Taking Frege at his Word 0198865473, 9780198865476

Table of contents :
Dedication
Preface
Acknowledgments
Contents
A Note on Citations, Translations, and Abbreviations
Representation of Frege’s Logical Symbols
PART I. NATURAL LANGUAGE AND THEORIES OF MEANING
1. Language and the Standard Interpretation
2. Frege’s New Logic and the Function/Argument Regimentation
PART II. METAPHYSICS AND THE STANDARD INTERPRETATION
3. Metaphysics and the Standard Interpretation
PART III. METATHEORY AND THE STANDARD INTERPRETATION
4. Soundness, Epistemology, and Frege’s Project
5. Reference, the Context Principle, and the Significance of Sentential Priority
6. The Context Principle, Sentential Priority, and the Pursuit of Truth
PART IV. PUTTING FREGE’S LESSONS TO WORK
7. Why Frege’s Apparently Absurd View Is not Absurd at All
8. Mathematical Knowledge and Sentential vs. Subsentential Priority
Bibliography
Index

Citation preview

Taking Frege at his Word

Taking Frege at his Word JOAN WEINER

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Joan Weiner 2020 The moral rights of the author have been asserted First Edition published in 2020 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2020939200 ISBN 978–0–19–886547–6 DOI: 10.1093/oso/9780198865476.001.0001 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

For Mark

Preface Although this book is not meant as an introduction to Frege, I have tried to make it as readable by nonspecialists as I can. Since I presuppose that the reader is acquainted with the basic outlines of Frege’s career, I begin this preface with a brief description of these outlines. Frege was engaged, for virtually all his career, in a single project: that of showing that the truths of arithmetic are truths of logic. He first mentions this project, and the gapless inferences it requires, in the preface to his 1879 monograph, Begriffsschrift. And he tells the reader that, in the process of trying to construct such proofs, he found natural language to be an impediment. He attempts to solve this problem in Begriffsschrift. In Begriffsschrift he offers us a logical language that is designed to be capable of expressing all content that has significance for inference. And he offers us a system of evaluation that is designed to make it a mechanical task—once the inference is expressed in his new logical language—to identify inferences either as correct and gapless or as requiring additional premises. The resulting new logic was a major advance. In particular, Frege’s logic, unlike Boole’s, allows us to evaluate mathematical inferences whose validity depends on polyadic relations and nested quantification. Having introduced these logical tools, Frege turns, in his 1884 monograph, Foundations of Arithmetic, to the task of giving a detailed, but non-technical, description of his project. In Foundations he also discusses his reasons for thinking that the truths of arithmetic are analytic, that is, belong to logic. The next step was to give actual gapless proofs, in the logical language, of basic truths of arithmetic from logical laws. In the process of trying to construct these proofs, however, Frege realized that substantial revisions to his logical system were necessary. In order to develop the new version of his logic, he needed to reconceive some of the basic notions of his logic, including the notions of function, concept, and object. The new version of his logic, and the descriptions of these reconceptions, are first presented in a series of three works, Function and Concept, “On Sinn and Bedeutung” and “On Concept and Object,” published in 1891 and 1892. In 1893 Frege published the first volume of Basic Laws of Arithmetic, which includes both an introduction to the new version of his logic and purely logical proofs of some basic laws of arithmetic. In 1903, when the second volume was in press, Frege received a now-famous letter from Bertrand Russell, in which Russell showed that the logical system of Basic Laws was inconsistent. The culprit was one of Frege’s new logical laws, Basic Law V, a law that was crucial to the proofs

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needed to establish that arithmetic belongs to logic. Although Frege tried a number of alternative strategies after the discovery of the contradiction, none was successful. He wrote in a diary entry near the end of his life, “My efforts to become clear about what is meant by number have resulted in failure.”¹ In spite of the dramatic failure of Frege’s general project, much of the work he did in the service of this project has had a profound influence on the subsequent development of philosophy in the analytic tradition. One part of this influence can be found in his new logic. Another can be found in his treatment of language, natural as well as logically perfect. Indeed, virtually all philosophers who are trained in the analytic tradition read Frege’s writings in the context of an introduction to the philosophy of language. Students are taught that, in his papers “On Sinn and Bedeutung” and ”Thoughts,” Frege means to be introducing the beginnings of a semantics or theory of meaning, a theory that—even if it is not widely accepted today—sets the agenda for much of the subsequent philosophical writing about language. These views are part of what I will call the Standard Interpretation. The central tenets of this interpretation are: (1) It was Frege’s aim to give a theory of meaning, or a theory of the workings of language, for (some version of) natural language. The kind of theory in question is one that provides an account of how the truth of a sentence depends on the semantic features of its parts. (2) On Frege’s view, the relevant semantic features of subsentential expressions (e.g., proper names, predicates) of a properly functioning language are determined by a reference relation that links them to extra-linguistic entities. And the reference relation is determined by something like the information content associated with the expression. A sentential expression can be said to have a truth-value only if its constituent proper names (predicates, etc.) already refer to extra-linguistic entities. (3) An important part of Frege’s logical theory is constituted by metatheoretical proofs. He defines a truth predicate for his logical language and uses this predicate to give a proto-soundness proof for his logical system: a metatheoretic proof that includes proofs of the basic laws and of the validity of the rules. He also offers a proof, by induction on the complexity of a Begriffsschrift expression, that all expressions of his logical language refer to extra-linguistic entities. (4) Frege is a Platonist. He has an ontological theory on which there is a distinction between functions and objects and there is a “third realm” containing such non-perceivable, non-spatio-temporal entities as numbers, functions, and thoughts.

¹ Diary entry March 23, 1924, NS, p. 282/PW, p. 263.

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Many philosophers take the Standard Interpretation to be, not so much an interpretation of Frege’s statements, as a straightforward, uncontroversial report of those statements. And many philosophers take the secondary literature to be of interest only insofar as it elaborates or deepens the Standard Interpretation. This book is devoted to setting out an interpretation that is deeply at odds with the Standard Interpretation. There are, however, aspects of the Standard Interpretation with which I have no dispute. Insofar as the enterprise of using natural language to introduce, discuss, or argue about features of a formal system counts as metatheory, there can be no debate: Frege does have a metatheory. What I deny is that Frege offers metatheoretic proofs that count as part of his logical theory. And I deny that he means to offer anything like a proto-soundness proof. Moreover, I do not deny that Frege is a realist in the following sense: he believes that truth is independent of us and our thought. What I deny is that (to use Michael Dummett’s words), it is, because the expressions we use have such extra-linguistic correlates [referents] that we succeed in talking about the real world, and in saying things about it which are true or false in virtue of how things are in that world.²

And, finally, I do not deny that Frege has much to teach us about natural language. What I deny is that he means to develop something like a compositional semantic theory. To deny these parts of the Standard Interpretation is to differ with this interpretation over, to use Dummett’s words “fundamental questions concerning what Frege was about.”³ Dummett has argued that given the clarity of Frege’s writing, there is no reason for such divergence in interpretation. And Dummett might have been right about this, had Frege explicitly stated, for instance, that he meant to give a theory of how the truth-value of a sentence is determined by semantic features of its constituents. But no such statement appears in Frege’s writings. In fact, the problem with the conception of the Standard Interpretation as straightforward reportage, I argue, is that much of the reportage that appears in the writings of supporters of the Standard Interpretation is not accurate. And many of the views the Standard Interpretation attributes to Frege conflict with his explicit, repeated statements about what his project is and what he is trying to do. Moreover, on the Standard Interpretation, a number of the positions that, Frege claims explicitly, are central to his view are not just wrong but obviously wrong and even absurd. Contemporary interpreters of Frege’s work have overwhelmingly responded to these apparently absurd statements by engaging in what seem

² Dummett (1981a), p. 198. The expressions Dummett has in mind are subsentential expressions, since he thinks it is a mistake to take sentences to have extra-linguistic correlates. ³ Dummett (1981b), pp. x–xi.

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on the surface to be acts of charity—either by treating these statements as incidental blunders that a proper interpretation of Frege’s works should ignore, or by treating the apparently absurd statements as statements that Frege cannot possibly have meant us to take literally, by treating them as statements that a proper interpretation of Frege’s works should reinterpret. An example is Frege’s view that sentences are names of truth-values. Dummett refers to the view that sentences are names of truth-values as a “gratuitous blunder” and a “ludicrous deviation.”⁴ He calls it an absurdity that had “a fatal effect on Frege’s theory of meaning.” This much is true: if, as the Standard Interpretation has it, Frege means to be giving a theory of the workings of some version of natural language, there is something wrong with the claim that sentences are names of truth-values. For sentences and proper names play different roles in natural language. But there is something very strange about the idea that this is an incidental blunder on Frege’s part or that it had a fatal effect on a theory that he meant to be promulgating. For “On Sinn and Bedeutung” is widely taken as one of the founding papers in the philosophy of language. Yet a large portion of this paper—nearly 18 of its 25 pages—constitutes, according to Frege, a defense of the view that sentences are names of truth-values.⁵ And in no part of this defense does Frege consider the kind of issue that so exercises Dummett. Indeed, Frege never describes, or says that he wants to develop, the kind of theory that Dummett attributes to him. Nor does anything in Frege’s lengthy defense of the view that sentences are names of truth-values address the issue of how the view might create problems for (let alone have a fatal effect on) a theory of the workings of language. That is, the Standard Interpretation would have us ignore an explicit statement that Frege makes repeatedly and regards as an important part of his logic on the grounds that—from the point of view of a project that he never explicitly describes or advocates in his writings—it is absurd. But can we infer, from the absence of a discussion of theory of meaning in Frege’s writings, that he was not engaged in developing such a theory? There are certainly philosophers who routinely work out the specific details of their theories without describing the grander picture to which these details are meant as a contribution. One might suspect that Frege was such a philosopher. But this is not true. Typically, he tells us explicitly what he hopes to accomplish in each of his writings. For example, he writes in Begriffsschrift that he means to introduce a logical language adequate for the expression and evaluation of inferences, something that is important for his attempt to show that the truths of arithmetic

⁴ These statements, as well as the statements mentioned in my next sentence, appear on p. 184 of Dummett’s (1981a). ⁵ After introducing the claim that sentences are names of truth-values, he goes on, on “On Sinn and Bedeutung” (SB), p. 36, to say that he is going to subject the claim that sentences are names of truthvalues to a “further test” and this further test takes up nearly all of the rest of the article.

