative answers may be considered: (i) the class of all classes with two members, (ii) the attribute of being a class with two members. Answer (i) is a special case of Russell’s general definition of numbers as classes of similar classes. Two classes are said to be similar if there is a one-one relation which relates every member of one class to a member of the other class, and a relation R is one-one if at most one entity has R to a given entity and a given entity has R to at most one entity (for example, being the immediate successor of; being the positive square root of; being the wife of, in a monogamous society). It is natural to think of a number as the common property of all mutually similar classes. But this definition is open to the objection that there is no guarantee that a set of similar classes have just one property in common.7 For example, if all individuals have a certain property f in common, then all possible pairs that could be formed out of them would have the common property “containing as members individuals with property f .” By the foregoing definition, therefore, the number 2 would fail to be unique. On the other hand, the

7 See

Russell 1938, §110.

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class of all pair-classes is necessarily unique, and this is why Russell preferred to define numbers as classes of similar classes. Nevertheless, there is at least one reason why it may be better to regard numbers as special kinds of attributes, in the sense of definition (ii). The formal diﬀerence between classes and attributes is that attributes which apply to exactly the same entities may still be distinct, whereas classes with the same membership are identical. Thus, we can conceive of a universe in which all round things are blue and all blue things are round. If such were the world, the class of blue things would be identical with the class of round things, yet the attributes blueness and roundness, the one a color the other a shape, would remain perfectly distinct. Or, to illustrate by a mathematical example, the class of natural numbers between 1 and 3 is the same as the class of even primes, but the corresponding attributes are clearly diﬀerent; we describe the number two diﬀerently if we describe it as the natural number that is less than 3 and greater than 1, or as the only even number that is a prime number. Again, many distinct attributes are not possessed by anything at all: being a unicorn, being a blue dog that can speak French, being a golden mountain, and so forth. But they all correspond to the same class, the null class. There cannot be more than one null class, for if there were two, one would have to have a member which the other does not have, and this contradicts the hypothesis that neither has any members. Now, to come closer to the defect of definition (i), which is of interest here, the arithmetical meaning of the term successor (as used, for example, in the definition of 1 as the successor of 0) is such that necessarily distinct numbers have distinct successors. This, in fact, is one of the five postulates upon which the Italian mathematician and logician Peano erected the arithmetic of natural numbers. And one of Russell’s avowed aims in “logicizing” the concepts of arithmetic was to so define Peano’s primitives (“zero,” “successor,” “natural number”) that all the postulates (and therewith the theorems) of uninterpreted arithmetic turn into logically necessary propositions—propositions that are formally deducible from the purely logical axioms of Principia Mathematica8 (chapter 1-3).9 Russell noticed, however, with considerable intellectual discomfort, that his definition of natural numbers as classes of similar classes of individuals does not make Peano’s axiom that distinct numbers have distinct successors logically necessary. For suppose (what is conceivable without self-contradiction) that the number of individuals10 in the universe were some finite number n. Then the number n+1, defined as the class of all classes with 8 See

Russell 1950. Hempel 1983. 10 The concept “individual” is but negatively defined in Whitehead and Russell 1925: anything that is neither a class nor an attribute nor a proposition. The question is thus left open whether the individuals are observable things, or postulated particles, or physical events, or whatnot. 9 See

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n+1 members, would be the null class, since there would be no classes with n+1 members. For the same reason, n+2 would be the null class. Hence, we would get n+1=n+2, although n, being a nonempty class, is distinct from n+1. For reasons connected with the theory of types (a theory devised as a solution of logical paradoxes, which we cannot and need not explain here), Russell saw no other escape from this diﬃculty than to postulate that the number of individuals is not finite (“axiom of infinity”; see Russell 1950, chapter 13). Now, we have seen that n+1=n+2 follows from the assumption that the number of individuals is n, together with the Russellian definition of numbers as classes of similar classes. But if the number n+1 is, instead, defined as the attribute (of a class) of having n+1 members, Russell’s conclusion does not follow. For though, on that assumption, both attributes would be empty (inapplicable), they would remain just as distinct as, say, the attributes of being a mermaid and of being a golden mountain, and for just the same reason: they are defined as incompatible attributes. For example, the expressions “A has two members” and “A has three members” are so defined in Principia Mathematica (see the foregoing sample contextual definitions) that it would be contradictory to suppose that the same class had both (exactly) two and (exactly) three members. Therefore, even if just one individual existed, the number 2 would retain its conceptual distinctness from the number 3; and this argument, obviously, can be generalized for any finite number n and its successor. Apart from the question of the axiom of infinity, it is, within the framework of the logical analysis of mathematics contained in Whitehead and Russell’s monumental Principia Mathematica, unimportant whether numbers be conceived as classes (of classes) or as attributes (of classes), for names of classes are, in either case, contextually eliminable, according to Russell’s theory of classes as “incomplete symbols.” According to this theory, a statement that ascribes an attribute F to a class that is defined as the totality of entities that have attribute G is analyzed as follows: there is a predicative function θ such that θ is formally equivalent to G and such that θ has F.11 In this context, function means, not numerical function, but propositional function, and for the present purposes propositional functions may be identified with attributes. Two attributes are “formally equivalent” if they apply to the same entities, and a predicative function is an attribute that does not presuppose, in its definition, a totality of attributes. For a full understanding of this theory of classes, familiarity with the theory of types, especially the “ramified” theory with its distinction between the type and the order of a function, would be required.12 But for the present discussion,

11 See

Whitehead and Russell 1925, 71f. Whitehead and Russell 1925, Introduction and chapter 2. More easily intelligible expositions of the ramified theory of types are Lewis and Langford 1951, chapter 13, and Copi 1954, appendix B.

12 See

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all that matters is that in the “primitive” notation of Principia Mathematica, no names of classes but, instead, variables ranging over attributes (propositional functions) occur. For a semantic nominalist holds that the language of mathematics involves an obscure Platonism as long as there are in its primitive notation either names of, or variables ranging over, abstract entities; and he regards attributes as abstract entities.13 Take, for example, the simple statement that two individuals, x and y, diﬀer in at least two respects. This means that there is an attribute f and an attribute g such that f is not equal to g and such that x has f and y does not have f and x has g and y does not have g, or y has f and x does not have f and x has g and y does not have g, and so on. But here f and g are variables ranging over attributes; therefore such an analysis is unacceptable to a semantic nominalist. Now consider the more complicated statement: 2 + 2 = 4. The logical analysis performed by Frege and Russell shows that we do not have to regard the symbols “2” and “4” as names of mysterious entities. Even the names “the class of all classes with two members” and “the class of all classes with four members” are, in principle, eliminable. But what cannot be got rid of, apparently, are variables ranging over attributes. For if, in conformity with Russell’s view that classes are logical fictions, not real entities, we define numbers as certain kinds of attributes of attributes (in the sense in which “being a color,” for example, is an attribute of attributes, namely, of blueness, redness, and so on), we obtain the following analysis of 2 + 2 = 4: for any attributes f , g, h, if just two individuals have f and just two individuals have g, and nothing has both f and g, and if an individual has h if and only if it has either f or g, then just four individuals have h (illustration: f = being a coin in my left pocket at time t0 , g = being a coin in my right pocket at time t0 , h = being a coin in one or the other of my pockets at t0 ). Is mathematics, then, irremediably Platonistic? Or is there some way of eliminating attribute variables so as to satisfy the demands of semantic nominalism? The issue is a highly technical one, but the following suggestion should be intelligible to those without any knowledge of symbolic logic. As was explained in earlier paragraphs, the nominalist insists that a predicate can be meaningful without being the name of anything. For this reason he refuses to transcribe (i) “x is red” into (ii) “x has the attribute redness.” From (ii) we may deduce (iii) “there is an attribute f such that x has f ,” but since “is red” has meaning only in context, it is not a name of a value of a variable. The deduction of (iii) from (i) is as legitimate as the deduction of (iv) “there is something which I am now thinking of” from (v) “I am now thinking of a unicorn.” The

13 What

I call “semantic nominalism” has been formulated by W. V. Quine, in Quine 1949 and in Quine 1947a. See especially Quine 1953a, chapters 1 and 6. The program of nominalistic reconstruction of mathematics is sketched by W. V. Quine and N. Goodman in Goodman and Quine 1947.

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transition from (v) to (iv) is tempting because it is similar to the perfectly valid deduction of (vi) “there is something which I am now eating” from (vii) “I am now eating an apple.” But, whereas we could not eat apples if there were no apples, we can think of unicorns in spite of there not being any. If we allow the deduction of (iv) from (v), we get entangled in a verbal contradiction, since the something which (iv) says I am thinking of is, of course, a unicorn, and then there must be unicorns after all! To resolve the contradiction, the Platonist may say that the object of my thought is the attribute of being a unicorn, not a concrete unicorn. The nominalist, on the other hand, criticizes the translation from (v) to (iv) on the ground that “unicorn” is not a name but a predicate (or part of the predicate “is a unicorn”). Correctly analyzed, (v), unlike (vii), does not assert a relationship between two independently existing entities but simply characterizes my thought as what we might call a “unicornish” thought. Similarly, “this is a picture of a unicorn” characterizes the picture as unicornish; it does not mean that this has the relation being a picture of to an independently existing entity—in the way that this is the meaning of “this is a picture of my son.” Bearing in mind the distinction between predicates (which may not be replaced by variables ranging over some kind of entities) and names, let us return to the logical analysis of 2 + 2 = 4. To satisfy the nominalist, we must replace “have f ,” where f represents a name of an attribute, with “are f .” But then f , g, and h become letters that represent predicates, not names of attributes; hence, they are not variables over which we may “quantify,” as the logicians say. The prefix (quantifier) “for any attributes f , g, h” must, accordingly, be cut oﬀ, for the letters f , g, and h do not represent names of attributes at all. What, then, is the law of arithmetic 2 + 2 = 4 in logical and, at the same time, nominalistic interpretation? The surprising answer is that there is not one such law, there is not a sentence containing, besides variables, only logical constants which we can point to and say “this sentence expresses the arithmetical law that 2 and 2 make 4,” for “if just two individuals are f and just two individuals are g ...” is a schema, not a statement; it does not express a proposition at all. In order to obtain propositions out of it, we must substitute specific predicates for the schematic letters in it, and there are as many such propositions as there are triplets of predicates applicable to individuals. If the language of arithmetic is reconstructed nominalistically, then, no laws of pure arithmetic can be formulated in the object-language—that is, the language which, whatever it may refer to, does not refer to symbols. Instead, one has to formulate in the meta-language (the language that is used to talk about the object-language) statements to the eﬀect that any substitution instance of such and such a schema is logically true. This conclusion, of course, applies not only to the laws of arithmetic and of higher mathematics but also to the basic laws of logic. For example, consider

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the law of medieval logic, “what is true of all is true of any,” which, in symbolic logic, reads as follows: for any y, if all x’s have attribute f , then y has f . A Platonistic logician can refer to one logical truth expressed by the sentence: for any attribute f and for any individual y, if all x’s have f , then y has f . A nominalistic logician must ascend to the meta-language and there declare that every statement of the form “for any y, if all x’s are f , then y is f ” is logically true.14

4.

What Do the Ontological Questions Mean?

To what extent a nominalistic reconstruction of mathematics is feasible is certainly an interesting technical problem that deserves investigation in its own right. But attempts at such reconstruction are usually motivated by a philosophic prejudice about what “really exists.” Being asked why he wants to get rid of variables ranging over attributes, the nominalist is likely to reply that he cannot admit the existence of attributes because the very notion of “attribute” is obscure to him. But even if the notion could be clarified to his satisfaction, he would probably remain reluctant to admit the existence of such entities. Thus, nominalists usually concede that the notion of “class” is quite precise, and yet they cannot countenance classes as “real entities.” The main argument in support of the accusation that the meaning of the word attribute (and of its synonyms in philosophic literature: universal, property, intension of a predicate) is obscure is that it is not clear under which conditions two expressions can be said to designate the same attribute.15 For example, is the attribute (of a number) of being equal to the product by itself the same as the attribute of being the successor of zero? Is the attribute of being a rectilinear closed figure with three sides the same as the attribute of being a rectilinear closed figure with three interior angles? We have here predicates which are interdeducible (logically equivalent), but it seems debatable whether they are synonymous. And the question of the criterion of synonymy is, indeed, diﬃcult and highly controversial in contemporary semantics.16 The same nominalists who frown on attributes find it easier to accept talk about classes precisely

14 Subtle

problems emerge once one asks whether, granted that the object-languages of logical and mathematical systems can be constructed in accordance with nominalistic restrictions, the meta-language also could be nominalistic. For example, surely we mean to assert the logical truth of all possible substitution instances of a given schema, not just of those that are actually written down somewhere and at some time. But it is even doubtful that the method described in the text makes possible a nominalistic rewriting of mathematical object-languages, for quantifiers that refer to attribute variables within the scope of another quantifier cannot be eliminated by such a simple device. The statement “there is no largest number” illustrates this complication if numbers are construed as attributes of attributes: for any attribute f , if f is a number-attribute of an attribute, then there is a number-attribute (of an attribute) g such that g is larger than f. 15 This argument has been used chiefly by W. V. Quine. See Quine 1951 and Quine 1947b. 16 See Goodman 1952b, Mates 1952, Naess 1949, Frege 1949, Carnap 1955, Quine 1951.

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because the condition of identity of classes is clear and uncontroversial: A and B are the same class if, and only if, they have the same members. Now, this particular argument against “Platonistic” discourse seems to me highly illogical. If “attribute” is held to be obscure on the ground that the identity condition for attributes is not clear, then one would expect that all is well with the identity condition for individuals—that is, those concrete entities, whatever they may be (material particles, observable things, qualitatively distinguishable events, space-time points), which alone really exist according to nominalist ontology. Yet, the identity condition for individuals that is usually accepted is Leibniz’s principle of the “identity of indiscernibles,” which in modern logic is formulated thus: x = y if, and only if, for every attribute f , x has f if and only if y has f . Clearly, if such is the definition of identity of individuals, and a given kind of entity is deemed obscure to the degree that its identity condition is not clear, then the alleged obscurity of “attribute” must infect the term individual too. Two replies are open to the nominalist: (i) Since we realize that an explicit definition of identity requires quantification over an attribute variable, we do not explicitly define this logical constant at all. Instead, we lay down an axiom schema for identity which does the same job as the explicit definition. It says, in eﬀect, that identicals may be substituted for each other without change of truth-value of the sentence into which the substitution is made, and contains a schematic predicate letter, not a quantifiable attribute variable: if F x, and x = y, then Fy. (Note that, in accordance with the explanation given earlier, a single axiom or definition is thus replaced by a family of axioms of the same form, namely, those that result from the substitution of a predicate for the schematic letter F). (ii) Identity is explicitly definable with the help of a class variable; hence, the intelligibility of “individual” exceeds that of “attribute” by at least as much as the latter is exceeded by the intelligibility of “class”: x = y, if and only if x and y are members of just the same classes of individuals. My rejoinder to (i) is that an exactly similar axiom schema can be laid down for identity of attributes. That is, it is postulated that from “ f is F” and “ f = g” we may deduce “g is F.” Here F is a schematic letter to be replaced by predicates that are applicable to attributes, not to individuals (for example, “is a color,” “is possessed by exactly two individuals,” “is a desirable attribute”). Consequently, attributes and individuals are still in the same boat with regard to the alleged obscurity of the identity condition. As rejoinder to (ii), obviously just the same kind of explicit definition of identity of attributes can be given. Attribute f is the same as attribute g if, and only if, f belongs to just the same classes of attributes that g belongs to. The argument from the identity condition, therefore, does not establish the superior intelligibility of nominalistic discourse. Independent arguments are needed. Perhaps the basic argument is the simple “common-sense” argument

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that the only way of proving the existence of anything is sense perception or inference from perceived things or events to unperceived things or events, in the manner of the physical scientist. But attributes cannot be perceived by the senses, nor does the postulation of the existence of attributes explain the nature of the perceived world in the way in which the theory of electrons, say, explains observed phenomena. Nevertheless, the assertion that attributes are “abstract,” not given in sense experience, should not be uncritically accepted. There are many simple qualities which recur in our sense experience and which, by this criterion, are universals. We see the same shade of blue in diﬀerent patches or in diﬀerent parts of the same patch, hear the same pitch in diﬀerent sounds, taste the same taste in diﬀerent portions of the milk we drink. It is likewise artificial to deny that we have direct experience of relational universals—for example, of the relation of temporal succession and the relation of spatial proximity. The counterargument that it is not, after all, certain that precisely the same quality recurs presupposes that one understands what it would be like to perceive exactly the same quality in diﬀerent contexts. But since the expressions “exactly the same shade of blue,” “exactly the same pitch,” and so on, can be explained only ostensively17 (you point, say, to adjoining parts of the same expanse and say “see, they, for example, are the same shade of blue—that is what I mean by ‘same shade of blue”’), this argument is really self-defeating. At any rate, the argument from the limits of sensory discrimination is hardly relevant to the question of the existence of mathematical attributes. For (natural) numbers may be regarded as attributes of attributes of individuals; that is, the number n is the attribute (of an attribute) of having n instances, and to deny that two attributes f and g have the same number is to deny that their instances can be matched one by one. But surely there are attributes that have the same number in this sense—for example, being a finger on my left hand and being a finger on my right hand. And if there are attributes that have the same number, does it not follow that there are attributes? What, then, is the argument between the nominalist and the realist all about? Indeed, it is diﬃcult if not impossible to attach any sense to the “ontological” question about whether there are attributes.18 If someone asks me this question, my natural reaction is, “why, of course there are: blueness, round-

17 It may, indeed, happen that a appears the same color as b, b appears the same color as c, yet a is distinguishable in color from c. Such cases suggest the following general definition of qualitative identity in terms of indistinguishability: x and y have the identical quality Q if, and only if, anything which is indistinguishable with respect to Q from x is also indistinguishable with respect to Q from y. But then, “indistinguishable (with respect to Q)” is the term that must be ostensively defined, and it will be self-refuting to deny that sensory appearances (many philosophers speak of “sense-data”) exhibit genuine universals. 18 The chief advocate of the logical positivist view that such ontological questions are “pseudo-questions,” devoid of cognitive meaning (a view I strongly incline to myself), is R. Carnap. See Carnap 1950a.

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ness, hardness, twoness, and millions more.” If he denies that these are examples of attributes, then he cannot mean by attribute what is ordinarily meant by the word, for one explains its meaning in terms of just such examples before entering into the more technical explanation of what, in general, distinguishes attributes from other types of entities, like individuals, or classes, or propositions. To deny that roundness is an attribute would be as absurd as to deny that 5 is a number, or that parenthood is a relation, or that the desk I am writing on is a thing. But if it is not denied, then one cannot deny, either, that there are attributes, since, according to elementary logic, any statement of the form “a is f ” entails “there is an x (at least one) such that x is f .” In the terminology of some contemporary analytic philosophers of language, the ontological statement is really a trivial analytic statement that cannot be denied if one understands the intended meaning of the type designation, like attribute, relation, class, and so on.19 To be sure, if one cannot countenance entities other than particular things because one cannot understand how something can “be” without having a tangible and visible existence at a particular place (one at a time!), then one’s intellectual discomfort is just self-imposed through the fallacy of “reification”: one is just naively associating with the expression “there are” images of spatially located things. One might, for example, be made intellectually uncomfortable by a mathematician’s statement that the number 0 exists. According to the Russellian class conception of numbers, the number 0 is a very peculiar class: the unit class whose only member is the null class. But how can a class be said to exist unless it has an existing member? And is not the null class, by this very criterion of class existence, nonexistent? Yet, in the sense of “there is” which is implicitly defined by the entailment (which was given in the last paragraph), “if a is f, then there is an x such that x is f ,” the statement “there is a class which is the null class” is just a trivial consequence of the logically true statement, “there is nothing which is distinct from itself” (in class terminology, the class of things that are not self-identical is null) as well as a trivial

19 It

is often overlooked by philosophers that a statement which is analytic in the sense that it cannot be seriously denied by one who understands it does not have to express a necessary proposition—that is, a proposition that cannot be conceived to be false. If this is overlooked, then one can easily refute the argument in the text by an analogy: You might as well argue that “there are men” is a trivial analytic statement, since it follows from “John Smith is a man,” which is analytic, since only a man could properly (that is, would conventionally) be named “John Smith.” But although it is a contingent fact, not a logical necessity, that there are men, one could not possibly deny that there are men if one grasps the conventional meaning of man, for this word has acquired its meaning through ostensive definition—that is, through uttering the word man while pointing at a man. Similarly, the meaning of attribute and other type designations must be explained by examples before one can proceed to construct an abstract definition; hence, a disagreement about whether there are attributes is bound to be as verbal as would be a disagreement, if one ever broke out, about whether there are men. (A comprehensive and detailed analysis of the meanings of analytic and necessary is contained in Pap 1958c. For an acute discussion of the meanings of ontological statements, see White 1956).

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consequence of any logically true statement of the form “there is nothing which both is and is not f .” The puzzle about how such shadowy, ghostly entities as the null class, and therewith the number 0, can exist arises only because one associates with the expressions “there is” and “there exists” images of spatial location, although they function purely as logical constants in the context of mathematical existence assertions: “there is something which is f ” means no more nor less than “not everything is not f .” Most existential statements that occur in everyday discourse refer to physical objects or events that exist, or occur, at some place and at some time—or at some place-time, in relativistic language. From this derives the habit of associating with the logical constant “there is” and its variants, images of spatial localization. It is only these associations, and the failure to understand the purely logical nature of such existential statements as “the null class exists,” “there are classes,” “there are attributes,” which produces intellectual discomfort. To be sure, some mathematicians and logicians reject an existential statement “there is an x such that x is f ” as meaningless in a case where it cannot be proved by deduction from a provable corresponding singular statement “a is f.” They are sometimes called “intuitionists,” sometimes “finitists.” They reject merely indirect proofs by the reductio ad absurdum method: there is an x such that x is f , because if you assume that there is no such x you get involved in a contradiction, whence it follows that the assumption is false; and if the contradictory of a proposition p is false, then p is true (law of the excluded middle). A well-known application of this “finitist” methodology of mathematics is the rejection of Zermelo’s axiom of choice. This is the following existence assertion: for any class K of mutually exclusive and non-null classes, there is a class which contains exactly one member from each member of K (“multiplicative class”). Now, in order to prove that a given class is a multiplicative class with respect to K, one must define it in terms of a method of selection of its members from the members of K. For example, let K be the class of all mutually exclusive pairs of integers. Here, a multiplicative class is easily defined as the class to be constructed by picking, say, the smaller integer from each pair. On the other hand, consider the class K which is formed as follows: we first take the class of proper fractions between 0 and 1, then the class of proper fractions between 1 and 2, then the class of proper fractions between 2 and 3, and so on. Obviously, the members of K are classes that have no members in common, since the integers that delimit the intervals are not themselves proper fractions. But now it is a little less obvious that one can describe a method of constructing a multiplicative class. The classes from which the members of the multiplicative class are to be selected have neither a first nor a last member, since between any proper fraction and the “closest” integer, there is another

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proper fraction. One might think of representing each class by its “mid-point,” perhaps—that is, by that fraction which equals the arithmetic mean between the integral limits. But this method of selection becomes unavailable for the following class K: the class of proper fractions between 12 and 32 , limits excluded, then the class of proper fractions between 32 and 52 , limits excluded, and so on. For here the “mid-points” are integers, not proper fractions. Perhaps there is still a method of selection with respect to this K, but do we have any guarantee that for any class of mutually exclusive non-null classes a method of constructing a multiplicative class can be described? Certainly not. Russell mentions, in his Introduction to Mathematical Philosophy (Russell 1950, 126), the class of all pairs of socks in order to illustrate that it is not self-evident that there is a “selector” for defining a multiplicative class with respect to a class of mutually exclusive non-null classes. A multiplicative class, here, could be defined only if there were a property which just one sock in each pair possessed, say being the left sock, or being the red sock, and so on. The finitist concludes that it is meaningless to assert the axiom of choice in complete generality. He holds that it is meaningless to assert that there is a multiplicative class of classes unless one can give a rule for constructing one. And this fits into his general methodological requirement that existence theorems be proved “constructively”—that is, by exhibiting an instance, or a rule for constructing an instance, that satisfies the existence theorem.20 But, however the methodological dispute between finitists and “realistic” mathematicians who wish to continue using nonconstructive proof methods (such as the reductio ad absurdum method by which Euclid discovered the irrationality of the square root of two) may be resolved, it really has no bearing on the ontological pseudo-questions, “are there classes?,” “are there attributes?,” and so on. For suppose we accept the finitist interpretation of “there is an x such that x is f ” as meaning “there is at least one constructively provable statement of the form ‘x is f ’,” where x ranges over abstract entities like numbers or classes. What should prevent one who has a penchant for asking ontological questions from asking whether there really are such abstract entities? The very asking of a question of the form “is there a number with property f ,” he might say, presupposes that there are numbers,21 whatever proof technique one may use to answer it. This presupposition, however, can be established quite eas-

20 See

Fraenkel 1928, par. 14, and Wilder 1965, chapter 10. finitists usually accept the natural numbers as, in some sense, “given”—in accordance with Kronecker’s often cited statement that God himself created the natural numbers, though he left the construction of all the other kinds of numbers and classes to man. Sometimes it is the “rule of construction” of natural numbers (adding one) that is said to be intuitively given rather than the natural numbers themselves. But it is not clear in what sense a natural number can be said to be a mental construction. Acquiring an idea of a number n by “counting up” to it is, of course, a mental process, but when we ascribe to a class a particular number, we ascribe an attribute to it that cannot plausibly be identified with the mental process of counting.

21 Indeed,

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ily by citing examples of numbers. To be sure, no philosopher who regards the existence of numbers and other abstract entities as problematic would be satisfied with the ironical answer: “Of course, there are numbers: 2, 5, 7, and many, many others.” But the trouble with the ontological question is just that no method of finding the answer has ever been described by those who keep asking it. And this means not that we do not know the answer but that we do not understand our own question. There is but one interpretation of the ontological question that makes it both intelligible and interesting, and this is an interpretation that turns it into that semantical-logical question which was touched on earlier in this article. What exactly did Russell mean by the assertion that classes are not real entities but are just “incomplete symbols”? The best way to discover the meaning of a statement is to look at the reasons oﬀered in support of it. Now, Russell’s reason for denying the reality of classes was that statements that contain names of classes or class variables are satisfactorily translatable into a symbolism which does not contain such expressions. (To take a simple example: “the class of dogs is included in the class of animals” means “for any x, if x is a dog, then x is an animal”). I propose to identify this reason for the denial of the reality of classes with the meaning of the denial. Similarly, the nominalist who denies the reality of any kind of abstract entities may be interpreted as aﬃrming that the meanings of statements that contain names of, or variables ranging over, abstract entities can be analyzed by means of a language that satisfies nominalistic requirements. Whether a nominalistic language is clearer, more intelligible, than a realistic language is, of course, debatable; indeed, this may be a matter of taste, of arational preference. But it must be admitted, I think, that the modern semantical interpretation of the time-honored (or time-dishonored) nominalism-realism dispute has the merit of making it scientifically meaningful, although it may be countered that, for this very reason, it is misleading to call this modern semantical-logical issue by the same name because one thus seems to accuse those who prefer the one or the other type of language of a subconscious addiction to realistic or nominalistic “ontology.”

Chapter 13 EXTENSIONALITY, ATTRIBUTES, AND CLASSES (1958)

In Principia Mathematica, an extensional system embodying the theory of types, Peano’s postulate “distinct (natural) numbers have distinct (immediate) successors” is not formally derivable from purely logical axioms. The axiom of infinity (“the number of individuals is infinite”) is required for the proof that there is no finite cardinal n such that n equals n + 1. Russell argued as follows:1 if only n individuals existed, then the number n + 1, being defined as the class of all classes that have n + 1 members (or that would have exactly n members if exactly one element were withdrawn from them), would be equal to the null class; for in that case no classes with n + 1 members would exist. But by parity of reasoning, the successor of n + 1 would also be equal to the null class; therefore n and n+1, which on the hypothesis made are distinct numbers, would have the same successor. The usual reaction to this argument is that, without abandoning Russell’s conception of numbers as classes of similar classes, we fortunately do not need to postulate the axiom of infinity after all. For there are other ways of solving the logical paradoxes besides the theory of types, and once our constructive eﬀorts are unimpeded by the latter, we can construct an infinite sequence of abstract entities without presupposing the existence of a single concrete individual: the null class, the unit class whose only member is the null class, the class whose members are the foregoing two classes, and so on. And once we have an infinite set of such abstract, though typically impure, entities, we can rest assured that no natural number will collapse into the null class. This is the approach of set theory, where such ghostly classes as the one just mentioned can be postulated to exist provided their definitions do not give rise to contradiction. However, I would like to re-examine Russell’s argument in order to see whether it is perhaps possible to get rid of the axiom of infinity without abandoning the (simple) theory of types.2 1 See

Russell 1950, 132.

2 Already in Carnap 1954a Carnap maintained that no axiom of infinity is needed if a coordinate-language is

employed. A coordinate-language, unlike a “thing-language” (or a “substance-language”), uses numerical

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Obviously Russell’s argument depends on (a) the definition of numbers as classes of similar classes and (b) the principle that classes with the same membership, including the degenerate case of no membership, are identical. Condition (b) can hardly be questioned since it is just this identity condition which is characteristic of classes in contrast to the attributes by which they are (intensionally) defined: attributes may be distinct though they determine the same class. But suppose we defined a natural number as a property3 common to all classes that are similar to a given class; for example, the natural number two is defined as the property common to all couples (of type 1) by virtue of which they are couples.4 Clearly, it could not be argued any more that if only 9 individuals existed, then 10 would be equal to 11, etc., for though both the class of all 10-membered classes and the class of all 11-membered classes would be the null class and hence identical, the numbers 10 and 11 would in the new conception be two perfectly distinct null properties, no more to be identified than the property of being a golden mountain and the property of being a mermaid. The validity of Peano’s axiom would then be independent of the contingency whether the world of individuals is finite or infinite. It should be noted that since Russell himself stipulated the formal deducibility from logical truths of Peano’s postulates as a criterion of adequacy for the definitions of the arithmetical concepts in terms of logical constants, and his definitions of numbers as extensions do not satisfy this criterion within a type-theoretical system, Russell was really committed to a rejection of them as inadequate. In The Principles of Mathematics (§110) Russell rejected the conception of numbers as common properties of similar classes on the ground that this kind of definition “by abstraction” would not guarantee the uniqueness of the individual numbers: how do we know that the property of having two members is the only property that is common to all couples?5 Suppose, for example, that all individuals have mass, as I suppose it should be according to materinames of space-time points, like “the 3rd position along the x-axis from position 0,” and ascribes qualities, resp. values of magnitudes, directly to these points, not to substrata by which they are (allegedly) occupied. Carnap shows that we can construct atomic sentences in such a language in such a way that the arguments are names of numbers; for example, instead of “the space-time point numbered 3 is blue” we can say “the number 3 is correlated to a blue space-time point” (cf. Carnap 1956a, 86). But I have never been able to understand how this trick is to make the axiom of infinity dispensable. Of course, the sentence “there are infinitely many natural numbers” is analytic in the Peano system of arithmetic, but this only means that it follows from the postulates of the system, not that it follows from purely logical axioms. It was in order to demonstrate its analyticity in the latter sense that Russell found himself driven to postulate the axiom of infinity; hence Carnap’s argument seems to me to beg the question. 3 I am using “property” and “attribute” synonymously, as is customary among logicians. Perhaps it would be better to avoid using “property” in this technical context, since in ordinary language a property is essentially a property of something, just as a brother is a brother of somebody, and it therefore sounds paradoxical to speak of a “null property.” 4 The symbolic expression of this definition is a property-abstract, not a class-abstract: 2 = (λ f )[(∃x)(∃y) (x y.(∀z)( f z ≡ (z = x ∨ z = y)))]. See Carnap 1956a, 115. 5 Apparently Russell did not suspect at that time that he would not be able to preserve the intuitive uniqueness of the numbers (i.e., there is but one number n, for any value of “n”) anyway, on account of the restrictions imposed by the theory of types on the use of “all.”

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alist ontology; or suppose that all individuals are qualitatively distinguishable space-time points or space-time regions, as presupposed by coordinate languages patterned after space-time theory. Then all couples would have the common property, not only of being couples, but also of containing members of the described sorts. Yet, could not this diﬃculty be circumvented by defining natural numbers as common properties of similar classes which (properties) are not defined in terms of any properties of the members of the classes or by reference to specific individuals that are members of the classes (the latter restriction would rule out such common properties as “containing individual a,” in the case of overlapping classes)? At any rate, no such diﬃculty arises if we speak of the number n as of the property of being an n-membered class, or an n-instanced property; we do not need to say that two is the property common to all couples any more than we need to say that redness is the property common to all red objects. It is sometimes supposed that the distinction between attributes and classes is a distinction that makes no diﬀerence with respect to extensional languages. This is a prima-facie plausible view, for the following reason: if F and G are distinct attributes though the corresponding classes the x’s such that Fx and the x’s such that Gx are identical, then there must be some context in which the substitution of “F” for “G,” or vice versa, leads to a change of truth-value. But if all contexts in which such an interchange might be made are extensional (e.g., “(∀x)(F x ⊃ Hx),” “Fa”), then by definition of “context which is extensional with respect to ‘F’, resp. ‘G,”’ no change of truth-value can occur. It is only in nonextensional contexts, for example, such modal contexts as “it is necessary that (∀x)(F x ≡ F x),” that the diﬀerence of extensionally equivalent attributes manifests itself. Such being the case, how can it make any diﬀerence at all whether we say “n is the class of all n-membered classes” or whether we say “n is the property of being an n-membered class,” as long as we stay within an extensional system like Principia Mathematica? Yet, have we not just shown that it makes an enormous diﬀerence, since it is only the latter analysis that permits the deduction of Peano’s critical axiom from purely logical truths in a type-theoretical system? This apparent contradiction can, however, be solved as follows. It looks as though Principia Mathematica could be shown to be not completely extensional after all. For consider the statement “if no more than n individuals exist, then the class of n + 1-membered classes is identical with the class of all n + 2-membered classes,” which is true. The statement which results from it by substituting for the class names the property names “the property of being an n + 1-membered class” respectively “the property of being an n + 2-membered class” is, however, false. For in order for property F to be identical with property G it must be necessary that F and G have the same extension, and since

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it would be a contingent fact, if a fact at all, that the mentioned properties are empty (and hence coextensive), they would still be distinct. Nevertheless, there is a subtle fallacy in the conclusion that therefore Principia Mathematica is not completely extensional. The fallacy is the tacit confusion of predicates, or sentential functions, with names of attributes. A sentence is said to be extensional with respect to a constituent expression E, if any replacement of E with an expression of equal extension leaves the truth-value of the sentence unchanged (the latter is then said to depend only on the extensions, not the intensions, of the constituent expressions). Now, the extension of a name is the entity named by it. Since the very distinction between names of classes and names of attributes (“ xˆ(F x)” and “F xˆ” in Principia Mathematica) presupposes the distinction between classes and attributes, it follows that a name of a property cannot have the same extension as a name of a class even if the named class is the extension of the named property. In the symbolism of Principia Mathematica,6 the extension of the class-name “ xˆ(F x)” is identical with the extension of the predicate “F”—or of the sentential function “F x”—but not with the extension of the property-name “F xˆ.” Nothing, therefore, prevents an extensional system from containing sentences of the form “ xˆ(F x) F xˆ,”7 even as theorems! A system, then, may be completely extensional in the explained sense (i.e., all its sentences are extensional with respect to all constituent expressions that can be said to have an extension), and nevertheless distinguish explicitly between classes and attributes by employing, like Principia Mathematica, diﬀerent kinds of names for them. Moreover, this diﬀerence is not a diﬀerence that makes no diﬀerence, as shown by my discussion of the question whether we need an axiom of infinity in a type-theoretical system.

6 For

the sake of typographical convenience I replace the Greek letters by Latin letters. confusion between names and predicates seems to me to underlie the statement by Russell and Whitehead (endorsed by Carnap in Carnap 1956a, 148) that the signs of identity and non-identity are not extensional in contexts like “ xˆ(F x) = F xˆ” and “ xˆ(F x) F xˆ.”

7 The

Chapter 14 A NOTE ON LOGIC AND EXISTENCE (1947)

In reference to Mr. E. J. Nelson’s recent article on “Contradiction and the Presupposition of Existence” (Nelson 1946), I want to comment upon a fundamental assumption involved in the author’s arguments which I find highly questionable. The following is the paradox for the solution of which Mr. Nelson elaborates a distinction between “the necessary conditions of the existence of a proposition” and “the necessary conditions of its truth (exclusive of its existence).” f a implies (∃x)[ f x ∨ ∼ f x], but ∼ f a, which is the apparent contradictory of f a, implies the same existential proposition. This common implicate, however, could conceivably be false, viz., in case no individuals at all existed. Hence, by transposition, both f a and ∼ f a could conceivably be false; and since contradictory propositions cannot conceivably be both false, f a and ∼ f a are not contradictories. Mr. Nelson’s solution of the paradox, if I have understood his arguments correctly, is that implication, as a logical relation, must be construed as a relation between what propositions assert (and propositions assert the necessary conditions of their truth); but f a does not, according to Mr. Nelson, assert that any individual (specifically, the individual a) exists, it only presupposes the existence of a as one of its constituents. Mr. Nelson’s conclusion seems to be that f a does not imply the existence of at least one individual in the same sense of “implies” in which, for example, (∃x) f x implies ∼(∀x) ∼ f x. The existence of at least one individual is rather “presupposed” by the existence of the proposition f a; it is not a truth-condition of f a in the sense in which, say, ga would be a truth-condition of f a if f a ⊃ ga were true. Now, as far as I can see, Mr. Nelson has not justified his assumption that, in asserting (∃x)[ f x ∨ ∼ f x], the existence of at least one individual is asserted and that, therefore, the above existential statement expresses a contingent truth which “is not certifiable on the grounds of logic alone.” Let us first consider a most puzzling consequence of Mr. Nelson’s assumption. If “(∀x) f x implies (∃x) f x” is, as in Russell’s logic, accepted as a transformation rule, then the debated existential statement follows from its corresponding universal which is tautologous: (∀x)[ f x ∨ ∼ f x]. But how can a contingent truth follow from a

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tautology? There seem to be three ways in which the consequence that contingent truths may be entailed by tautologies could be avoided: (1) To abandon the stated transformation rule. (2) To interpret logical truths, in Mill’s fashion, as simply the most extensive empirical generalizations. (3) To deny Mr. Nelson’s assumption that (∃x)[ f x ∨ ∼ f x] asserts the existence of individuals, and to maintain that this existential statement, like its corresponding universal, is purely tautologous. It seems to me that (3) is the most plausible and least burdensome alternative to take. We only have to translate the statement “(∃x)[ f x ∨ ∼ f x]” (where “ f ” is supposed to be a predicate constant) into the language of the class calculus to see its purely tautologous character. For the corresponding statement in the class calculus is: the universal class (i.e., the class which includes every class)1 has members. But to say that the universal class has members is simply another way of saying that it is diﬀerent from the null class, which is surely tautologous. The belief in the contingent character of an existential statement of the sort under discussion might arise as follows. Any existential statement can be formally exhibited as the implicate of a singular statement which verifies it. Thus f a ∨ ∼ f a implies (∃x)[ f x ∨ ∼ f x]. The transformation rule here used is: f z implies (∃x) f x (where “z” is the name of some unspecified individual). However, any existential statement may also be exhibited as the implicate of a (universal statement, with the help of the transformation rule: (∀x) f x implies ∃x) f x. Now, in formal logic, the predicate variable “ f ” may take on as values either empirical predicates (such as “red”) or logical predicates (such as “red or not-red”). Obviously, those existential statements whose predicate is vof the empirical variety cannot be known to be true in any other way than by erifying at least one of the singular statements each of which entail it; unless at least one zebra had been observed, the hypothesis “there are zebras” could not be asserted as true. This is the kind of consideration which naturally leads to the translation of existential statements into logical sums (finite or infinite) of singular statements. However, it is equally obvious that an existential state(ment whose predicate is of the logical variety (examples: (∃x)[ f x ∨ ∼ f x], ∃x) ∼( f x . ∼ f x)), is not asserted as true on the ground of the observation of individuals which exhibit it. It is asserted as a mere corollary of the corresponding universal statement which is a logical truth and in which descriptive empirical predicates have a purely vacuous occurrence (i.e., the truth-value of the statement would not change if any other empirical predicate were substituted instead). 1

It should be understood that the reference is to classes of individuals, i.e., classes designated by class symbols of type 1.

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That existential quantification, in formal logic, cannot be regarded as indicative of the assumption that at least one individual exists, is furthermore evident from the following consideration. Suppose we construct the concept of a kind which, as far as we know, has no instances. We may then draw analytic consequences from our arbitrary definition and formulate them as conditional statements. To illustrate, let “ f x” stand for “x is a goblin,” and “gx” for “x is imponderable.” Then we may assert (∀x)[ f x ⊃ gx] as an analytic truth, assuming that imponderability is a defining characteristic of goblins. But now it may occur to us to subdivide the empty class of goblins into two likewise empty classes, viz., the variety of pale goblins, say, and the variety of pink goblins. Let “hx” stand for “x is pale,” so that we can assert: ∼ (∀x)[ f x ⊃ hx]. But this universal statement is equivalent to the existential statement: (∃x)[ f x . ∼ hx]. By the principle of simplification, it follows that (∃x) f x; in other words, from an arbitrary definitional construction of an empty class and likewise empty subclasses thereof, we seem to have proved that the defined class is not empty after all: there are goblins! A clever theologian might exploit arguments of this sort in order to show that there is nothing logically objectionable in the assumption of the “ontological” argument that “essentia” may involve “existentia.” However, the proper conclusion to be drawn is that existential quantification, in formal logic, is a purely formal operation which has no “ontological” import whatever. As the above sample argument shows, “(∃x) f x” cannot be interpreted to assert that the property f has empirical instances. f might be a definitionally constructed property determining a subclass of a class determined by a more abstract or less complex property. Thus, “(∃x) f x” might assert that there are isosceles triangles—besides other varieties of triangles—without carrying the implication that any empirical instances of the concept “isosceles triangle” exist. Hence, if the individual constants “a,” “b,” etc., are regarded as proper names of empirical particulars, “(∃x) f x” can hardly be said to be synonymous with the logical sum “ f a ∨ f b ∨ . . . ∨ f n.” Mr. Nelson argues that “a exists” is a common implicate of f a and ∼ f a. Whether the former proposition be regarded as contingent or necessary, he continues, at any rate it implies “(∃x)[ f x ∨ ∼ f x],” which is itself contingent. Langford’s interpretation of this situation is alleged to have been that f a and ∼f a are not contradictories and that, indeed, singular propositions have no formal contradictories at all. Mr. Nelson, instead, concludes that “a exists” cannot be said to be implied by what “ f a” asserts. There are three final comments I wish to make. (1) Mr. Nelson seems to assume throughout his discussion that the sentence “a exists” expresses a proposition. But which are the constituents of this curious proposition? Presumably the individual a and the property of existence. However, we simply misuse language if we ascribe existence as a property to an individual; we can properly ascribe it only to a kind of individuals. It makes

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sense to say “zebras exist,” but if someone pointed to an individual and said “this zebra exists,” he could at best mean that what he sees is a real zebra, and that he does not suﬀer from an hallucination. But even so, he would not be ascribing a property, named “reality” (which may or may not be taken as identical with existence), to the individual he sees; he would rather be making the meta-linguistic assertion that the assertion “this is a zebra” (where the designatum of “zebra” is, of course, the class of physical zebras, not the class of mental images of zebras) is true. Mr. Nelson does not seem to have considered Russell’s demonstration that the logical analysis of sentences having “existence” as their grammatical predicate leaves you with propositions which do not contain existence as a constituent. Thus, in Russell’s analysis, “zebras exist” would mean “the sentential function ‘x is a zebra’ is true for some values of x” or “the class of zebras is non-empty.” (2) Why not argue that (∃x)[ f x ∨ ∼f x] must be a tautology just because it is implied by contradictory premises? Mr. Nelson aﬃrms that “no two propositions are contradictories if they have a common implicate, regardless of the status of that implicate.” But is it not an established theorem of formal logic that a tautology is implied by any proposition, just as a self-contradictory proposition distinguishes itself from self-consistent propositions by the fact that it implies any proposition? To say that p entails q is to say that p. ∼ q involves a self-contradiction. Hence, if both p entails q and ∼ p entails q, p. ∼ q, as well as ∼ p. ∼ q, must involve contradictions. But if q is a tautology, then ∼ q is a contradiction hence the condition of entailment is satisfied in both cases. (3) Mr. Nelson might reply that, if apparently contradictory premises have a common implicate which is contingent in the sense of being conceivably false, then either they are not really contradictory (Langford’s conclusion which Mr. Nelson is “reluctant” to accept), or they do not imply that contingent proposition in the ordinary sense of “implication,” i.e., the sense in which implication is a logical relation between assertions. This seems, indeed, to be Mr. Nelson’s own conclusion. The alternative, however, which he has not considered, is that the common consequence (∃x)[ f x ∨ ∼ f x] could not be false, being a logical truth. I shall now present my final argument for this alternative. According to Mr. Nelson, the above existential statement implies that individuals exist, which could be false. But what is the logical status of the statement “individuals exist,” what is its logical form? It seems to have the same form as the statement, say, “lions exist,” and since the latter would in Russell’s logic, which does not recognize “existence” as a predicate, be rendered as (∃x)(lion(x)), “individuals exist” might be formalized in the same fashion: (∃x)(individual(x)). It thus looks like a contingent truth, since, just as we might find no individual verifiers for (∃x)(lion(x)), we might, as it seems, find no x’s which have the property of being individuals. Thus, “a is an individual” looks like a contingent truth, of the same character as “a is a lion.” But upon closer inspection it turns out to be an example of Carnap’s “pseudo-object-sentences”; the “paral-

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lel syntactic sentence,” from which it is inferable without any factual inquiry into the extra-linguistic referents of language, is: “a” is the name of an individual. If I assign the name “a” to an individual, it still remains logically possible that a should turn out not to be a lion. But it is logically impossible that it should fail to be an individual. But if “a is an individual” is not contingent, then its implicate “individuals exist” cannot be contingent either. It is, indeed, natural to ask: Is it not logically possible that individual constants should remain without referents? But this question arises from a confusion between individual constants (which are proper names) and predicates (which are general names). Predicates are names of properties (or relations) which may or may not have individual instances, but individual constants are, by definition, names of individuals. The same kind of objection may be urged against Mr. Nelson’s assertion that (∃P)[Pa ∨ ∼ Pa] could be false, on account of asserting the existence of properties. “ f is a property,” like “a is an individual,” is a pseudo-object sentence; its “syntactic parallel” is: “ f ” is a predicate. A world of individuals devoid of properties is a logically impossible world. We may be mistaken in believing that a given individual has a given property; it may turn out that it has a diﬀerent property and, indeed, that no individual at all possesses the first property. But it is logically impossible that an individual should fail to have any properties at all. For “a has no properties” implies that a has the property of having no properties, which is self-contradictory. In other words, if there is no way of describing a given individual, it can at least be described as undescribable.

Chapter 15 THE LINGUISTIC HIERARCHY AND THE VICIOUS-CIRCLE PRINCIPLE (1954)

Most contemporary logicians seem to be in agreement that Russell’s hierarchy of orders of functions and of orders of propositions, dictated by the vicious-circle principle “whatever is defined in terms of a totality cannot be significantly said to belong to that totality,” is unnecessary. For, so the argument goes, the simple theory of types suﬃces for the solution of the logical paradoxes; the rest of the paradoxes, however, which Russell proposed to solve (or avoid, whichever term be more suitable) at one stroke by his comprehensive vicious-circle principle are semantical paradoxes which can be overcome by due observance of distinctions of levels of language. It is, of course, understandable that logicians would eagerly embrace the Tarski-Ramsey-Carnap method of solving the nonlogical paradoxes (liar paradox, Grelling paradox, Berry’s paradox, etc.), since the vicious-circle principle led to the embarrassing dilemma “either accept the axiom of reducibility or reject large portions of classical mathematics.” It is worth noting, however, that logicians who take seriously the distinction between sentences and propositions, in Carnap’s sense of “designata” of sentences, may nevertheless be faced with nonlogical paradoxes that call for some form of a vicious-circle principle. In order to show this, I shall formulate a paradox which is perfectly analogous to the liar paradox except that it refers to propositions, not to sentences. Instead of assuming that the only sentence inscribed in a given space S is false, and that the foregoing sentence is itself the sentence inscribed in S (or that the only sentence uttered by A within a specified time interval t is false, and the foregoing sentence is itself the sentence uttered by A within t), let us assume that the only proposition believed by A is false, and that the proposition thus described is the very proposition that the only proposition believed by A is false. Denoting the proposition in question by “P,” we can prove that P is both true and false, as follows:

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(1) (∀p)[bel(A, p) ⊃ F(p)] H1 (= P) (2) (∀p)[bel(A, p) ⊃ p = P] H2 (3) bel(A, P) H3 (4) bel(A, P) ⊃ F(P) from H1 (5) F(P) from H3 , 4 (6) (∃p)[bel(A, p). ∼ F(p)] from5, H1 (7) (∃p)[p = P. ∼ F(p)] from 6, H2 (8) ∼ F(P) from 7 The distinction of language-levels is not relevant to this paradox since falsity is predicated of propositions, not of sentences. If the bound variable in “P” were a sentential variable, then “P” would belong to a higher language-level than the sentences which are substitutable for the variable. Step (4) would accordingly be a violation of the prohibition of “semantically closed” languages. But the bound variable is meant as a propositional variable; all the sentences constituting the proof belong to the “non-semiotic” part of the meta-language, in Carnap’s phrase. A “non-semiotic” paradox even more closely related to the liar paradox in its usual semantic formulation can be obtained by substituting for the utterance of a sentence an assertion of a proposition. To be sure, one asserts a proposition by uttering a sentence, but nevertheless “A asserts the proposition that . . . ” is not synonymous with “A utters the sentence ‘. . . ’.” The sentence “A asserts that the only proposition which he asserts within time interval t is false” belongs to the same language-level as whatever sentence expresses the proposition described by “the only proposition which . . . ” However, since the distinction between believing a proposition and uttering a sentence expressing the proposition is even more obvious than the distinction between asserting a proposition and uttering a specific sentence expressing the proposition—if someone is so confused as to suppose that “A asserts that p” entails “A says ‘p’,” he still may not be so confused as to suppose that “A believes that p” entails “A says ‘p”’—I have constructed the paradox above about belief. Now, if we accept the vicious-circle principle, the solution of the paradoxes, which are at once nonlogical (in that nonlogical constants, “believing” and “asserting,” occur in them) and non-semiotic, is obvious: step (4) is a violation of this principle since the proposition P, being of higher order than the values of “p,” lies outside the range of significance of the function “bel(A, p).” This is but another way of saying that according to Russell’s principle hypotheses 1 and 2 are significant only if the quantifier “for every p” is restricted to “for every p of a given order.” However, one might object that it is not necessary to question any deductive step leading to the contradiction “F(P). ∼ F(P).” For, since the hypotheses are all of them contingent propositions, what is to be concluded is simply that they are incompatible, that at least one of them is false. (In Hilbert and Ackermann 1949, 120-21 it is claimed that for just this

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reason the liar paradox is no genuine paradox at all.) While this objection may be valid with respect to the belief-paradox, I think that it is invalid with respect to the assertion-paradox. For, as is pointed out by Lewis and Langford in their discussion of the paradoxes (Lewis and Langford 1951, chapter XIII), it is clearly possible that someone should assert, within t, that the only proposition asserted by him within t is false, and should assert no other proposition within t. Such a possibility can be established empirically by actually uttering a sentence expressing this proposition within t and uttering no other sentence within t. It is true that from “A asserts within t that the only proposition he asserts within t is false” it does not follow that A utters within t the sentence “the only proposition I assert within t is false”; he might utter another sentence which expresses the same proposition. Indeed, the whole distinction between the semiotic and the non-semiotic version of the liar paradox, here emphasized, hinges on this fact. But what is here essential is rather that the converse inference is justified: from the utterance of the sentence we can infer the assertion of the proposition. This inference is perhaps not logically valid, since a sentence which is commonly used to express one proposition may in a given instance be used to express a diﬀerent proposition. But if the evidence of linguistic utterance establishes as much as a probability for the hypothesis that the proposition in question was asserted, that is all that is needed to prove the possibility of the hypothesis. If so, the vicious-circle principle seems to be the only acceptable solution of the paradox after all. It should be noted that the vicious-circle principle, interpreted as a general restriction of the significance-ranges of functions, solves all the paradoxes which can be solved by avoiding “semantically closed” languages, while the converse does not hold, as just illustrated in terms of the non-semiotic paradoxes of belief and of assertion. Thus, consider the much discussed Grelling paradox about the heterologicality of “heterological.” The definition of “het” involves universal quantification over a property-variable: het(‘P’) =d f (∀P)[Des(‘P’, P) ⊃∼ P(‘P’)]. In the derivation of “het(‘het’). ∼ het(‘het’),” the property het is taken as a value of the property-variable “P,” which is a clear violation of the viciouscircle principle. In this case, indeed, the solution in terms of stratification of expressions coincides with the solution in terms of stratification of extralinguistic entities: if it is insignificant to say of a designation of a second-order property, like “het,” that it has the second-order property het, then the predicate “het” cannot have the same meaning in the sentences “het(‘het’)” and “het(‘long’).” In Russell’s terminology, the semantic phrase “het” is systematically ambiguous. But it would be a serious mistake to suppose that, just because the stratification of extralinguistic entities (properties, propositions) entails the systematic ambiguity of logical and semantical constants, there-

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fore what can be accomplished by this “ontological” stratification can always be accomplished by the stratification of expressions as well. For the latter is irrelevant to paradoxes that are derivable within what Carnap calls the “nonsemiotic part” of a meta-language. To be sure, the entire problem which is here submitted for further consideration would disappear if statements of belief and assertion were translatable into extensional statements about expressions. But, as particularly Church has shown (Church 1950), it is highly doubtful whether Carnap’s attempts at such translation have succeeded. Further, for reasons partly similar to Church’s, I think that the absolute concept of truth (truth as predicable of propositions) is not reducible to the semantic concept of truth (truth as predicable of sentences).1 At any rate, the foregoing considerations show that there is a connection between the question whether all nonlogical paradoxes are soluble without the vicious-circle principle and the question whether propositions are reducible to sentences.

1 Cf.

the arguments I present in chapter 6.

V

PHILOSOPHY OF MIND

Chapter 16 OTHER MINDS AND THE PRINCIPLE OF VERIFIABILITY (1951)

1.

The Principle of Verifiability as Generator of Philosophical Theories

The principle of verifiability is that famous criterion of propositional significance whose discussion, I venture to guess, has occupied more space in the publications of contemporary analytic philosophers than discussion of any other topic. It is not the purpose of this discussion to add a little more water to the ocean of literature on the precise meaning or formulation and the general implications of this principle, which is often referred to as the very soul of the movement called variously “logical positivism” and “logical empiricism.” A comprehensive review of this kind has, indeed, been made unnecessary by C. G. Hempel’s excellent contribution to a recent issue of Revue Internationale de Philosophie dedicated to the theme “logical empiricism.” My purpose in this paper is to seize upon fairly clear standard formulations of the principle and, without commitment pro or con the principle itself, scrutinize their implications for one of the basic problems of the philosophy of psychology (or “philosophy of mind,” as I dare unashamedly call it): the analysis of statements about other minds. Specifically, I shall attack the wide-spread view that acceptance of the principle logically compels a physicalistic interpretation of statements about other minds. But before proceeding to this specific task, I shall discuss the concept of a “philosophical theory dictated by the principle of verifiability” in terms of a crucial instance from contemporary analytic philosophy. In this way the reader may be led to appreciate the necessity of coming to grips with this semantic principle if one wishes to discuss intelligently any fundamental issue in philosophical analysis, or at least in that branch of philosophical analysis which occupies itself with empirical discourse in everyday as well as scientific language. By the principle of verifiability will here be understood the liberal version laid down in (Carnap 1937): a sentence is cognitively meaningful if and only

249

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if it is in principle confirmable. The criterion is intended to apply only to those declarative sentences (contrast “declarative” with imperative, interrogative, optative, etc.) which are “putatively factual,” in Ayer’s phrase, i.e., the predicates “confirmable” and “non-confirmable” are so meant that they do not significantly apply to logically decidable (analytic and self-contradictory) sentences. The qualification “cognitively” is used in order to indicate which of the various senses of the ambiguous term “meaningful” is intended as the explicandum: it is the sense of “meaningful” in which “S is meaningful” entails “S is either true or false” or, in the technical terminology of some semanticists, “S expresses a proposition.” Thus it would be an irrelevant test of the verifiability principle if one adduced a sentence heavy with emotive and pictorial meaning and asked rhetorically “well, is it confirmable?” Now, in this formulation the principle purports to tell us whether or not a sentence is cognitively meaningful but it does not tell us how one is to find out what a sentence cognitively means. Nevertheless, once one has assented to the principle one will with little hesitation accept the following rule which may be used to discover what the cognitive meaning of a sentence is: a sentence means the conditions (or states of aﬀairs) whose ascertainment would establish its truth. It is important that “establishing its truth” here be understood in the sense of conclusive verification, not in the sense of (partial) confirmation: the ascertainment of bloodstains on the clothes of a man suspected of murder confirms the proposition “he committed the murder,” but nobody would maintain that this condition is even part of the meaning of the proposition. The distinction between logical entailment and factual implication is of paramount importance in this context. If there is a law by means of which S 1 implies S 2 , we would ordinarily say that the state of aﬀairs described by S 2 is evidence for the truth of S 1 , but no reputable version of the verifiability principle has ever maintained that evidence in this sense has anything to do with the meaning of S 1 . It is only if S 2 is analytically entailed by S 1 , that the state of aﬀairs described by S 2 would be called a truthcondition, in the Wittgensteinian sense, for S 1 and that it, consequently, would be identified with part of the meaning of S 1 . It will be assumed that the reader is familiar with the crucial distinction between theoretical and practical possibility of confirmation (instead of “theoretical confirmability” one also speaks of “confirmability in principle”), and is aware that “confirmable,” in this context, means “theoretically capable of confirmation.” But one further ambiguity should be cleared away before proceeding since it is crucial with regard to the application of the principle to the language of psychology which the ensuing discussion is to focus on. What is meant by an “ascertainable (verifiable) condition”? What constitutes an admissible method of verification? There have been and are opponents of logical positivism (e.g. A. C. Ewing) who gratuitously take the positivists to identify verification with sense-perception and hence arrive at the conclusion that for the positivist sentences which can be conclusively established by introspection only, such as “I remember to have had scrambled eggs for breakfast,” are

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cognitively meaningless—or at least are so unless translated into physicalistic language. But if logical positivism is defined by the liberal principle of verifiability as stated, it is neutral with respect to the issue between introspectionist psychology and methodological behaviorism: introspection is a proper method of verification. As a matter of fact, a positivist might make the following use of his semantic principle, which would show clearly that he does not intend to restrict the term “verification” to the procedure of establishing a statement by sense-perception. A so-called ethical intuitionist claims that an ethical sentence of the form “x is wrong” is either true or false, yet not verifiable or refutable by sense-perception: wrongness, he would say, is an objective property of certain actions, but it is “non-natural,” inaccessible to sense-perception. He likewise disclaims that ethical sentences are psychological. The positivist asks: what experience, then, constitutes verification of such a judgment? Answer: a feeling of disapproval in the mind passing the judgment. Conclusion forced by the principle of verifiability: the ethical sentence means either (a) the speaker disapproves of x, or (b) any person meeting specified requirements (e.g., enlightened about the probable consequences of such an act) would feel that way about x. According to translation (a) the self-proclaimed “intuitionist” is reduced to a disreputable “subjectivist” (according to the somewhat undiscriminating terminology of Ewing, in Ewing 1949); according to (b) he is an “objectivist,” since wrongness is analyzed as a dispositional property of actions whose presence is independent of the actual occurrence of feelings of disapproval (if nobody were properly enlightened about the act, everybody might approve of it even though everybody would disapprove of it were he properly enlightened). But according to either analysis “x is wrong” would be an empirical statement, contrary to the intuitionist claim. The fact that the principle of verifiability can, and has been, used that way shows that the accusation of a “narrow” identification of verification with sense-perception is unfounded: a feeling of disapproval cannot be known to occur through sight, touch, smell etc., but only through introspection. Phenomenalism, as a theory of the meaning of statements about physical objects, is an excellent illustration of a philosophical theory closely connected with the principle of verifiability. What do we mean by a statement like (p) “there is a table here-now”? The principle of verifiability gives us the following direction: determine the verifiable consequences of this statement: the state of aﬀairs expressed by the totality of these verifiable (logical) consequences is what the statement means, no more and no less. But how should we determine the consequences? Is (q) “somebody here-now has a sense-impression of a table”1 for example, a logical consequence of the statement? It cannot be denied that verification of (q) is relevant to the truth or falsity of (p), but 1 This

is intended as a statement in the so-called sense-data language. “Sense-impression of a table,” therefore, should be replaced by a complex phrase containing the names of the kind of sense-data normally produced by tables.

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before we would be justified in saying that q is part of the meaning of p we would have to show that p logically entails q. As a matter of fact, it does not, since it is not self-contradictory to suppose that p is true and q is false. It must be admitted, I think, that this decision, viz. that a certain conjunction of statements (p and not-q) is not self-contradictory, is arrived at by philosophers quite independently of the principle of verifiability. Indeed, this seems to be demonstrably the case, since there are both positivists and non-positivists who hold that p does not entail q and that Berkeley’s “esse est percipi,” therefore, is a sheer semantic mistake. The claim, then, that the principle of verifiability functions as a guide to the meaning of a statement must be taken with a grain of salt. Before I can decide whether a verifiable consequence of p forms part of the meaning of p, or is only indirect evidence for p (in the sense of “indirect evidence” in which the bloodstains, in my previous example, were indirect evidence for the statement “he committed the murder”), I must know whether it is a logical consequence or follows from p only with the help of additional factual premises. And to this question the principle of verifiability seems to be irrelevant. Thus all philosophers except confused disciples of Berkeley will agree that in order to deduce q from p we require the further premise (r) “somebody is here-now, fulfilling such and such conditions (suitably placed, open eyes, perceptually normal etc.).” The only logical consequence of p, then, is the conditional statement joining r as antecedent to q as consequent. Since we are here discussing phenomenalism only by way of illustration, we may pardonably disregard all the subtleties involved in an adequate formulation of this theory, and roughly define it as the claim that p means no more and no less than “if r, then q.” The question before us is: in which sense can it be said that phenomenalism is dictated by the principle of verifiability? We have already seen that this principle does not tell us how to find out what a statement does or does not logically entail and thus what it does or does not (cognitively) mean. The instruction “find the conditions whose ascertainment would establish the truth of the statement” is not helpful, since “ascertainment of C would establish the truth of p” means “the statement expressing C is logically equivalent to p,” and we have not been provided with a method for determining what the logical consequences of a statement are.2 Positivists and nonpositivists alike decide such questions, in their analytic practices, intuitively by asking themselves whether a given conjunction of statements (like “there is a table here-now but it is not perceived by anybody”) sounds self-contradictory. The principle of verifiability exercises only a negative function, in this sense: if a set of statements q1 , . . . , qn is admitted to exhaust the verifiable conse-

2 It

should be kept in mind throughout this discussion that the relevant concept of “logical consequence” is the concept applied to natural languages, not the formal concept applied to deductive systems. (See on this point Pap 1949d.)