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are analytic.⁶ He writes in Foundations that he means to make it seem probable that the laws of arithmetic are analytic.⁷ And he writes in Basic Laws that its purpose is to establish the epistemological nature of the truths of arithmetic.⁸ Thus it is difficult to believe that, in order to understand Frege correctly, we need to take his explicit statements to be contributions to elaborate, but unstated, projects. Moreover, as with most of his writings, Frege does tell us explicitly what the purpose of “On Sinn and Bedeutung” is. There is no mention of theories of meaning or of the workings of language. This article, he tells us, is supposed to solve a puzzle about identity statements. To read Frege as the Standard Interpretation would have us read him, of course, is to align his concerns with contemporary concerns. But do we really do Frege a favor by reading him in this way? To my mind, we do not. Such interpretations are born of the conviction that, if Frege says something out of step with what we say now, he is the one who is wrong, he is the one who needs instruction. I believe that we owe it to Frege, and to ourselves, to explore the views that result if we take seriously, and in most cases literally, the statements that appear repeatedly, and in prominent places, in his corpus. The purpose of Taking Frege at his Word is to offer an interpretation that does just that; that takes him at his word. And there is, I argue, much to be learned from the view that emerges from taking him at his word. This is not to say that on my interpretation, unlike the Standard Interpretation, it is possible to give a story on which everything Frege says is true. As I discuss in Chapters 1 and 3, this is not possible. It is not just that Frege, like most philosophers, sometimes makes contradictory statements. It is also that he explicitly tells us that he does not mean us to take all his statements to be statements of literal truths. It is, I argue in Chapter 2, Frege’s understanding of functions—and his understanding of predicates as function-names—that enables his monumental advance in logic. But, notoriously, his explicit description of this understanding also leads to the “concept horse” problem: many of his actual statements turn out, on his own account, to be either false or to miss his thought.⁹ On the Standard Interpretation, it is difficult to see how Frege could say such things. For, on the Standard Interpretation, his statements about the nature of functions, concepts, and objects belong to his logical theory. I argue in Chapter 3 that this is another instance in which the Standard Interpretation goes wrong. The statements in question are not meant as part of Frege’s logical theory but as elucidations— they belong, not to the theory, but to its propaedeutic. It is often thought, however, that such a response hardly redounds to Frege’s credit. It cannot help, one might think, for Frege to relegate his doctrines of concept and object to the propaedeutic of his logical theory if these doctrines ⁶ Begriffsschrift (BS), preface. ⁷ Foundations (FA), §87. ⁸ Basic Laws (BLA) vol. i, pp. vii, 1. ⁹ “On Concept and Object” (CO), p. 204.

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themselves are incoherent. But to say this is to miss the point. For what I argue is that Frege does not have doctrines of concept and object. Still, one might think, the objection must have merit. Isn’t coherence a standard of evaluation that should be applied to anything a philosopher says? The answer, which I discuss in Chapter 3, is that it is not. Frege writes repeatedly that what he is saying about concepts and functions is, strictly speaking, either false or nonsense. So I would not be taking him at his word if I were trying to explain how the statements really are true or coherent. But, given the centrality of these statements to Frege’s understanding of logic, it would also be a mistake to ignore them. Why would he choose to make these statements? I argue that he means to use his statements about functions and concepts as non-literal hints to get his readers to understand how his logic works. But how can this be an acceptable role for these statements to play? In order to answer this question, it is useful to notice that there are other purposes for which non-literal hints (hints that need not be, although they can be, true or coherent) can be of value to us. Here is an example: a flute player of my acquaintance was told to store some of the air in her feet when she breathes in and to let the air up out of her feet when she comes to the end of a particularly long phrase. The value of such a heuristic is not to be determined by whether or not it can be made true or makes sense. Given that this heuristic was helpful to my acquaintance, it would be misguided to object to its use on the grounds that it makes no sense because air cannot be stored in the feet. In this particular case, the heuristic had value because it enabled a flute player to play a problematic phrase. Similarly, it is misguided to object to Frege’s non-literal hints about how to understand his logic and logical language on the grounds that they are not coherent or not literally true. Coherence and truth do not belong among standards of evaluation for such non-literal hints. When I talk about providing an interpretation that takes Frege at his word, I mean to be providing an interpretation on which those statements that, Frege tells us, are important—no matter how absurd they may sound to some of us today—are neither ignored nor offered elaborate reinterpretations. My interpretation in this book is organized thematically around some of Frege’s prominently stated—but, many have thought, absurd—views on three general topics: language, metaphysics, and metatheory. Chapters 1–6 are largely devoted to textual exegesis. In Chapters 7 and 8, however, I discuss some of the consequences of the views that I have attributed to Frege. And I argue that many of these views are not just prima facie defensible, but teach us important, and exciting, things about how to approach issues that are of concern to us today. On the interpretation that I offer in this book, it is not that Frege has nothing to say about language, but that his views about language are very different from the views attributed to him by the Standard Interpretation. It is not just that, as I argue, Frege does not mean to be offering us an account of how the truthvalues of sentences are determined by semantic features of their constituents. It is also that a look at Frege’s actual practices illustrates a problem with this kind of

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account. For, as I argue in Chapters 7 and 8, one consequence of the kind of semantic theories that are based on referential relationships between subsentential expressions and extra-linguistic objects is that many actual, and apparently unexceptionable, ways in which language is used in some of our scientific inquiries should be rejected. This does not, in itself, constitute an objection to such semantic theories; philosophical arguments can surely elucidate problems with a scientific practice. But in this case, I argue, it is not the scientific practices but the semantic theories that should be rejected. I have tried to show that, if we truly want an account of the workings of language, then Frege’s insights—in particular, many insights that are not familiar to us—will be an indispensable tool.

Acknowledgments This project has been in the works for some time. My first attempt at describing it was in a proposal for funding for a sabbatical grant for the 2000–2001 academic year. I am immensely grateful to the Guggenheim Foundation and the American Philosophical Society for having awarded me fellowships that enabled me to spend two years (2000–2001 and 2002–2003) working full-time on the project. I want to thank the University of Wisconsin-Milwaukee, for granting me a sabbatical leave for the 2000–2001 academic year, as well as the Philosophy Programme Fellowship, University of London, School of Advanced Study for hosting me during that period and the Bogliasco Foundation Liguria Study Center for the Arts and Humanities for a residency during the spring of 2001. I am also grateful to Indiana University for granting me a research leave for the academic year 2002–2003. During these periods, I was able to work out many of the details that are central to the case I make here. However, the project I had mapped out in my proposal turned out to be too ambitious to complete in that time period. Two subsequent sabbaticals from Indiana University in 2009 and 2015–2016, as well as a fellowship from the National Endowment for the Humanities for the calendar year of 2016 helped me to finish an initial draft of the complete manuscript. Over the years I have published many articles in which I set out pieces of the project. This book includes large chunks from a number of these articles. They include: “Why does Frege care whether Julius Caesar is a number? Section 10 of Basic Laws and the context principle,” Ebert and Rossberg, eds., Essays on Frege’s Basic Laws of Arithmetic, Oxford University Press (2019); “Frege on indirect speech: where the standard interpretation goes wrong,” Paradigmi (2013); “What’s in a Numeral? Frege’s Answer,” Mind (2007); “Science and Semantics: the Case of Vagueness and Supervaluation,” Pacific Philosophical Quarterly (2007) Vol. 88 Issue 3; “Semantic Descent,” Mind (2005) Vol. 114, No. 454; “What Was Frege Trying to Prove? A Response to Jeshion,” Mind (2004) Vol. 113, No. 449; “Section 31 Revisited: Frege’s Elucidations,” E. Reck, From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy, Oxford University Press (2002); “Frege and the Linguistic Turn,” Philosophical Topics (1997) Vol. 25, No 2; “Has Frege a Philosophy of Language?” W. W. Tait (ed.), Early Analytic Philosophy: Essays in Honor of Leonard Linski, Open Court Press (1997); “Burge’s Literal Interpretation of Frege,” Mind (1995) Vol. 104, Issue 415;“Realism bei Frege: Reply to Burge,” Synthese, March (1995). The articles listed are published with permission, where necessary. Because this manuscript has been so long in the making, I have not been able to keep track of the many debts I owe for discussion and correspondence to the

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colleagues, friends, and scholars with whom I have been in communication. I regret that I am not able to list them all. However, among them are Jan Harald Alnes, Gisela Bengsston, Patricia Blanchette, Robert Brandom, Philip Ebert, Luca Ferrero, Dirk Greimann, Jeremy Heis, Jim Hutchinson, Juliet Kennedy, Richard Lawrence, Øystein Linnebo, Fraser MacBride, Robert May, Marco Panza, Chris Pincock, Eva Picardi, Thomas Ricketts, Marcus Rossberg, Ian Rumfitt, Stewart Shapiro, Sanford Shieh, Scott Sturgeon, Jamie Tappenden, Gabriele Usberti, Nicla Vassallo, Kai Wehmeier, and Crispin Wright. I would also like to thank two anonymous referees at Oxford University Press for helpful suggestions and, especially, Peter Momtchiloff for encouragement and support. I owe special thanks to Gary Ebbs and Mark Kaplan for reading and commenting on numerous versions of this manuscript and for providing invaluable philosophical wisdom and advice. Although the approach I take to contemporary issues in the final chapters of this book comes primarily from my reading of Frege, this part of the project has also been influenced by some of Hilary Putnam’s writings, particularly his writings about the significance our deference to experts should have for our understanding of the workings of language. It is one of my profound regrets that by the time I had sorted out my views on this topic, it was too late to hash them out with him.

Contents A Note on Citations, Translations, and Abbreviations Representation of Frege’s Logical Symbols

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PART I. NATURAL LANGUAGE AND THEORIES OF MEANING 1. Language and the Standard Interpretation I. Reportage and the Theory of Meaning Ia. Reportage and the Treatment of the Logical Language in Basic Laws Ib. Reportage and “On Sinn and Bedeutung”

II. Some General Remarks about Interpretation III. The Standard Interpretation and “On Sinn and Bedeutung” IIIa. Dummett and Truth-values as Objects IIIb. Logical Language and Natural Language IV. A Theory of Meaning for What Language? IVa. Natural Language IVb. Must the Improved Version of Natural Language be Free of Logical Imperfections? IVc. Could Frege Want a Theory of Natural Language as it Is? IVd. Why Should Ambiguity be a Problem? IVe. Why Does Frege Analyze (some) Imperfect Parts of Natural Language? V. So Where Are We Now?