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quences of a statement p, and the consequences are held to be logical, i.e., to follow from p alone, without the aid of additional premises, then the principle says: p means what this set of statements means, and no more. In other words, if p is a putatively factual statement, it is held to be synonymous with the total set of its verifiable logical consequences. It follows that if a putatively factual statement has no verifiable consequences, it has no cognitive meaning.3 Now, let us, omitting niceties which certainly ought to be discussed if this were a paper on phenomenalism, define phenomenalism as the thesis that a statement like p is synonymous with a finite set of sense-data statements like “if r, then q.” The correctness of this thesis cannot, for the reasons stated, be deduced from the principle of verifiability. What can, however, be deduced from the latter is that if that set of sense-data statements constitutes the totality of verifiable logical consequences of a statement about physical objects, then phenomenalism is correct. A philosopher could consistently accept the principle of verifiability and reject phenomenalism on the ground that none of those sense-data statements is logically entailed by a physical statement. On the other hand, if he could be brought to admit that if physical statements have any verifiable logical consequences, then these are the mentioned sensedata statements, the verifiability principle would, indeed, force him into the dilemma of either accepting phenomenalism, which he finds intuitively an unsatisfactory meaning analysis, or confessing that he has no way of intelligibly communicating what according to him physical statements mean (I think the latter alternative adequately characterizes the position of many opponents of phenomenalism). But actually he need not make the admission on which this dilemma is contingent: he may simply say that the only logical consequences of physical statements are, not pure sense-data statements, but statements containing likewise names of physical objects and events. The main point of this preliminary discussion was to show that the principle of verifiability by itself can never dictate a specific analysis of a statement or class of statements, but only relatively to the assumption that if the statement or class of statements to be analyzed have a verifiable content at all, they must be synonymous with the analysans proposed. Once this assumption is granted, the principle of verifiability leads to the conclusion that the proposed analysis is a correct analysis of the only cognitive meaning the analysandum has. It will be shown now that one may consistently accept the principle of verifiability and

3 If

the requirement were stated simply in terms of “verifiable consequences” instead of “verifiable logical consequences,” it would be demonstrably so liberal as to be wholly ineﬃcient in the exclusion of nonsense. On the other hand, it has been argued that if the stronger requirement is adopted, then abstract scientific theories would be condemned as meaningless since no testable consequences are deducible from them without the use of additional premises which connect the abstract terms of the theory with observables. The diﬃculty can perhaps be solved by acknowledging reduction sentences, along with explicit definitions, as semantic rules, but a thorough investigation of this problem is beyond the scope of this discussion.

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repudiate physicalism or logical behaviorism, since the assumption, explicitly or implicitly made by all logical behaviorists, that behavioristic translations alone give statements about other minds a verifiable content, is false.

2.

The Behaviorist’s Confusion about the Notion of Verifiability

This section will be opened by a representative quotation from an early defense of physicalism by Carnap (Carnap 1932a). It is only fair to Carnap, however, to remind the reader that the physicalistic thesis maintained by Carnap and his followers at the present time diﬀers from the earlier thesis in a very important respect: the claim that psychological sentences are synonymous with physicalistic sentences, however complex, has been abandoned, and all that is asserted is reducibility of psychological terms to physicalistic terms (“physicalistic” is used broadly to cover both physiological and behavioristic terms). Whether Carnap, after having modified the physicalistic thesis in this fashion, would still agree with his former argument to be quoted, I do not know. I shall argue that if he does, then he is inconsistent with his own confirmability criterion of meaning; and if he does not, then physicalism as asserted by him and his school is not subject to philosophical debate, is neutral with respect to any “theory of mind,” since it is merely a statement analytic of the meaning of “intersubjective language” (see section 4). Arguments from analogy are, indeed, not demonstrative, but they are legitimate as probable arguments. Let us consider an example of an argument from analogy from everyday life. I see a box of definite shape, size, color; I notice that it contains steel-point pens. I find another box of similar appearance; I infer by analogy the probable conclusion that it likewise contains steel-point pens. The objector means that his inference to the other mind has the same logical form. If this were the case, his inference would indeed be justified. But this is not the case; here the conclusion is meaningless, a mere pseudo-statement. For, as a statement about another mind which is not to be interpreted physicalistically, it is in principle untestable. (My translation.)

This quotation shows clearly that the principle of verifiability led to the physicalist analysis of statements about other minds, because the assumption was made, to quote my own statement, “that if the statement or class of statements to be analyzed have a verifiable content at all, they must be synonymous with the analysans proposed.” But it is easy to show that this assumption is erroneous, that the verifiability of statements about other minds, in the relevant sense of “verifiable,” does not require their translatability into physicalistic language. This relevant sense of verifiable is, of course, “in principle capable of confirmation.” In the early days of the history of logical positivism, the principle of verifiability had, indeed, the “intolerant” form: a sentence is cognitively meaningful (i.e., a statement, or proposition) if and only if it is in

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principle capable of conclusive verification. I shall argue later that even if this strong criterion of propositional significance were adopted—and it has largely been abandoned since (Carnap 1937)—it would not follow that the cognitive significance of statements about other minds could be rescued only by logical behaviorism. For the moment I shall simply demonstrate that statements about other minds are, in the “mentalistic” sense intended by old-fashioned, unabashed psychophysical dualists (and those conservative people you can meet in the street even though you will shock them if you call them by that name), confirmable; and that, since the legitimacy of an argument from analogy only requires the confirmability of its conclusion, there is nothing wrong, from the point of view of the present-day positivist theory of meaning, with the traditional argument from analogy as a justification of beliefs about other minds. Let m2 be a mental state of mind M2 which I infer from observation of a physical state b2 of the body B2 , where B2 is the body of the person with the mind M2 . An inductive justification of this inference requires, of course, a psycho-physical law (= any law correlating mental and bodily or behavioral states) for which there is confirming evidence. The probability of the occurrence of m2 relatively to the evidence that b2 occurred is then, roughly speaking, proportional to the probability of the law asserting that an event of the same kind as m2 is either a suﬃcient or a necessary condition for events of the same kind as b2 . Let this psycho-physical law, formulated for M2 and B2 , be denoted by “m2 —b2 ” and the corresponding law, formulated for my own mind M1 and own body B1 , by “m1 —b1 .” Clearly, the only way the logical behaviorist could deny that the inference to m2 , which we may suppose to be drawn by M1 , is capable of inductive justification, would be by claiming that the relevant psycho-physical law which would have to enter into the argument is incapable of confirmation. Superficially, it looks as though his case could be constructed as follows: the only psycho-physical law for which M1 can obtain a confirming instance is m1 —b1 ; the law required as a premise, however, is m2 —b2 , and to suppose that M1 could verify a conjunction of singular statements that would count as a confirming instance of m2 —b2 (like “he is afraid now and his hands tremble now”) is to suppose that a person could be directly aware of a mental state not his own, which is a contradiction in terms. This whole argument, however, breaks down once we make a distinction between direct and indirect confirming evidence for a law. Consider the special case of the general gas law which we obtain by replacing “for any ideal gas” with the more restricted quantifier “for any instance of nitrogen gas under ideal conditions.” Suppose that we had never experimented with nitrogen gas, but nevertheless predicted, from the analogy of experiments with other gases, that the volume of a mass of nitrogen gas kept at constant temperature would be halved if the pressure upon it were doubled. Would anybody deny the induc-

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tive justifiability of this prediction on the ground that we had not, as yet, any confirming evidence for the law “if kept at constant temperature, nitrogen gas behaves in accordance with the formula PV = k”? Rather elementary rules, whether explicit or implicit, of inductive logic would lead one to say that there is indirect confirming evidence for this special law, inasmuch as the latter is a deductive consequence of a universal law for which there is confirming evidence (let us assume that the universal gas law was obtained as an induction from experiments with a series of gases excluding nitrogen). In just the same way the universal psycho-physical law “whenever a person satisfying such and such conditions is in a state of anxiety, his hands tremble, his pulse accelerates etc.” is obtained as an induction (how reliable such an induction may be is an important problem of the methodology of psychology, but irrelevant to the logical issue under debate) from the special case m1 —b1 , for which M1 has direct confirming evidence, and since m2 —b2 is a deductive consequence of the universal psycho-physical law, it is indirectly confirmed by the very evidence which directly confirms m1 —b1 . Surely the following rules of inductive logic, invoked in my defense of the psycho-physical use of the argument from analogy, could not be questioned: if e confirms p, and p entails q, then e confirms q; if p entails q and e confirms q, then e confirms p. These rules are part and parcel of accredited scientific procedure.4 In case the reader should mistrust my argument on account of my speaking of “minds,” a terminology that reeks of disreputable metaphysics, it may be well to add that the argument does not depend on any commitment to any “theory of mind.” The argument would not be aﬀected, for example, by a materialistic translation of “mental event m1 is owned by mind M1 ” into “m1 is directly caused by an event in the cerebro-neural system of body B1 .” The very possibility of such a materialistic translation disposes of an argument against the psycho-physical use of the argument from analogy, put forth by Carnap in the same article, which unlike the argument already quoted is independent of 4 It

cannot be denied that these rules of inductive logic would require considerable refinement before being acceptable as valid in general. Specifically the second rule leads to a paradoxical consequence in case the relation of p to q is that of a conjunction of unrelated conjuncts to one of its conjuncts: e confirms B, (A and B) entails B, therefore e confirms (A and B); actually e may be wholly irrelevant to A, in which case one would not admit that the conjunction had been confirmed. But the case we are considering is diﬀerent: the universal generalization can, indeed, be interpreted as the conjunction of the sub-generalizations referring to the sub-classes of the whole class, but then the conjuncts have predicates in common and are thus not wholly unrelated. In any case, the only rule needed to justify the above argument for the indirect confirmability of propositions about other minds is this: let A1 and A2 be subclasses of A, where A is a suﬃciently proximate genus with respect to these sub-classes (e.g., I would say that the genus “organisms” is not suﬃciently proximate to the sub-class “negroes” to warrant analogical inference from negroes to, say, oak trees); and let e confirm a generalization about A1 ; then e also confirms the corresponding generalization about A2 . To the extent that the expression “suﬃciently proximate genus” is vague, the rule itself is, indeed, vague. But if this vagueness be held as a reason against the use of the rule for purposes of inductive justification, purely physical arguments from analogy will likewise suﬀer, and it could not be said that the psycho-physical use of this type of argument presents a unique diﬃculty.

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the principle of verifiability. It is worth quoting in full (my translation from the original German): The predicative linguistic form ‘I am angry’ does not adequately express the intended state of aﬀairs. It expresses that a given object has a given property. But all that is given is an experienced feeling of anger. A correct linguistic formulation of this feeling would be, say, ‘now anger’. But once this correct formulation is used, the very possibility of an argument from analogy vanishes. For then the premises become: whenever I, i.e. my body, exhibit symptoms of anger, then anger occurs; the other person’s body resembles my body in many respects; the other person’s body now exhibits symptoms of anger. Now the conclusion cannot be formulated any more, since the sentence ‘now (or: then) anger’ contains no more ‘I’ that could be replaced by ‘the other’. If, on the other hand, one wanted to formulate the conclusion by refusing to make a replacement and simply retaining the form of the premise, the resulting conclusion would indeed be meaningful, yet evidently false: ‘therefore now anger’; for this means, put in ordinary language: ‘I am now angry’. (Carnap 1932a, 120)

This argument is vitiated by a curious inconsistency. Carnap begins by claiming that “I am angry” can mean only “anger occurs now” (Russell makes a similar claim in criticizing Descartes for starting from the premise “I think” as though it expressed an experiential datum and nothing more). This claim is obviously untenable since it leads to the consequence that “I am angry” as spoken by myself and “I am angry” as asserted by another person are synonymous statements! But even if Carnap’s initial claim were valid, the conclusion of his argument contradicts that very claim: if Carnap says that the conclusion “therefore now anger” is evidently false, he evidently presupposes that “I am angry” and “you are angry” are not synonymous. It is surely conceivable that a feeling of anger did occur at the time referred to by the conclusion of the analogical argument, hence if Carnap is sure of the falsehood of the conclusion it must be because he clearly distinguishes the sense of “I am angry” from the sense of “you are angry.” And, indeed, this diﬀerence of meaning can be recognized pre-analytically, i.e., independently of any analysis of such sentences. Carnap might reply that the only way the diﬀerence of meaning between “I am angry” and “you are angry” could be analyzed on empiricist ground, i.e., without postulating metaphysical substances (Christian “souls,” Cartesian “res cogitantes”), would be by translating them into physicalistic sentences about diﬀerent bodies. But apart from the circumstance that such translations would obviously be incorrect since I can know that I am angry while being blissfully ignorant of the bodily symptoms by which my feeling is manifested, the reply may be countered by indicating two (vastly diﬀerent) methods of translation which should be equally acceptable to an empiricist as conforming to the maxim “refrain as much as possible from postulating inferred entities.” The first is none other than the materialist analysis mentioned above: “I am angry” (as spoken by body B1 ) = “anger occurs, and this mental event is directly caused by a brain-event in B1 .” The diﬀerence between the

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singular premise and the singular conclusion of the argument from analogy is then perfectly clear; but the translation is not physicalistic since the psychological term “anger” reappears in the analysans without physicalistic interpretation. The essential point is that, as illustrated by this analysis, it is possible to distinguish the sense of the predication “I am angry” from the sense of the existential statement “anger occurs now” without postulating a metaphysical “self” objectionable to empiricists. But actually this materialist analysis is no more acceptable an account of what is meant by “I am angry” than a physicalist analysis which pretends to do away with “mental” events in the sense of events inaccessible to public observation, for this obvious reason: if it is conceivable that I should know about my anger without knowing about the bodily symptoms of my anger, it is still more easily conceivable that I should know about my anger without knowing anything about the connection of feelings with brain-events. And in fact it is not a priori impossible that my feelings should be directly caused by events in another body than the body whose back I cannot see with my unaided eyes. For this reason what has come to be known as the “phenomenalist” or “logical construction” theory of the self, which is equally inspired by the desire to do without metaphysical substances and whose historical ancestor is Hume, is far more plausible. According to this analysis “I am angry now” is a statement of class-membership: the present mental event is asserted to be a member of the class of mental events which is what the pronoun “I” refers to. Just what the relation between two successive mental events (each element of the temporal series, the mental history, may itself be a complex event, just like instantaneous states of a physical system described in terms of several state variables) is by virtue of which they are members of the same series, it is not easy to tell. If we are the kind of philosophers who rest satisfied with analogies, we can compare successive states of a mind to points on a line, the mind to the line, and then say that just as two points belong to the same line if their coordinates satisfy the same equation, so two mental events belong to the same mind if their descriptions satisfy the same “equation.” It is, of course, highly obscure what could be meant by the “equation” defining a mental history, and moreover this analogy leads to the questionable consequence that a mental event could be a state of several minds, a point of intersection of mental histories, as it were. A detailed discussion of the logical construction theory of the self, which Hume adumbrated, would not be to the present purpose. My purpose was merely to show that Carnap was wrong in suggesting that if “I am angry now” is to be considered as a phenomenological statement free from metaphysical commitments it must be translated into the impersonal form “anger occurs now.” Nevertheless I might briefly indicate, before leaving the topic, that the phenomenalist theory of the self can do better than suggest obscure analogies, such as the equation of a line in analytic geometry. One can at least state,

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in literal language, a straightforward suﬃcient condition for the truth of the proposition “x and y are events owned by the same mind (or ‘states’ of the same mind)”: this will be the case if x is the event of remembering y or y the event of remembering x.5 What makes the problem diﬃcult, however, is the fact that this is not a necessary condition; it would normally be held that there are innumerable past experiences of mine which I do not now remember, which statement would be a contradiction if the above criterion were an analysis of the notion of mind-ownership.

3.

Are Statements about Other Minds Conclusively Verifiable?

There is a school among present-day analytic philosophers (consisting mainly of followers of Wittgenstein) according to which it does not make sense to speak of increasing confirmation of a statement if the limit approached, but never reached, by this process is so conceived as to be in principle beyond attainment. They would therefore argue that as long as it is conceded to be logically impossible to verify statements about other minds conclusively, it does not help to prove their confirmability, and we will still have to interpret them physicalistically if we wish to rescue their cognitive significance. Now, in the first place, this semantic principle, that it does not make sense to speak of increasing approach to a limit if it is logically impossible to reach the limit, is highly dubious. The statement, e.g., that certainty, in the sense of maximum probability (probability equal to unity), is the limit approached by the probability of a generalization as the number of confirming tests of the generalization increases without limit (which implies that certainty would be reached only after an infinite number of confirming tests had been completed), is perfectly analogous to limit statements in mathematics and mathematical physics, and it would be interesting to see how the Wittgensteinians would prove their meaninglessness. Second, this principle is radically inconsistent with the central idea of Carnap 1937, which is to construct a meaning criterion that is compatible with the significance of unrestricted generalizations, where unrestricted generalizations are statements that are confirmable, yet in principle incapable of conclusive or complete verification. Third, it is hard to see how the validity of this semantic principle, if it were valid, could be a good reason for adopting a physicalistic interpretation of statements about other minds: for few logical behaviorists would be prepared to say that the meaning of such a statement would be exhausted by a finite set of physicalistic sentences, and if the set is infinite,

5 The

above presupposes that remembering is never directly a relation to a physical object or event even though our language frequently suggests this: we say “I remember that house” but we would accept “I remember to have seen that house” as a translation of the intended meaning.

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then adoption of that very semantic principle would lead to the conclusion that the statement had not been made any more significant by the translation.6 But fourth, the questions just discussed are, within the context of our problem, purely academic. They would be to the point only if statements about other minds, as intended by common sense, were in fact beyond the logical possibility of conclusive verification. And I proceed to give a simple proof of their conclusive verifiability. If Mr. A says with regard to himself “I am angry,” he is, as will readily be conceded, making a statement which is conclusively verifiable, by introspection. Suppose, now, that Mr. B says to Mr. A “you are angry.” Here we have a statement about another mind which is said to be in principle unverifiable if it is intended as a non-physicalistic statement, like Mr. A’s statement “I am angry.” But this position is, as it stands, glaringly inconsistent since the two statements which diﬀer only in the personal pronoun are strictly synonymous. They are synonymous by virtue of asserting the same fact, in the same way in which “it is raining today” and “it rained yesterday,” spoken a day later than the first sentence, assert the same fact. We must not confuse, of course, propositions that are inferable from the occurrence of a given token of such a statement with propositions that are inferable from the meaning of the statement. From the statement “a token of the sentence-type ‘you are angry’ was spoken” we can infer with some probability that the speaker of the token referred to a person other than himself, and from the statement “a token of the sentence-type ‘I am angry’ was spoken” we can infer with some probability that the speaker of the token referred to himself. Which proves that these statements, describing linguistic events, have diﬀerent meanings, but not that the statements which they are about have diﬀerent meanings. Indeed, if the claim that “you are angry” means observable behavior and physiological states were taken seriously, it would follow that when asked “are you angry?” and then answering the question after a moment’s introspection, the answer we give is not an answer to the question we were asked. This proof is so simple that it may easily provoke an accusation of simplemindedness: “You have naively overlooked the crucial distinction between ‘verifiability by me’ and ‘verifiability by anybody (public verifiability)’. What inspired logical hehaviorism was the desire to eliminate from psychology statements which are not publicly verifiable.” The criterion of public verifiability here proposed evidently comes to this: a statement is cognitively meaningful if and only if it is in principle possible for anybody to verify it conclusively. The task to be confronted, then, is to prove that statements about other minds, as intended by common sense, are even publicly verifiable in this sense. I shall

6 Even

if one could argue in favor of translatability into a finite conjunction of physicalistic test-sentences, some of these test-sentences would undoubtedly be themselves unrestricted generalizations, e.g., “if anybody had insulted John at that time, John would have become violent.”

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follow more or less old lines of argument used by Ryle and Ayer7 to the effect that what holds with respect to public verifiability for statements about the past holds also for statements about other minds. The strategy, as it were, of the argument will be this: if statements about the past are conceded to be publicly verifiable, statements about other minds will have to be conceded to be publicly verifiable in the same sense; if statements about the past are held to be publicly unverifiable, the same will have to be said of statements about other minds; but, on this alternative the version of the criterion of verifiability under discussion will simply have to be repudiated, since nobody in his senses would want to lay down a law which leads to the elimination of history along with the elimination of metaphysics; therefore, whichever alternative be true (whether statements about other minds are publicly verifiable or publicly unverifiable), no acceptable criterion of significance can make logical behaviorism inevitable. Consider the statement “exactly 50,000 years ago, it rained where I am now standing.” Is it logically possible that I should have conclusive evidence for this statement? The question reduces to the question: is it logically possible that I should have been alive 50,000 years ago? Now, the statement “I was alive 50,000 years ago” would be self-contradictory only if “I,” in this sentence, were a synonym for the description “the person who was born x years ago” (where x, of course, is less than 50,000) or for a description which entailed information about the person’s birth date. But this could be the case only if a sentence like “I was born 30 years ago” were analytic. But surely such a sentence conveys factual information. One cannot establish its truth by analyzing the meaning of “I,” for “I” is an indicator term like “now,” “this,” that has no intensional (connotative, conceptual) meaning to be analyzed. In this respect a statement of the form “I have property P” is like a statement of the form “this object has property P.”8 In order to get an analytic statement out of the latter statement-form, one would have to substitute for “P” the trivial predicate “being the object referred to by this sentence” (and then one obtains a self-referential sentence which some semanticists would rule out as meaningless). Similarly, to get an analytic sentence out of “I have property P” one would have to substitute for “P” a trivial predicate like “being the per-

7 See

Ryle 1936 and Ayer 1947, chapter III, section 15. first glance this statement may seem to conflict with the possibility of constructing a theory of the self, where “theory of the self” is interpreted, in the formal mode of speech, as a rule for translating sentences of the form “I am in state Q” into sentences not containing the pronoun “I.” But actually there is here no more conflict than there is between the claim that “this” has no connotation and the claim that sentences of the form “this physical object has quality Q” could be analyzed. What is analyzed, in the latter case, is the statement-form “physical object O has quality Q,” and here the analysanda are stated without indicator terms (synonyms for “indicator term” in current literature are “egocentric particulars,” used by Russell, and “token-reflexive terms,” used by Reichenbach). 8 At

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son referred to by this sentence.” But we fortunately need not prove any such general proposition as that all predicate-substitutions of a certain kind in the statement-form “I have property P” yield factual statements. All we need is the admission that a statement of the form “I was born at time t” (where t is a time preceding the time of utterance of the sentence) is synthetic. Then it follows that it is not self-contradictory to suppose that I should have witnessed, and thus conclusively verified, an event which occurred before my birth. At first sight it seems as if the verifiability-by-me of a statement about your mental state constituted a radically diﬀerent problem from the verifiabilityby-me of a statement about an event that preceded my birth. For is it not self-contradictory to suppose that I could be you; and is this not the only condition that would enable me to obtain the required conclusive evidence? Ayer’s answer to this question is worth quoting at some length: although it is a necessary fact that the series of experiences that constitutes my history does not in any way overlap with the series of experiences that constitutes the history of any other person, inasmuch as we do not at present choose to attach any meaning to statements that would imply the intersection of such series, nevertheless, with regard to any given experience, it is a contingent fact that it belongs to one series rather than another. And for this reason I have no diﬃculty in conceiving that there may be experiences which are not related to my experiences in the ways that would be required to constitute them elements in my empirical history, but are related in similar ways to one another . . . This does not mean that any experience can actually be both mine and someone else’s; for I have shown that that possibility is ruled out by the conventions of our language. It means only that with regard to any experience that is in fact the experience of a person other than myself, it is conceivable that it should have been not his but mine. (Ayer 1947, 168-169)

The discussion of this subtle matter may be simplified by the introduction of a few symbols. Let ‘M2 ’ refer to your mind, which according to the logical construction theory of the self, evidently adopted by Ayer, is a series of “experiences” to be denoted by ‘(m2 , ma2 , mb2 , . . . , mn2 )’; and similarly ‘M1 ’ refers to my mind, and is short for the series-symbol ‘(m1 , ma1 , mb1 , . . . , mn1 )’. Ayer interprets “you are in mental state mi2 ” to mean that mi2 belongs to the series M2 by virtue of standing in certain relations to the other elements of that series which it does not bear to any elements of the series M1 . He does not, indeed, tell us what those relations are in terms of which minds or “mental histories” are to be logically constructed, but it is not necessary to work the logical construction theory of the self out in detail in order to deal with the issue at hand. It is a contingent fact, says Ayer, that mi2 should bear these relations to the other elements of M2 rather than to the elements of M1 (presumably in the same sense in which it is a contingent fact that a stone which belongs to a given heap of stones belongs to that heap and not to another heap). And if it had in fact stood in these relations to the elements of M1 , then this very fact, expressed in our

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symbolism by “mi1 occurred,” would have constituted conclusive verification of the statement ascribing this mental state to M2 . I have deliberately restated Ayer’s argument in such a way that the non sequitur, as it were, springs before one’s eyes. If the sentence “mi1 occurred,” which expresses what according to Ayer might conceivably have been the case and would, if it had been the case, have constituted conclusive verification of the statement about the other mind, is translated back into non-symbolic English it reads: “I was in this mental state,” and is thus not a statement about the other mind at all. It remains true, therefore, that to suppose conclusive verification, by the speaker, of a statement about another mind is to suppose the contradiction that a statement be verified which is at once about the speaker’s mind and not about the speaker’s mind. But while Ayer’s argument is invalid, it nonetheless makes the point it was intended to make, viz. that statements about other minds are, with respect to conclusive verifiability by the speaker, in exactly the same boat with statements about the past. What spoils Ayer’s argument is that if the speaker had experienced, say, the toothache which he predicated of the other mind, and at that moment had been asked “which statement is conclusively verified by your present experience?,” he would have replied “the statement ‘I have a toothache now’,” not “the statement ‘he has a toothache now’.” But in just the same way, if I had been standing 50,000 years ago where I am standing now and had observed that it rained, the proposition verified by my observation would then have been formulated by the sentence “It is raining here now,” not by the sentence “It rained here 50,000 years ago.” It could therefore be argued that once a statement about the past was conclusively verified by the speaker it would cease to be a statement about the past. Should we say, then, that public verification, i.e., verification by any observer of any age, including the speaker, of statements about the past is logically impossible? I submit that it is logically arbitrary which way we answer this question. The main point is that, whichever answer we decide upon, we should give the corresponding answer to the corresponding question concerning the public verifiability of statements about mental events not owned by the speaker. The remainder of my argument has already been stated in my “strategic outline” above. Before leaving this question of the logical possibility of direct verification, by the speaker, of a statement about another mind, something may profitably be said about the irrelevance of telepathy to this issue. If it could be shown that instances of telepathy are instances of direct knowledge of other people’s mental states, the possibility of such direct knowledge would of course be established empirically. To take a typical case of alleged telepathic cognition, let A be the telepathic knower, B a person who at this moment thinks of a given card drawn at random from a shuﬄed pack of cards, screened oﬀ from A, and suppose that the frequency of correct answers given by A as to the card B is

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thinking of, is significantly above chance expectation. Suppose we interpret this situation by saying A is directly aware of B’s mental state since his belief evidently could not have been arrived at by inference from observations of B’s body and behavior. What could be meant here by “direct awareness”? If it means that A’s belief about B’s mind is not the result of inductive inference, this fact is irrelevant to this question of the possibility of direct knowledge: A does not have direct knowledge of B’s act of attention in the sense in which B has such direct knowledge. One could, indeed, define a sense of “direct knowledge” in which it would be true to say that A had direct knowledge of B’s mental state: “I directly know that p” = “I believe that p without the use of induction, and p is true.” But this is hardly a relevant sense of “direct knowledge” since in this sense there is no doubt that even direct knowledge of propositions about the past and of unrestricted generalizations is possible: it is, of course, possible that utterly groundless beliefs about the past, about generalizations, about other minds, should be true. The only relevant sense, then, in which it could be said that A has direct knowledge about B’s mind would be: A believes without the use of induction that B’s mind is in this state, his belief is true, and A has grounds for his belief. But once the direct-knowledgeclaim is analyzed in this way, it turns into a self-contradiction! What grounds could A have for his belief if not inductive grounds! The only alternative interpretation of “direct awareness” that remains is: A is directly aware of B’s mental state in the sense in which B is directly aware of it, i.e., introspectively. Evidently this interpretation implies that telepathic cognition is an “intersection of mental histories” since A is thus asserted to own the numerically same mental event which is owned by B’s mind. As Ayer points out, in the quoted passage, such a statement oﬀends against our present linguistic rules according to which the expression “two people owning an identical experience” has no significant usage. It may be added that this interpretation presupposes that at least the mental state of the telepathic cognizer, when telepathic communication occurs, is of the same kind as the mental state cognized; which is not true in general, since the cognized mental state may, for example, be a state of desire, and if the knower at the moment of telepathic cognition shared that desire his mental state could not be described as cognitive at all. If such a coincidence of desires in two minds occurred, and did not admit of ordinary psychological explanation, one might look for a special kind of causal explanation, but one would not describe the coincidence as a case of cognition at all.

4.

Physicalism as an Analytic Thesis

So far we have been concerned exclusively to show that no plausible version of the principle of verifiability could be said to entail physicalism or logical behaviorism, meant as the thesis of the synonymity of psychological sentences

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with sentences mentioning only physical events or states. As was mentioned, however, since the recognition of reduction sentences as indispensable tools of meaning specification, this earlier thesis has been abandoned anyway, and present-day physicalists merely assert the reducibility of psychological terms to physical terms (specifically to terms of the “thing-language,” which is the non-technical part of the physical language chosen as a common reduction basis for the language of psychology and the language of physics).9 The point to be argued in this concluding section is that physicalism in the specified sense is no “theory of mind” at all; that it provides no answer to the question what is meant by “mind” and “mental event”; that it is merely an analytic consequence of what is understood by “intersubjective language”; that it, therefore, cannot be in conflict with any philosophical theories concerning the relation of mind to body; that, in particular, it would be a gross misunderstanding to suppose that physicalism is somehow materialism in the semantic mode of speech and a refutation of psycho-physical dualism. As a preamble to the proof of these assertions, let us again quote Carnap, this time from a later publication with which he is probably still in substantial agreement: We do not at all enter a discussion about the question whether or not there are kinds of events which can never have any behavioristic symptoms, and hence are knowable only by introspection. We have to do with psychological terms, not with kinds of events. For any such term, say, ‘Q’, the psychological language contains a statement form applying that term, e.g., ‘The person . . . is at the time . . . in the state “Q”.’ Then the utterance by speaking or writing of the statement ‘I am now (or: I was yesterday) in the state “Q,”’ is (under suitable circumstances, e.g., as to reliability, etc.) an observable symptom for the state Q. Hence there cannot be a term in the psychological language, taken as an intersubjective language for mutual communication, which designates a kind of state or event without any behavioristic symptom. Therefore, there is a behavioristic method of determination for any term of the psychological language. Hence every such term is reducible to those of the thing-language. (Carnap 1950b)

Carnap here explicitly leaves it an open question whether there might not occur mental events without any behavioristic symptoms, such as, I suppose, thoughts and wishes of the deceased which are not communicable to the survivors, except perhaps in a s´eance. Notice that this admission already involves a repudiation of the sort of logical behaviorism which Gilbert Ryle, in his provocative book The Concept of Mind (Ryle 1949a), plays up as an eﬀec9 Reducibility

of terms, it should be noted, does not insure reducibility of laws. Even if the terms of science S 1 were explicitly definable with the help of terms of science S 2 , it would not follow that the laws of S 1 are deducible from the laws of S 2 . For example, terrestrial mechanics and astronomy operate with the same set of primitive terms, but it does not follow that their respective laws are inter-deducible; as a matter of fact, they would have remained separate sciences if it were not for Newton’s unifying discovery of the law of universal gravitation.