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10 13 13 16 20 20 24 28 29 30 32

2. Frege’s New Logic and the Function/Argument Regimentation I. Why Does Frege Need a Microscope?

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Ia. What Are Logical Truths? The Begriffsschrift Answer II. The First Version of Frege’s Microscope IIa. The Begriffsschrift Notion of Function IIb. Functions and How We Regard an Expression IIc. Do Functions and Arguments Differ in Kind? IId. Some Problems with the Linguistic-expression View of Function III. Two Problems with the Begriffsschrift Account of Identity IIIa. The First Identity Problem IIIb. A Second Identity Problem IV. Frege’s Introduction of Sinn and Bedeutung as a Solution to the Identity Problems

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IVa. The Modified Begriffsschrift View of Identity IVb. The New View of Identity V. Functions, Arguments, and Objects VI. Frege’s View of Sentences as Truth-value Names VIa. Sentences in Begriffsschrift VIb. Sentences in “On Sinn and Bedeutung”

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PART II. METAPHYSICS AND THE STANDARD INTERPRETATION 3. Metaphysics and the Standard Interpretation I. Frege’s Logical Notion of Function Ia. Familiar Function-expressions and Functions Ib. Unsaturatedness and some Unfamiliar Function-names and Functions Ic. Concept and Object

II. Function and Object as Metaphysical or Ontological Categories IIa. Functions and the Problem of Predication IIb. Platonism and Frege: Burge’s “Literal” Interpretation IIb.i. What Does Frege Say about Non-spatio-temporal Entities? IIb.ii. Does Frege Qualify these Remarks? IIb.iii. Can There be a Literal Statement that Functions Are Atemporal? IIc. What Work Does Frege’s Function/Object Distinction Do? IId. “Function” and “Object” in Statements of Literal Truths IIe. Elucidation, its Uses and Evaluation IIf. The Concept Horse Revisited

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85 87 92 93 95 97 100 102 104 109

PART III. METATHEORY AND THE STANDARD INTERPRETATION 4. Soundness, Epistemology, and Frege’s Project I. Logical Laws and Metatheory Ia. Quine’s Argument for Semantic Ascent Ib. Schemata and Logical Laws

II. Modus Ponens, Logical Laws, and Metatheory IIa. The Justification of Modus Ponens IIb. Concepts as Functions; Why “is the True” Is not a Truth Predicate IIc. Metalinguistic Variables IId. Modus Ponens: Metatheory and Elucidation IIe. Elucidation and the Justification of Modus Ponens and Basic Law I III. Soundness and Frege’s Epistemological Project IIIa. Does Frege Need a Soundness Proof? IIIb. Epistemology, Best Proof, and Generality

115 117 117 119

122 122 123 128 130 133 135 135 137

 IIIc. What is Required of a Primitive Logical Law? IIId. Basic Law V, Epistemology, and Semantic Proof IIIe. An Alternative Epistemological-semantic Story: Analyticity as Truth by Virtue of Meaning IIIf. But did Frege Give a Semantic Proof of Basic Law V? IIIg. Is Basic Law V a Primitive Logical Law?

IV. Soundness and the New Science: Did Frege Envision a Soundness Proof? V. Conclusion

5. Reference, the Context Principle, and the Significance of Sentential Priority I. Realism and Extra-linguistic Entities II. The Metaphysical Requirement, the Context Principle, and the Sentential Priority View IIa. Identity and the Metaphysical Requirement IIb. The Context Principle and the Sentential Priority View in Foundations IIc. The Sentential Priority View and Basic Laws IId. §10 and the Significance of Identity Statements

III. §§28–31 of Basic Laws IIIa. §31 and Metatheory IIIb. The Inductive Proof Interpretation and §29 IIIc. §§28–31 and the Circularity Puzzle IIId. The VR (value-range) Function-name IV. §10 of Basic Laws IVa. §10 and the Standard Interpretation: Three Difficulties IVb. Solving the Three Difficulties of §10 IVc. The Sentential Priority View, Realism, and Personal Epistemology

6. The Context Principle, Sentential Priority, and the Pursuit of Truth I. Changing the Subject and the Logicist Project: What makes Logicism about our Arithmetic? Ia. Why Define the Number One and Concept Number? Ib. Why Frege Would Reject the Apparently Obvious Faithfulness Requirement Ic. What Are Frege’s Actual Faithfulness Requirements?

II. How Does this Square with the View that the Sentences Expressing the “Well Known Properties of the Numbers” are True? IIa. The Sentential Priority View Revisited IIb. Methodology: Natural Language and Inquiry

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170 170 173 173 178 181 185

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211 213 216 220 226

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PART IV. PUTTING FREGE’S LESSONS TO WORK 7. Why Frege’s Apparently Absurd View Is not Absurd at All I. Vagueness Ia. Vagueness and Deference to Experts Ib. Vague Predicates, Semantics, and Empirical Investigation

II. Precisification IIa. Supervaluationism, Precisification, and the “Homophonic” Objection IIb. “Obese”: A Case Study IIb.i. Is “Obese” a Technical Scientific Term? IIb.ii. Precisification and Empirical Studies IIb.iii. The Binaries Objection IIb.iv. Changing the Subject—the Homophonic Objection Revisited III. Explication and Vagueness as a Logical Defect

8. Mathematical Knowledge and Sentential vs. Subsentential Priority I. The Logical Notion of Objecthood II. Metaphysical Objecthood and Subsentential vs. Sentential Priority IIa. Predicates and Carving Nature at the Joints IIb. Another Example: The Case Definition of AIDS

III. Frege’s Alternative IIIa. Benacerraf ’s Problem and Subsentential Priority IIIb. Frege’s Realism and the Significance of Sentential Priority IIIc. Sentential vs. Subsentential Priority IV. Conclusion

Bibliography Index

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270 271 276 277 280 284 284 285 289 292

297 305

A Note on Citations, Translations, and Abbreviations Frege introduced a number of technical terms in his writings. For reasons that I go into in this book (see especially Chapter 5), I have decided to use some of these terms, untranslated and unitalicized, as if they were English words. These include the term “Sinn” and the term “Bedeutung” and its cognates. I also use Frege’s name for his logical language, “Begriffsschrift,” which has no natural translation, untranslated and unitalicized. It is worth noting that, while some writers and translators use the term untranslated, others use the term “Concept-script.” However, when I write about the 1879 monograph in which Frege first introduced his new logic, I use the italicized expression “Begriffsschrift”. Many of Frege’s writings, particularly his most influential, have been republished in a number of different volumes and different translations. In most of the publications of writings that were originally published in Frege’s lifetime, the pagination of the original publication is included. In order to make passages easy to find, I have used the original pagination, when available, in my citations rather than citing a particular translation. Since most publications of Begriffsschrift and Foundations of Arithmetic do not include original pagination, I have cited section numbers rather than page numbers. For writings that were unpublished in Frege’s lifetime, as well as correspondence, I offer both English and German citations from the two collections, Nachgelassene Schriften (trans. as Posthumous Writings) and Wissenschaftlicher Briefwechsel (trans. as Philosophical and Mathematical Correspondence). In footnotes, I use the following abbreviations: AIMCN

BLA vol. i BLA vol. ii BLC

BS

“Über den Zweck der Begriffsschrift,” Jenaische Zeitschrift für Naturwissenschaft, 16 [1883], supplement pp. 1–10. Trans. as “On the aim of the conceptual notation,” CN, pp. 90–100. Grundgesetze der Arithmetik Band I, H. Pohle, Jena (1893). Trans. in Ebert and Rossberg (2013). Grundgesetze der Arithmetik Band II, (1903a). Trans. in Ebert and Rossberg (2013). “Booles rechnende Logik und die Begriffsschrift” (1880/1881) NS, pp. 9–52. Trans. as “Boole’s logical Calculus and the Concept-script,” PW, pp. 9–46. Begriffsschrift. Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. (1879). Trans. by S. Bauer-Mengelberg. Halle: Nebert. In van Heijenoort (ed.) (1970).

xxii

   , ,  

CO

CSB

CT

FA

FB

FTA

LI

LM Logik NS OFG I

OFG II

On Schoenflies

OSJ

PCN

“Über Begriff und Gegenstand,” Vierteljahrsschrift für wissenschaftliche Philosophie, 16 (1892). Trans. as “On Concept and Object,” in McGuinness (1984), pp. 192–205. “Ausführungen über Sinn und Bedeutung,” in Nachgelassene Schriften pp. 128–36 (1892–95). Trans. as “Comments on Sense and Meaning” in PW pp. 118–25. “Gedankengefüge,” Beiträge zur Philosophie des deutschen Idealismus 3, Band (1923). Trans. as “Compound Thoughts,” in McGuinness (1984), pp. 36–51. Grundlagen der Arithmetik, Breslau: Wilhelm Koebner (1884). Trans. by J. L. Austin as The Foundations of Arithmetic, Evanston, Il: Northwestern (1980). Funktion und Begriff, Jenaischen Gesellschaft für Medicin und Naturwissenschaft, H. Pohle: Jena (1891). Trans. as Function and Concept in McGuinness (1984), pp. 1–31. “ ‘Über Formale Theorien der Arithmetik,” Sitzungberichte der Jenaischen Gesellschaft für Medizin und Naturwissenschaft (1885). Trans. as “On Formal Theories of Arithmetic,” in McGuinness (1984), pp. 112–22. “Über das Trägheitsgesetz,” Zeitschrift für Philosophie und philosophische Kritik, 98 (1891) pp. 145–61; Trans. as “On the Law of Inertia,” CP, pp. 123–36. “Logik in der Mathematik” (1914) in Nachgelassene Schriften, pp. 219–72. Trans. as “Logic in Mathematics,” in PW pp. 203–50. “Logik” (1897) in Nachgelassene Schriften pp. 137–63. Trans. as “Logic” in PW pp. 126–51. Hermes, Hans et al. (eds.), Nachgelassene Schriften, Hamburg: Meiner Verlag (1983). “Über die Grundlage der Geometrie,” Jahresbericht der Deutschen Mathematiker-Vereinigung 12, Band (1903). Trans. as “On the Foundations of Geometry: First Series,” in McGuinness (1984), pp. 319–24; 368–75. “Über die Grundlagen der Geometrie,” Jahresbericht der Deutschen Mathematiker-Vereinigung, 15, Band (1906). Trans. as “On the Foundations of Geometry: Second Series,” in McGuinness (1984), pp. 293–309; 377–403; 423–30. “Über Schoenflies: Die logischen Paradoxien der Mengenlehre,” NS 191–99. Trans as ‘On Schoenflies: Die Logischen Paradoxien der Megenlehre’, PW, pp. 176–84. “Über des wissenschaftliche Berechtigung einer Begriffsschrift,” Zeitschrift für Philosophie und philosophische Kritik, 81 (1882) pp. 48–52. Trans. as “On the Scientific Justification of a Conceptual Notation”, in Bynum (1972). “Über die Begriffsschrift des Herrn Peano und meine eigene,” Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der

   , ,  

xxiii

Wissenschaften zu Leipzig. Mathematisch–Physische Klasse. 48, Band (1896), pp. 361–78. Trans. as “On Mr. Peano’s Conceptual Notation and My Own,” in McGuinness (1984), pp. 234–48. PMC Gabriel, G., et al., McGuinness, B. (eds), Kaal, H. (trans.) Gottlob Frege: Philosophical and Mathematical Correspondence, Chicago: The University of Chicago Press (1980). PW Hermes, Hans, et al. (eds.), Peter Long and Roger White (trans.) Posthumous Writings, Chicago: University of Chicago Press (1979). Review of Husserl Rezension von E. Husserl, Philosophie der Arithmetik, Zeitschrift für Philosophie und philosophische Kritik 103, pp. 313–32 (1894). Trans. as Review of E. G. Husserl, Philosophie der Arithmetik I, in McGuinness (1984), pp. 195–20). SB “Über Sinn und Bedeutung,” Zeitschrift für Philosophie und philosophische Kritik, 100 (1892). Trans. as “On Sense and Meaning” in McGuinness (1984), pp. 25–50. T “Der Gedanke”, Beiträge zur Philosophie des deutschen Idealismus, I, 58–77 (1918/1919). Trans as “Thoughts,” in McGuiness (1984), pp. 351–72. WB Gabriel, G., (ed), (1976) Gottlob Frege: Wissenschaftlicher Briefwechsel, Felix Meiner Verlag.