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tive antidote against the dualistic “Cartesian Myth,” and according to which to talk about mental events is to talk about behavioral dispositions if not overt behavior. It is true that the opening sentence of the above quotation could be interpreted as a polite hint that the “question” is regarded by the author, like most philosophical questions formulated in the “material mode of speech,” as a pseudo-question. But this interpretation would be in conflict with Carnap’s explicit statement, made in the same paragraph, that a proposition like “I am angry” is clearly knowable by introspection without observation of behavior. As a matter of fact, I would maintain that the fundamental reason behind the substitution of reduction sentences for explicit definitions as bridges from the psychological language to the physical language is not the reason given by Carnap himself for the use of reduction sentences, but a consideration which brings him into substantial agreement with psycho-physical dualism. If we gave an explicit definition of a dispositional predicate “P,” Carnap argued, of the form (1) “Px ≡ (Ox ⊃ Rx)” (where “O” refers to a test operation and “R” to a test result), we would get the paradoxical result that “P” would be predicable of x provided the test operation is never performed upon x. This paradox derives, of course, from the definition of material implication according to which a material implication is true provided its antecedent is false. This paradox is avoided by using the reduction sentence (2) “Ox ⊃ (Px ≡ Rx)” which, not having the form of an equivalence, does not enable elimination of the predicate “P.” But what if a satisfactory definition of the nomological meaning of “if-then” (Reichenbach’s term for the meaning of this connective in statements of natural laws or causal connections) were at hand which allowed us to restate (1) in terms of nomological implication instead of material implication? Would this destroy the raison d’ˆetre of reduction sentences? It is easily seen that it would still be impossible to construct an adequate explicit definition of psychological sentences in terms of sets of behavioristic test-conditionals (conditional sentences of the form “if stimulus S acts on organism O, then response R occurs”) or any kind of physicalistic sentence at all. For according to the standard method of testing the adequacy of explicative definitions, practiced by analytic philosophers however divergent their theories about philosophical analysis may be, such a definition would imply that at least one physicalistic sentence is logically entailed by a psychological sentence like “I experience now such and such a color image.” But it is evidently a question of fact which behavioral and physiological symptoms are correlated with which mental states of an organism. This is the fundamental reason why reduction sentences, which are sentences with factual content though they have a semantic function,10 have to be employed. If by “psycho-physical dualism” 10 It

is, indeed, a pressing question just what precisely is conveyed by the antithesis factual-analytic, as applied to non-formalized languages, if a reduction sentence is said to be in a sense both factual and a

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is meant the logical (or semantic) thesis that for any mental event m and for any physical event p it is logically possible that m should occur without p and p without m, there is nothing in physicalism to conflict with it. We must now examine the meta-linguistic formulation of physicalism contained in the above quotation from Carnap. Carnap maintains that if “Q” is a psychological term whose meaning is communicable, then there must be at least one reduction sentence connecting the mental state Q with a behavioristic symptom, since “the utterance by speaking or writing of the statement ‘I am now (or: I was yesterday) in the state Q’ is (under suitable circumstances, e.g., as to reliability, etc.) an observable symptom for the state Q.” But it turns out that in order to obtain a plausible reduction sentence of this kind a “suitable circumstance” will have to be specified which makes the reduction circular. The reduction sentence “if x is in state Q at time t, then, if x is asked at t ‘in what state are you now?’, x will (under conditions C1 , . . . , Cn ) reply ‘I am in state Q”’ is halfway plausible only if the set C1 , . . . , Cn contains the condition “x knows that ‘Q’ means Q”! But thus the reduction basis for the term “Q” contains not only a name of the term “Q” but the term “Q” itself, which makes the test-sentence ineﬀective in just the same sense in which a circular definition is ineﬀective as a semantic rule of application. This criticism, however, hits only an unfortunate formulation of the proof of physicalism, not the central idea of the proof. I propose the following alternative formulation. What does it mean to communicate the meaning of a psychological word, say, “happy”? One might think that such communication involves a theoretical diﬃculty which does not confront us if the problem is to communicate the meaning of a physical term, say, “hard.” The latter problem is solved by the method of ostensive definition, of pointing at objects to which the predicate is applicable; but how, so it may be asked rhetorically, can one “point at” a mental state, such as happiness? Little analysis, however, is required to see that essentially one and the same method of conditioning is used in both cases. If I expect that by uttering the word “hard” while you touch hard objects I can get you to understand what I mean by the word, it is because I expect that similar tactile sensations will be produced in you when you touch objects which produce such sensations in me (our psycho-physical argument from analogy again!) and that you will in time associate a memory-image of that kind of sensation with the verbal stimulus. Similarly, in order to communicate what I mean by “happy” I must catch you in, or get you into, a situation in which I have inductive reasons to believe you are in a state of happiness, and produce a corresponding habit of association in you. Now, if there were no reliable correlations between feelings of happiness and behavioral symptoms of partial specification of meaning. In my opinion this is one of the most urgent problems of semantic analysis, stimulated but far from solved by Carnap 1937.

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happiness—if, in other words, observations of a certain kind of behavior would never justify our inference “probably he feels happy now”—we just could not use this method of conditioning in order to communicate our meaning; we would have no way of telling when a feeling of happiness is likely to occur in a child to whom we wish to communicate the meaning of “happy.” It follows that the proposition “there are inductive reasons to believe that ‘Q’ is intersubjectively meaningful”11 entails the proposition “there are inductive reasons to believe that there are reliable behavioral symptoms of the mental state Q.” If there are such behavioral symptoms, their names can be used as a reduction basis for the psychological term “Q.” Hence the proposition “if ‘Q’ belongs to an intersubjective language, then there is a behavioristic reduction basis for ‘Q”’ is purely analytic. And evidently an analytic statement cannot be incompatible with a speculative existential statement like “there are mental states (states for which there are no intersubjectively meaningful designations!) that are entirely uncorrelated with bodily states.” I conclude, with my eye on those who feel that logical positivism, if true, shatters their fondest hopes, that the postulate of the reducibility of intersubjectively meaningful terms to the prosaic terms of the “thing-language,” like the postulate of verifiability, is completely neutral with respect to the traditional speculative question concerning the causal possibility of disembodied minds.

11 To

be quite accurate, we should say “intersubjectively meaningful and unambiguous,” since it is conceivable that I communicate a psychological meaning of “Q” to you but not the meaning I intended to communicate.

Chapter 17 SEMANTIC ANALYSIS AND PSYCHOPHYSICAL DUALISM (1952)

In his highly significant and provocative book, The Concept of Mind, Gilbert Ryle undertakes to show that the Cartesian theory of two worlds, the physical world characterized by publicity and the mental world characterized by privacy, inaccessibility to all but one, is a “myth” created by misunderstandings of language. His method is an excellent example of the Wittgensteinian method of diagnosing the origin of puzzling philosophical theories as pointless, confusing departures from ordinary language. If, for example, a man puzzles how on earth it is possible ever to verify a proposition about the future since, after all, one cannot observe an event that has not yet occurred, his puzzle is of the kind which can be eﬀectively dissolved by the Wittgensteinian treatment. The purpose of the following discussion is not a comprehensive critical review of Ryle’s book. It is rather to show that his crucial arguments against the “privacy theory” of sensations—which is the aspect of psycho-physical dualism most relevant to epistemology—fail to establish what they are supposed to establish; and that they fail mostly because they exhibit a kind of exploitation of ordinary usage for purposes of criticism of philosophical theories which is characteristic of the Wittgensteinian method and which I consider to be an unwholesome feature of an otherwise most sanitary movement of semantic hygiene. Ryle’s basic thesis is that the theory of mental acts like believing, knowing, aspiring, results from the failure to see that sentences containing such psychological verbs are statements about dispositions, not about events or processes. I do not wish to add here to the literature pro and con the theory of mental acts; for, as stated, my specific purpose is to examine Ryle’s arguments, set forth in chapter 7 (“Sensation and Observation”), against the Sense-Datum theory of Descartes, Locke, and Hume, which survives vigorously in the epistemological writings of Russell, Broad, and Moore. I cannot refrain, however, from expressing the opinion that Ryle’s discussion is inconclusive because he has failed to formulate and discuss the basic semantic questions at issue between behaviorism and dualism: (1) Are any behavioristic sentences analytically entailed by such “mentalistic” sentences as “I remember to have seen this person

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before,” i.e. are there any behavioristic sentences with which a sentence of the latter type is logically incompatible; or is the relation between the mentalistic and the behavioristic language only that of reducibility in Carnap’s sense? If the latter is the case, as maintained by the contemporary form of physicalism, the most enlightened semanticist can go on believing in “Cartesian ghosts” since dualism is perfectly compatible with the public confirmability of mentalistic statements. (2) Is the above formal clarification of the issue really so clarifying, considering that the distinction between L-implication and Fimplication is clearly defined for formalized language-systems but not for natural languages? If this skepticism with regard to the very tools of non-formal semantic analysis, which is becoming more and more vociferous, should be unanswerable, it would be anything but clear what Ryle is saying when he says that statements which the dualists interpret as referring to “ghostly” mental acts are really about behavioral events or behavioral dispositions. I think the following is a fair summary of Ryle’s basic criticism of the SenseDatum Theory. Sensations are wrongly regarded as ways of observing, as though to see, to hear, to touch, to smell, were necessarily to observe something. The making of observations involves, indeed, the having of sensations, but it involves, moreover, an active element, the attempt to find something out, and above all, what one can properly be said to observe (where proper usage = ordinary usage) is always a physical object, event, or process—where “physical” is used broadly so as to cover anything capable of being publicly witnessed, including, e.g. purposive behavior. “. . . the ordinary use of verbs like ‘observe’, ‘espy’, ‘peer at’ and so on is in just such contexts as ‘observe a robin’, ‘espy a ladybird’, and ‘peer at a book”’ (Ryle 1949a, 224). Ryle thus implies that the verb “to observe” (or synonyms like “to watch”) is misused if it is used to describe an apparition in a dream or a state of hallucination—a contention surely open to debate though it will not be debated now. We may, I think, formulate Ryle’s central criticism of dualistic epistemology as follows: Knowledge is a state of mind (= behavioral disposition?) terminating a successful inquiry; where there is no attempt to find something out, there is no occasion for knowledge to eventuate (this, of course, is again a statement about proper usage). It follows that the mere having of sensations does not constitute knowledge and that to speak of absolutely certain, immediate knowledge, expressed by such sentences as “I see red,” “I taste a bitter taste,” is to misuse the word “knowledge” (notice that hyper-semanticist Ryle and anti-semanticist Dewey shake hands here). The Sense-Datum Theory postulates a curious kind of performance, called “pure sensation,” which diﬀers from observation ordinarily so-called in that it does not logically involve physical objects. But since an act requires an object (at least if it is expressed by a transitive verb, as are the verbs “seeing,” “hearing,” “touching,” etc., in ordinary usage), and pure sensations by contrast to perceptions have no physical objects, the theory invents an appropriate kind of object called “Sense-Datum.” With the help of

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this device the epistemologist constructs a pure sense-datum language whose sentences have the same subject-verb-object structure as sentences describing perceptions of physical objects: “I taste a sour sense-datum” resembles “I taste a lemon,” “I see a round sense-datum” resembles “I see a round penny,” etc. It must be conceded, I think, that Ryle correctly diagnosed the linguistic origin of one form of the Sense-Datum Theory, viz. the form which splits pure sensations into an act and a non-physical object called variously “sense-datum” or “sensum” or “sensibile.” I am thinking particularly of Broad’s theory of Sensa, which involves the postulation of non-physical entities which, like Platonic essences (except that they are “sensible” rather than “intelligible”) are somehow there to become objects of the kind of awareness called “pure sensation”: “It is quite certain that there is a diﬀerence between the two propositions: ‘This is a red round patch in a visual field’ and ‘This red round patch in a visual field is intuitively apprehended by so-and-so’. Even if as a matter of fact there are no such objects which are not intuitively apprehended by someone, it seems to me perfectly certain that it is logically possible that there might have been” (Broad 1949, 209-10). If all that Ryle had intended to establish by revealing that sensation is not a form of observation had been that it is misleading to speak of sensations as (mental) acts, I would have no further quarrel with him—except remarking what is perhaps obvious, viz. that one can consistently repudiate the act-theory of sensation and still believe in other kinds of mental acts, such as deciding, inferring, etc., as events defying definitional reduction to publicly observable actions. But evidently this accomplishment would not be enough to lay the Cartesian ghosts of traditional dualistic epistemology. Ryle claims moreover to dispose of the myth of private facts (or events), called “sensations,” which constitute the subject-matter of immediate, absolutely certain knowledge. But if it is linguistic confusion to suppose, with Broad, that there are mysterious non-physical objects, called “sensa,” ready to be sensed, it does not in the least follow that it is linguistic confusion to suppose that there are events, called “sensations,” which are not publicly observable in the way physical events are publicly observable. How does Ryle propose to show that it is nonsense to speak of sensations as events whose occurrence can be said to be known, yet never publicly known, i.e. known by anybody except the person owning the sensation? As far as I am aware, he uses two main arguments, which may be called: (1) the argument from the impossibility of a pure sense-datum language, (2) the argument from the impropriety of speaking of observing sensations. (1) Before entering into explicit discussion of the first argument it might be pointed out that by no means all philosophers who speak the language of sense-data are guilty, like Broad and like Moore at the time of (Moore 1903) of the crime commendably lamented by Ryle, to split sensation-events into act and non-physical object. In Ayer’s sense-datum language, for example, “I see a red sense-datum” is simply an artificial way of saying “I seem to see

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something red,” the latter sentence involving the ordinary perceptual use of “to see,” i.e., the use in which “I see x” entails that x is a physical object ready to be seen. (Strictly speaking, what is here called the perceptual use is not the ordinary usage but at best the most frequent usage. It is proper to speak of what one saw in a dream. If the perceptual use were the only proper use, a man describing his dream in the “I saw . . . ” idiom rather than the “I seemed to see . . . ” idiom would be speaking either improperly or falsely—which I doubt.) A philosopher who finds it convenient to speak of sense-data, therefore, is not necessarily committed to the analysis of perceptual situations into a relative product involving over and above the observer and the physical object an intermediate entity: (x perceives y) = (∃z)[(x directly senses z) and zRy], where the interpretation of “R” is the big headache for this kind of Sense-Datum Theory. But assuming that Ryle would grant this point, we may formulate argument (1) as follows. Let “S p ” denote a sentence reporting a perception, like “I see a robin,” a sentence which could not be true, that is, unless a physical fact existed (“there was a robin at that time and at that place”); and let “S s ” denote a sentence which is just like an S p except that it has no physical implication— understanding “implication” here in the sense of “analytic implication,” since by virtue of psycho-physical laws (correlations of sensations with brain-events) any S s may factually imply physical facts. Ryle’s argument then is that no S s can be constructed, but that the moment one attempts to construct such a sentence, it inevitably turns into an S p . “I see blue,” e.g., can only be explained by translating into “I see the color usually presented by such objects as A, B, C,” and thus implies by its very meaning the physical proposition that there are (in the tenseless sense of “there are”) objects of kind A, B, C. Now, it is nonsensical to assert the occurrence of events (events described by S s sentences) which it is in principle impossible to describe: “Whereof one cannot speak, thereof one must be silent” (the concluding sentence of Wittgenstein’s Tractatus (Wittgenstein 1933) which latter, by the author’s own ironical confession, is one big sin against this precept). Now, I shall argue that what Ryle has shown is that the concept of an S s sentence with a communicable meaning is self-contradictory—and in that sense meaningless—but has not shown what needs to be shown in order to refute the theory of private sensations, viz. that the supposition of the occurrence of pure sensations unconnected with physical situations is meaningless. Suppose I taste an apple-like taste (I mean this sentence in a way in which it does not logically imply the possession of taste-organs). The only way I can explain the meaning of “apple-like taste” to another person is by directing him to perform the physical operations which I believe, on the basis of analogical reasoning, will produce a similar taste-sensation in him. If the occurrence of taste-sensations were entirely uncorrelated with such operations as biting into an apple and thus were entirely unpredictable, we simply would lack the means

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of developing an intersubjective language of tastes. But the very fact that this statement concerning the necessary condition of an intersubjective language of sensations is intelligible proves that the hypothesis of a world in which sensations are entirely uncorrelated with definite physical situations, a world in which one would never be in a position to say “it is probable that he is experiencing such and such a sensation now,” is meaningful (cf. on this point, Schlick’s article “On the Relation between Physical and Psychological Concepts” (in Feigl and Sellars 1949, German original in Schlick 1938), especially section VII. That ordinary language contains no “neat” sensation-vocabulary, i.e., a vocabulary usable for constructing sensation-reports wholly devoid of physical implications, simply proves that ordinary language is intersubjective. Naturally, there is no other way of communicating to a child the meanings of such words as “sour” and “red” but to direct him to perform the kind of physical operations (tasting lemon juice, looking at blood) which one expects, be such expectations logically justifiable or not, will cause him to experience the kind of sensations designated by the words.1 Perhaps the following simile will clarify my argument. Suppose there existed a few circles to which I alone had privileged access, and that in my soliloquies I occasionally talked about those circles, using the word “circle.” Assuming that no other circles exist and that I alone could see the circles that exist, how could I explain to other people what I mean by “circle”? Well, I could give a so-called causal definition by saying “a circle is the kind of figure which you would produce if you rotated a stretched thread kept fixed at one end.” Suppose furthermore that a law holds in this hypothetical universe according to which a circle once produced by the described method is visible only to the person who produced it. It is clear that in this universe the meaning of “circle” could not be communicated by pointing to public circles but only by reference to straight-lines in rotational motion (and, incidentally, it would be theoretically impossible to make sure that one had communicated one’s meaning, since, by hypothesis, one could not verify whether the experiment leads to the same result when it is performed by other people). Would it make sense for those people to speak of a possible universe in which there were circles but no straight-lines and no rotational motions? It surely would be a mean-

1 In the “Afterthought,” at the end of Ryle 1949a, chapter 7, Ryle confesses to a bad conscience about having

adopted the epistemologist’s “sophisticated” use of the word “sensation.” Ordinary language, he contends, contains no such expressions as “visual sensation” and “auditory sensation” (where ordinary language is contrasted with both philosophical and scientific language). I confess that this bad conscience impresses me as just another example of that pathological oversensitivity, cultivated in some circles of analytic philosophy, to philosophers’ departures from ordinary language. What philosophers find epistemologically interesting about painful sensations (in one’s leg or throat or eye), viz. the “privacy” of these events, is just a property which likewise characterizes the events described by such sentences as “I seem to see a red patch.” It is for this reason that they extend the term “sensation” to these events. Why should philosophers feel guilty about such departures from ordinary usage any more than scientists?

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ingful hypothesis, I should think, even though they would have to admit that in such a universe the word “circle” would lack a communicable meaning. I think that Ryle’s argument from the impossibility of a neat sensation-vocabulary is parallel to the argument of that apostle of ordinary language who got up and accused his speculating philosopher friend of contravening the ordinary usage of the word “circle” in speaking of circles isolated from straight-lines and rotational motions. It might be added, before proceeding to examination of the second argument, that Ryle’s arguments from ordinary usage against the epistemologist’s language are largely irrelevant because they overlook that the epistemologist is engaged in what has been called logical reconstruction of empirical knowledge, and in this process inevitably and quite consciously develops an artificial language. No philosopher, to my knowledge, has ever claimed that atomic sentences, i.e. sentences which are logically simple in the sense that any two such sentences are logically consistent and logically independent, occur in ordinary language, nor that they describe isolated events—pure sensations actually occurring outside of perceptual contexts. If a sentence on the atomic level, like “I see red now,” is to be construed as a sentence of ordinary English which may be true or false, it must be translated into “I see something red now” and thus ceases to be atomic: it has turned into an existential statement describing a perceptual situation2 involving both sensation and a physical object. True, but irrelevant, for it is the epistemologist’s purpose to reveal a level of language which is implicit in ordinary language though never spoken. A brief analogy may serve to remove the air of paradox from this concept of implicit language. It is impossible to imagine a sound-intensity divorced from any definite soundpitch, and accordingly a sentence describing an experience of sound-intensity isolated from an experience of sound-pitch could hardly be found in ordinary descriptions of sound phenomena. Still, such sentences may have to be constructed if one wanted to make the complexity of the meaning of ordinary sound-language explicit (“a loud tone was heard” would not be suﬃciently atomic, since “tone” already designates a complex of pitch, intensity, and quality). As a matter of fact, it might be conjectured that some epistemologists have fallen into the confusions justly criticized by Ryle just because they did not go far enough in divorcing their artificial language of logical reconstruction from ordinary speech forms: “I taste sour” does not sound grammatical (unless it is a lemon that is speaking), so the epistemologist says “I taste a sour sensedatum” in order to preserve the subject-verb-object form of ordinary sentences

2 My

use of “perceptual situation” in this paper diﬀers from Broad’s usage in that hallucinations, i.e., situations in which one seems to see an object which does not exist, are not perceptual situations at all according to my usage, while according to Broad they constitute a species of perceptual situation, called “totally delusive perceptual situations.”

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describing perceptual situations (“I taste a sour fruit”). And in some instances, as already admitted, these sentences of the epistemologist’s own making have led to the mythical belief in sense-data as sensible, yet non-physical entities. (2) Ryle undertakes to show not only that the mistaken assimilation of sensation to observation led semantically confused epistemologists to the invention of sense-data, but also that it is nonsense to think of sensations as possible objects of observation. “To observe a sensation,” he holds, makes no more sense than “to spell a letter” (Ryle 1949a, 206). If this is so, then the assertion that sensations, unlike physical events, cannot be observed by other minds is, indeed, of the same order as the assertion that letters cannot be spelled. The tacit implication of this comparison is evidently that the theory of absolutely certain knowledge of private sensations constituting the infallible basis of all our fallible beliefs about the physical world and other minds, vigorously defended again by Russell in (Russell 1948), is as devoid of significance as would be the emphatic assertion that only words, not their constituent letters, can be spelled. Now, Ryle’s unquestionably superb knowledge of the English language denies me the right to throw doubt on his assertion that it is not proper usage to speak of “observing” or “witnessing” one’s headache the way one may speak of “observing” or “witnessing” a car accident, while it is proper to use the verb “noticing” in this context (Ryle 1949a, 206). But what of it? The psychophysical dualist, I am sure, will gladly accommodate his language to the rules of good Oxford English and henceforth speak of sensations as events which cannot be noticed by more than one person at a time, unlike such physical events as car accidents. If Ryle allows the propriety of speaking of a “noticed headache,” he should also allow the propriety of the sentence “I know that I have a headache—though you may doubt it”; but this is an excellent instance of private knowledge in the sense in which according to the dualist all knowledge of sensations is private. But in fact Ryle’s subtle investigation of the proper usage of such English verbs as “observing,” “witnessing,” “noticing,” seems to me to be wholly irrelevant to the epistemological problem. The question is whether it makes sense to speak of sensations as events whose occurrence can be known with certainty by one person but not with the same doubt-precluding certainty by other people. If this question is answered aﬃrmatively, the “Cartesian ghosts” have not been eﬀectively banished. And the only relevant semantic question in this connection is whether it is proper to speak of “knowledge” of the occurrence of a sensation. We could easily imitate Ryle’s method of observing carefully the speaking habits of educated English or American people, and cite in support of the propriety of such a usage of “knowledge” such sentences as “I know that these patches look all the same color to me whatever they may look like to you—you can only know what they look like to you.” But it is more important to point out that Ryle is logically committed by his own statements

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to acceptance of the propriety of the expression “knowledge of sensations.” Observation of a robin, we are told, entails the having of sensations, though it involves more than that. Put in the formal mode, this means that the statement “I observed a robin” could not be true unless some statement reporting a sensation were true. But if it makes sense to speak of the truth of statements reporting sensations it must surely also make sense to speak of knowledge of the events reported. I shall make some concluding remarks about the recent tendency of analytic philosophers practicing therapeutic positivism,3 a practice brilliantly exemplified by Ryle’s The Concept of Mind, to regard the theory of the logical privacy of the mental as merely the upshot of arbitrary linguistic conventions. Consider the statement “it is logically impossible for one person to notice directly (notice that ‘notice’ has been substituted for ‘observe’) another person’s headache, the way he could notice directly another person’s sneeze; that’s what makes the former event mental and the latter physical.” The usual reaction of therapeutic positivists to this assertion of the impossibility of direct knowledge about other minds is to ask: “Well, what would it be like to notice (directly) another person’s headache? Would this not amount to noticing a headache which is yours and not yours at the same time? But then the event which you say is impossible is simply a self-contradiction, just like the event of spelling a letter (= decomposing into its constituent letters what is not composed of letters). Your talk about the ‘logical privacy of the mental’, then, is as pointless as would be talk about the impossibility of spelling letters.” I would be the last one to deny that this kind of analysis of the expression “logical privacy of the mental” is immensely clarifying. It is important to realize that it would be self-contradictory, in terms of current meanings of the expressions “thought,” “direct knowledge,” to suppose that A could have direct knowledge of B’s thoughts in the sense in which B could have such knowledge. One important consequence of this semantic fact is that telepathy should not be interpreted as direct knowledge of another mind’s mental states but rather as direct belief, i.e. belief not derived from physical observations, about another mind which stands a high chance of being correct. This may be demonstrated as follows: if we credited A with direct knowledge of B’s present thought in the sense in which we credit B with such knowledge, then we would accept the following two premises as conclusive evidence for the proposition describing B’s mental state: (1) A asserts that B is in this mental state, (2) A’s assertion is honest, i.e. A believes what he says. But the fact is that while we would accept

3 I am not using this label so as to imply the conception of philosophy as resembling closely psychoanalysis in method and purpose. By a therapeutic positivist I mean a philosopher who conceives of (good, worthwhile) philosophy as having primarily a therapeutic purpose, viz. the dissolution of typically “philosophical” perplexity by revealing the linguistic confusions which cause such perplexity.

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these premises as providing conclusive evidence if A and B were one and the same person (not that I mean to imply that we could have conclusive evidence for the truth of these premises in the sense of “conclusive evidence” in which the premises, if true, would be conclusive evidence for the conclusion), we don’t if they are diﬀerent persons: we would first ask B directly about his mental state before we accepted A’s telepathic belief as an instance of knowledge. The point I am arguing can be made particularly evident if we suppose that the mental state of B which A is alleged to have direct knowledge of is a state of belief. For in that case the claim would either mean that A is introspecting his own belief which happens to be shared simultaneously by B, or that the state of belief is literally a common constituent, a point of intersection as it were, of the mental histories of A and B. In the first case there is no telepathic situation at all, in the second case we are violating the ordinary usage of the expression “mental state” in somewhat the same way as we would violate the ordinary usage of “particle” if we spoke of the possibility of the same particle occupying simultaneously diﬀerent places. Yet, I submit that after all the virtues of this kind of semantic analysis have been recognized, one is left with an enormous non sequitur when one turns to the claim that the dualist’s concept of the mental defined by logical privacy is nonsense. All that has been shown is that the assertion of the impossibility of certain, immediate knowledge of unowned mental states is an analytic consequence of the meanings of the expressions “certain, immediate knowledge” and “unowned mental state,” and not a factual thesis like the impossibility of catching a certain thief. The further premise, which would have to be added and substantiated before the argument of Ryle and associates could be accepted, is that those meanings are gratuitously stipulated by dualists in sheer violation of ordinary usage. To illustrate such gratuitous deviationism in linguistic conduct: if a philosopher arbitrarily defines “there is evidence for p” to mean “p is logically demonstrable” and then announces that it is impossible to produce evidence for empirical propositions referring beyond present experience; or, if he defines “seeing a physical object” to mean “seeing simultaneously the whole surface of the object as well as all its past and future states” and then announces that common sense is mistaken in believing that physical objects could be seen; or if he defines “seeing the same object A again” to mean “seeing A again with exactly the same properties it had previously” and derives from his definition the paradoxical consequence that it is impossible to see the same object twice—in cases of this sort philosophers are rightly blamed by therapeutic positivists for building perplexing philosophical theories by means of nonsensical (i.e. self-contradictory) concepts misleadingly associated with meaningful everyday expressions. The question before us is: Is a philosopher similarly guilty of arbitrary departure from ordinary usage when he says “I can

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be aware only of my feelings, not of anyone else’s; I cannot know for certain, though I may conjecture, what other people’s thoughts are”? That we do, in everyday discourse, speak of awareness and certain knowledge of other people’s feelings and thoughts, is admitted. In fact, such sentences as “I am clearly aware of her jealousy; I know he thinks I do not know about his aﬀair with Mrs. X” are, in the sense in which they are intended, frequently known to be true. The philosopher’s claim that such sentences cannot be known to be true or to be false seems, therefore, like the same sort of willful paradox, brought about by surreptitious departures from the ordinary meanings of words. Yet, the semantic situation here diﬀers in an important respect from the kind of “philosophical perplexity” illustrated above. In the cases I adduced, the tricky philosopher convicts the man in the street of cockily holding unjustifiable beliefs, by substituting for his vague but applicable concepts, precise but self-contradictory (and so inapplicable) concepts. But this idle game must be distinguished from the fruitful game of revealing ambiguities in ordinary language which are unnoticed by the man in the street. Suppose I asked the ordinary mortal for whom knowledge of other minds is simply a fact, no problem at all: “Are you aware of your friend’s loneliness in the same sense of ‘awareness’ as you might be aware of such a feeling in yourself?” Surely, Mr. Common Sense need not be analytical above average to come to see, after some reflection, that there is a big diﬀerence between introspective awareness of a feeling, which is in a sense not easy to analyze infallibly, and inference to similar feelings in others, which is in an obvious sense fallible. The essential point is that this concept of direct, introspective knowledge of a mental event is a concept in actual use, not a concept willfully constructed by philosophers intent on belittling common sense. The dualist’s statement “it is impossible to know that another person feels lonely the way one can know the occurrence of such a feeling in oneself” is, indeed, analytic of the relevant meaning of “knowledge.” But the statement’s analyticity would convict it of triviality only if the concept analyzed were one artificially constructed and never used by non-philosophers. If the advocates of common speech whose attack on psycho-physical dualism I have criticized were to use their method consistently, they would, I suspect, approach practically any philosophical analysis of a concept in use with the same cynical attitude. Take, e.g., Hume’s denial of a necessary connection between cause and eﬀect. This denial, of course, is exactly the same kind of “violation of ordinary usage” as the dualist’s denial of the possibility of knowing for certain what another person is thinking or sensing or feeling: we do say, quite often, “this eﬀect must necessarily follow. . . ,” and in the sense in which this kind of statement is intended we frequently have excellent grounds for asserting it. But the ordinary man will quickly cease to be puzzled by Hume’s statement if the latter is clarified in the usual fashion, viz. as the denial

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of the logical demonstrability of causal laws. The fact that the ordinary man now discovers that he never really held the belief just demolished by the subtle philosopher does not convict the philosopher of having launched a Quixotic attack on windmills. For, as Moore has taught us, the purpose of analytic philosophy is not to refute common-sense beliefs nor to declare them as unfounded, but to clarify them. Thus Hume, whatever his own confusions may have been, has the merit of having clarified the diﬀerence between inductive and deductive inference, empirical and logical connection, a diﬀerence obscured by the ambiguous use of the words “reason,” “ground,” “consequence” (even “cause,” in the Aristotelian tradition) in both senses. The concept of the mental as the logically private, in contrast to what is publicly observable, is similarly the result of analysis of confused meanings. And I submit that if by psycho-physical dualism be meant no more than the conception of the mental here defended against Ryle’s ingenious assault, there is no reason to suspect psycho-physical dualists of metaphysical backwardness isolated from the progressive tools of modern semantic analysis.

VI

PHILOSOPHY OF SCIENCE

Editors’ Note: Throughout this part Pap uses a notational convention adopted from Carnap 1937, which is described in full in the Introduction, at page 18.

Chapter 18 THE CONCEPT OF ABSOLUTE EMERGENCE (1951)

I understand the business of the philosophy of science to be painstakingly careful analysis of concepts, principles, and methods used in science. This broad statement fails, of course, to diﬀerentiate such analysis of concepts as inevitably occurs in a developed science itself (e.g. analysis of the concepts “simultaneity,” “absolute motion,” “energy,” etc. in physics) from such analysis as is more likely to be the professional concern of the philosopher of science as philosopher. The most natural object of distinctively philosophical analysis concerned with science would seem to be the very activity, or class of activities, defining science in general, rather than some one specific science. As it goes without saying that one such activity characteristic of science is prediction, the analysis of the concept of predictability is a vital task of the philosophy of science. Ability to predict is a virtue marking a good scientist; ability to analyze clearly the concept of predictability is a virtue marking a good philosopher of science. Now that permanent limits are set to the scientist’s ability to predict by “the very nature of things” (if I may be permitted to use loose language in paraphrasing a doctrine propounded primarily by philosophers who speak without precision) has been one of the inspiring themes of the doctrine known as the theory of “emergent evolution.” The best known emergent evolutionists in the English tradition are probably S. Alexander, the author of Space, Time, and Deity (Alexander 1920), and C. L. Morgan, the author of Emergent Evolution (Morgan 1923) and, more recently, The Emergence of Novelty (Morgan 1933). Their central idea was that the process of evolution produces more and more complex “levels,” like the atomic level, the level of chemical compounds, the biological level, etc.; and that on each level new qualities emerge which are absolutely unpredictable on the basis of the laws applying to the lower levels. Perhaps the best way to impress upon the reader the urgency of analyzing the meaning of the doctrine before either embracing or rejecting it, is to present a sample or two of the language through which Alexander expresses his metaphysical insight:

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The Concept of Absolute Emergence (1951) The higher quality emerges from the lower level of existence and has its roots therein, but it emerges therefrom, and it does not belong to that lower level, but constitutes its possessor a new order of existent with its special laws of behavior. The existence of emergent qualities thus described is something to be noted, as some would say, under the compulsion of brute empirical fact, or, as I should prefer to say in less harsh terms, to be accepted with the “natural piety” of the investigator. It admits no explanation. (Alexander 1920, vol. 2, 46) A being who knew only mechanical and chemical action could not predict life; he must wait till life emerged with the course of Time. A being who knew only life could not predict mind, though he might predict that combination of vital actions which has mind . . . Now it is true, I understand, that, given the condition of the universe at a certain number of instants in terms of Space and Time, the whole future can be calculated in terms of Space and Time. But what it will be like, what qualities it shall have more than spatial and temporal ones, he cannot know unless he knows already, or until he lives to see. (Alexander 1920, vol. 2, 327-328)

I have italicized the phrases which cry out most loudly for analysis. The following semantic discussion of the problem of emergence will not, however, make any further reference to Alexander. My point of departure will be, instead, a similar view expressed with far greater precision by a far more lucid philosopher than Alexander: C. D. Broad, in The Mind and its Place in Nature (Broad 1949). My purpose is to shed some light on the old question of emergent qualities, thrown into prominence mainly through the vitalism versus mechanism issue in the philosophy of biology, by using a semantic line of analysis which, to my knowledge, has been neglected by both parties to the dispute. It is customary to discredit the belief in absolutely unpredictable qualities on the ground that what scientific theories known today do not permit us to predict, scientific theories known tomorrow may well bring within the bounds of predictability. Thus there was a stage in chemistry when no general laws correlating molecular structure and sensible properties of compounds were known which would enable one to predict sensible properties of a hitherto unobserved compound on the basis of its molecular structure; but such laws are now known.1 To speak of absolute unpredictability, unpredictability once and for all, convicts one, in fact, of metaphysical obscurantism, motivated perhaps by a subconscious hostility against the faith in the omnipotence of science. Indeed, I would not wish to deny that those who, following Alexander, recommend “natural piety” in the face of absolute novelty, most likely have no clear idea as to what they mean by such “absolute novelty.” It seems to me, however, that this vague notion of absolute emergence can, with the help of semantic concepts, be explicated in such a way that whether a quality is emergent is independent of the stage of scientific knowledge, but rather depends on the question whether certain 1 For

a precise statement of this “relativistic” theory of emergence or novelty see Henle 1942 and, more recently, Hempel and Oppenheim 1948, §5.

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predicates are only ostensively definable. Specifically, my purpose is to show that a law correlating a quality Q with causal conditions of its occurrence can, without obscurantism, be argued to be a priori unpredictable if the predicate designating Q is only ostensively definable. The concept of “a priori unpredictability” here used will be defined in due time. My starting-point is a distinction elaborated by Broad with great analytical eﬀort though questionable success in the chapter “Mechanism and its Alternatives” of Mind and its Place in Nature: the distinction between an emergent (or “ultimate”) law and a non-emergent (or “reducible”) law. An example by means of which Broad explains his notion of emergent law is the law connecting the properties of silver-chloride with those of silver and of chlorine and with the (presumably molecular) structure of the compound: if we want to know the chemical (and many of the physical) properties of a chemical compound such as silver-chloride, it is absolutely necessary to study samples of that particular compound. It would of course (on any view) be useless merely to study silver in isolation and chlorine in isolation; for that would tell us nothing about the law of their conjoint action . . . The essential point is that it would also be useless to study chemical compounds in general and to compare their properties with those of their elements in the hope of discovering a general law of composition by which the properties of any chemical compound could be foretold when the properties of its separate elements were known. (Broad 1949, 64)

Let us notice that according to the definition of an emergent law, implicit in the quoted passage, at least a necessary condition (but I suspect likewise a suﬃcient condition) of emergence of a law of the form “if C1 , . . . , Cn , then R” (where the antecedent refers to a set of interacting components, and the consequent to a resultant of this interaction) is that instances of R must be observed before the law could be known with some probability. More exactly, if L is an emergent law in Broad’s sense, then it cannot be confirmed indirectly, by deduction from more general laws, before direct confirming evidence is at hand. For short, let us say that an emergent law is deducible only a posteriori, or unpredictable a priori. The meaning of this condition will best be grasped if we consider Broad’s illustration of reducible laws, i.e. laws that could be theoretically predicted with the help of a general composition law before any direct observational evidence exists. Consider the law of projectiles according to which the trajectory of a projectile is, under ideal conditions, a parabola. It is true that direct observational evidence for this law was at hand before Galileo deduced it, with the help of the parallelogram law of forces (Broad’s example par excellence of a general composition law), from the law of freely falling bodies and the law of inertia. But it is clearly conceivable that the law should have been reached by deduction from those premises concerning the eﬀects of isolated force components before instantial evidence was obtained (in fact, this was the case with regard to a special case of the law, namely

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the flight of high-speed cannonballs). This, then, is what Broad would call a reducible law: it is a priori predictable in the sense that it is capable of prior confirmation through deduction by means of a general composition law before any confirming instances are observed. For the present purpose we may be satisfied with a denotative definition of “general composition law” as the kind of deductively fertile composition law illustrated by the parallelogram law. Notice that the general composition law is not claimed to be itself a priori predictable; it is rather claimed to make special composition laws, like the law of projectiles, a priori predictable. Now, this concept of reducible law is no sooner defined than it provokes the question: how could it ever be shown that a given law is absolutely irreducible? In order to show this, one would have to prove that no general composition law could conceivably have been known which would have enabled a skilled scientist to predict the law a priori. Broad himself seems to recognize the relativity of such irreducibility to the stage of scientific knowledge at least in the case of chemistry, for the quoted passage concerning the properties of silver chloride is followed by the statement “so far as we know, there is no general law of this kind.” Indeed, it is easily describable what such a general composition law of chemistry might be like: if a metal combines with an acid in solution, there results a salt and free hydrogen. This law may have been inductively derived by observing interactions of metals M, M , M with acids A, A , A respectively, and then be used to predict that if M should react with A a salt will result of which no instances have yet been observed. Broad indeed admits that “mechanistic” progress in chemistry is possible to the extent that a deduction of R from C1 , . . . , Cn with the help of general composition laws might be accomplished if R is a physical disposition of compounds, like ready solubility in water. But he holds such deduction to be in principle impossible if R is a secondary quality, i.e. a disposition to produce a sensation of a certain kind, like a pungent smell (Broad 1949, 71). Does Broad have a point? I shall concede that in claiming absolute emergence for laws correlating secondary qualities with microscopic physical conditions he is inconsistent with his own definition of emergence; but I shall nevertheless argue that he came close to making a valid point overlooked by the “relativists.” Broad claims that not even the “mathematical archangel” (that is, the Laplacian calculator turned to physical chemistry) could predict what NH3 would smell like unless someone (not necessarily himself) had smelled it before. If Broad asserts the impossibility of theoretically certain prediction, his assertion is true but trivial: even the probability which is conferred on a special law of dynamics by deduction from the parallelogram law falls short of a maximum, since the parallelogram law itself is still capable of falsification as long as not all of its deductive consequences have been tested. But if he asserts the impossibility of “prediction” in the only sense in which prediction is ever possible, he is

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clearly wrong: just suppose that chemists had evidence suggesting the generalisation “whenever two gases chemically combine in the volume proportion 1:3, the resulting compound has the smell S .” If the original evidence for this general composition law does not include observations upon the formation and properties of ammonia, the special composition law “NH3 has smell S ” could well have been a priori predicted by somebody less than a mathematical archangel before anybody had smelled that gas. It is conceivable, incidentally, that Broad confused the proposition he did assert, and which has been shown to be false, with the undeniable but irrelevant proposition that “this gas has smell S ” cannot be logically deduced from the premise “this gas has such a molecular structure” alone, without the use of an additional premise asserting the correlation between structure and secondary quality. However, there is a logical diﬀerence between the hypothetical general composition law just mentioned and the parallelogram law, which will prove to be crucial for the problem of emergence. The deduction of a special law made from the former took simply the form of deriving a substitution-instance and the predicate referring to the predicted quality was explicitly contained in the general premise. But the parallelogram law does not contain the concept of a specific type of compound motion, such as circular motion or motion along a parabola; it only contains the concept of a specific form of functional dependence of the direction and magnitude of a compound motion upon the directions and magnitudes of the component motions. A simply way of putting the diﬀerence is this: one could understand the parallelogram law without thinking of the determinate quality of motion it may be used to predict, and therefore without ever having witnessed an instance of the predicted quality. But since the general law correlating microscopic conditions with sensations of quality Q contains the very same concept of Q as the derived substitution-instance, and Q is a simple quality of which, in Hume’s language, one cannot have an “idea” without “antecedent impression,” the law cannot even be understood unless an instance of the predicted quality has at some time been witnessed.2 In this sense the deductions made from such a general law do not lead to “novelty”; when we test the deduction empirically we do not encounter a new quality the way physicists would acquaint themselves with a new quality of motion if they tested their prediction of circular motion from the parallelogram law which latter, as we might suppose, they had inductively derived from observations of rectilinear motions only. Notice that if a predicted quality Q fails to be novel in the specified sense, it does not follow that the special law “if C1 , . . . , Cn , then Q” must be directly confirmed before it could be indirectly confirmed by deduction from a general composition law; it only follows that instances of Q,

2 The

assumption, here involved, of only ostensively definable predicates is discussed in the sequel.

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which may be associated with other complexes than C1 , . . . , Cn as well, must be observed before indirect confirmation is possible. Let me clarify the point in terms of the laws correlating wave motions of the air with sound phenomena. One might be inclined to think that a general law correlating frequencies and pitches could easily be formulated which would enable a priori prediction of hitherto unexperienced sound phenomena in just the way in which the parallelogram law enables the prediction of so far unobserved forms of motion. Thus, let X be the highest pitch so far heard, which we shall assume to be not the highest audible pitch, and suppose that comparisons of various pitches led to the well-known law “the higher the frequency, the higher the pitch.” With the help of this simple law we can easily predict that the pitch corresponding to a frequency higher than the frequency corresponding to X will be higher than X, before ever having heard such a pitch. If now the question should be raised whether this deduction is just like the discussed case of deduction by simple substitution, namely, a prediction of a quality which must already have been observed if the general premise is to be intelligible, the answer will have to be somewhat qualified. Strictly speaking, the quality which is being predicted is “higher in pitch than X,” which quality is not explicitly mentioned in the general premise and therefore need not have been experienced in order for that general premise to be understood. Yet, the reason why such a priori prediction is possible is that the predicate designating the quality is complex and made up of parts whose meanings are understood through ostensive definition: the meaning of the relational predicate “higher pitch” is understood because some, though not all, instances of this relation have been experienced, and the meaning of the proper name “pitch X” is understood, let us say provisionally, because pitch X has been heard. We might generalize from this example, and lay down the following principle: if a novel (that is, so far unobserved) quality Q is to admit of a priori prediction, then it must be complex in the sense that the expression describing it contains sub-designators (predicates and/or proper names), and these sub-designators, being understood through ostensive definition only, designate old qualities. If this principle is correct, then it follows that if there are qualities which admit of a priori prediction, there must also be qualities, less complex ones, which do not admit of a priori prediction. In terms of our illustration: one would, indeed, make a perfectly defensible claim if one said, a` la Broad, that no amount of physical and physiological information could enable one to predict that a frequency increase would produce a sensation of rising pitch, if the relational predicate “higher pitch” admitted only of ostensive definition; since in that case one would not know what quality one is predicting and the deduced statement would acquire its meaning only after verification, which is absurd. It is, however, of the utmost importance to realize that the concept of a priori unpredictability, as here analyzed, is absolute only relatively to the assumption

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that certain descriptive predicates admit only of ostensive, not of verbal, definition. If this semantic premise fails in a given instance, the claim of a priori unpredictability likewise breaks down. Take, for example, the question whether it could be a priori predicted that a definite frequency would produce that definite pitch named “X.” According to the present analysis, this question reduces to the question whether the meaning of the proper name “X” could be understood, in other words, whether the designated quality could be imagined, by someone who had never experienced the quality. Conceivably the pitch might be described in terms of interval-relations to already heard pitches (say, as the pitch one third higher than pitch Y, where the expression “one third higher” would itself be defined as meaning “such that if Y and X occur simultaneously, the interval named ‘third’ is heard”). If such a relational description enabled one to imagine the as yet unheard pitch, one would know what one was predicting before verification of the prediction in terms of immediate experience. A “relativist” with regard to the problem of emergence might now think that his position remains unconquered after all, since it is always conceivable that a given quality-designation be understood by description rather than by ostentation. Whether a given proper name be only ostensively definable or verbally definable by means of a synonymous definite description is, indeed, not a logically decidable question but a question of psychology, specifically concerning possibilities of imagination. Just as in one logical calculus the logical constants C1 and C2 may be primitives and the logical constants C3 , C4 , C5 defined, while in an alternatively constructed calculus, say, C4 and C1 are taken as primitives and the rest defined; so in a descriptive language, say, phenomenological acoustics, the set of ostensively defined proper names3 (what Russell calls “logically proper names,” contrasted with proper names introduced as abbreviations for descriptions) is not uniquely determined. Thus, using “C” as ostensively defined proper name, and “higher pitch” (and its converse “lower pitch”) and “third” as ostensively defined relational predicates, we could introduce the proper names “E,” “G,” “B,” by verbal definition (for the sake of simplification, I assume the C-major scale as the field of the relation so as to be able to neglect the distinction between major and minor intervals). This is assuming the psychological possibility of imagining an as yet unsensed quality in the field of a relation R some instances of which have been sensed, on the basis of sensed qualities in the same field. If we further provide an ostensive definition for the relational predicate “equidistant” (pitches), we might even be able to introduce the names of all the missing pitches in the C-major scale

3 In

calling such quality-designations as “B-flat” proper names, I do not mean to imply, of course, that the designated pitches are particulars. I am using “proper name” as a term relative to a given language-level, such that the descriptive terms of lowest order in language L (terms occurring as grammatical subjects but not, in L, as grammatical predicates) are called “proper names” relatively to L.

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by description. But an alternative construction of the language is clearly conceivable: we might take “second” and “higher pitch” as ostensively defined relational predicates, “G” as ostensively defined proper name, and then all the other proper names and interval-designations might be introduced by description without the use of the relational predicate “equidistant.” It may perhaps be doubted whether the description “the complex pitch resulting if a pitch a second higher than the pitch a second higher than G is sounded simultaneously with G” would enable one to get an auditory image of a third if one had never heard a third before; but this is a psychological question of fact. If we call our first model of a language of phenomenological acoustics ‘L’ and our second model ‘L ’, we can now make the following assertions: a law correlating the pitch G with a frequency is a priori unpredictable in L , and so is the law correlating a definite frequency-ratio with the pitch-interval called “second”; but those same laws are a priori predictable in L, if we assume that the meanings of verbally defined expressions in such languages are intelligible in the sense that the verbal definition can produce an image of the quality or relation defined, independently of any previous experience of the latter. If to say that quality Q (or relation R) is absolutely emergent is to say that the law correlating Q (or R) with quantitative physical conditions is a priori unpredictable, it follows that absolute emergence is relative to a system of semantic rules. In this respect the concept of absolute emergence turns out to be surprisingly analogous to the concepts of indefinability and indemonstrability. Is the relativist, then, wrong in denying the existence of absolutely emergent qualities? He is wrong if he denies the semantic truism that some descriptive terms must be given meaning by ostensive definition if it is to be possible to give meaning to any descriptive terms by verbal definition. Perhaps he is right, on the other hand, in his claim that no descriptive term is, by some obscure kind of necessity, definable by ostentation only. Even Hume, whose principle that every simple idea must be preceded by a corresponding impression is equivalent to the semantic principle that predicates designating simple qualities can become meaningful only through ostentation, allowed for the famous exception, the missing shade of blue—in fact, it could be argued that the same logic which forced him to admit this one exception should consistently have led him to allow for an infinite class of similar exceptions. However, I would like to conclude with the tentative suggestion that for every sense-field there is a general ordering relation, instances of which could not possibly be imagined antecedently to being sensed. I am referring to the relation “higher pitch” for the auditory sense-field, the relation “brighter color” for the visual sensefield, and analogous transitive and asymmetrical ordering relations for other sense-fields or other dimensions of the same sense-fields. Whenever a verbal definition is given for a term designating an element in the field of such an ordering relation R, the relational predicate “R” is itself used in the definiens,

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together with one or more names of other elements in the field. Thus Hume’s missing shade of blue, which has not been seen yet, would be verbally defined as the shade equidistant from, say, b4 and b6 , where these are separated by a larger distance than the other consecutive elements in the series of increasingly dark shades. To say that b5 (the missing shade) is equidistant from b4 and b6 evidently means that it is just as much darker than b4 as b6 is darker than it. But then the meaning of “darker than” must be understood by ostentation, and, on pain of circularity, the method of relational description by which “b5 ” was verbally defined is unavailable. Indeed, I do not have the faintest notion what an analysis of such a simple relational concept could be like. If so, then a law correlating quantitative changes in physical conditions with such changes in sensed qualities as are expressed by the terms “darker,” “louder,” “higher in pitch,” etc., is absolutely emergent after all. And limits would be set to the possibility of a priori prediction, not by the stage of scientific progress, but by the limits of semantic analysis.

Chapter 19 REDUCTION SENTENCES AND OPEN CONCEPTS (1953)

Since the time when Carnap published his classical contribution to the analysis of scientific language, “Testability and Meaning” (Carnap 1937), it has come to be universally recognized by competent philosophers of science that the concept of explicit definition is inadequate for the analysis of specifications of meaning going on in science. Instead the term reduction sentence has been incorporated into the essential terminological furniture of the meta-language in which scientific procedures are described. In Carnap 1937, Carnap initially explained the need for reduction sentences in connection with the problem of defining dispositional predicates. His argument was that if an explicit definition of dispositional predicate “P,” of the form “P = (if Q, then R)” (where “Q” refers to a test operation, “R” to the corresponding test result) is used, one is faced with the paradoxical consequence that “P” would be vacuously predicable of any substance upon which the test operation is never performed. As this paradox altogether depends upon the interpretation of “if, then” in the sense of material implication, Carnap’s argument might conceivably fail to be persuasive, on account of the following objection. Employment of reduction sentences is necessary only as long as we fail to analyze the meaning of “if, then” in nomological conditionals, in other words, fail to analyze the concept of necessary connection involved in contrary-to-fact conditionals. For, as among others Reichenbach has emphasized (Reichenbach 1947, chapter 8), one of the necessary conditions of adequacy of an analysis of nomological conditionals is just that the truth of the conditional should not follow merely from the falsity of its antecedent. In view of such an objection, it would certainly be lamentable if Carnap 1937 should have conveyed the impression (in all probability contrary to the author’s intention) as though the mentioned paradox of material implication were the basic reason, or even the only reason, for the construction of reduction sentences as means of specifying the meanings of scientific terms. It is conceivable that, once we have a correct analysis of the nomological conditional, we could analyze dispositional predicates of the thing-language (like “soluble,” “fragile”) in terms of nomological conditionals,

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or sets of such, connecting test operations and test-results, so that we could revert from the bilateral reduction sentence “if Q, then (if and only if P, then R”) to the explicit definition “P = (if Q, then R).” Carnap’s argument merely shows the need for reduction sentences in a language of logical reconstruction which suﬀers from the shortcoming that no causal statements of ordinary language and scientific language are translatable into it; and in such a language we had better not attempt to talk about dispositions and causal connections in the first place. However, I shall argue in the following that reduction sentences will remain indispensable, whatever the outcome of the logic of nomological conditionals now in the making may be. The other argument for reduction sentences, likewise to be found in Carnap 1937, but whose cogency is independent of whether our language of logical reconstruction is extensional or not, may be called the argument from surplusmeaning. It is often asserted that the meaning of a construct like electrical current cannot be exhausted by a single test-conditional, or even a finite conjunction of such, in the language of observables; and for this reason explicit definition of a construct in terms of observable predicates is said to be impossible. Carnap points out that the physicist implicitly uses the method of partial meaning-specification by means of reduction sentences since he wishes to leave his concepts “open” for application to new contexts, contexts which the reduction sentences so far stated do not legislate for, as it were. I believe, however, that the characterization of reduction sentences as application-rules for open concepts conflicts to some extent with their characterization as introductions of terms without antecedent meaning. And since I wish to make the argument from surplus-meaning as convincing as possible, it will be useful to reveal the mentioned inconsistency in Carnap’s own discussion of reduction. A reduction pair [(p1 ⊃ (p2 ⊃ p3 )); (p4 ⊃ (p5 ⊃∼ p3 ))] is said to have factual content, expressible by a sentence which does not contain the introduced term, viz. ∼ (p1 .p2 .p4 .p5 ). A point of great importance which will be emphasized in the sequel might as well be touched on right now. Carnap and the logical empiricists use the expression “p has factual content” in such a way that if and only if p has factual content (or is “synthetic”), then p is empirically refutable. Now, since the conjunction of the two members of a reduction pair has the mentioned factual consequence, it must be regarded as itself a factual statement. But a conjunction cannot be factual unless at least one conjunct is factual. Yet, if p3 is first given a meaning by a given reduction sentence then that reduction sentence is empirically irrefutable. Hence a reduction pair could have no factual content if both members were meaning rules in a sense in which this implies empirical irrefutability. We could, of course, arbitrarily designate one member of the pair as a meaning rule and the other member as a factual statement in the “object-language”: it is not self-contradictory to suppose that an instance of p1 .p2 . ∼ p3 be found if ∼ p3 can be inferred from p4 .p5 , and it is not self-contradictory to suppose that an instance of p4 .p5 .p3 be

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found if p3 can be inferred from p1 .p2 . But this way of splitting the reduction pair into a refutable factual statement of the object-language and a (partial) semantic rule of the meta-language (with regard to which it does not make sense to speak of “refutation”), is clearly contrary to the spirit of Carnap’s theory of reduction. If so, then the concepts “having factual content” and “being factually empty” which logical empiricists, including Carnap, have always intended as contradictories are not applicable in the original sense to sentences whose predicates acquire meaning through reduction instead of explicit definition; and it is, in that case, misleading to apply the same semantic concepts to such sentences. It is true that Carnap explicitly extends the meanings of “analytic” and “synthetic” in such a way as to make these semantic concepts applicable to sentences whose predicates admit of reduction but not of explicit definition. Yet, this extension of the meaning of “analytic” is confusing since according to the original meaning of the term it is characteristic of analytic sentences that they are empirically irrefutable, while according to the generalized meaning an analytic sentence may be empirically disconfirmable. This will become clear as we turn our attention to the special case of reduction pairs called by Carnap “bilateral” reduction sentences. In the case of a bilateral reduction sentence (Rb ) the “representative” sentence degenerates into ∼ (p1 .p2 .p1 . ∼ p2 ), which is a tautology, and hence this kind of reduction sentence is said to be factually empty. Now, this sounds plausible enough if we think of an isolated Rb ; for to refute p1 ⊃ (p2 ≡ p3 ) we require a case of p1 .p2 . ∼ p3 or of p1 . ∼ p2 .p3 , and if the only basis for predicating p3 is p1 .p2 , and the only basis for predicating ∼ p3 is p1 . ∼ p2 , then neither of those cases can occur. However, Carnap himself recognizes that, apart from the paradox of material implication which precludes explicit definability of disposition concepts in an extensional language, the main reason for reduction sentences is the desire the leave concepts “open” for application in new contexts. Isolated occurrence of a reduction pair or of a Rb is therefore the exception, and occurrence within a system of convergent R-sentences (converging, that is, to the open concept) is what the very purpose of R-sentences would lead one to expect. As a matter of fact, it would seem that if an Rsentence occurs isolated, like the famous R-sentence for “soluble” and similar R-sentences for dispositional predicates of the thing-language, this indicates that we have to do with a closed concept serving merely the purpose of shorthand, which would be explicitly definable were we only permitted to use the concept of causal implication in the definiens. Carnap sees clearly that what may be called the “systemic” occurrence of R-sentences is the rule, as the following quotation shows: in most cases a predicate will be introduced by either several reduction pairs or several bilateral reduction sentences. If a property or physical magnitude can be determined by diﬀerent methods then we may state one reduction pair or one bilateral reduction sentence for each method. The intensity of an electric current

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Like a reduction pair, such a set of convergent Rb -sentences is said to have factual content because factual statements not containing the “introduced” term are deducible from it (if the set consists, e.g., of the two Rb -sentences: p1 ⊃ (p2 ≡ p3 ), and p4 ⊃ (p5 ≡ p3 ), then such a consequence is: ∼ (p1 .p2 .p4 . ∼ p5 )). But it should now be obvious that the “theorem” of the factual emptiness of Rb sentences holds only for isolated Rb -sentences, and so only for those of them that really involve closed concepts. If “p1 ⊃ (p2 ≡ p3 )” is accompanied by “p4 ⊃ (p5 ≡ p3 ),” it makes good sense to say “even though p1 .p2 was verified, it is probable that ∼ p3 ,” for the probability judgment could be supported by the evidence p4 . ∼ p5 . And since a set of convergent Rb -sentences is never constructed (I did not say reconstructed) all at once, but, as Carnap himself observes, new R-sentences for the same term are laid down as new discoveries are made, reduction sentences are rarely introduction-sentences at all. Indeed, it is just because the reduced term does have antecedent meaning that the Rsentence can without paradox be said to function both as meaning rule and as expression of an empirical law. Before proceeding with the elaboration of the positive thesis of this paper, I wish to clear away a possible objection to the view that an isolated Rb -sentence, such as the one for “soluble,” is not a partial meaning rule at all, and hence could easily be replaced by a (nominal) explicit definition in a non-extensional descriptive language. Let such an explicit definition have the form: S (x, t) ≡d f if O(x, t), then R(x, t). Then “S ” is clearly predicable if, on performing O, R occurs, and “non-S ” is clearly predicable if, on performing O, R fails to occur. But what about the x’s upon which O has not been performed? Do they or don’t they have the defined disposition? According to the argument which I wish to refute, the definition leaves the question whether those x’s have the disposition undecidable; in other words, the definition does not permit us to apply the law “(∀x)(S x ∨ ∼ S x)” since, on the contrary, we have: (∀x)(∼ O(x, t) ⊃ (∼ S (x, t) . ∼ (∼ S (x, t))). Thus Carnap, referring to the Rb sentence for “soluble,” writes: “If a body b consists of such a substance that for no body of this substance has the test-condition—in the above example: ‘being placed in water’—ever been fulfilled, then neither the predicate nor its negation can be attributed to b” (ibid.) Carnap is of course right since he is referring to an extensional reduction sentence, not to an explicit definition in terms of causal implication. From O(x, t) we can deduce, with the help of

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the Rb -sentence, the equivalence S (x, t) ≡ R(x, t), but since it is impossible to decide whether R is predicable without performing O, it is on the mentioned supposition strictly undecidable whether S is predicable. But in the case of the contemplated explicit definition no such undecidability arises, for the following reason. If O has not been performed upon x, then it is either conceivable that it might be or it is inconceivable that it might be performed on x. If, e.g., x is the wooden match, mentioned by Carnap, which was just burnt up, then the first alternative holds: it is clearly conceivable that the match had not been lit and had instead been thrown into water. “Conceivable” is here used in the sense of “it is meaningful to suppose that O be (or had been) performed on x” (no commitment to any theory of meaning is necessary in the context of this argument). If so, then we simply do not know whether or not “S ” is predicable of this x, at least not with certainty—an obvious qualification in view of the possibility of indirect confirmation by means of laws of the form “if one instance of kind K has disposition D, then all instances of K have D.” On the second alternative, it does not make sense to suppose that O be performed upon x. For example, the sentence “a hydrogen atom is thrown into water” is presumably meaningless. If so, the sentence “hydrogen atoms are soluble in water” is itself meaningless, and in that sense neither true nor false. But thus it appears that the argument according to which the explicitly defined dispositional predicate does not fall under the law of the excluded middle merely shows that the definiendum has a limited range of significant application. Since this holds presumably for any predicate, it would follow that complete meaning-specification is in all cases impossible. I proceed now to formulate the argument from surplus-meaning against explicit definability of such concepts as “electric current,” “mass,” “temperature” (and it will turn out that a similar argument holds for qualitative concepts of kinds of things). If only we take a close look at the thesis of explicit definability, at the consequences entailed by it, we shall convince ourselves of its untenability. Suppose, then, that out of the n laws of functional dependence1 involving the quantity Q as variable, one be selected as definition of Q (and thus shifted from the object-language to the meta-language), while the others are interpreted as empirical propositions about Q (not about the symbol “Q”). To fix our ideas, suppose we accepted Mach’s definition of mass in terms of the ratio of the accelerations mutually produced in two interacting particles, which turns the third law of motion, via the second law, into a definitional truth. Let “p” stand for a predication of the defined functor (such as “the

1 It

might be noted that such functional laws have precisely the form of bilateral reduction sentences since on the condition which is not expressed in the equation itself a determinate change of x is a necessary and suﬃcient condition for a determinate change of f (x). E.g., on the condition of constant gas pressure, a doubling of gas temperature is a necessary and suﬃcient condition for a doubling of gas volume.