Representation of Frege’s Logical Symbols Because Frege’s actual two-dimensional notation is both cumbersome and difficult for nonspecialists to read, I have chosen, for the most part, to use contemporary symbols in this book. However, both versions of Frege’s logical language make use of symbols that have no correlates in contemporary logical notations. Among these are the judgement stroke and the symbol that he calls the “content stroke” (in the first version, which appears in Begriffsschrift) and the “horizontal” (in his later writings). For convenience, in this description of Frege’s symbols I will use the expression “horizontal” when I am talking about both versions of his logical language. The judgement stroke is a short vertical line that (in both versions of Begriffsschrift) appears at the beginning of any line in a proof. It is attached via a horizontal to the rest of the formula. So, for example, each line of a proof will begin with the following composite symbol “⊢”. (Since the length of a horizontal is variable, a line of a proof might begin with something that looks more like this, “⊢”). The horizontal also appears prefixing, not only entire lines of a proof, but sentential expressions that are constituents of the lines of the proof. The mechanics of his language do not permit his symbols for negation, conditional, and the quantifier simply to be prefixed to a formula. They must be attached to horizontals. This works as follows: The representation of denial, in both versions of Frege’s logic, requires two horizontals as well as a short vertical stroke, which he calls a “negation stroke.” A Begriffsschrift expression that represents the denial of a formula would look something like this:

⊤A (For Frege’s explanations, see Begriffsschrift §7, and Basic Laws, vol. i, p. 10). Similarly, although Frege introduces an expression that he calls the “conditional stroke,” a conditional cannot be represented by the conditional stroke alone but requires horizontals as well. The representation of a conditional with antecedent A and consequent B would look something like this:

B A This expression can be viewed as containing three horizontals, one prefixed to the bottom of the conditional stroke, and the other two on each side of the top of the

xxvi

   ’   

conditional stroke. (For Frege’s explanations see Begriffsschrift §5 and Basic Laws vol. i, p. 20). Frege’s quantifiers require a symbol called the “concavity.” The variable that is bound by the quantifier sits on top of the concavity and the concavity is flanked by two horizontals. The concavity looks like this: a . This use of horizontals is an important kind of punctuation. For example, it is a part of Frege’s technique for representing the scope of quantifiers. I have used the concavity, in quotations and in discussions of the quotations. In other contexts, I use contemporary quantifier notation. In the first version of Begriffsschrift, there is a sense in which the horizontal symbols act solely as a kind of punctuation. For, unlike the negation stroke, or conditional stroke, the horizontals (or content strokes) are not given any content.¹⁰ The punctuation roles that the content stroke plays in the first version of his logical language are captured by different kinds of punctuation and conventions in contemporary logical notations. Thus, given that the content stroke does not seem to have any content, I sometimes use contemporary notation (e.g., “~A”, “A !B”) to represent denials and conditionals in the first version of Frege’s logical language. For convenience, when I am talking about logical regimentation but not about Frege’s particular logical languages, I also use contemporary notation. However, the situation with the second version of Frege’s logical language is different. One of Frege’s key departures from traditional logics was to abandon the subject/predicate analysis of statements in favor of function/argument analysis. Frege’s conception of function in Begriffsschrift is somewhat problematic (for discussion of these problems, see Chapter 2) and, I have already suggested, the content stroke does not appear to be a function-name. Indeed, it is not clear that in the first version of his logic, any of the logical symbols are to be understood as function-names. However, when Frege introduces the second version of his logic, he bases his function/argument analysis on a new notion of function. And in the second version of the logic, Frege introduces the horizontal as a functionname. The symbols used to express negations and conditionals have constituent expressions that are not punctuation marks or logical connective signs but, rather, function-names. To use an expression like “~A” instead of “⊤A” then, is to leave out two function-names. Because it is sometimes important to know how many horizontals appear in a formula and where they appear, I have chosen at some points to use a hybrid notation that combines contemporary logical symbols with Frege’s horizontals. For example, “–~–A” and “–(–A!–B)” “ a g(a).” In the second version of Begriffsschrift, Frege also introduces a new expression for a second-level function that takes first-level functions as arguments and gives, as value, the value-range of the argument (for an explanation of this function, ¹⁰ When he introduces this symbol, no content is associated with it. Frege writes only that the content stroke “combines the signs that follow it into a totality.” Begriffsschrift, §2.

   ’   

xxvii

see Chapter 3). This function-symbol, which I call the “VR function-name” is constructed from Greek vowels along with a first-level function-symbol. For example, in the expression “ ἀ ΦðαÞ”, “Φ” appears in the argument place and the rest of the symbols belong to the representation of the VR function. I follow Frege in using the symbol “≡” for the identity sign of the first version of Begriffsschrift and the symbol “=” for the identity sign of the second version of Begriffsschrift.

PART I

NATURAL LANGUAGE AND THEORIES OF MEANING

1 Language and the Standard Interpretation Although there are many areas of disagreement among those who write about Frege, for many years there was near consensus that certain views can be attributed to Frege. These include views about natural language—for example, that Frege means to develop a (part of a) theory of meaning for natural language; about metaphysics—for example, that Frege has a metaphysical theory on which there is a third realm whose denizens include numbers, thoughts, and functions; and about his logical language—for example, that Frege offers a truth definition for his logical language. Indeed, the view that Frege was interested in giving a theory of meaning is often articulated as if it were not interpretation but merely straightforward reportage. In this book, my aim will be to introduce a different kind of interpretation, an interpretation on which many of what are taken, by the Standard Interpretation, to be obvious truths about Frege’s views are not true of Frege’s views at all. As I mention in the preface, my purpose in discussing the Standard Interpretation is clarificatory. It is no part of my aim to defend a sociological story about what views philosophers hold about Frege. I do not, in particular, claim that everyone (or, almost everyone) who writes on Frege agrees with the Standard Interpretation. Indeed, in the post-1980s flowering of Frege exegesis, there have been a number of challenges, mine among them, to various aspects of the Standard Interpretation. Why, then, do I place so much emphasis on the Standard Interpretation in this book? My emphasis on the Standard Interpretation is an attempt to ward off a kind of misunderstanding that frequently occurs in my conversations about Frege. I often discover—well into such a discussion—that I am speaking with people who take the Standard Interpretation to be so obviously true that they have not even entertained the possibility that I might be offering an interpretation that conflicts with it. It is in hopes of warding off such confusion that I begin this chapter with a discussion of some parts of the Standard Interpretation that are typically taken to be so obvious as not to require textual defense. One of these is that it was Frege’s aim to give a theory of meaning for (some version of) natural language. For example, Michael Dummett writes that Frege “wanted to give a general account of the workings of language.”¹ He continues, “An account of the working of language is a theory of meaning, for to know how

¹ Dummett (1981a), p. 83.

Taking Frege at his Word. Joan Weiner, Oxford University Press (2020). © Joan Weiner. DOI: 10.1093/oso/9780198865476.003.0001

4

    

an expression functions, taken as part of the language, is just to know its meaning.” According to Dummett, an important part of such an account is “the description of the structure of the sentences of this language” and a theory of truth. And he writes that an account of the way in which their truth-values were determined, and the rules of inference laid down were then seen to be justified by the rules governing the assignment of truth-values.²

Donald Davidson provides a similar description of Frege’s interest in language. He writes, Frege saw the importance of giving an account of how the truth of a sentence depends on the semantic features of its parts, and he suggested how such an account could be given for impressive stretches of natural language. His method was one now familiar: he introduced a standardized notation whose syntax directly reflected the intended interpretation, and then urged that the new notation, as interpreted, had the same expressive power as important parts of natural language.³

According to Davidson, such an account can be characterized as a theory of truth and also a theory of meaning. Let us consider the claim that one of Frege’s aims was to develop a theory of meaning in the sense just described.

I. Reportage and the Theory of Meaning Given the prevalence of this view in contemporary writings, one might suspect that the philosophers from whose writings I have just quoted are simply engaged in reportage. One might suspect Frege explicitly claims to be giving an account of how the truth-values of sentences belonging to a certain part of natural language depend on semantic features of their parts. One might suspect that Frege explicitly claims to be giving such an account for the sentences of his logical language, Begriffsschrift. But Frege does not explicitly claim to be doing any of these things. Indeed, the expressions “theory of meaning,” “theory of truth,” “theory of the workings of language, “semantics” are not translations of any expressions Frege uses. According to Dummett, Frege’s term for the theory of the workings of language or theory of meaning is “logic.” But Frege never says that one of the

² Dummett (1981a), p. 81.

³ Davidson (1977), in Davidson (1984), p. 202.

    

5

tasks of logic is to explain the workings of natural language. Nor does Frege ever claim that it is part of the task of logic to provide some sort of theory for the logical language—for example, a theory of how the truth-value of a sentence of the logical language is determined by interpretations of its constituents. This does not show, of course, that the Standard Interpretation is incorrect. It could be that Frege does mean to be giving a theory of meaning of the sort Dummett and Davidson describe, but simply chooses not to announce this project. But what it does show is that this aspect of the Standard Interpretation is not reportage but interpretation. And, supposing the Standard Interpretation is right, Frege’s reticence is, at least, odd. For throughout his writing, Frege takes great care to make sure that his audience understands his larger projects, as well as the roles played by particular discussions in those projects. He explicitly tells us the purpose of Begriffsschrift: it is to introduce a logical language adequate for the expression and evaluation of inferences.⁴ This, he tells us, will be used to produce gapless proofs of the truths of arithmetic, so that we can determine whether they are analytic or synthetic. He explicitly tells us the purpose of Foundations: he hopes to make it seem probable that the laws of arithmetic are analytic.⁵ The purpose of Function and Concept, he writes, is to tell us about some “supplementations and new conceptions” that require alterations of the original version of his logic and, in particular, to provide an explanation of the notion of function that he deems too elaborate to put into Basic Laws without alienating some readers.⁶ The purpose of Basic Laws, he tells us, is to establish the epistemological nature of the truths of arithmetic.⁷ Discussions in his earlier works explain how it is that the proofs in Basic Laws can establish the analyticity of the truths of arithmetic.⁸ Frege is also typically explicit about the purposes of many of his shorter discussions. His monographs are divided into parts, the parts into groups of sections. And his tables of contents include descriptive titles for each section, each group of sections and each part. Nor are Frege’s descriptions of his projects typically at all obscure. Indeed, Dummett writes, “Frege is one of the clearest of all philosophical writers.”⁹ Thus there is, at least, a puzzle for supporters of the Standard Interpretation. Were it an important part of his project to give an account of how the truth-values of sentences (whether of natural language or of his logical language) are determined by semantic features of their constituents, we would expect him to have announced and described this enterprise. Yet no such announcements or descriptions can be found in Frege’s writings. It is perhaps even more puzzling that there is no mention of such a project in his later writings. In 1902, when the second volume of Basic Laws was in press, ⁴ BS, preface. ⁵ FA, §87. ⁶ FC, p. i. ⁷ BLA vol. i, pp. vii, 1. ⁸ The whole of FA is, arguably, devoted to this topic. But see, especially, Part V of that work. ⁹ Dummett (1981), p. xi.