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mass of particle A is m”), and “E” for the evidence which, by the definition, is logically equivalent to p (such as “the ratio of the acceleration imparted to A by unit particle B to the acceleration imparted by A to B is equal to m”). Then Prob(p/E) = 1. This means that the outcome of any further tests of p, based on contingent laws connecting mass with other quantities, would be irrelevant to the question of the truth of p if only E is accepted as certain. And if all these other tests yielded a value for the mass of A inconsistent with m, then all the contingent laws would have to be abandoned while p would remain unshaken. Thus, suppose that A and B had equal mass by Mach’s definition (i.e. m = 1), but that we found that A and B produced unequal strains on a spring scale, further that they did not balance on a beam-balance, further that they exerted unequal gravitational attractions on a given mass at a given distance (since the law of gravitation is an independent assumption of mechanics, not deducible from the third law, this is a logical possibility), and so forth. We still would be logically compelled to stand by our hypothesis. For the method of explicit definition here criticized implies that, where E1 , . . . , En is the sum-total of evidence confirming p by virtue of contingent laws, Prob(p/E) = Prob(p/E.E1 . . . . . En ) = Prob(p/E. ∼ E1 . . . . . ∼ En )! I submit that this is highly counterintuitive, and that no scientist would act in accordance with such peculiar equations.2 The following illustration should clarify the point at issue. Suppose a scientist, finding that each metal has a unique melting-point, decides that it is best to define metals by their melting-points. The question before the philosopher of science is how to interpret the meaning of “definition” as used in this context. My thesis is that it should not be interpreted as a declaration of synonymy (such as “ ‘iron’ is a synonym for ‘substance with melting-point M’ ”) but rather as an assignment of great probability-weight to a selected reduction sentence for the name of the metal. Suppose, then, that while according to past experience 2 In

a recent article, Bergmann 1951b, Gustav Bergmann argues that a psychological concept like “seeing green” is definable (in use) in terms of a stimulus-response sequence in just the same way in which a physical construct like “electrical field” is definable (in use) in terms of a subjunctive conditional “if an electroscope were placed at P at t, then the leaves of the electroscope would diverge.” Bergmann is aware of the argument from surplus-meaning against such explicit definitions of physical constructs, the argument “. . . that it (the definition) does not do justice to all we mean by ‘electric field’ . . . ; that there are many other ‘tests’; that the description of any one of these could equally well serve for R (read: the definiens); and that this very fact, together with other laws about electric fields, belongs to the meaning of ‘electric field’ ”(Bergmann 1951b, 99). But his answer is anything but convincing: the opposition is simply accused of the elementary confusion between a (meta-linguistic) statement about the meaning of “electric field” (that’s what the definition is) and physical statements about electric fields. Bergmann gives no evidence of being aware of the really serious reason behind the objection from the “surplus-meaning” of constructs: that, on the formalization of the theory of electrical fields proposed, the degree of confirmation of the existential hypothesis about the electric field relatively to the electroscopic evidence would be a maximum, so that a negative outcome of all other possible tests could have no tendency whatever to diminish the credibility of the existential hypothesis but would instead logically compel us to abandon all the physical laws about the electric field defined as proposed.

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any substance with melting-point M had properties P1 , . . . , Pn (for which reason M was selected as a reliable indicator of these properties), the scientist is suddenly confronted with a specimen having M but lacking all of these properties. Would he really insist that the specimen is an instance of the metal in question and that the generalization “all instances of this metal have properties P1 , . . . , Pn ” had simply been refuted? Should such anomalous specimens turn up frequently, I think it likely that he would be frankly “illogical” and say “probably all these specimens ought to be classified as the metal in question, and we’d better give up the definition in terms of melting-point”: such a statement, be it noted, sounds illogical if we attach to the word “definition”3 the meaning customarily attached to it by logicians who like things to be neat, but not if we interpret it as a reduction sentence. For, as we have seen, the very systemic occurrence of a reduction sentence makes it possible for it to be corrected through its associates. If it be asked whether, according to the view here presented, a generalization about a natural kind, like “all iron has melting-point M,” is analytic or synthetic, it must be replied that this dichotomy is inapplicable to propositions involving open concepts. If an explicit definition of “iron,” in the sense of a statement of synonymy, were at hand, then the question would be appropriate; but this is precisely the presupposition which is here denied. Suppose the question were raised “is it self-contradictory to suppose that a lemon tasted just like an apple, in other words, does ‘x is a lemon’ analytically entail ‘x does not taste like an apple’?” Clearly the question cannot be answered since the line between the defining properties of lemons and those properties which lemons have contingently just is not clearly drawn. This is what I mean by calling such a concept open; an alternative terminology would be to say that such class-names as “lemon” or “iron” are intensionally vague4 . It is not de-

3 The

fact that not only the admittedly unprecise class-concepts of everyday language, but likewise the precisely “defined” class-concepts of science are really open concepts, was clearly seen by Waismann, in (Waismann 1945). Waismann makes the point so well that he deserves to be quoted at some length: “The notion of gold seems to be defined with absolute precision, say by the spectrum of gold with its characteristic lines. Now what would you say if a substance was discovered that looked like gold, satisfied all the chemical tests for gold, whilst it emitted a new sort of radiation? ‘But such things do not happen’. Quite so; but they might happen, and that is enough to show that we can never exclude altogether the possibility of some unforeseen situation arising in which we shall have to modify our definition. Try as we may, no concept is limited in such a way that there is no room for any doubt. We introduce a concept and limit it in some direction; for instance, we define gold in contrast to some other metals such as alloys. This suﬃces for our present needs, and we do not probe any farther. We tend to overlook the fact that there are always other directions in which the concept has not been defined. And if we did, we could easily imagine conditions which would necessitate new limitations.” (Waismann 1945, 122-23) 4 I contrast intensional vagueness with extensional vagueness, the sort of vagueness besetting terms like “hot,” “bald,” “blue,” since intensional vagueness is predicable only of complex predicates, predicates whose meaning can be explained by verbal description. Intensional vagueness entails, I suppose, extensional vagueness, since if we are not certain about the intension of a term we cannot be certain about its extension either, but the converse entailment does not hold since unanalyzable predicates may be extension-

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nied that the semantic rules for a word like ‘lemon’ are determinate to some extent; a man who applied the word “lemon” to, say, a glass could promptly be accused of semantic sin. But while it may be easy to agree on the genus of an explicit definition, trouble will quickly come when the diﬀerentia is to be specified. If a man should confidently declare, e.g., that sourness forms part of the meaning of the term, his confidence could easily be shaken by asking him “would you unhesitatingly, then, classify a thing as a non-lemon if it resembled lemons in all respects except that it tasted like a sweet apple?” The point is that the semantic rules governing such class-names are not suﬃciently determinate to force a clear-cut decision between the classification “non-lemon” and the classification “apple-tasting lemon.” It may be asked at this point how this admitted phenomenon of intensional vagueness that characterizes a qualitative thing-language bears on the question, here primarily at issue, of the explicit definability of scientific concepts. Prima facie, it is diﬀerent with a scientific language, like the language of quantitative physics, since here we do find explicit definitions, carefully constructed by the scientists. But as pointed out before, if a physicist like Mach speaks of a “definition” of “mass” in terms of the third law of motion he does not attach to the word “definition” the meaning underlying the analytic-synthetic distinction. I would like to make the point quite clear by adding as illustration physical definitions of such a fundamental concept as “time-congruence.” As is well known, such a definition consists in the selection of a standard-clock (where “clock” has the generalized meaning of “physical system in periodic motion”), such as the rotating earth, and if interpreted as a declaration of synonymy, turns the statement that the selected standard clock goes at a uniform rate into a tautology. Thus, relatively to the convention to designate as equal time intervals during which the earth rotates through equal angles, the statement that the earth rotates uniformly would be a tautology. Now, that such a definition was never intended by physicists as a declaration of synonymy should be evident from the fact that quite early in the history of post-Copernican physics it was recognized that the earth’s rotation cannot be strictly uniform, for various physical reasons. It might be replied that when such a statement was made a tacit shift to a new standard-clock must already have occurred, such that “uniform motion” in the statement “the earth does not rotate uniformly” had a diﬀerent meaning from its meaning in the tautology (relatively to the original convention) “the earth rotates uniformly,” and that therefore an interpretation of the definition as a declaration of synonymy does not really force us into the absurd claim that subsequently physicists contradicted a tautology. But the reply is unconvincing since recognition of the non-uniformity of the original standard-clock ally vague and it does not make sense to distinguish between the defining and the accidental properties in the case of simple concepts.

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motivates the physicist to look for a better standard-clock, which presupposes that the original standard-clock is rejected as non-uniform before a better one has been selected. Actually, what leads to such a rejection is the fact that the original “definition” together with physical principles implies physical consequences that are not borne out by observational evidence and it is found more convenient to change the “definition” and retain the physical principles. Thus it turns out that if time is measured in terms of the earth’s rotation, then the centripetal acceleration of the moon as calculated by the inverse square law is discrepant with the same acceleration as calculated on the basis of the law of 2 uniform circular motion: a = vr , where v is the linear velocity with which the particle revolves around the center, and a is the acceleration directed toward the center. Since the latter law is a direct consequence of the general laws of motion, this discrepancy might have led Newton to abandon the law of universal gravitation. Instead, realizing that the calculation of the moon’s velocity of revolution was based on the use of the earth’s rotation as time-measure, he suspected that consistency could be restored by assuming suitable variations in the earth’s velocity of rotation. The moral of the illustration is that a statement like “the earth rotates uniformly” or “light rays move, in vacuo, with constant speed” actually functions like a physical hypothesis subject to correction even though it is called a “definition” or “convention.” By Einstein’s definition of “distant simultaneity,” it would be meaningless to question whether two light-rays emitted from places A and B toward one another, and reaching an observer placed midway simultaneously, really were emitted simultaneously from A and B—the question, that is, would be just as meaningless as the question “is this husband really married” if the definition were a true statement of synonymy. And since the question is equivalent to the question whether the two light-rays really moved toward the observer with the same speed, it would also be meaningless to question the constancy of the speed of light in this context. But suppose, now, that the same observers who perceive the light-signals emitted from A and B simultaneously perceived sound-signals, emitted from A and B simultaneously with the light-signals, in succession, and suppose that this discrepancy occurred regularly. Suppose furthermore that the most refined measurements led to the conclusion, long ago, that the velocity of sound relatively to the sound-source in a given still medium at constant pressure and temperature is constant. Assuming these ideal atmospheric conditions and further strict simultaneity of the emissions of light- and sound-signals from a given place, the hypothetical discrepancy could be resolved in two and only two ways: either we must assume that the sound-waves were propagated from A and B with unequal speeds, or we must make the analogous assumption for the light-rays. Now, according to the theory of physical definition here criticized, it is meaningless to make the latter assumption, while the former assumption makes perfectly good sense. I con-

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tend that if one assumption makes sense so does the other. Such a contention conflicts, indeed, with a famous statement made by the great Einstein himself, but I derive courage from the reflection that this is a statement he made qua philosopher of physics, not qua physicist: That light requires the same time to traverse the path A → M as for the path B → M is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own free will in order to arrive at a definition of simultaneity. (Einstein 1931, 27-28)

The fact is that, as just illustrated, situations are conceivable which would make the question significant whether the velocity of light is really constant, and since the possibility of encountering facts which throw doubt on p is just what marks p as an empirical hypothesis, there is no reason why what Einstein calls a “stipulation” should be sharply distinguished from a physical hypothesis. Now, if such a “definition” of distant simultaneity be interpreted as a (systemic) reduction sentence for an open concept, the paradox of the statement “the physical law itself defines the physical concept” vanishes. Suppose we constructed a pair of Rb -sentences for “distant simultaneity” as follows: (1) if two light-rays are emitted from places A and B when events Ea and Eb happen at those places (so far only contiguous simultaneity has been mentioned!), then Ea and Eb are simultaneous if and only if the light-rays reach an observer stationed midway simultaneously. (2) if two sound-waves are emitted from A and B when events Ea and Eb happen at those places, and if the medium is still (i.e., at rest relatively to the sound-sources) and uniformly dense, then Ea and Eb are simultaneous if and only if the sound-waves reach an observer stationed midway simultaneously. It is, now, conceivable that this pair of convergent laws should be inconsistent in the sense that there are distant events which are simultaneous according to (1) but not according to (2), or vice versa. And if careful experiments revealed such an inconsistency, the physicist would reject either (1) or (2) as probably false even though he may have declared them as operational “definitions.” The designation, therefore, of one out of several convergent laws, call it L, as a “definition” is best interpreted as a declaration to the eﬀect that L will be maintained if inconsistencies of the described kind turn up until all other remedies fail. The theory of open concepts will now be further supported by turning to the tricky problem of the fields of significant application of physical concepts. Carnap points out (Carnap 1937, 449) that what motivates the scientist to specify meanings by reduction sentences instead of laying down explicit definitions is the desire to leave open the field of application of the term beyond the field already investigated. For example, if we laid down an explicit definition by which statements about temperature are synonymous with statements about reactions of mercury thermometers when brought into contact with the sub-

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stance to which temperature is ascribed, we would preclude application of the term “temperature” to, say, the sun. Carnap’s technical explanation, however, of the disadvantage of such explicit definitions seems inadequate. Now we might state one of the following definitions: Q3 ≡ (Q1 .Q2 ) (D1 ) Q3 ≡ (∼ Q1 ∨ Q2 ) (D2 ) If c is a point of the undetermined class, on the basis of D1 ‘Q3 (c)’ is false, and on the basis of D2 it is true. Although it is possible to lay down either D1 or D2 , neither procedure is in accordance with the intention of the scientist concerning the use of the predicate ‘Q3 ’. The scientist wishes neither to determine all the cases of the third class positively, nor all of them negatively; he wishes to leaves these questions open until the results of further investigations suggest the statement of a new reduction pair; thereby some of the cases so far undetermined become determined positively and some negatively. (Carnap 1937, 449).

Carnap’s argument is clearly based on the concept of material implication, and thus the impression is again created as though reduction sentences were a temporary device which could be disposed of once we have an analysis of nomological conditionals. If a physicist were to lay down the definition (temp(x, t) = y◦ ) ≡d f if a mercury thermometer is brought into contact with x at time t, then the top of the mercury will coincide with the mark y at t,

he would obviously intend “if, then” in the sense of nomological implication. If so, the predicability of the defined functor cannot be inferred from the nonfulfilment of the antecedent of the definiens. If for x we substitute the sun and for t any time at all, the antecedent turns into a description of a physical impossibility; but this does not convert the conditional into a true statement (nor, of course, into a false statement, since there is no diﬀerence between material and nomological implication as far as the falsity-conditions are concerned). What would follow from such a definition is rather that it is insignificant to apply the defined term in a situation in which the condition described by the antecedent cannot, by virtue of logical or physical laws, be fulfilled. Perhaps a more telling illustration would be a definition of “the temperature of gas G at t is uniform” in terms of “if V1 and V2 are any two volume elements of G, then the average molecular velocities in V1 and V2 at t are equal.” If now the question were raised whether all the individual molecules of G have the same temperature at t, it would surely be dismissed as meaningless. At any rate, the situation is this: when the physicist speaks of the temperature of the sun, he surely does not mean to predict the result of hypothetical operations with mercury- or gas-thermometers; this will be admitted regardless of whether or not one holds it to be meaningless to speak of what would happen if such a thermometer were brought into contact with the sun. It follows that if such an operational definition explicates the meaning with which “T ” is used in contexts of limited T -ranges, then in extrapolating beyond those limited ranges one uses the same symbol either without meaning or with a diﬀerent meaning.

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This dilemma becomes especially apparent if we turn from extrapolation to high temperatures to extrapolation to low temperatures, and take a fresh look at the old question whether it is meaningless to speak of a temperature below the absolute zero. If the operational definition of temperature in terms of the gas thermometer were a true statement of synonymy, then the hypothesis that a substance has a temperature lower than −273◦ C would assert that if thermal contact were established between the substance and a gas thermometer the latter would be found in a state of negative volume and negative pressure—which is surely nonsense. But on the other hand, strong reasons can be adduced for the meaningfulness of the hypothesis of temperatures lower than the absolute zero. It is, after all, a contingent fact that the pressure coeﬃcient (or the coeﬃcient of volume expansion, which has the same value) for gases has the value it 1 ) has been verified only for a limited range has. And at any rate this value ( 273 of temperature variation. It would seem significant, then, to entertain the possibility that complete shrinkage of a gas has not yet occurred when it reaches −273◦ C.5 But then again, one might plausibly argue that in entertaining such a possibility we must be thinking of some other way of measuring T than with the gas thermometer as presently calibrated. Therefore a strict operationalist who holds, with Bridgman, that diﬀerent operations define diﬀerent concepts, would be justified in saying that in the hypothesis “T might sink below the absolute zero” the symbol “T ” must have a diﬀerent meaning from the meaning explicated by the gas laws. But that this second horn of the dilemma, the view that in extrapolating a physical law beyond the experimental range of its variables we change the meanings of the variables and thus really talk about diﬀerent physical quantities, is no easier to take than the first, is not diﬃcult to show. Consider the hypothesis about the sun’s temperature again. This temperature is usually calculated by means of the Stefan-Boltzmann law relating the absolute temperature of a surface to its intensity of heat-radiation (I) in conjunction with the law that I is inversely proportional to the square of the distance of the surface from the heat-source. That is, after the value of I at the sun’s surface has been calculated by means of the latter law, the Stefan-Boltzmann law is applied to calculate the value of T at the sun’s surface. But as this value of T is outside the range to which the “operational” definition of “T ” refers, the “T ” in the conclusion of the mathematical deduction has a diﬀerent meaning from the “T ” in the premises (both the proposition of measurement about the temperature of the earth’s surface and the Stefan-Boltzmann law) of the mathematical deduction. Now, since the terms in the conclusion of a valid argument must

5 We

may disregard in this discussion the fact that there would be no gas anyway at such a low temperature, on account of condensation; for the question is what suppositions are meaningful, and it is surely meaningful to suppose that a substance could survive the gaseous state at such a low temperature.

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have the same meaning which they have in the premises, we are led to this consequence: if the conclusion has any physical significance at all, then it does not follow from the premises which are the sole ground of its credibility! Our problem, then, is to formulate a criterion of identity of meaning of a physical functor in diﬀerent contexts of application; in other words, a criterion for determining whether a symbol like “T ” stands for the same physical quantity in diﬀerent contexts of application. Carnap’s theory of reduction sentences suggests this solution: if the symbol is a “nodal point,” to use a suggestive metaphor, of a system of convergent reduction sentences, then it has the same meaning in all the contexts described by the various members of the system provided the system is consistent. Thus, suppose that on the basis of measurements of the pressure of a thermometric gas, we obtain the result: T (A, t0 ) > T (A, t1 ), and that on the basis of measurements by means of a mercury thermometer we obtain the same result. Then we have confirmed the consistency of the reduction-system converging to “T ,” and thus we have (partially) justified our identification of the meanings of “T ” in these diﬀerent contexts, or our claim that what is measured by these diﬀerent methods is the same quantity. Similarly, if a physicist finds in terms of Mach’s actionreaction test (measurement of accelerations) that two bodies have equal mass, and subsequently verifies that they produce equal strains on a spring-scale kept at constant height (which proves equality of the gravitational forces acting on the bodies), he confirms the hypothesis that the quantity measured in these two ways (in terms of impact-forces in one experiment, in terms of gravitational forces in the other experiment) is one and the same. Now, the method of extrapolating numerical laws involves the employment of the same physical functor in contexts of measurement and in contexts of calculation. But how could it be maintained that the functor has the same meaning in the two kinds of contexts? It must be conceded that to say, with Bridgman, that it still has “operational” meaning in the contexts of calculation since calculations, after all, are also operations, is tantamount to reducing the operationalist theory of meaning to a truism. Nevertheless, if we closely observe the reasons for which a physicist believes that in calculating the values of the sun’s temperature and mass he comes to know the values of the same quantity as is measurable in other contexts (though these calculations cannot be verified by measurements), we shall find that the same criterion of physical consistency is presupposed. If the extrapolated law leads to consistent calculations, the extrapolation is considered justified, and the variables of the equations are said to represent the same quantity no matter whether their values can be determined by measurement or by calculation only. In this connection it is important to distinguish two kinds of test of a physical equation which may be called the correspondence-test and the consistency-test respectively. The correspondence-test of the physical equation y = f (x) consists in

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the three steps: measurement of x, calculation by means of the equation of the corresponding value of y, measurement of y for the purpose of testing the calculation. But if the calculated value of y lies outside the experimental range of y, the correspondence-test is obviously unavailable. All that can be done is to determine whether diﬀerent calculations of y from diﬀerent bases yield consistent results. Thus the mass of the sun was calculated, in Newtonian astronomy, on the basis of the centripetal acceleration of a revolving planet, according to the equation (deducible from the second law in conjunction with the law of . Now, it is logically conceivable that as the calculation gravitation): a = G·M r2 is repeated on the basis of diﬀerent values of a and r, corresponding to the diﬀerent orbits and periods of revolution of the various planets, inconsistent values of M are obtained. If this happened, the law of gravitation would have failed the consistency-test (provided, indeed, that the trouble is not blamed on the general laws of motion), and the extrapolation of the concept of mass to celestial bodies would be suspect. If, on the other hand, calculations of M on the basis of optical data, viz. spectral displacements of the light emitted by atoms on the surface of the sun (which according to the general theory of relativity are due to the sun’s gravitational field), should corroborate the calculations based on mechanical data, the extrapolation would be further justified. The same holds, of course, for extrapolations to microscopic contexts, as when we speak, e.g., of the mass of an atom, or of an electron (think of the extrapolation of the law of conservation of mechanical energy to the motion of electrons, which led to the calculation of the mass of an electron). The main point is that a concept like mass is, by the physicist, left open not only for further experimental contexts but also for calculational contexts in which no correspondence-test is feasible; and that the attribution of an identical property (uniformly called “mass”) is justified to the extent that the extrapolation of experimentally confirmed laws survives the described consistency-test. In recent years the suspicion has been growing on the part of some analytic philosophers that all is not well with the analytic-synthetic distinction as applied to natural, unformalized languages. Whether or not the reasons by which their skepticism is nourished are cogent, the sketched theory of open concepts clearly implies that the analytic-synthetic distinction is not applicable to propositions involving open concepts. This is overlooked by many philosophers of science who conceive it as a major task of “logical reconstruction” to make the language of science logically tidy by sharply segregating object-linguistic sentences which are empirically refutable statements “about reality” and metalinguistic sentences which express stipulations concerning the use of symbols and thus cannot significantly be said to be refuted by facts. They are the ones who say “either ‘F = ma’ is a definition of force, in which case it expresses no physical law; or it expresses a physical law, in which case the meaning of ‘force’ must be determined independently of this equation (say, in terms of

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the spring-balance)”; and since either alternative seems unsatisfactory to some people some of the time, we have the interminable controversy about the logical status of the second law of motion. In order to leave no doubt that this pattern of semantic analysis is ill fitted to the language of physics, let us take a look at one more illustration. The principle of the conservation of heat says, qualitatively, that the amount of heat lost by one part of a thermally isolated system is equal to the amount of heat gained by the remaining part, and quantitatively: m1 c1 (t2 − t1 ) = m2 c2 (t1 − t0 ); for example, the system may consist of a hot piece of iron immersed in a volume of water, in which case the c’s are the specific heats of water and iron respectively, m1 the mass of the piece of iron, m2 the mass of the water, t2 the initial temperature of the piece of iron, t0 the initial temperature of the water, and t1 the temperature at which equilibrium is reached. Since c2 here is unity, by the definitions of specific heat and of the unit of heat (the calorie), c1 can easily be calculated in terms of this equation on the basis of measurements of the masses and the temperatures. But if we adhere to the principle that an equation does not express a physical law unless each quantity involved in it is measurable independently, i.e., without the calculational use of that equation, it becomes doubtful whether the heat-equation could be interpreted as a physical law. For how could c1 be determined without presupposing conservation of heat? The usual definition of specific heat as amount of heat required to raise the temperature of unit mass by unit degree hardly suggests a non-circular method of measurement: “quantity of heat” absorbed by m grams of a substance S as S is heated one degree C is itself defined as the product of m times the specific heat of S . Indeed, the elementary experimental method of determining specific heats is just the method of mixtures, a method altogether based on the assumed validity of the conservation-equation. Most physicists, however, would regard the latter as expressing a physical law even if they had to admit that the method of mixtures is the only method for determining specific heats. For though no correspondence-test could be carried out, the consistency-test would remain possible and this possibility is suﬃcient for bestowing physical content on the equation. Conceivably thermal equilibrium is reached at a diﬀerent temperature the next time the mixture experiment is performed with the same masses of the same substances at the same initial temperatures, and in that case the method of mixtures would fail to yield unique values of specific heat. And the physicist is inclined to say that the equation has physical content inasmuch as its consistency, in the sense illustrated, can be established only by experiment. Those who are relentless in enforcement of the analytic-factual dichotomy will no doubt reply: The analytic statements of physical theory can be disentangled from its factual statements easily enough. The heat-equation is nothing more than a definition of specific heat relatively to a conventional unit of specific heat. But the gen-

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Reduction Sentences and Open Concepts (1953) uinely factual proposition is the proposition that the defined property is a constant for a given substance. Mach made just this logical point in connection with the concept of mass. According to his definition of mass, the equation of conservation of momentum (which, by the second law of motion, is equivalent to “action is equal and opposite to reaction”) is analytic. But that the same value of the mass of a given body is obtained no matter how the conditions of the interaction-experiment are changed, this is a contingent fact, not deducible from the definition.

This argument would, indeed, be convincing if the equations which it is proposed to move into the meta-language of physics6 could be interpreted as explicit definitions of closed concepts. But since, as has been shown, the word “definition” refers in this context to nothing but a reduction sentence selected as reliable indicator, it only looks as though the tidy pattern of analysis had won out. We have seen that a systemic reduction sentence has factual content inasmuch as it is open to correction through the other members of the system, but at the same time is a (partial) meaning rule. It is surprising that so little cognizance has been taken of this unorthodox implication of Carnap’s theory of reduction sentences by the advocates of the either-or pattern of semantic analysis. It will be shown, now, that a close look at the meaning of the “if, then” occurring in reduction sentences brings into sight a hitherto neglected concept of semantics which I choose to label “quasi-semantic probability implication.” Consider again the simple implication “if x is a lemon, then x is sour.” It is not analytic, for, as we have seen, circumstances are conceivable under which we would be strongly inclined to classify x as a lemon even though x is not sour. According to what I called the either-or pattern of semantic analysis, one would conclude that “sour” forms no part of the meaning of “lemon” nor is deducible from the definition of “lemon”—in short, that the implication is synthetic. But the obvious trouble with this conclusion is that it presupposes that “lemon” is a precisely defined class-term, that we can draw a sharp line between those properties diﬀerentiating lemons from other kinds of fruit that are “logically connoted” by the class-term and those which lemons have contingently. Since, on the contrary, such class-terms are intensionally vague, the sharp concept of logical connotation had better be replaced by a continuous concept “term T connotes property P to degree x.” We might ask, for example, whether people would more readily refuse application of the term “lemon” to a fruit which, though mature, is not yellow than to a fruit which is not sour; 6 This

type of reconstruction of the language of physics seems to be recommended, e.g., by Philipp Frank, in Frank 1946. Thus he says that the second law of motion could be regarded as an operational definition of mass: “the ratio of two masses is inversely proportionate to the accelerations which they get from one and the same force,” but adds: “This definition is only unambiguous if we obtain the same value of mass whatever force we apply” (Frank 1946, 14). The physical law, then, is not the equation, but the statement that the defined quantity is constant with respect to a specified group of variations.

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and if the answer is aﬃrmative, “lemon” might be said to connote yellowness to a higher degree than sourness.7 But if it is possible for a lemon to be non-sour, in the sense that the complex of qualities which is usually compresent with sourness might be present in some instance without sourness, then “if x is a lemon, then x is sour” ought to be interpreted as a probability implication. Similarly, a term like “furious” stands for a correlation of an emotion with bodily symptoms of the emotion, such as fiery eyes, but situations are conceivable in which we would apply the term “furious” to a smiling person; a definite kind of facial expression may be connoted by this psychological term to a high degree, yet it is not connoted to the maximum degree. Therefore, “if x is furious at time t, then x does not smile at t” ought to be interpreted as the same kind of probability implication. Yet, it is only by means of such probability implications that the diﬀerential meanings of such terms can be explicated (“diﬀerential” meanings are contrasted with “generic” meanings, such as “fruit” in the case of “lemon,” “emotion” in the case of “furious”). I call such implications “quasi-semantic,” in analogy to Carnap’s term “quasi-syntactic,” since they convey semantic information and yet belong to the object-language. If the highly pertinent question should be raised how the concept of probability involved in such implications which state, not the truth-conditions, but the probability-conditions of a sentence, is to be explicated, negative answers can, at this stage, be given more readily than positive answers. It can be identified neither with Reichenbach’s frequency concept nor with Carnap’s logical concept of degree of confirmation, and for exactly the same reason: If “Prob(F x/Gx) = p” is an empirical statement about a frequency, then it can presumably be determined whether an object falls into the reference-class G independently of ascertaining whether it has property F. Even if it should not be possible to state a suﬃcient condition for the truth of “Gx” and only criteria could be specified which make it more or less probable that the predicate “G” applies, those criteria surely could not involve a reference to the very property whose relative frequency in class G is to be determined (substitute, for illustration, “suﬀering from tuberculosis” for “G,” and “living another ten years” for “F”). If the above formula, on the other hand, expresses an assignment of a degree of confirmation in Carnap’s sense, it is still more obvious that it 7 The

idea, here advanced, of replacing the clear-cut relation of meaning with a continuum of degrees of meaning may be regarded as an extension of the theory of vagueness put forth long ago by Max Black in the study Black 1937. Black there attempts to generalize the laws of logic (specifically, the law of the excluded middle) so as to make them applicable to vague predicates. In order to do so, he replaces the function “L is applicable to x” with the more complicated function “L is applicable to x with degree mn ” which is defined as follows: the limit approached by the ratio of the number of applications of L to x(m) to the number of applications of non-L to x(n) as the number of discriminations of x with respect to L and the number of observers increases indefinitely, is mn . This gradation of the relation of denotation (predicability) naturally suggests a similar gradation of connotation as here called attention to.

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diﬀers from the kind of probability implication here presented for further examination. For Carnap’s degree of confirmation is a measure of the amount of overlap of the ranges of two sentences, where the range of a sentence is defined as the class of state-descriptions in which the sentence would be true. These ranges, however, are determined by the semantic rules of the language, and the semantic rules for sentences have the form of statements of truth-conditions. The idea of probability-conditions as meaning-specifications seems, therefore, to be foreign to Carnap’s treatment of logical probability.8 As just remarked, a probability implication “Ax −− Bx” in Reichenbach’s sense (or, for classes of successive events, “Axi −− Byi ”) cannot be regarded as a meaning-specification of “A” in terms of “B,” since Reichenbach defines the concept of probability in such a way that it must be possible to determine whether an element belongs to the reference-class A independently of determining whether it, or the corresponding element, falls into the attribute-class B. However, if we take the definition of “meaning” in terms of psychological (or behavioral) dispositions seriously, we might nevertheless analyze the statement “if A, then with probability p, B,” intended as a meaning-specification, as a statistical generalization about verbal behavior, and thus, to coin a new term for a hitherto neglected concept, as a quasi-semantic probability implication: If a person (of specified description, such as “making a truthful statement”) verbalizes “A,” then with probability p he expects, or is disposed to believe, B—where p is a relative frequency. Notice that the reference class of this statistical law of behavioristics is the class of persons of specified description verbalizing “A,” and not the class which “A” designates; it can therefore be maintained without contradiction that membership of this reference-class can be determined independently of membership of the corresponding attributeclass. At least this tentatively suggested analysis works in some cases, if not in all. Consider, e.g., the question whether “having a back” is part of the meaning of “chair,” so that it is self-contradictory to call a stool a “chair,” or whether stools may be regarded as a species of chair. The fact is, of course, that the relevant linguistic habits are by no means fixed. All we can say is that the fact that x is called a chair makes it probable to some degree that x is believed to have a back, where the probability in question may be interpreted as a frequency. But notice that the quasi-semantic probability implication in the object-language 8A

sketch of the program of generalization of the concept of connotation (designation, intension), somewhat analogous to the generalization of the concept of implication in Reichenbach’s probability-logic, was presented by A. Kaplan, in Kaplan 1946. The sketch has recently been worked out in formal details, see A. Kaplan and H. F. Schott (Schott and Kaplan 1951). The central idea is the replacement of the concept of “necessary and suﬃcient condition” for membership in a given class with the continuous concept of indicators, of varying probability-weights, of membership. Probability implications are to replace analytic implications as means of specifying the meanings of class-terms. However, the above considerations make it doubtful whether the concept of probability involved in such partial meaning rules could be identified either with Reichenbach’s frequency concept or with Carnap’s concept of logical probability.