6

    

Russell sent Frege a now-famous letter showing that the logic of Basic Laws was inconsistent. And, ultimately, Frege came to believe that his logicist project had ended in ruins—in one diary entry, he wrote, “My efforts to become clear about what is meant by number have resulted in failure.”¹⁰ But that is not to say that Frege regarded all his efforts to have ended in failure. He mentions a number of his other accomplishments.¹¹ Surely, providing the beginnings of a theory of truth, either for Begriffsschrift or for a fragment of natural language would count as an accomplishment. Thus, had he seen himself as doing such a thing, one would expect to find, in his post-contradiction writings, some mention of this as one of his successes. There is, however, no talk in his unpublished fragments or in his later publications about either the success or failure of a project having to do with giving any sort of theory of truth for any fragment of natural language.

Ia. Reportage and the Treatment of the Logical Language in Basic Laws Even so, it may seem that it requires little more than reportage to support this aspect of the Standard Interpretation. For it has seemed to many that, regardless of Frege’s actual descriptions of what he is doing, there is no question about what he does. That is, he offers us something that clearly is, if not a comprehensive theory of meaning, a large part of such a theory for a fragment of (some version of) natural language. The writing that seems most clearly to demonstrate an interest in natural language is his essay “On Sinn and Bedeutung.” Something like the Standard Interpretation seems to be supported by the evident fact that Frege did discuss natural language; by the fact we can use this view to explain the point of some of the individual discussions of “On Sinn and Bedeutung” and, in particular, by the role of compositionality in these discussions. But, for all that, in “On Sinn and Bedeutung” Frege does not spell out the kind of theory that the Standard Interpretation attributes to him. The evidence that he did mean to set out such a theory is the combination of these discussions of natural language with Frege’s treatment of his logical language. It is in his treatment of the logical language, according to the Standard Interpretation, that we get the introduction of a logically perfect language and its semantics. And this semantics for the logical language is meant, according to the Standard Interpretation, to be used in our understanding of natural language.¹² ¹⁰ Diary entry March 23, 1924 NS, p. 282/PW, p. 263. Neo-Fregean logicists may disagree, but the neo-Fregean assessment of the importance of Frege’s logicist project differs from Frege’s own assessment. ¹¹ See, e.g., NS, p. 200/PW, p. 184. This note is dated August 5 1906—that is, after he had decided that his logicist project had failed. ¹² See, e.g., Davidson (1984), p. 202.

    

7

Dummett offers us a two-page summary of both the construction of the logical language Frege introduces in Basic Laws and its semantics.¹³ This summary is not presented as an interpretation—no passages from Frege are quoted, discussed, or even footnoted—but simply as reportage. And, Dummett remarks, the procedure that Frege carries out is, save for a few details, “exactly the same as the modern semantic treatment of the language of predicate logic.”¹⁴ Were Dummett’s report correct, it would be entirely clear that Frege thought it important to give—and did give—an account of how the truth values of Begriffsschrift sentences are determined by semantic features of their constituents. And it would be pretty safe to say, as Dummett does, that there would be no reason for interpretations to differ, at least on this fundamental issue.¹⁵ The problem is that, far from being straightforward reportage, many fundamental features of Dummett’s descriptions of Frege’s treatment of the logical language in Basic Laws are factually false. For example, Dummett writes, The analysis which Frege gave constituted the invention of the language of (higher-order) predicate logic, as we have it now: a language in which the atomic sentences are formed from individual constants (i.e. simple proper names), and primitive functional expressions, predicates and relational expressions, and in which complex sentences are formed by means of the truth-functional sentential operators, and quantifiers of fixed type and level, as required.¹⁶

The highlighted part of this passage is false. First, Frege’s logical language, Begriffsschrift, has no individual constants. Indeed, since Begriffsschrift has no individual constants, it cannot contain the kind of atomic sentences familiar from contemporary logical notations—that is, a single predicate letter followed by (the right number of) individual constants. Second, Frege does not have a category of atomic or simple sentences.¹⁷ Dummett then goes on to explain that Frege’s interest in the analysis of language was not solely in the analysis of a logical language but also in the analysis of natural language. And Frege’s analysis of his logical language is to be used in our understanding of natural language. Dummett writes, Frege’s notion of reference is best approached via the semantics which he introduced for formulas of the language of predicate logic. An interpretation of

¹³ Dummett (1981a), pp. 89–90. ¹⁴ Dummett (1981a), p. 90. ¹⁵ Dummett (1981b), p. xi. ¹⁶ Dummett (1981a), p. 81, emphasis added. ¹⁷ In §29 of vol. i of BLA, Frege gives a list of classifications of Begriffsschrift expressions. Not only is there no category for atomic sentences, there is no category for sentences. We will see why this is later. There is also another problem with Dummett’s presentation. While Dummett does not say explicitly that Frege is giving a syntactic definition of sentence, this is suggested in the passage. As we will see shortly, this is not right.

8

     such a formula (or set of formulas) is obtained by assigning entities of suitable kinds to the primitive non-logical constants occurring in the formulas.¹⁸

That is, Dummett goes on to explain, Frege’s interpretation assigns to each individual constant an object; to each primitive non-logical, unary functionsymbol, a unary function that is everywhere defined; to each primitive, nonlogical one-place predicate a property defined over all objects, etc. Again, this is factually false. First, as I have just mentioned, there are no individual constants in Begriffsschrift. Hence, supposing Frege offers a semantics for his language, it cannot involve assignments of objects to individual constants. Second, Frege’s logical language contains no primitive non-logical constants at all. Nor does Frege give the familiar inductive definition of something like “formula” or “sentence.” Indeed, Frege’s exposition of his logical language does not even contain terms that could be translated as “formula” (or, for that matter “sentence”). Dummett’s report of Frege’s Basic Laws treatment of his logical language cannot provide support for the Standard Interpretation, for the simple reason that it is false. But, even if Dummett’s report is false, it may seem that there is another kind of reportage that will do the job. For the obviousness of the Standard Interpretation is usually taken to come, not from Frege’s treatment of his logical language, but from his treatment of natural language in “On Sinn and Bedeutung”. Indeed, nearly every student who is introduced to Frege’s writings is introduced to them via a reading of this paper.

Ib. Reportage and “On Sinn and Bedeutung” There is no question that Frege writes about natural language in “On Sinn and Bedeutung.” There is, indeed, no question that he discusses different sentential constructions and how their truth-values are determined. And there is no question that these discussions have had an enormous impact in the development of contemporary philosophy of language—in particular, the development of what Dummett refers to as “theory of meaning” in the passages quoted above. Given that “On Sinn and Bedeutung” so manifestly is a contribution to this kind of theory of meaning, it may seem a very small step to infer that—even if he did not explicitly say so—Frege meant to be developing part of such a theory in “On Sinn and Bedeutung.” There are problems with this apparently small step, however. The sentential constructions that Frege discusses in “On Sinn and Bedeutung” belong to natural language. But, as both Dummett and Davidson acknowledge, the kind of theory of

¹⁸ Dummett (1981a) , p. 89.

    

9

meaning at issue cannot be given for an actual natural language—the defects of natural language make this impossible. The demands of a theory of meaning, then, require an improved version of natural language. And both Dummett and Davidson conceive of Frege’s logical language as a formalization of an improved version of natural language. The problem with this, as we will see shortly, is that Frege himself denies that his logical language should be conceived in this way. There are several important ways in which Frege’s conception of the workings of his logical language make it unsuitable to play the role that, according to Dummett and Davidson, it is meant to play. In fact, one of the surprising features of the Standard Interpretation is that it requires us to explain away, not only many of Frege’s statements about what he means to be doing, but also some of what he does. Another problem is that, on this conception of the aim of “On Sinn and Bedeutung,” Frege’s own account of its purpose is simply incorrect. Frege begins “On Sinn and Bedeutung” with a puzzle about identity. Nathan Salmon, for instance, takes it merely to be a rhetorical device for introducing the topic. Salmon writes, “pace Frege, it is not a puzzle about identity. It has virtually nothing to do with identity.”¹⁹ One might suppose, of course, that Frege decided that, for rhetorical purposes, it would be useful to begin with the puzzle about identity. But this still does not explain why, supposing Frege’s real concern is with semantic theory for natural language and with the importance of Sinn or of information content, he never actually says so. Even if he chose, for rhetorical purposes, to begin with a puzzle that does not address his main concerns, we would expect him to explain what those concerns are at some point in the paper. We might, for instance, expect him to end the paper by explaining the significance of Sinn or information content for a theory of meaning. Yet Frege not only fails to say that his real concern is with information content, he ends the paper with a discussion of how the puzzle about identity statements has been solved. “On Sinn and Bedeutung” simply does not offer straightforward support—the support offered by reportage—for the Standard Interpretation. Even so, one might still think that, in the end, no other interpretation could explain Frege’s interest in how the construction of sentences of natural language figures in determining

¹⁹ Salmon (1986) p. 12. Although, as far as I know, Dummett does not explicitly say that Frege’s puzzle is not really a puzzle about identity, he does characterize Frege’s concern in distinguishing Sinn and Bedeutung as a concern with information content. Also, while he acknowledges Frege’s use of identity in the introduction of the distinction, there is little talk of identity in Dummett’s discussion of Frege’s need for the notion of Sinn. See, e.g., Dummett (1981a) ch. 6 and (1978) 116–44. Moreover, in the latter paper, Dummett argues that Frege’s puzzle about identity can be generalized to affect all atomic sentences, provided that every sentence is either true or false. The view that it is not really a problem about identity is widespread and, in particular, not limited to supporters of the Standard Interpretation, however. See Taschek (1992) p. 771. In contrast, Michael Kremer, does see Frege as trying to solve a problem with identity, although Kremer’s characterization of the identity problem is somewhat different from the one I give here. See Kremer (2010).

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their truth-values. But, as I shall argue in Chapter 2, there is a perfectly good interpretation of “On Sinn and Bedeutung”, an interpretation on which Frege is doing exactly what he claims to be doing—solving a problem about identity. There is no need, on this interpretation, to deny Frege’s explicit statements. And, if we try to interpret “On Sinn and Bedeutung” in the context of its role in Frege’s project, we can see exactly how this problem about identity arises for Frege and why it was so important to him.