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“if x is a chair, then with probability p, x has a back” diﬀers in kind from a probability implication like “if x is a human male, then with probability p, x is white.” In the first place, it would be incorrect to interpret the latter probability implication as semantic, since its validity obviously does not depend on the existence of language-habits: if no language at all existed, it might still be true that a certain proportion of human males are white. Second, there is no connection at all between the meanings of “human male” and “white,” even though the corresponding classes overlap. According to the sketched analysis, “‘S ’ means D with probability (or degree) p” is a statistical generalization of descriptive semantics. It would therefore be meaningless, on the basis of this analysis, to ask to what degree “S ” means D in a single instance of verbalization. For this reason, it may be prudent to give consideration to an alternative analysis which is compatible with the meaningfulness of such questions. “Meaning to degree p” might be interpreted as designating a given strength of expectation (or, of conditioned response, if “mentalistic” terms must be avoided), where strength of expectation is measured in some appropriate way to be determined by psychometricians. Assuming that “S ” represents a declarative sentence and “D” a state of affairs, degree of meaning D would be related to reluctance of withdrawing S as follows: the greater the degree with which “S ” means D, the smaller the reluctance with which “S ” will be withdrawn (or “not-S ” will be asserted) in case D is disbelieved. To illustrate, consider the following possible properties of a chair: (a) having at least three legs, (b) capable of seating just one person, (c) having a back. Correspondingly let us construct the functions: “chair” means (for a fixed individual) (a) to degree p1 ; “chair” means (b) to degree p2 ; “chair” means (c) to degree p3 . It is reasonable to conjecture that p1 is the highest and p3 the lowest degree. In that case the reluctance with which the individual will call an object “chair” if it is believed to lack (a) is greater than the reluctance with which he will call it “chair” if it is believed to lack (b), and still greater than the reluctance with which he will predicate the word if it is believed to lack (c). It is evident that systemic reduction sentences are just such quasi-semantic probability implications, a fact which is concealed by their formulation as material implications. For, suppose that, using “if p1 , then (if p2 , then p3 )” as a rule of inference, we conclude p3 on the basis of verification of p1 and p2 . If this inference were deductive, then, if all the negative reduction sentences for the same open concept (i.e., those formulating conditions under which ∼ p3 may be asserted) led to the contradictory conclusion, we would simply have to throw them out on the ground that p3 had already been established conclusively. This would be tantamount to treating the negative reduction sentences as refutable and the positive ones as analytic—an absurd asymmetry. A system of reduction sentences is rather a system of reciprocally confirming hypothe-

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ses: the evidence p1 .p2 confirms the hypothesis p3 , and the other members of the system point to further tests which may further confirm or disconfirm the hypothesis. The very applications of the method of reduction of terms which Carnap mentions show that the connections between the reduced terms and the terms of the reduction basis are probability connections and not deductive ones. This emerges clearly from his discussion of behavioristic reduction of psychological terms, in Foundations of the Unity of Science. Speaking of the possibility of behavioristic reduction of a term like “angry,” Carnap writes: It is suﬃcient for the formulation of a reduction sentence to know a behavioristic procedure which enables us—if not always, at least under suitable circumstances— to determine whether the organism in question is angry or not. And we know indeed such procedures; otherwise we should never be able to apply the term ‘angry’ to another person on the basis of our observations of his behavior, as we constantly do in everyday life and in scientific investigation. A reduction of the term ‘angry’ or similar terms by the formulation of such procedures is indeed less useful than a definition would be, because a definition supplies a complete (i.e., unconditional) criterion for the term in question, while a reduction statement of the conditional form gives only an incomplete one. But a criterion, conditional or not, is all we need for ascertaining reducibility. (Carnap 1934, 419)

Carnap here says that we know behavioristic symptoms which are suﬃcient conditions, but none which are suﬃcient and necessary conditions, for a state of anger and a state of non-anger respectively. But clearly we cannot assert with any greater confidence that the presence of a given behavioristic symptom is invariably accompanied by the presence of anger than that the absence of such a symptom is invariably accompanied by the absence of anger. Since the correlation between behavioral states and mental states is neither one-many nor many-one (i.e., many-many), probability implications are all we can establish no matter whether a suﬃcient condition or a necessary condition for a given mental state is in question. Just as the possibility of strong self-control, mentioned by Carnap, makes it diﬃcult to find a behavioristic term which expresses a strictly necessary condition for anger, so the possibility of putting on an act makes it diﬃcult to find one that expresses a strictly suﬃcient condition. The theory of open concepts has a particularly noteworthy implication for the psycho-physical problem, which may just be touched upon in conclusion. It was hinted above that just in the way class-terms like “lemon” connote a correlation of qualities, so names of mental states like “furious” connote a correlation of a feeling with bodily expressions and behavior. According to the tidy pattern of semantic analysis, we ought to distinguish clearly the meaning of “furious,” which is an introspectable emotion, from the logically contingent accompaniments of the mental state, such as a red face and a trembling voice. The psycho-physical law “whenever x is in mental state M, then x exhibits bodily and behavioral expressions B” is, on this view, clearly a contingent proposition. But the question whether this is a contingent proposition may be

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like the question whether “all lemons are sour,” or “all lemons are yellow” is a contingent proposition. As we saw, instances are conceivable which we would be inclined to classify as lemons even though they are not sour, or not yellow: in this respect the propositions are contingent. But if we were pressed to mention diﬀerentiating defining properties of lemons, we would be likely to mention just such properties as these: in this respect the propositions are analytic. The trouble, of course, comes from treating “lemon” as a closed concept and hence attempting to apply the dichotomy analytic-contingent whereas some such continuous concept as “degree of connotation” ought to be applied. Similarly, the question about the logical status of the psycho-physical law presupposes that a term like “furious” either is, or is not, definable in terms of behavior. But clearly such terms are not “introduced” into a natural language through verbal definition. They come to mean what they mean by learned association with both introspectable (mental) states and bodily and behavioral expressions. We should not therefore ask whether the latter are logically connoted but instead to what degree they are connoted by the psychological terms. It may be that the degree to which the introspectable feeling is connoted is much higher than the degree to which any given “expression” of the feeling is connoted. Thus it may be that I would not retract the statement “I am furious” even if it were pointed out to me that I exhibit all the symptoms of a happy, contented man, since the emotion is undeniably present. On the other hand, there are names of emotions, like “love,” which connote public symptoms to a higher degree; this means that if one is convinced by others that one’s behavior diﬀers from the characteristic behavior of people who are “in love,” one would more readily retract the introspective judgment. Thus the inconclusiveness of the behaviorism-dualism dispute is ultimately due to the uncritical application to a natural language involving open concepts of a semantic meta-language, involving the Carnapian dichotomy “Limplication-F-implication,” which is suitable only to language-systems. If space permitted it, I would argue that the uncritical use of this dichotomy lies also at the root of another time-honored inconclusive controversy, that of phenomenalism vs. realism: do statements about physical reality mean sensedata? Perhaps analytic philosophy has come to the stage where it is ripe for a “Copernican revolution” analogous to the one credited to Kant: we have to scrutinize the semantic tools with which we explore semantic reality; before proceeding with asking questions of the sort “does p really entail q, and is r really self-contradictory,” we ought to give more serious attention to the question “what do we mean by ‘entailment’ as applied to statements involving open concepts?”

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Appendix [Editors’ note: We include in this appendix Pap’s reply to a brief discussion of chapter 19 by A. Caracciolo and V. Somenzi. Both the discussion and reply follow the article reprinted here as chapter 19 in Methodos vol. V, no. 17. (1953), at 43-4.] In chapter 19 I made the point that a physicist’s selection of a law as a “definition” of a physical quantity (as e.g., Mach’s definition of mass in terms of the third law of motion) is best interpreted as the assignment of a great probability-weight to one member of a system of reduction sentences. This interpretation was contrasted with the interpretation of such a definition as a rule of substitution of symbols which is devoid of “factual content,” i.e., such that it is meaningless to speak of its empirical confirmation or disconfirmation. Now, I am quite uncertain whether my critics really disagree with me on this point. Their explicit denial that such a selection “entails attribution to the selected reduction sentence of a greater probability-weight than to other possible indicators” suggests that they do. But they are silent about their reasons for denying it. My reason for aﬃrming this is, briefly, the following: suppose we consider, for simplicity’s sake, a reduction-system for “mass” consisting of just two members, viz. the third law of motion and Hooke’s Law in the special form “bodies of equal mass suspended successively on a spring at constant level produce equal strain,” and suppose that this system is discovered to be inconsistent in the sense explained in chapter 19. Assuming that the source of the inconsistency is theoretical error and not experimental error (there is no doubt, we may suppose, that the bodies imparted equal accelerations to each other and that they produced unequal strains), one will have to conclude that one or the other of the two laws is false. But once this inevitable conclusion has been drawn, the question will arise: which is more likely to be false? Now, on the interpretation of such a “definition” (like Mach’s definition of mass) as a logical equivalence— or more exactly, a meta-linguistic rule of substitution relatively to which the object-linguistic statement “equal masses impart to each other equal accelerations” is a tautology—it does not even make sense to say that on the experimental evidence the definition is probably false. Since I have repudiated this interpretation (for reasons to be found in chapter 19), the only alternative interpretation that remains is that the law which is called the “definition” is that law which would be considered less likely to be false in case such an inconsistency were to arise. The concept of “probability-weight” here involved is a pragmatic concept (in the terminology of Morris and Carnap) since it refers to the attitudes of scientists toward their theories when faced with the necessity of revising one or the other of them; although a scientists’s degree of reluctance to abandon a given law is, of course, closely related to the objective degree of confirmation (Carnap’s semantic concept) which he assigns to it on the basis of past experience. My critics seem to prefer what I chastisingly referred to as the “tidy (either-or) pattern of semantic analysis,” according to which a given law is in a given context of inquiry either analytic (definitional) or synthetic, though, as they point out, it is up to the inquirer to make the choice. But thus they simply dodge the main argument of chapter 19, which was that since physical laws involve open concepts the analytic-synthetic dichotomy is not applicable to them; that a reduction sentence is, by virtue of its systemic occurrence, both a (partial) meaning rule and a factual statement; and that accordingly the analytic-factual dichotomy should be replaced by a continuous concept (concept admitting of degrees) which I choose to call “quasi-semantic probability implication.”

Chapter 20 EXTENSIONAL LOGIC AND LAWS OF NATURE (1955)

By an extensional logic I mean a logic satisfying the following two requirements: a) all connectives are extensional, i.e., the truth-values of compound statements formed by means of them depend only on the truth-values, not the meanings, of the component statements; b) any two predicates of equal extension, no matter how diﬀerent their meanings, are mutually substitutable salva veritate in any context. The example par excellence of such a logic is, of course, Principia Mathematica, where the concept of “material” implication satisfies the first, and the concept of “formal” implication the second requirement. The insuﬃciency of this logic to deal with the logical modalities (in particular the concept of deducibility, since “q is deducible from p” is obviously not a truth-function of p and q) has been realized for a long time, ever since C. I. Lewis constructed his system of “strict implication,” and in recent years logicians have been intensively occupied with the problems of modal logic. At the same time, however, much attention has been given, especially by the analytic philosophers in the United States and in England, to the problem of interpreting an important class of contingent statements, which prima facie violate the postulate of extensionality, by means of the extensional language of Principia Mathematica (supplemented, perhaps, by a meta-language containing the syntactic concept of derivability): I am referring to the subjunctive conditionals and statements of laws of nature, which abound in both everyday language and scientific language. Among these analysts there are some who continue to believe that this language (or language-structure) is adequate for the expression of all genuine propositions. As against them, I am going to argue that these eﬀorts to reconstruct the mentioned type of statements in terms of material implication plus derivability are bound to fail, and that some kind of intensional implication must be accepted. As the subjunctive conditional first raised its ugly head when the problem of explicitly defining disposition predicates was first investigated (by Carnap, in Carnap 1937), it will be convenient to begin with a brief consideration of

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this problem. Since material conditionals are capable of vacuous truth (truth due to the falsity of the antecedent), the reconstruction of “if x were immersed in water, then x would dissolve,” which is a natural analysis of “x is soluble in water,” as a material conditional leads of course to the paradoxical consequence, pointed out by Carnap, that a match which is kept dry until it is burnt up, is soluble in water. But the reduction sentence: (∀x)(∀t)(Q1 (x, t) ⊃ (Q3 (x) ≡ Q2 (x, t)))

(R)

which avoids this paradoxical consequence of the explicit definition: Q3 (x) ≡d f (∀t)(Q1 (x, t) ⊃ Q2 (x, t)

(D1 )

has in turn counterintuitive consequences, in the face of which the desire to return to explicit definitions, of a more complicated character, which avoid the shortcomings of both (D1 ) and (R), is quite natural. That (R) has counterintuitive consequences is easy to show. As Carnap explicitly pointed out, it attributes a meaning to Q3 only within the class determined by Q1 ; for individuals outside this class the question whether or not they have the dispositional property is undecidable, and since this undecidability is due to the fact that the disposition predicate is not defined for non-members of that class, it is a theoretical, not just a practical, undecidability. In other words, with respect to such individuals it is meaningless to say either that they have or that they do not have the disposition in question. But thus it appears that Carnap’s device throws us from Scylla into Charybdis: explicit definitions were discarded because they entail that any individual upon which the relevant kind of experiment is never performed has the disposition, but now we are faced with the no more palatable result that it is meaningless to attribute the disposition to such an individual. Carnap tried to escape from this consequence by conceding that it is still meaningful to attribute a disposition Q3 to an individual b outside the class determined by Q1 , provided that b belongs to a natural kind (such as wood, in the case of the match mentioned above) which overlaps that class. Thus we have confirming evidence for the law “all wood is insoluble in water,” and therefore the question “is this match, which has never been, and never will be, immersed in water, soluble in water?” is still significant on Carnap’s theory. But two serious objections remain. First, assume a disposition which is unique to a particular individual. For example, a particular human being may have the disposition to feel nauseated when exposed to the smell of an orange, and it is logically possible than no other organism has that disposition and that the disposition is never actualized for the simple reason that the individual concerned is never exposed to the fatal smell. To be sure, as long as the class determined by Q1 (in this example, the class of organisms exposed at some time to the smell of an orange) is not empty, confirming or disconfirming evidence with respect to the statement “if this individual were exposed to the

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smell of an orange, he would feel nauseated” is still obtainable. But what if no oranges existed? Then the class Q1 would be empty and hence the above statement would definitively be meaningless by Carnap’s theory. And this is strange, since intuitively the truth, and a fortiori the significance, of the nonexistential conditional “for any x and y, if x were an orange and y smelled x, then y would feel nauseated” is compatible with “there is no x such that x is an orange.” Second, (R) is constructed on the assumption that solubility is a permanent disposition, as evidenced by the universal quantification over the time variable. But there are, of course, also transient dispositions, like electrical charge. Suppose, now, that x is the very first body whose electrical charge has been ascertained by a human observer. Then the statement that x is electrically charged at t1 , the time of our supposed first experiment, would be meaningful, yet the statement that x is electrically charged at some earlier time t0 would be meaningless.1 An adequate explicit definition of a disposition predicate Q3 should have the advantage over (R) that it rescues the significance of the statement Q3 (b), where b is an object upon which the test operation has never been performed and possibly belongs to a natural kind K such that the test operation has not been performed upon any member of K, and over (D1 ) that it does not allow vacuous predication of Q3 . The following explicit definition satisfies these requirements:2 Q3 (x) ≡d f (∃ f )[ f (x).(∃y)( f (y).Q1 (y)). (∀z)( f (z).Q1 (z) ⊃ Q2 (z))]

(D2 )

Unfortunately (D2 ) is likewise open to serious objections. First, it entails that it is self-contradictory to say “a is soluble but there are no liquids” (if there are no liquids, the class of immersed objects—ˆyQ(y)—is empty); but the intuitive sense of “a is soluble” is “if there were a liquid, and a were immersed in it, then a would dissolve.” Second, (D2 ) involves a confusion between the grounds for believing a proposition and the analysis of the proposition believed. For, according to (D2 ), the subjunctive conditional which is condensed into Q3 (a) asserts not only the existence of a law in accordance with which 1 In

connection with such time-dependent disposition predicates (“elastic, “irritable” are other examples) there arises a further diﬃculty for Carnap’s theory. Let Q3 (x, t) represent such a disposition with respect to which an individual may change, and let (∀x)(∀t)(Q1 (x, t) ⊃ (Q3 (x, t) ≡ Q2 (x, t))) be the bilateral reduction sentence by which this predicate is introduced. Carnap argues that this sentence is analytic if the disposition predicate has no independent meaning, specified by other reduction sentences. If so, then the singular sentence Q3 (a, t0 ) can be analytically inferred from the singular sentences Q1 (a, t0 ) and Q2 (a, t0 ). But this result is irreconcilable with the plausible view that to assert a subjunctive conditional is to make an implicitly general statement, an assertion of causal connection which goes beyond a purely descriptive “post hoc” report. 2 This definition is copied from Wedberg 1944, 237, who cites it for purposes of criticism from Kaila 1967. Kaila later improved the definition, in his book Kaila 1941 (in answer to an objection by Carnap), and finally, by the time he wrote Kaila 1945, abandoned this kind of definition altogether.

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Q2 (a) is predictable from Q1 (a), but moreover the existence of confirming evidence—(∃y)(Q1 (y).Q2 (y))—for that law. Yet, to suppose that “there exist no grounds for believing p” entails “p is false,” is to be guilty of confusing truth with knowledge of the truth.3 The third objection leads us straight to the line of analysis of subjunctive conditionals whose futility has been demonstrated especially by Chisholm and Goodman4 . It is that values of the predicate-variable f can be constructed such that objects to which we do not want to ascribe Q3 will again, as in the case of D1 , vacuously satisfy the definiens. Thus, we might set f (x) = (x = a ∨ x = b), where Q1 (b) is true yet Q1 (a) false.5 If, to avoid this impasse, f is restricted to general properties (i.e. properties whose definition does not involve reference to a particular individual), trivialization is still possible in terms of such predicates as Q1 ⊃ Q2 .6 And if this variable is still further restricted, to something like “intrinsic” properties, properties determining natural kinds (“wooden,” “golden,” “metallic,” etc.), we run the danger of defining per obscurius. For, apart from the consideration that the concept of “natural kind” stands in at least as much need of analysis as—and may even logically presuppose—the concept of “disposition,” to say of a property P that it is intrinsic to an individual x is meaningless unless it is elliptical for P is intrinsic to x qua member of class K. Thus even color could be intrinsic to a given individual relatively to a description of the individual in terms of color predicates; on the other hand, if a given match is described as an instrument for lighting cigarettes which I now hold in my hand, being wooden may be argued to be an extrinsic property of it, since it is not entailed by the description by which it was referred to. If the intuitive notion of a property without which an individual “would not be what it is” is analyzed in this way, it appears that one cannot significantly divide properties into intrinsic and extrinsic ones. Now, the program of precluding trivial satisfaction of the definiens by imposing suitable restrictions on the range of its bound major variable, is precisely what Chisholm and Goodman have shown to face insuperable diﬃculties. If we analyze “if it were the case that p, then it would be the case that q” (regardless of whether it is in fact the case that p)7 as meaning “there is a true

3 This

objection applies to Thomas Storer’s explicit definition of “soluble,” which is very similar to D2 (Storer 1951). I think that the analysis of counterfactual conditionals given by B. J. Diggs (Diggs 1952), subtle and careful as it is, is likewise open to it.—For a lucid warning against the confusion of truth with knowledge of the truth, see Carnap 1949. 4 Chisholm 1946 and Goodman 1947. 5 This counterexample is due to Carnap. 6 Cf. Wedberg 1944 7 “The Problem of Subjunctive Conditionals” is a better terminology than “The Problem of Contrary-to-Fact Conditionals,” precisely because in asserting subjunctive conditionals one just as often implies nothing with respect to the truth-value of the antecedent as one implies that the antecedent is false.

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proposition R such that q is entailed by p.R (and by no less than that),”8 we must exclude material or formal implications which are vacuously true from the range of “R”; otherwise incompatible subjunctive conditionals could be both true.9 But this restriction has been shown to be insuﬃcient because of the possibility of taking as R a true, yet not vacuously true, formal implication such that again incompatible subjunctive conditionals turn out to be both true: such formal implications are the so-called accidental universals. Thus one could prove both “if you were to go to the concert, you would spend an enjoyable evening” and “if you were to go to the concert, you would not spend an enjoyable evening” from the falsity of the antecedent, by taking as R alternatively the accidental universal “(∀x)( f (x) ⊃ (x goes to the concert ⊃ x spends an enjoyable evening))” or “(∀x)( f (x) ⊃ (x goes to the concert ⊃ x does not spend an enjoyable evening)),” provided the person denoted by “you” is the one person satisfying “ f (x).”10 It was pointed out by both Chisholm and Goodman that accidental universals are intuitively distinguished from laws of nature by their inability to support subjunctive conditionals: it is unreasonable to infer from “all the people in this room are tall” that if my little boy were in this room he would be tall; it would be more reasonable to infer from this hypothesis that some people in this room would not be tall. On the other hand, it is alleged to be reasonable to infer from “all ravens are black” that if my little boy were a raven he would be black. Clearly, if this were all that could be done to distinguish these two types of universal statements, the explored analysis of subjunctive conditionals would be circular. Let us see, however, whether we can explain this intuitive diﬀerence in terms of a characterization of accidental universals which makes no use of the notion of the subjunctive conditional. It may seem that the diﬀerence is simply that between a universal statement about an extensionally defined (and hence “closed”) class and a universal statement about an intensionally defined (and hence “open”) class. This is actually the position taken by Popper (Popper 1949), who blames the whole “problem” here under review on the confusion of extensional and intensional interpretation of class-terms. However, this simple solution of what Popper calls a “modern riddle” misses the mark for two reasons. First, if x ∈ A meant x = a ∨ x = b ∨ . . . ∨ x = n (extensional definition), then the inference from “if y (which is not an A) were an A” to “then y would be a B” would be perfectly valid, for the simple reason that its premises would

8 It

may be remarked en passant that even if this analysis were adequate it is doubtful whether it accomplishes the aimed at reduction to extensional concepts, since entailment itself is prima facie an intensional function. 9 Notice that “if it were the case that p, then it would be the case that q” diﬀers from p ⊃ q precisely in that it is incompatible with “if it were the case that p, then it would not be the case that q,” whereas p ⊃ q is compatible with p ⊃∼ q. 10 Cf. Chisholm’s example, Chisholm 1946, 491.

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be self-contradictory (inferences from self-contradictory premises are bound to be valid). Therefore the proposed interpretation does not account for the specified intuitive diﬀerence. Second, the class-terms in accidental universals have intensional meanings, just like the class-terms in laws, since one cannot deduce from an accidental universal about A, by mere analysis of the meaning of “A,” which individuals belong to A (e.g. we can all understand the statement “all the advisers of Eisenhower speak English” without knowing who the advisers of Eisenhower are). But while this simple consideration, viz. that one can understand an accidental universal without knowing which objects fulfill it, is a strong argument against the interpretation of accidental universals as finite conjunctions of singular statements, we still are driven to recognize the distinction between closed and open classes as crucial, as the following dialectic will reveal. Why is it that from the (counterfactual) hypothesis “my boy is in this room” I am entitled to infer “not all people in this room are tall?” Evidently because I tacitly entertain the further premise “my boy is not tall.” But if I similarly make use of the premise “my boy is not black,” then the statement “if my boy were a raven, then not all ravens would be black” is as defensible as the statement “if my boy were in this room, then not all people in this room would be tall.” On the other hand, if we are not permitted to make any assumptions about the individual which we are considering as a possible member of A, then the statement “if that individual were an A, then some A’s would not be B’s” is never defensible, no matter whether “all A are B” be accidental or lawlike. Now, if in raising this vexing question what would be the consequence if an individual, call it c, which as a matter of fact has properties incompatible with being an A, were diﬀerent from what it is and were an A, one is not to suppose anything about c except membership of A, how can one be inhibited from inferring “c would be B” if supplied the information “all A are B”? And the simple answer is of course: if A is a closed class excluding c, such that c just is not an instance to which the generalization applies. I wish to show, however, that the distinction between closed and open classes, here invoked, can itself be explained only in terms of the subjunctive mood. In the first place, the distinction in question surely is not the distinction between finite and infinite classes. Even if we include in a biological species, e.g., individuals yet to be born, we consider it unlikely (though logically possible) that their number is infinite, and nevertheless a true generalization about such a species is regarded as a law. To this one might reply that even though the classes determined by the predicates are finite in either case, the diﬀerence is that in the case of laws only the variables of quantification have an infinite range. And since one could not plausibly maintain that in asserting a law we commit ourselves to the belief in an infinite universe of individuals, this interpretation amounts to populating the range of the variable in “for any x, if

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x ∈ A, then x ∈ B” with possible individuals. And once we permit ourselves this way of speaking—which is reminiscent of Meinongian ontology—we are led to conceive of “open” classes as of classes of actual or possible individuals. We have come here upon Lewis’ distinction between the “denotation” and the “comprehension” of a class-term: the comprehension of “raven,” e.g., includes not only all actual ravens, but also all “consistently thinkable” ravens, and insofar as “all ravens are black” is a law it asserts an inclusion of comprehensions. But while Lewis was no doubt on the track of an important distinction, his formulation of the distinction is most unfortunate. For, in the first place, the most natural interpretation of the statement “there are no consistently thinkable ravens that are not black” is “it is not consistently thinkable that something should be a raven without being black.” But on this interpretation the supposed law of nature turns into an analytic truth! Second, Lewis overlooks that whether or not it is consistently thinkable that a given individual have a given property depends on our way of referring to the individual. For example, if the individual who is now speaking is described as the individual who was born in Z¨urich on Oct. 1 1921, then he is a consistently thinkable raven, for it involves no contradiction to suppose that one and only one individual was born at that time and place and that it happened to be a raven. Indeed, in defining the comprehension of Φ as the totality of the x’s such that it is consistently thinkable that Φx (cf. Lewis 1946, 63), he implicitly ascribes to all self-consistent predicates the same, viz. universal, comprehension. For, for any individual a, some description “( x)ψx” can be found such that “Φ( x)ψx” is self-consistent (as a last resort, “( x)(x = a)” could be taken). These diﬃculties seem to me to arise from the conception of an “open class” as a totality comprising, along with actual entities, possible entities. But to say that there exist, besides the actual members of A, also possible members of A, can only mean that there are individuals which might have property A although they actually do not have it. Thus A, in the universal implication “for any x, if x ∈ A, then x ∈ B,” may be an open class even if the range of x is a finite set a, b, . . ., n. The openness of A can mean nothing else than that the connective “if-then” is meant as a subjunctive connective. In other words, in saying that the universal implication refers to an open class, we are saying that it involves statements about unactualized possibilities, such as “if Arthur Pap, instead of being a man, were a raven, he would be black.” This entails, of course, that the extensional conception of universal statements as being conjunctions of purely descriptive singular statements11 is valid only for accidental universals.

this is not meant the equivalence of (∀x)(x ∈ A ⊃ x ∈ B) to (a ∈ A.a ∈ B).(b ∈ A.b ∈ B).. . ..(n ∈ A.n ∈ B), where a, b, . . ., n are all the members of A, which must be rejected not only on the ground that the class A may be empty, but also on the ground that one cannot deduce from the universal statement which 11 By

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I hope to have shown that it is futile to think one can avoid the subjunctive “if-then,” as an irreducibly non-extensional connective, by defining a law as a true, and synthetic, formal implication about an open class. But further, in order to answer the obvious objection from the incompatibility of subjunctive conditionals with common antecedent and incompatible consequents (e.g. “all sugar is soluble” and “all sugar is insoluble”), one would have to qualify that “(∀x)(x ∈ A ⊃ x ∈ B)” expresses a law only if either A is nonempty or it is deducible from a true formal implication which is not vacuously true.12 By virtue of this qualification one could maintain, e.g., that “(∀x)(sugar(x).immersed(x) ⊃ dissolves(x)))” expresses a law even relatively to a universe devoid of liquids, because it is deducible from the non-vacuous implication “(∀x)(sugar(x) ⊃(immersed(x) ⊃ dissolves(x))).” But since relatively to such a universe “(∀x)(sugar(x) ⊃ (immersed(x) ⊃∼ dissolves(x)))” would likewise be true, one would have no basis for defending the plausible opinion that, even relatively to such a hypothetical universe, “all sugar is soluble,” and not “all sugar is insoluble,” expresses a law. Perhaps this diﬃculty, and related ones, can be solved by first defining a “fundamental law of nature” as a true, synthetic proposition expressed by a formal implication none of whose predicates are empty and which contains no individual constants, and then defining a “law of nature” as a synthetic universal proposition deducible from a fundamental law of nature. But I cannot refrain from adding at once three objections to this proposal: 1. Notice that relatively to a universe devoid of liquids, the sentence “all sugar is soluble” would be meaningless and hence would not express a law; for, “soluble” can be introduced into an extensional language only by a reduction sentence, but, as we have seen, if there are no immersed objects, the reduction sentence endows “soluble” with no meaning at all; hence we could not meaningfully say “even in a universe devoid of liquids, it would be a law that all sugar is soluble.” Mutatis mutandis, this holds for laws concerning such dispositions as vaporization points, melting points, freezing points, relatively to universes lacking those temperatures at which in this universe substances change their states of aggregation. 2. While it is, indeed, a characteristic feature of accidental universals that they contain individual constants essentially, and it is therefore tempting to stipulate that a sentence expresses a law only if no individual constants occur essentially in it, this stipulation is nonetheless counterintuitive. Suppose, for example, that Galileo’s law of freely falling bodies and Kepler’s three laws, which contain such individual constants as “the earth” and “the sun,” had never

individuals belong to A (cf. Lewis 1946, 65); but the usually accepted equivalence to (a ∈ A ⊃ a ∈ B).(b ∈ A ⊃ b ∈ B).. . ..(n ∈ A ⊃ n ∈ B), where a, b, . . ., n are all the individuals over which the variable ranges. 12 For an analysis of the concept of law along this line, cf. Reichenbach 1947, chapter VIII.

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been explained by deduction from such (apparently) purely general propositions as the law of gravitation and the principles of Newtonian mechanics. Would it not be perfectly proper to refer to them as “laws,” albeit unexplained laws (what Mill called “empirical” laws) just the same? 3. Think of all the laws of mathematical physics that refer to ideal conditions and entities, like motions under the sole influence of gravity, ideal gases, frictionless engines, point-particles, etc. Would it not be bold to claim that they are deducible from statements whose predicates denote empirically given objects and situations? Yet, if these be not “fundamental laws of nature,” which are? However, my main argument against the possibility of expressing laws of nature by means of the symbolism of extensional logic is best stated by returning to the inquiry, commendably started by Lewis, into the range of the individual variables. It may be obscure to say that it consists of all actual individuals, since this involves, by contrast, the conception of “possible individuals.” But the idea can be expressed in the “formal mode of speech” by saying that the substitutable constants a, b, etc. are what Russell calls “logically proper names,” i.e. names in the sense in which “Pegasus,” “Hamlet,” etc. are not names. If a formal implication, then, is logically equivalent to the conjunction of its substitution-instances, it makes an assertion only about observed cases (as was emphasized by Lewis in Lewis 1946).13 It may seem that a predictive content can nevertheless be secured for formal implications by allowing descriptions as substituends for “x”: as Russell pointed out, by means of ( x)xRa we can make significant statements about unobserved individuals. But in the language of Principia Mathematica descriptions in turn are defined (contextually) in terms of statements involving variables, and thus we are led back to a, b, etc. as the primitive names of the entities over which the variables range. If, therefore, we want to avoid the presupposition of “possible entities” as included in the range of the variables, we are driven to the subjunctive mood after all: for any (at some time actual) x, if x were an A, then x would be a B. But is it significant to suppose that an individual which is, say, a stone, might instead be, say, a raven? Clearly, such a supposition is significant only if the notion of a substratum is significant. A name a denotes a substratum if P(a) is a self-consistent, contingent statement regardless of what property P is, provided only that it is a self-consistent property of the first level (notice that this condition would not be fulfilled if a were a description). Curiously, then, the substratum (or a multitude of numerically distinct, though indistinguish-

13 Even before Lewis, Russell became aware of this diﬃculty, in Russell 1940, 255: “There is thus a hypothetical element in any general proposition; ‘ f (x) is true of every x’ does not merely assert the conjunction f (a). f (b). f (c). . . where a, b, c, . . . are the names (necessarily finite in number) that constitute our actual vocabulary. We mean to include whatever will be named, and even whatever could be named.”