II. Some General Remarks about Interpretation As I shall argue shortly, “On Sinn and Bedeutung” raises some problems for the Standard Interpretation. Before looking at these problems, however, it will be helpful to say a bit about some of the interpretive principles I am employing in this work. The first of these is that Frege’s actual choice of words matters. It may seem that this could go without saying or that, having said so, there would be nothing further to say. But this is not the case. For, one symptom of Frege’s enormous influence on contemporary philosophy is that we are sometimes so sure that we know what he said in a particular passage, and why he said it, that we do not realize that the actual words on the pages say something quite different. In my discussions, I will often refer to well-known passages—both from “On Sinn and Bedeutung” and from other works—and yet insist that their significance is not what it has typically been taken to be. What I want to emphasize in advance is that in many cases this is not because I plan to take familiar words and read them in a nontraditional way. Rather, in many cases my reading hinges on the difference between the words that actually appear on Frege’s pages and the words that many contemporary philosophers believe are on Frege’s pages. The fact that there are such differences can be documented by comparing contemporary philosophers’ reports of the words that appear on Frege’s pages with Frege’s actual words. For, in fact, mis-reportings of the words on Frege’s pages are common. Above, I have given one example of a mis-report of the words on Frege’s pages: Dummett’s inaccurate description of Frege’s logical language. One might suspect that this is an isolated example; that the reports of the words that appear on Frege’s pages—especially in more recent years when more attention has been paid to Frege exegesis—are accurate. Although I do not want to spend a great deal of time cataloguing mis-reportings, it may be useful to give an example that is not— at least, given the case I want to make in this book—philosophically loaded. This is the widely accepted and widely repeated statement that Frege claims that the Sinn of an expression is its mode of presentation. This statement appears in publications too numerous to mention. Among the writers who have said this in print are:

    

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Tyler Burge,²⁰ Gareth Evans,²¹ Kevin Klement,²² Ruth Millikan,²³ Thomas Ricketts,²⁴ and Nathan Salmon.²⁵ (It also appears in the current Wikipedia page on Frege.) Surely, one might think, this is a matter of reportage. But it is not. Frege writes, in the passage cited in support of this view, It is natural, now, to think of there being connected with a sign (name, combination of words, written mark), besides that which the sign designates, which may be called the meaning of the sign, also what I should like to call the sense [Sinn] of the sign, wherein the mode of presentation [die Art des Gegebenseins] is contained.²⁶

There is a difference between the claim attributed to Frege (that the Sinn of a sign is the mode of presentation) and Frege’s actual claim: that the Sinn contains the mode of presentation. If we take Frege’s actual words seriously, it is consistent with what he says that Sinn contains other things as well as mode of presentation. On its own, the passage in question does not support the claim that Sinn is mode of presentation. As I have indicated, this is a very minor point. Indeed although, as far as I know, Frege never says that Sinn is mode of presentation, it is arguable that Frege’s statements about Sinn in “Thoughts” do imply that Sinn can be identified with something like mode of presentation.²⁷ My purpose in dwelling on this very minor point is not to challenge a reading of Frege’s notion of Sinn. Rather, my purpose is to remind the reader that the words that appear on Frege’s pages are not always the words that we are accustomed to believe appear on his pages. I write this in the hope that I can count on readers who are willing, when the need arises, to look again at a familiar passage and realize that there really is a difference between what they thought Frege wrote and what Frege actually wrote. Of course, even if we agree on the importance of accuracy, that is not to say that every word on Frege’s pages can be given serious weight. After all, Frege cannot have meant that mode of presentation is literally contained in Sinn. The term “contained” can have, at most, metaphorical significance. Thus we not only need to be careful about reading the actual words on Frege’s pages, as opposed to the words that we think are on his pages, we also need to be careful about which words are to be taken literally. And we need to be explicit about points at which we are going beyond the text.

²⁰ Burge (1990) reprinted in Burge (2005), p. 243. ²¹ Evans (1982), p. 26. ²² Klement (2002), pp. 9–10. ²³ Millikan (1991). ²⁴ Ricketts (1986a). ²⁵ Salmon (1986), p. 47. ²⁶ SB, pp. 26–7. ²⁷ I say “something like mode of presentation” rather than “mode of presentation” because the actual words from SB (die Art des Gegebenseins) do not, as far as I know, appear anywhere else in Frege’s writings. It is clear that Frege—unlike some contemporary philosophers—does not mean to be using this phrase as a technical term. The passage from “Thoughts” (pp. 65–6) to which I’ve just alluded are passages in which he seems to conflate Sinn with how something is given (gegeben ist).

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One might think (consider, e.g., the quotation from Salmon above), however, that accuracy, on its own, is not always a virtue. After all, surely our interpretations should be constrained by the principle of charity: surely the best reading of a philosophical figure is the one that makes that figure into a source of wisdom rather than a source of philosophical mistakes. There is something right about this conception of the principle of charity. Interpretation invariably requires us to go beyond the words on a philosopher’s pages and, insofar as we do this, we need to use our own judgment about what makes sense and what does not. Moreover, even the greatest philosophers make mistakes. And in Frege’s case we know that he made one very large mistake—the introduction of Basic Law V—a mistake that he himself ultimately acknowledged as a mistake. Thus one might think that, in light of the obvious wisdom available in his writings, it is reasonable to pursue a pick-and-choose strategy in our attempts to give a good interpretation of his work. We can reject his mistakes without sacrificing his insights. Indeed, there is a near consensus about a number of ways in which Frege was mistaken. There is, for example, a near consensus that he was wrong to say that the concept horse is not a concept. There is a near consensus that his conception of functions as unsaturated—a conception that he defended at length and from which he never wavered—is mistaken. But how good a strategy this is, depends on the nature of our interest in the writer in question. It is often thought that, by taking Frege’s “incorrect” views seriously, we rob ourselves of a philosophical hero. And some have thought that our interpretive task should be that of picking out and elaborating on Frege’s views (or, more generally, those of a historical figure) that are actually true, while rejecting the views that are actually false. This strategy may give us a way of coming up with what is (by our lights) a good theory of the subject matter studied by Frege. But many of the views that are almost universally rejected are views that Frege characterizes as central to his conception of logic. If we take ourselves to be the real arbiters of what is correct and what is incorrect then we are not taking Frege to be a philosophical hero at all—we are taking ourselves to be the philosophical heroes, philosophical heroes who can reach back and save Frege from his mistakes. On this way of thinking about Frege—given the many statements in his writings that are widely taken to be absurd—he looks to be less a philosophical hero than a philosophical blunderer who happened, also, to have had some deep insights. This is not to say that the problem with the pick-and-choose strategy of interpretation is that we use our own judgment in our interpretation of a writer’s writings. If we cannot use our own judgment at all, it is difficult to see what we can do by way of interpretation, other than collecting quotations. Clearly, if we want to understand a writer, we must do more. We cannot take every statement of that figure as a statement of her/his considered philosophical views—we need to use our judgment to make this kind of discrimination. We need to use our judgment if

    

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we want to answer questions about the views that the figure we are studying has not explicitly answered. And we need to use our judgment in order to deal with inconsistencies that may crop up in the writing of a philosophical figure. What could be wrong, then, with the Standard Interpretation’s strategy of singling out Frege’s insightful statements from those that are obviously wrong? Above, I have suggested that this attitude might explain many of the mis-reportings of the words on Frege’s pages. But that is not the only problem. If, like Salmon, we think that we can find in Frege’s writings an important puzzle that “has virtually nothing to do with identity,” there is nothing wrong with investigating this puzzle. But if we want to understand Frege’s views, why would we choose to ignore his statement that there is a puzzle about identity rather than trying to understand what he thinks that puzzle is and how he thinks the views of “On Sinn and Bedeutung” address this puzzle? What I want to suggest is that, while we need to go beyond the text in the interpretation of a historical figure, it is a mistake to take it as a fundamental interpretive principle that the figure in question must share with us an understanding of the correct way to think about a particular issue. It should not be impossible for us to attribute to a historical figure—even one as familiar to us as Frege is—a perspective that is in important respects at odds with our own. I propose, as a principle of interpretation, that we take Frege at his word both in passages in which he tells us what he means to be doing and in passages in which he tells us that a particular view is fundamental to his project. I shall argue in Chapter 2 that pace Salmon, there is a puzzle about identity that stems from the views of Begriffsschrift and that this puzzle is solved by his introduction of the Sinn/Bedeutung distinction. But what about the statements in Frege’s writings that we “know” are wrong? What about his views that sentences are names of truth-values; that functions are unsaturated? Given the roles these views play in his thought, it is, in my view, a mistake to ignore them—even if the result is to conclude that much of what Frege writes has only antiquarian interest. However, as I shall argue, the significance of these views is not antiquarian. In Chapters 7 and 8, I shall argue that some of Frege’s unfamiliar views yield solutions to puzzles that bother philosophers today— including apparently intractable puzzles about vagueness and about numbers.

III. The Standard Interpretation and “On Sinn and Bedeutung” IIIa. Dummett and Truth-values as Objects I want to turn next to a problem that exemplifies some of the issues mentioned above. This has to do with Frege’s claim, in “On Sinn and Bedeutung”, that there

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are two truth-values, the True and the False, and that sentences are names of these objects. All true sentences name the True and all false sentences name the False. Dummett, who reads “On Sinn and Bedeutung” as a contribution to a theory of meaning, writes that Frege’s view that sentences are names of truth values is an “absurdity” and “ludicrous deviation” that had a “fatal effect upon Frege’s theory of meaning.”²⁸ But if this is so, why did Frege himself not realize it? One might think that it is Dummett who is making the mistake here. For, one might respond, it is not as if the view that sentences are names of truth-values is anathema to anyone who studies language. Irene Heim and Angelika Kratzer, for example, consider the view that truth-values are denotations of sentences in their discussion of what a systematic semantics for natural language should be like.²⁹ And, while they acknowledge the oddness of the claim that sentences denote truth-values, they also say that—as long as the result is a compositional theory that pairs sentences with their truth conditions—there is nothing wrong with this claim. If we think that what we want, when we want a theory of meaning or a theory of the workings of language, is a compositional semantic theory of this sort, there is no reason to regard the view of sentences as names of truth-values as fatal to a theory of meaning. But Dummett’s objection is not simply to the statement that sentences name truth-values. Rather, Dummett’s objection is to taking truth-values (or the denotations of sentences) to belong to the logical category of objects rather than to constitute a separate logical category. To see the significance of Dummett’s worry, it is important to consider its ramifications for the workings of a logically perfect language. On Frege’s view, the result of filling in the subject-slot of a first-level predicate with an object-name (any object-name) should be a meaningful sentence. How does natural language measure up? Let us consider the predicate “is green.” One might be inclined to say that the only use of the predicate “is green” is to talk about ordinary physical objects. To say that the number 2 is green would be pointless or silly. But that has to do with our purposes for using that predicate, not with the sentences it can be used to form. There is nothing wrong with the sentence “2 is green.” It seems to be a meaningful, albeit false, sentence. Numbers are not colored objects, hence 2 is not green. Similar wellformed, if somewhat absurd, sentences can be constructed by using other names. But while we can put other object names in the subject slot of “is green,” that is not to say that putting any well-formed expression in that slot of should yield a sentence. After all, the result of filling the slot with another predicate (e.g., “is prime is green”) does not (nor should it, on Frege’s view) yield a sentence. With that in mind, let us consider the significance of taking sentences to be names of truth-values and truth-values to be objects. Consider, for example, the

²⁸ Dummett (1981a) p. 184.

²⁹ Heim and Kratzer (1998), ch. 2.