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able substrata) lurks behind the individual variables of the usual algorithm of symbolic logic, and makes its embarrassing presence felt especially if we try to express subjunctive conditionals in that algorithm. Let me call attention, in conclusion, to the alternative type of language which contains only names of properties, no names of substrata, and which is advocated by Russell himself as an alternative to the usual subject-predicate type of language.14 In such a language we could presumably express the law “all ravens are black” without subjunctive mood, by asserting simply the compresence of the property being a raven with the property being black.15 This would, of course, be a realistic language which nominalistically-inclined logicians, like Quine and Goodman, who want to admit only names of individuals and treat all other descriptive constants as syncategorematic, will repudiate. But it is worth thinking about the question whether the desire to avoid the obscure notion of the substratum, and its symbolic counterpart, the Russellian “logically proper names,” does not force such a realistic language upon us after all.

14 Russell

1940, chapter VI. See also Ayer 1952a. Chisholm 1946: “This suggests that the terms of ‘non-accidental’ connections are the properties of things. And if we cannot get rid of the subjunctive by any other means, we can define it in terms of these ‘connections’.” 15 Cf.

Chapter 21 DISPOSITION CONCEPTS AND EXTENSIONAL LOGIC (1958)

One of the striking diﬀerences between natural languages, both conversational and scientific, and the extensional languages constructed by logicians is that most conditional statements, i.e., statements of the form “if p, then q,” of a natural language are not truth-functional. A statement compounded out of simpler statements is truth-functional if its truth-value is uniquely determined by the truth-values of the component statements. The symbolic expression of this idea of truth-functionality, as given in Principia Mathematica, is p ≡ q ⊃ ( f (p) ≡ f (q)). That is, if “ f (p)” is any truth-function of “p,” and “q” has the same truth-value as “p,” however widely it may diﬀer in meaning, then “ f (q)” has the same truth-value as “ f (p).” Clearly, if I am given just the truthvalues of “p” and “q,” not their meanings, I cannot deduce the truth-value of “if p, then q”—with a single exception: if “p” is given as true and “q” as false, it follows that “if p, then q” is false, provided it has a truth-value at all. On the contrary, the knowledge that matters for determination of the truth-value of a “natural” conditional—let us call them henceforth “natural implications,” in contrast to those truth-functional statements which logicians call “material conditionals” or “material implications”—is rather knowledge of the meanings of the component statements. In the case of simple analytic implications like “if A has a niece, then A is not an only child” such knowledge of meanings is even suﬃcient for knowledge of the truth of the implication; at any rate knowledge of the truth-value of antecedent and consequent is irrelevant. In the case of those synthetic natural implications which assert causal connections, knowledge of meanings is not, indeed, suﬃcient, but it is necessary, and knowledge of the truth-values of the component statements is not presupposed by knowledge of the truth-value of the implication.1 Consider the conditional (which 1 In

this context “knowledge” is used in the weak sense in which “p is known to be true” entails that there is evidence making it highly probable that p, not the stronger claim that there is evidence making it certain that p.

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may or may not be “contrary-to-fact”): if I pull the trigger, the gun will fire. It would be sad if belief in such an implication were warranted only by knowledge of the truth of antecedent and consequent separately, for in that case it would be impossible for man to acquire the power of even limited control over the course of events by acquiring warranted beliefs about causal connections. Notice further that frequently presumed knowledge of a causal implication is a means to knowledge of the truth, or at least probability, of the antecedent; if this is an acid then it will turn blue litmus paper red; the reaction occurred; thus the hypothesis is confirmed. Knowledge of the consequences of suppositions is independent of knowledge of the truth-values of the suppositions, no matter whether the consequences be logical or causal. The diﬀerence between material implication and natural implication has been widely discussed. The logician’s use of “if p, then q” in the truthfunctional sense of “not both p and not-q,” symbolized by “p ⊃ q,” is fully justified by the objective of constructing an adequate theory of deductive inference, since the intensional meaning of “if, then,” be it logical or causal connection, is actually irrelevant to the validity of formal deductive inferences involving conditional statements. This is to say that such conditional inference-forms as modus ponens, modus tollens, and hypothetical syllogism would remain valid in the sense of leading always to true conclusions from true premises if their conditional premises or conditional conclusions were interpreted as material conditionals asserting no “connections” whatever. The so-called, and perhaps misnamed, paradoxes of material implication, viz., that a false statement materially implies any statement and a true statement is materially implied by any statement, are not logical paradoxes. The formal logician need not be disturbed by the fact that the statements “if New York is a small village, then there are sea serpents” and “if New York is a small village, then there are no sea serpents” are, as symbolized in extensional logic, both true; for since this is due to the falsity of their common antecedent, modus ponens cannot be applied to deduce the contradiction that there both are and are not sea serpents. No contradiction arises. However, it is in the application of extensional logic for the purpose of precise formulation of empirical concepts and propositions that serious diﬃculties arise. The “paradoxical” feature of material implication that the mere falsehood of the antecedent ensures the truth of the implication leads, not to formal inconsistency, but to grossly counterintuitive factual assertions when extensional logic is applied to the language of empirical science. This becomes particularly evident if one tries to formalize so-called operational definitions by means of extensional logic. For the definiens of an operational definition is a conditional whose antecedent describes a test operation and whose consequent describes a result which such an operation has if performed upon a certain kind of object under specified conditions. A concept which is operationally defined in this sense may be called a “disposition concept.” Suppose, then, that a disposition concept is defined by a material conditional as follows:

Disposition Concepts and Extensional Logic (1958)

D(x, t) ≡d f (O(x, t) ⊃ R(x, t))

329 (1)

The question might be raised whether the time-argument could be omitted from the disposition predicate, so that the definition would look as follows: Dx ≡ (∀t)(O(x, t) ⊃ R(x, t)). Which form of definition is suitable depends on inductive considerations. If the disposition is “intrinsic” in the sense that a generalization of the form (∀t)(∀x)[x ∈ K ⊃ (O(x, t) ⊃ R(x, t))] has been highly confirmed (where K is a natural kind), a time-independent disposition predicate is appropriate. Examples of such intrinsic dispositions are solubility and melting point (the latter is an example of a quantitative disposition whose operational definition accordingly would require the use of functors, not just of qualitative predicates). On the other hand, the symbol “D(x, t)” is appropriate if D is such that for some objects y both “(∃t)(O(y, t).R(y, t))” and “(∃t)(O(y, t). ∼ R(y, t))” holds; for example, being electrically charged, elasticity, irritability. Now, as Carnap pointed out in “Testability and Meaning,” a definition of the form of (1) has the counterintuitive consequence that any object has D at any time at which it is not subjected to O, and that any object on which O is never performed has D at all times.2 There is a close analogy between the interpretation of the Aristotelian A and E propositions as generalized material implications (or “formal implications,” in Russell’s terminology) and the extensional interpretation of operational definitions, in that both have the consequences that intuitively incompatible statements are compatible after all. If “all A are B” means “(∀x)(Ax ⊃ Bx)” and “no A are B” means “(∀x)(Ax ⊃ ∼ Bx),” then both may be true, since both would be true if nothing had the property A, which is logically possible. Thus the student introduced to extensional symbolic logic learns to his amazement that both “all unicorns live in the Bronx zoo” and “no unicorns live in the Bronx zoo” are true statements—for the simple reason that there are no unicorns, from which it follows that there are no unicorns of any kind, neither unicorns that live in the Bronx zoo nor unicorns that don’t live in the Bronx zoo. Similarly, suppose a physical functor like “temperature” were operationally defined as follows: temp(x, t) = y ≡d f a thermometer is brought into thermal contact with x at t ⊃ the top of the thermometric liquid coincides with the mark y at t + dt. Then the clearly incompatible statements “temp(a, t0 ) = 50” and “temp(a, t0 ) = 70” would both be true, on the basis of this definition, if no thermometer were in contact with a at t0 ; indeed a would have all temperatures whatsoever at any time at which its temperature is not measured.3 2I

have slightly changed Carnap’s way of putting the counterintuitive consequence, in accordance with my using “D(x, t)” instead of “Dx.” 3 There seems to be fairly universal agreement now among philosophers of science that the simple kind of explicit definition of disposition concepts in terms of material implication is inadequate, precisely because we want to be able to say of an object which is not subjected to the test operation by which a disposition D is defined that it does not have D. One exception to this trend might, however, be noted: Gustav Bergmann maintains (Bergmann 1951a) that such explicit definitions nevertheless provide adequate analyses of the

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Some philosophers have suggested that the reason why counterintuitive consequences result if material implication is substituted for natural implication is that a material implication is true in cases where the corresponding natural implication has no truth-value. If the antecedent of a natural implication is false, they suggest, then the natural implication is “undetermined”; it is true just in case both antecedent and consequent are true, and false in case the antecedent is true and the consequent is false.4 Now, the combinations F F and F T do, indeed, leave the truth-value of a natural implication undetermined in the sense that they leave it an open question which its truth-value is. But the same holds for the combination T T. It is not the case that every true statement naturally implies every true statement. If it should be replied that nevertheless the joint truth of antecedent and consequent confirms a natural implication, it must be pointed out that if so, then the joint falsehood of antecedent and consequent likewise confirms it, by the principle that whatever evidence confirms a given statement S also confirms whatever statement is logically equivalent to S :5 if “p and q” confirms “if p, then q,” then “not-q and not-p” confirms “if not-q, then not-p,” and therefore confirms “if p, then q.” Or, to put it diﬀerently but equivalently: if “p and q” confirms “if p, then q” then it also confirms “if not-q, then not-p,” but this is to say that a natural implication is confirmable by an F F case. To illustrate: suppose I say to a student “if you study for the course at least one hour every day, then you will pass the course.” If this conditional prediction is confirmed by the fact that the advised student put in at least one hour for the course every day and passed the course, then the same fact ought to confirm the equivalent prediction formulated in the future perfect: “if you will not pass the course, then you will not have studied for it at least one hour every day.” But further, it just is not the case that no truth-value is ordinarily assigned to a natural implication whose antecedent is false. Everybody distinguishes between true and false contrary-to-fact conditionals. In particular, the belief that disposition concepts—in a sense of “adequate analysis” which is obscure to me. Referring to Carnap’s example of the match which is burned up before ever being immersed in water and therefore would be soluble by the criticized definition of “soluble,” he says “I propose to analyze the particular sentence ‘the aforementioned match is (was) not soluble’ by means of two sentences of the ideal schema, the first corresponding to ‘This match is (was) wooden,’ the second to the law ‘No wooden object is soluble.”’ In what sense do these two sentences provide an analysis of “soluble”? Bergmann is simply deducing “the match is not soluble” from two well-confirmed premises, and is therefore perhaps giving a correct explanation of the fact described by the sentence, but since “soluble” reappears in the major premise—as it must if the syllogism is to be valid!—its meaning has not been analyzed at all. It is one thing to give grounds for an assertion, another thing to analyze the asserted proposition. 4 See O’Connor 1951, 354. Also, the Finnish philosopher E. Kaila once attempted to escape from Carnap’s conclusion that disposition concepts are not explicitly definable by proposing that “Dx” be taken as neither true nor false in case x is not subjected to O (which proposal, incidentally, is consonant with Carnap’s proposal of introducing dispositional predicates by reduction sentences, as we shall see later); see Kaila 1945. 5 This has been called the “paradox of confirmation.” See Hempel 1945a, Hempel 1945b, and Carnap 1950b, section 87.

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an object has a certain disposition may motivate people to subject it, or prevent it from being subjected, to the corresponding test operation; we are, for example, careful not to drop a fragile and valuable object because we believe that it would break if it were dropped. What we believe is a proposition, something that is true or false; to say that it only becomes true when its antecedent and consequent are confirmed, is to confuse truth and confirmation.6 Let us see, now, whether perhaps a more complicated kind of explicit definition of disposition concepts within the framework of extensional logic can be constructed which avoids the shortcoming of (1): that an object upon which test operation O is not performed has any disposition whatsoever that is defined by means of O. Philosophers who follow the precept “to discover the meaning of a factual sentence ‘p’ reflect on the empirical evidence which would induce you to assert that p” might arrive at such a definition by the following reasoning. What makes one say of a wooden object that it is not soluble in water even before testing it for solubility in water, i.e., before immersing it in water? Obviously its similarity to other objects which have been immersed in water and were found not to dissolve therein. And what makes one say of a piece of sugar that it is soluble before having immersed it? Evidently the fact that other pieces of sugar have been immersed and found to dissolve. In general terms: the evidence “(x1 ∈ K . Ox1 . Rx1 ) . (x2 ∈ K . Ox2 . Rx2 ) . . . . (xn ∈ K . Oxn . Rxn )” led to the generalization “(∀x)(x ∈ K ⊃ (Ox ⊃ Rx))” from which, together with “x0 ∈ K,” we deduce “Ox0 ⊃ Rx0 .” The latter conditional is not vacuously asserted, i.e., just on the evidence “∼ Ox0 ,” but it is asserted on the specified inductive evidence. Such indirect confirmability7 of dispositional statements seems accurately reflected by the definition schema:8 Dx ≡d f (∃ f )[ f x.(∃y)(∃t)( f y.O(y, t)). (∀z)(∀t)( f z.O(z, t) ⊃ R(z, t))]

(2)

If we take as values of “ f ” alternatively “being wooden” and “being sugar,” then it can easily be seen that on the basis of such a definition, involving application of the higher functional calculus to descriptive predicates, wooden objects that are never immersed in a liquid L are not soluble in L, whereas pieces of sugar can with inductive warrant be characterized as soluble in L even if they are not actually immersed in L. 6 For

a lucid warning against this confusion, see Carnap 1949. confirmation of a conditional is distinguished from (a) direct confirmation, consisting in the verification of the conjunction of antecedent and consequent, (b) vacuous confirmation, consisting in the verification of the negation of the antecedent. 8 D is here assumed to be an intrinsic disposition in the sense explained above. The above schema is, with a slight alteration, copied from Anders Wedberg’s “The Logical Construction of the World” (Wedberg 1944, 237), who cites it for purposes of criticism from Kaila 1967. A variant of this definition schema has more recently been proposed by Thomas Storer: Storer 1951. 7 Indirect

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Unfortunately, however, the undesirable consequences of (1) reappear if certain artificial predicates are constructed and substituted for the predicate variable “ f .” Thus Carnap pointed out to Kaila that if “(x = a) ∨ (x = b) ,” where a is the match that was burned up before ever making contact with water and b an object that was immersed and dissolved, is taken as the value of “ f ,” “Da” is again provable. This seemed to be a trivial objection, since evidently “ f ” was meant to range over “properties” in the ordinary sense of “property”: who would ever say that it is a property of the match to be either identical with itself or with the lump of sugar on the saucer? But if “ f ” is restricted to general properties, i.e., properties that are not defined in terms of individual constants, the undesirable consequences are still not precluded. As Wedberg pointed out (Wedberg 1944), vacuous confirmation of dispositional statements would still be possible by taking “O ⊃ R” as value of “ f .” Nevertheless, I doubt whether the objection from the range of the predicate variable is insurmountable. To be sure, it would lead us to a dead end if we defined the range of “ f ” as the class of properties that determine natural kinds. For our philosophical objective is to clarify the meaning of “disposition” by showing how disposition concepts are definable in terms of clearer concepts. But I suspect that we need the concept of “disposition” for the explication of “natural kind,” in the following way: if a class K is an ultimate natural kind (an “infima species,” in scholastic terminology), then, if one member of K has a disposition D, all members of K have D. If “ultimate natural kind” could be satisfactorily defined along this line, “natural kind” would be simply definable as “logical sum of ultimate natural kinds.” To illustrate: would a physicist admit that two samples of iron might have a diﬀerent melting point? He would surely suspect impurities if the two samples, heated under the same standard pressure, melted at diﬀerent temperatures. And after making sure that the surprising result is not due to experimental error, he would invent names for two subspecies of iron—that is, he would cease to regard iron as an “ultimate” kind—and look for diﬀerentiating properties other than the diﬀerence of melting point in order to “account” for the latter. But be this as it may, it seems that vacuous truth of dispositional statements could be precluded without dragging in the problematic concept of “natural kind” by the following restriction on the range of “ f ”: we exclude not only properties defined by individual constants, but also general properties that are truth-functional compounds of the observable transient properties O and R.9 There remains, nevertheless, a serious objection relating to the second conjunct in the scope of the existential quantifier: there is a confusion between the meaning of a dispositional statement and the inductive evidence for it. To

9 The

latter restriction has been suggested to me by Michael Scriven.

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see this, just suppose a universe in which the range of temperature is either so high or so low that liquids are causally impossible in it. If so, nothing can ever be immersed in a liquid, hence, if “O(y, t)” means “y is immersed in L at t,” “(∃y)(∃t)( f y.O(y, t))” will be false for all values of “ f .” But surely the meaning of “soluble” is such that even relative to this imaginary universe “sugar is soluble” would be true: in using the dispositional predicate “soluble” we express in a condensed way the subjunctive conditional “if a sample of sugar were immersed in a liquid, then it would dissolve,” and this just does not entail that some sample of sugar, or even anything at all, is ever actually immersed in a liquid. True, no mind could have any evidence for believing a proposition of the form “x is soluble” if nothing were ever observed to dissolve; indeed, it is unlikely that a conscious organism living in our imaginary universe (for the sake of the argument, let us assume that the causal laws governing that universe are such that conscious life is possible in it in spite of the prevailing extreme temperatures) would even have the concept of solubility. But it does not follow that the proposition could not be true just the same. The idea underlying (2) is obviously this: the evidence on which a contraryto-fact conditional is asserted—if it is a confirmable, and hence cognitively meaningful, statement at all—is some law that has been confirmed to some degree; therefore the conditional is best analyzed as an implicit assertion of the existence and prior confirmation of a law connecting O and R.10 Now, I agree that the existence of some law in accordance with which the consequent is deducible from the antecedent is implicitly asserted by any singular counterfactual conditional, though the asserter may not be able to say which that law is (formally speaking, he may not know which value of “ f ” yields a universal conditional—the third conjunct of the definiens—which is probably true). To take an extreme example: if I say, “if you had asked your landlord more politely to repaint the kitchen, he would have agreed to do it,” I have but the vaguest idea of the complex psychological conditions that must be fulfilled if a landlord is to respond favorably to a tenant’s request which he is not legally obligated to satisfy, yet to the extent that I believe in determinism I believe that there is a complex condition which is causally suﬃcient for a landlord’s compliance with such a request.11 But that there is confirming evidence for the law whose existence is asserted—and more specifically instantial evidence—is causally, not logically, presupposed by the assertion of the dispositional state-

10 For

example, the painstaking attempt made by B. J. Diggs, in Diggs 1952, to achieve an extensional analysis of the counterfactual conditional is guided by this idea. 11 One might, though, take the more moderate view that warranted assertion of counterfactual conditionals merely involves statistical determinism, i.e., belief in the existence of a statistical law relative to which the consequent is inferable from the antecedent with a probability suﬃciently high to warrant practical reliance on the conditional. But on either view singular counterfactual conditionals derive their warrant from a law, whether causal or statistical.

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ment. A proposition q is causally presupposed by an assertion of proposition p, if p would not have been asserted unless q had been believed; in other words, if the acceptance of q is one of the causal conditions for the assertion of p;12 whereas q is logically presupposed by the assertion of p, if p entails q. To add an illustration to the one already given: consider the singular dispositional statement “the melting point of x is 200◦ F,” which means that x would melt at any time at which its temperature were raised to 200◦ F (provided the atmospheric pressure is standard). Surely this proposition is logically compatible with the proposition that nothing ever reaches the specified temperature. That there should be instantial evidence for a law of the form “any instance of natural kind K would, under standard atmospheric pressure, melt if it reached 200◦ F” is therefore not logically presupposed by the dispositional statement, though very likely it is causally presupposed by its assertion. Could our schema of explicit definition, then, be salvaged by extruding the existential clause? Only if “(∀z)(Fz.Oz ⊃ Rz)” (where “F” is a constant predicate substituted for the variable “ f ”) were an adequate expression of a law. But that it is not follows from the fact that it is entailed by “∼ (∃z)(Fz.Oz).” Thus, if “F” means “is wooden,” and it so happens that no wooden thing is ever immersed in a liquid, it would be true to say of a match that it is soluble. It may well be that to ascribe D to x is to ascribe to x some intrinsic property f (however “intrinsic” may be explicated) such that “Rx” is deducible from “ f x.Ox” by means of a law; but this, as most writers on the contrary-tofact conditional have recognized, leaves the extensionalist with the tough task of expressing laws in an extensional language. The view that every singular counterfactual conditional derives its warrant from a universal conditional is sound—though one cannot tell by a mere glance at the predicates of the singular conditional which universal conditional is presupposed13 —but it should not be overlooked that universal conditionals that are accorded the status of laws by scientists may themselves be counterfactual. There are no finite physical

12 Notice

that while “A says ‘p”’ does not entail, but at best confers a high probability upon “A believes that p,” the latter proposition is entailed by “A asserts that p,” according to my usage of “assert” as an intentional verb. I am not denying, of course, that there may be a proper purely behavioristic sense of “assert”; nor do I deny that “A asserts that p” may properly be so used that it is compatible with “A does not believe that p.” My usage may be explicated as follows: A believes that p and utters a sentence expressing the proposition that p. 13 This seems to be overlooked by O’Connor, who, following Broad, concludes his analysis of conditional sentences (O’Connor 1951) with the claim that “a particular contrary-to-fact conditional has exactly the same meaning as the corresponding universal indicative statement.” The examples given by him indicate that by the universal statement corresponding to the “particular” contrary-to-fact conditional he means the universal conditional of which the latter is a substitution instance. Obviously, it might be true to say “if the trigger of the gun had been pulled, the gun would have fired” though there are exceptions to the generalization “any gun fires if its trigger is pulled.” The singular conditional is elliptical; in asserting it one presupposes the presence in the particular situation of various causal conditions which the antecedent does not explicitly mention. (See, on this point, Pap 1952b; Pap 1955a, chapter IV A; and Chisholm 1955.)

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systems that are strictly closed, isolated from external influences, but the law of the conservation of energy says that if there were such a system its total energy would remain constant; there are no gases that are “ideal” in the sense that their molecules do not exert “intermolecular” forces on one another, but the general gas law says that if there were such a gas it would exactly satisfy the equation “PV = RT ;” there are no bodies that are not acted on by any external forces (indeed, the existence of such bodies is incompatible with the universal presence of gravitation), but the law of inertia says that if there were such a body it would be in a state of rest or uniform motion relative to the fixed stars.14 If such laws were formulated extensionally, as negative existential statements, like “there are no ideal gases that do not satisfy the equation: PV = RT ,” they would be vacuously true; anything whatsoever could be truly asserted to happen under the imagined unrealizable conditions. And it could hardly be maintained that, in analogy to the process of validation of singular contrary-to-fact conditionals, such laws are asserted as consequences of more general laws that have been instantially confirmed. If the general gas law, for example, is asserted as a deductive consequence of anything, then it is of the kinetic theory of gases, whose constituent propositions are surely not the kind of generalizations that could be instantially confirmed.15 But further, there is the much-discussed diﬃculty of distinguishing extensionally laws from universal propositions that are but accidental. It may be the case that all the people who ever inhabited a certain house H before H was torn down died before the age of 65. The statement “for any x, if x is an inhabitant of H, then x dies before 65” would then be true, yet nobody would want to say that it expresses a law. As Chisholm and Goodman have pointed out, if it were a law, then it would support a counterfactual conditional like “if Mr. Smith (who is not one of the inhabitants of H) had inhabited H, he would have died before 65.” Now, an extensionalist might try the following approach: what distinguishes laws from accidental universals is, not an obscure modality of existential necessity (as contrasted with logical necessity), but their strict universality. That is, a universal statement expresses a law only if either it contains no individual constants or else is deducible as a special case from well-confirmed universal statements that contain no individual constants. The predicates of the fundamental laws, i.e., those that contain no individual constants, should be purely general.

14 It

might be objected that the law of inertia can be formulated in such a way that it is not contrary to fact: if no unbalanced forces act on a body, then it is at rest or in uniform motion relative to the fixed stars. But when the law is used for the derivation of the orbit of a body moving under the influence of a central force, it is used in the contrary-to-fact formulation since the tangential velocities are computed by making a thought experiment: how would the body move at this moment if the central force ceased to act on it and it moved solely under the influence of its inertia? 15 For further elaboration of this argument against the extensional interpretation of laws, see Pap 1963b.

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However, a serious criticism must be raised against this approach. Just suppose that H were uniquely characterized by a property P which is purely general in the sense that it might be possessed by an unlimited number of objects.16 P might be the property of having a green roof; that is, it might happen that H and only H has P. In that case the accidental universal could be expressed in terms of purely general predicates: for any x, if x is an inhabitant of a house that has a green roof, then x dies before 65.17 It may be replied that although the antecedent predicate is purely general it refers, in the above statement, to a finite class that can be exhausted by enumeration of its members, and that it is this feature which marks the statement as accidental. Admittedly, so the reply may continue, it sounds absurd to infer from it “if y were an inhabitant of a house that has a green roof, then y would die before 65,” but this is because we tacitly give an intensional interpretation to the antecedent predicate. If instead it were interpreted extensionally, viz., in the sense of “if y were identical with one of the elements of the actual extension of the predicate,” the inferred subjunctive conditional would be perfectly reasonable. To cite directly the proponent of this explication of the distinction under discussion, Karl Popper: “the phrase ‘If x were an A . . . ’ can be interpreted (1) if ‘A’ is a term in a strictly universal law, to mean ‘If x has the property A . . . ’ (but it can also be interpreted in the way described under (2)); and (2), if ‘A’ is a term in an ‘accidental’ or numerically universal statement, it must be interpreted ‘If x is identical with one of the elements of A’ ” (Popper 1949). But this just won’t do. For “x is one of the elements of A” would, in the sense intended by Popper, be expressed in the symbolism of Principia Mathematica as follows: x = a ∨ x = b ∨ . . . ∨ x = n, where a, b, . . ., n are all the actual members of A.18 But if Popper were right, then, if “all A are B” is accidental, it could be analytically deduced from it that such and such objects are members of B, which is surely not the case. To prove this formally for the case where the actual extension of “A” consists of just two individuals a and b: (∀x)(x = a ∨ x = b ⊃ x ∈ B) is equivalent to (∀x)[(x = a ⊃ x ∈ B).(x = b ⊃ x ∈ B)], which is equivalent to the simple conjunction: a ∈ B.b ∈ B. But surely it can be supposed without self-contradiction that, as a matter of accident, all the inhabitants of houses with green roofs die before 65, and yet individual a, or individual b, survives the age of 65. What is logically excluded by the accidental universal is only the conjunctive supposition that a is an inhabitant

16 Notice

that a property may fail to be purely general in this sense even if it is not defined in terms of a particular object, e.g., “being the highest mountain.” 17 This criticism applies to C. G. Hempel and P. Oppenheim’s explication of “law” relative to a simplified extensional language system, in Hempel and Oppenheim 1948. R. Chisholm makes the same criticism, in Chisholm 1955. 18 He could hardly mean it just in the sense of “(∃y)(y ∈ A.x = y) ,” for this says nothing else than “x ∈ A,” and so does not amount to one of alternative interpretations of “x ∈ A.”

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of a house with a green roof and survives the age of 65. It is not denied that “a ∈ B.b ∈ B,” where a and b happen to be the only objects that have property A, is the ground, indeed the conclusive ground, on which the accidental universal “all A are B” is asserted; what is denied is that any atomic statements, or conjunctions of such, are analytically entailed by a universal statement, regardless of whether it is accidental or lawlike. The same confusion of the meaning of the universal statement with the ground on which it is asserted is involved in the following interpretation: (a ∈ A).(a ∈ B).(b ∈ A).(b ∈ B).. . ..(n ∈ A).(n ∈ B).(∀x)(x ∈ A ≡ x = a ∨ x = b ∨ . . . ∨ x = n). For clearly none of the atomic statements are entailed by the universal statement “all A are B.” But suppose that accidental universals were characterized pragmatically rather than semantically, in terms of the nature of the evidence which makes them warrantedly assertible. Thus P. F. Strawson suggests that only the knowledge that all the members of A have been observed and found to be B constitutes a good reason for asserting an accidental universal “all A are B.”19 Now, Strawson cannot mean conclusive evidence by “good reason,” since as long as there remain unobserved members of the subject class the evidence for a lawlike20 generalization is not conclusive either. He must therefore be making the more audacious claim that observations of a part of the subject class of an accidental universal cannot even make it probable that its unobserved members are likewise positive instances. He is then taking the same position as Nelson Goodman, who holds that if “all A are B” is accidental, it does not make sense to say that the evidence that observed members of A are B’s confirms the prediction that unobserved members of A are likewise B’s. But this criterion is highly counterintuitive. If 10 apples are picked out of a basket filled with apples and are found to be rotten without exception, it will be inductively rational to predict that the next apple that will be picked is likewise rotten. Yet, it may be just an accidental fact that all the apples in the basket are rotten. It is not necessary to assume that somebody deliberately filled the basket with rotten apples, though the circumstances may make this hypothesis plausible. It is possible, for instance, that somebody who made random selections (with closed eyes) of apples from a larger basket in order to fill up a smaller basket had the misfortune to get nothing but rotten ones though there were quite a few good specimens in the larger basket. An attempt to define the law-accident distinction in pragmatic rather than semantic terms, i.e., in terms of the kind of evidence leading one to assert the respective kinds of propositions, while retaining extensional logic for the formulation of the asserted propositions, has likewise been made by R. B. Braithwaite (Braithwaite 1953, chapter 9). He says as much as that the assertion of a 19 Strawson 20 A

1952, 199. lawlike statement is a statement which expresses a law if it is true.

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contrary-to-fact conditional causally presupposes acceptance of an instantially confirmed law from which the conditional component (the other component is the negation of the antecedent) is deducible, but that the truth-condition21 of the contrary-to-fact conditional is expressible in extensional logic: (p ⊃ q). ∼ p. There are two major objections to this approach: In the first place, contraryto-fact conditionals with identical antecedents and contradictory consequents (e.g., “if he had come, he would have been shot,” “if he had come, he would not have been shot”) are logically compatible on this analysis; whereas one should think that their logical incompatibility is a guiding criterion of adequacy for the semantic (not pragmatic) analysis of contrary-to-fact conditionals.22 Braithwaite in fact is saying that all contrary-to-fact conditionals whatever are true, though not all of them would be asserted by people confronted with the choice between asserting or denying them. But if a person honestly denies “p” and is familiar with the conventional meaning of “p,” then he does not believe the proposition expressed by “p”; yet, if the proposition expressed by “if A had happened, B would have happened” is simply t