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sentence “2 is prime.” This sentence is a name of an object, the True, hence eligible to be inserted in the subject-slot of “is green.” The result is “2 is prime is green.” But this string of words is not an English sentence. Thus a Fregean “theory of meaning,” insofar as it classifies sentences as object-names, cannot be given for English. English does not work that way. As Dummett reads Frege, this is a serious problem. In particular, as Dummett reads him, Frege thinks that any defect of natural language—any way in which the natural language does not fit his theory of meaning—can (and should) be fixed. Dummett writes, [N]o systematic theory of meaning will fit our linguistic practice as it actually is: but so much the worse for our linguistic practice, which ought to be revised so as to accord with such a theory.³⁰

Thus, on Frege’s view that sentences are names of objects, according to Dummett, a revision of English is required. But surely it is no defect of English that “2 is prime is green” is not a sentence. Dummett concludes that Frege has made a mistake in claiming that sentences are names of objects—as opposed, say, to some other sort of entity that occupies a different logical category. What makes Frege’s mistake so egregious, on Dummett’s view, is that it could so easily be avoided. Supposing, for the moment, that all this is right, we have a puzzle. How could Frege have made such an obvious mistake? One might suspect that the answer is that the assimilation of sentences to proper names was a view that Frege introduced only in passing, without much examination. But this is not correct. When he first introduces the view, in Function and Concept, he acknowledges that it may seem arbitrary and artificial. It will be established, he tells us, in his forthcoming essay “On Sinn and Bedeutung.” What does he do in “On Sinn and Bedeutung,” by way of establishing that sentences are proper names of truthvalues? The discussion of this issue begins on the seventh page, where Frege asks after the Sinn and the Bedeutung of an assertoric sentence. He then gives a brief argument that sentences are names of truth-values.³¹ One might think—indeed, many do—that this exhausts Frege’s defense of the view. Indeed, most of the remainder of the essay consists of a long discussion in which Frege analyzes a variety of natural language sentences. If we read “On Sinn and Bedeutung” in this way, it is natural to think that, after a very short argument that sentences name truth-values, Frege devotes the rest of the essay to his real interest—a demonstration of how the Sinn/Bedeutung distinction helps us begin to formulate a theory of the workings of language.

³⁰ Dummett (1981) p. 30.

³¹ SB, p. 32.

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But such a reading leaves out Frege’s description of the point of his lengthy evaluation of natural language sentences. For, after making the initial argument that sentences name truth-values, he proposes to put to a further test “the supposition that the truth value of a sentence is what it means [dass der Wahrheitswert eines Satzes dessen Bedeutung ist].”³² This further test, which requires the analyses of natural language sentences, takes up nearly all of the remainder of the essay until, on the final page, Frege writes, “let us return to our starting point” and turns, again, to the puzzle about identity with which the article begins. In fact, the defense of the view that sentences are proper names takes up nearly 18 of the 25 pages of “On Sinn and Bedeutung.” Clearly, then, Frege takes the view in question both to be important and to require substantial defense. There is an interpretive puzzle here. On the one hand, “On Sinn and Bedeutung” contains a long discussion in which Frege analyzes various natural language sentences. This appears to support Dummett’s claim that Frege means to be developing (part of) a theory of the workings of (something like) natural language. On the other hand, Dummett is adamant that sentences are not used as proper names in natural language. Yet the whole point of this long discussion— as Frege explicitly states—is to test his view that sentences are a kind of proper name. It should be evident that, if we are to sort this out, we need to pay attention to Frege’s stated views about language, both natural language and his logical language.

IIIb. Logical Language and Natural Language Frege is explicit about what he wants of logic and of a logical language. In his 1879 monograph, Begriffsschrift, he introduces the first version of his new logic and new logical language, Begriffsschrift. What is the point of introducing a new language? The need arose, he tells us, from his attempt to show that the truths of arithmetic can be given proofs that, disregarding the particular characteristics of objects, depends solely on those laws upon which all knowledge rests.³³

Or, as he also says, proofs whose only support is “those laws of thought that transcend all particulars.” This kind of proof, he continues, belongs to logic. It should not be surprising that one of the demands of this project is the construction of proofs that are gapless. For suppose we have a proof of a truth of arithmetic whose only official premises are laws upon which all knowledge rests—or, logical

³² SB, p. 36.

³³ BS, preface.

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laws—but which in fact relies on hidden presuppositions. It may be that one of those hidden presuppositions is, not a law of logic, but a truth that must be supported by (to use Frege’s expression) facts of experience. Thus he writes, To prevent anything intuitive from penetrating here unnoticed, I had to bend every effort to keep the chain of inferences free of gaps.³⁴

And it is his attempt to do just this that led him to formulate his logical language. For, he writes, In attempting to comply with this requirement in the strictest possible way I found the inadequacy of language to be an obstacle; . . . This deficiency led me to the idea of the present ideography. Its first purpose, therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed, so that its origin can be investigated.³⁵

This purpose of Begriffsschrift is also mentioned in his Foundations description, where he writes, It is designed to produce expressions which are shorter and easier to take in, and to be operated like a calculus by means of a small number of standard moves, so that no step is permitted which does not conform to the rules which are laid down once and for all. It is impossible, therefore, for any premiss to creep into a proof without being noticed.³⁶

The importance of eliminating unnoticed presuppositions is also mentioned in Basic Laws, where he writes, Herein, no presupposition can remain unnoticed; every required axiom must be uncovered. For it is precisely the presuppositions that are made tacitly or without clear awareness that bar insight into the epistemological nature of a law.³⁷

The logical language is intended to be a tool for evaluating the legitimacy of any inference on any subject and for preventing any presupposition from sneaking into an inference unnoticed. Once the primitive laws of logic are listed and our inferences are expressed in Begriffsschrift, it is supposed to be a mechanical task to determine whether or not an inference is correct and gapless, or whether it requires an unstated premise.³⁸ We should be able to see by inspection whether ³⁴ BS, preface. ³⁵ BS, preface. ³⁸ See FA, §91 and PCN, pp. 364–5.

³⁶ FA, §91.

³⁷ BLA vol. i, p. 1.

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or not a statement is a primitive logical law; whether or not the transition from one statement to another follows by Frege’s rule of inference. How does Frege understand the relation between his logical language and natural language? A logical language must be adequate for expressing all content that has significance for the inferential sequence, which Frege labels “conceptual content.”³⁹ One might be inclined to suspect that Begriffsschrift is meant to be a version of natural language from which the extra content that obscures inferential connections is removed. But this is not Frege’s view. He writes, I believe that I can best make the relation of my ideography to ordinary language clear if I compare it to that which the microscope has to the eye. Because of the range of its possible uses and the versatility with which it can adapt to the most diverse circumstances, the eye is far superior to the microscope. Considered as an optical instrument, to be sure, it exhibits many imperfections, which ordinarily remain unnoticed only on account of its intimate connection with our mental life. But, as soon as scientific goals demand great sharpness of resolution, the eye proves to be insufficient. The microscope, on the other hand, is perfectly suited to precisely such goals, but that is just why it is useless for all others.⁴⁰

A microscope does not filter out extraneous details from the images we see. What we see looking through the microscope is something we do not see when we look with the naked eye; the sharpness of resolution and magnification enable us to see what cannot be seen at all with the naked eye. Similarly, there is reason to believe that Frege means Begriffsschrift to have expressive power that is not available in natural language. For example, some of the proofs in Part III of Begriffsschrift require his definition of a property’s being hereditary in a sequence. These proofs are expressed in Begriffsschrift. And he says, about a property F’s being hereditary in an f-sequence, “it can become difficult and even impossible to give a rendering in words if very involved functions take the places of F and f.”⁴¹ Furthermore, while we might not see, from a natural language expression of a proof, that the proof requires an unstated premise, this should be impossible to miss in the Begriffsschrift expression of the proof. There are, then, respects in which Begriffsschrift can express more than natural language can express. But there are also respects in which Begriffsschrift has less expressive power than natural language. Frege tells us, for instance, that he has decided to forgo expressing anything that is without significance for the inferential sequence.⁴² For example, the logical language will not express the difference between what is

³⁹ BS, §§3–4.

⁴⁰ BS, preface.

⁴¹ BS, §24.

⁴² BS, preface.

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expressed in natural language by the use of “and” and “but.”⁴³ This, too, is a sense in which Begriffsschrift, like the microscope, is a scientific tool. While the microscope is perfectly suited to certain goals, Frege also says, “that is just why it is useless for all others.”⁴⁴ Similarly, he says that his logical language is “a device invented for certain scientific purposes, and one must not condemn it because it is not suited to others.”⁴⁵ For certain scientific purposes, natural language is defective. But these logical defects, Frege says, are necessary if natural language is to serve its purposes. In one of Frege’s early articles about his logical language, he compares natural language to the hand. He says, We build for ourselves artificial hands, tools for particular purposes, which work with more accuracy than the hand can provide. And how is this accuracy possible? Through the very stiffness and inflexibility of parts the lack of which makes the hand so dextrous.⁴⁶

It is a mistake to describe Frege’s logical language as a properly constructed version of natural language.⁴⁷ If our interest is in something that will serve the purposes of natural language, then Begriffsschrift is defective. Begriffsschrift is meant to be, not a perfect language, but a logically perfect language. It should be clear from this that we cannot simply read “logic” as Frege’s expression for theory of meaning and that we need to separate out Frege’s views about logical (and logically perfect) language from views about natural language. This is especially important given that Frege thinks that special requirements must be imposed on a language if it is to fulfill the tasks that he wants a logical language to fulfill. There is no reason to think that a properly functioning natural language will (or should) fulfill these logical requirements. Suppose, then, we abandon the assumption that Frege’s logical language is designed to be a kind of model for properly functioning natural language. And consider, now, the requirement that every result of putting an object-name in the subject-slot of a predicate must be a meaningful sentence of the language. It is no problem for Frege that this requirement, which is satisfied by his logical language, is not satisfied by natural languages. ⁴³ BS, §7. This kind of content, he claims, is used to hint that what follows is different from what one might at first expect. He also gives, as an example of a difference in content that is not a difference in conceptual content in the following pair of sentences: The Greeks defeated the Persians at Platea and The Persians were defeated by the Greeks at Platea. BS §3. ⁴⁴ BS, preface. ⁴⁵ BS, preface. ⁴⁶ p. 52. ⁴⁷ As Dummett does. See, e.g., Dummett (1981a) pp. 166–7, (1981b), pp. 30–1. He writes, The distinction he draws is not between two utterly different modes of expressing thoughts, but between language as functioning properly and as misfunctioning (1981b, p. 31).

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IV. A Theory of Meaning for What Language? IVa. Natural Language I have presented two problems for supporters of the Standard Interpretation. But these, one might think, give us no reason to give up on the Standard Interpretation. For it may seem implausible that Frege could be doing anything other than contributing to a theory of the workings of natural language in “On Sinn and Bedeutung.” For example, he writes, It is natural, now, to think of there being connected with a sign (name, combination of words, written mark), besides that which the sign designates, which may be called the Bedeutung of the sign, also what I should like to call the sense [Sinn] of the sign, wherein the mode of presentation is contained.⁴⁸

At this point the only names Frege has explicitly mentioned are “the point of intersection of a and b” and “the point of intersection of b and c” (where the letters in question are names of particular lines). These are natural language names and Frege seems to be drawing our attention to features of natural language names: that they both designate particular objects (in these cases, particular points) and are connected with something else (Sinn) that seems to be a kind of content— something that, he says, is grasped “by everybody who is sufficiently familiar with the language.”⁴⁹ And there is no question that this is meant to apply to natural language, since he writes that, in natural language “one must be content if the same word has the same Sinn in the same context.”⁵⁰ Moreover, Frege goes on to use this Sinn/Bedeutung distinction to explain other features of natural language, among them something he calls “indirect speech.” Some examples of indirect speech are statements that someone believes (regrets, approves, hopes, fears, etc.), for example, that the Evening Star is a planet. In the sentence (a)

Alice believes that the Evening Star is a planet

the italicized sentence (“the Evening Star is a planet”) appears indirectly. There is an apparent problem with indirect speech. Suppose that the name “the Evening Star” that appears in (a) in indirect speech designates a particular object (Venus). Then, it would seem, (a) tells us something about that object. The object in question, of course, has different names (among them, “Venus,” “the Morning Star,” “the Evening Star”) and, if what we want to do is to say something about this

⁴⁸ SB, pp, 26–7.

⁴⁹ SB, p. 27.

⁵⁰ SB, p. 28.

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object, it should not matter what name we use. That is, if (a) tells us something about this object, then (b)

Alice believes that the Morning Star is a planet

should tell us the same thing about the same object. It seems, then, that (a) and (b) must have the same truth-value. But this conflicts with our understanding and evaluation of natural language statements. For suppose that Alice does believe that the Evening Star is a planet but does not know that the Morning Star is the Evening Star. Suppose, in fact, that she believes that the Morning Star is a star (and not a planet). Then (a) must be true and (b) false. Frege’s analysis of indirect speech offers us a solution to this problem. He tells us that an expression appearing in indirect speech does not designate what it customarily designates. Rather, it designates its customary Sinn. That is, (a) does not tell us anything about Venus. It tells us, rather, something about the customary Sinn of “the Evening Star.” And thus there is no reason to think that (a) and (b) should have the same truth-value. For the names “the Morning Star” and “the Evening Star” give us the same object via distinct modes of presentation. Thus, since mode of presentation is (at least) a large part of Sinn, these names have distinct customary Sinn. Thus, unlike (a), (b) tells us nothing about the customary Sinn of “the Evening Star.” There is no problem with the claim that (a) can be true while (b) is false. Given all this—and given that there is no such thing as indirect speech in Begriffsschrift—it may seem difficult to imagine that Frege could have had any reason for writing about indirect speech other than an interest in developing a semantics for natural language. Moreover, he spends a good deal of time in “On Sinn and Bedeutung” examining how to determine the truth-values of sentences of natural language that are formed using a variety of other constructions that can have no analogue in Begriffsschrift. Thus, it may seem that, while “On Sinn and Bedeutung” presents us with some interpretive puzzles, there is no interpretive question about what Frege does in that paper. To say that Frege is developing (at least a part of) a semantics for natural language, one might think, is not very far from straightforward reportage. But this is not quite right. There is a question about what Frege does in the paper. To see why this is so, we need to think about what is required of a theory if it is to be a theory about natural language. The predominant view is that Frege does not mean to be giving a theory of natural language with all its defects but, rather, an improved version of natural language. Davidson, for example, writes, Frege saw the importance of giving an account of how the truth of a sentence depends on the semantic features of its parts, and he suggested how such an account could be given for impressive stretches of natural language. His method

22

     was one now familiar: he introduced a standardized notation whose syntax directly reflected the intended interpretation, and then urged that the new notation, as interpreted, had the same expressive power as important parts of natural language. Or rather, not quite the same expressive power, since Frege believed natural language was defective in some respects, and he regarded his new language as an improvement.⁵¹

Dummett is even more emphatic that Frege’s interest is in an improved version of natural language, rather than natural language as it is. He writes, The use of language, if it is to have the point it is intended to have, must be a practice capable of being codified; but natural language resists such codification at several places. We therefore need to revise our practice, at least when we are seriously concerned with the attainment of truth, so as to make such codification possible. To express the matter in terms of a theory of meaning, no systematic theory of meaning will fit our linguistic practice as it actually is: but so much the worse for our linguistic practice, which ought to be revised so as to accord with such a theory.⁵²

According to Dummett, Frege’s view is that natural language requires revision. Why should Frege be interested in an improved version of (or non-defective fragment of ) natural language rather than natural language as it is? One of the defects of natural language is that it has names that do not name anything. And it is not difficult to see why Frege’s theory should not apply to sentences that exhibit this defect if (as Davidson claims) Frege wants to give an account of how semantic features of the constituents of a sentence determine its truth-value. Consider, for example, the sentence “Odysseus was set ashore at Ithaca while sound asleep.” Frege writes, [A]nyone who seriously took the sentence to be true or false would ascribe to the name “Odysseus” a meaning [eine Bedeutung], not merely a sense [einen Sinn], for it is of what the name means that the predicate is affirmed or denied.⁵³

Frege seems to be telling us here that the relevant semantic feature of a proper name is the object it names.⁵⁴ If it does not name anything, it makes no contribution to determining the truth-value of the sentence. Why?

⁵¹ Davidson (1977), in Davidson (1984), p. 202, emphasis added. ⁵² Dummett (1981b), p. 30, emphasis added. ⁵³ SB, pp. 32–3. ⁵⁴ Note, however, that this is not what Frege says explicitly. We will return to this passage and its significance later (see Chapter 6, part IIa).

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Here is one way to think about it, a way of thinking that is in line with the Standard Interpretation: The sentence purports to tell us something about a particular object. Assuming, as we do, that there is no one that “Odysseus” names, there is no object whose character can be either described or misdescribed in the sentence. And Frege rejects the idea that there is a third truth-value: a truthvalue of a sentence that is neither true nor false. This gives us one account of why, supposing Frege to be giving such a theory of the truth conditions of sentences in a language, he would have rejected any account of language that contains proper names that fail to refer to anything.⁵⁵ Thus one ‘improvement’ on natural language is the elimination of proper names that do not name anything. In line with this, Dummett writes that, in a properly constructed language, it must be impossible to form a name that does not have a referent.⁵⁶ What else can we say about the improved version of natural language for which Frege, on these views, wants a theory? One might suspect that the improved version of natural language is actually Frege’s Begriffsschrift. But this is not plausible. As we have already seen, Frege explicitly claims that Begriffsschrift is a tool that, like the microscope, is invented for specific scientific purposes.⁵⁷ Just as the microscope cannot replace the eye, Begriffsschrift cannot replace natural language. For Begriffsschrift is both odd and inflexible. That is why Frege tells us we should not condemn it if it is not suited for other purposes. Indeed, as we shall see shortly, while the content of some of the sentences Frege discusses in “On Sinn and Bedeutung” might, in theory, be expressible in Begriffsschrift, there is no obvious way to express them in the version of Begriffsschrift that Frege gives us, nor does Frege suggest a way that Begriffsschrift might be modified to express such things. This is perhaps why Dummett is so adamant that Frege envisioned two sorts of revisions to linguistic practice: the adoption of Begriffsschrift for its peculiar purposes and the revision of natural languages so as to eliminate their defects. ⁵⁵ This explanation is the one that makes the most sense on the Standard Interpretation, where Frege’s concern is primarily with language. But, as I argue in this chapter, this is a highly problematic way to interpret Frege. His primary interest is in the expression and evaluation of gapless inference. So, it is worth noting that there is an important reason, stemming from his logical concerns, for Frege to reject a language with proper names that do not name anything. One important feature of Frege’s logic, that is constant throughout his career, is his rejection of subject/predicate analyses of sentences in favor of function/argument analyses. Frege means the “function” part of this analysis literally. The notion of function, of course, is not (as he tells us in BS, §10) the familiar mathematical notion of function used in analysis but a wider notion, which includes the functions of analysis. In a simple predication, the predicate is a function name and the subject a name of the argument. The sentence names the value of the function on the argument (a truth-value on the revised version of Frege’s logic). The result of putting a symbol that doesn’t name anything in the argument place of a function name cannot name a value of the function. Hence a sentence with a proper name that doesn’t name anything cannot have a truth-value. ⁵⁶ Dummett (1981a) p. 167. ⁵⁷ I have discussed this issue at greater length in Weiner (1997b) and (2007).

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Frege’s theory of meaning, Dummett claims, is meant to apply both to Begriffsschrift and to improved natural language.⁵⁸ On Davidson’s interpretation, also, it seems that the language in question must be, not Begriffsschrift, but a fragment or improved version of natural language. For Davidson talks of Begriffsschrift (the standardized notation) not as being an improved version of natural language but rather as having the same expressive power as (an improved version of) natural language. A theory of truth conditions for Begriffsschrift, then, might also be taken to provide a theory for an improved (but still natural, rather than formal) version of natural language. On the Standard Interpretation, the kind of language for which Frege wants a theory is a cleaned up version of natural language. But, once again, it is worth noting that this is one of the philosophically sophisticated projects that Frege himself never enunciates.

IVb. Must the Improved Version of Natural Language be Free of Logical Imperfections? Let us suppose, then, that “On Sinn and Bedeutung” is meant to set out the beginning of a theory of truth conditions for an improved version of natural language—a language other than Begriffsschrift. Given that this improved version of natural language is not Begriffsschrift, might it include some imperfections?⁵⁹ According to Dummett, what counts as an imperfection is determined by what is required of accounts of language. He writes, I do not mean that Frege abstained from giving a coherent account in order to make such features appear as imperfections of language: rather, it is because he thought he saw that no coherent account is possible that he regarded them as imperfections which had to be remedied when we devised a language fully apt for the expression of thoughts and the unassailable execution of deductive argument.⁶⁰

On Dummett’s view, then, it is simply not possible to give a coherent account of the truth-conditions of sentences of an imperfect language. The improved ⁵⁸ Dummett (1981b), p. 31. ⁵⁹ In order to answer this question, it is important to think about Frege’s understanding of imperfection. As we saw earlier, in the preface to Begriffsschrift Frege writes, not about a perfect language, but a logically perfect language. And in his later writings, as well, he tends to talk, not about perfect language but about logically perfect language; not about defects, but about logical defects. On Dummett’s view, of course, that “logic” is just Frege’s term for theory of meaning. On such a view, talk about logical defects might be talk about defects from the point of view of the workings of language in general. But, as I have argued above, such a view conflicts with Frege’s explicit statements about the difference roles of his logical language, which is a tool for specific scientific purposes, and natural language. ⁶⁰ Dummett (1981b) p. 32.

    

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language must be free of imperfections. But is the language for which, according to Dummett, Frege does give an account actually free of imperfections? One of the cornerstones of Frege’s account of language is supposed to be his account of indirect speech. And Dummett seems confident that the account of indirect speech is coherent. For he includes a substantial explanation of how the apparent problems with Frege’s treatment of indirect speech can be solved.⁶¹ But among the features of natural language that Frege regarded as defects are (in Dummett’s words), ambiguity of construction; ambiguity of words, whether or not resolved, even systematically, by context; vagueness; the use of predicates and functors not everywhere defined; and the occurrence of empty proper names.⁶²

Thus we have a problem. For, Frege’s explanation of indirect speech is an explanation on which every proper name is systematically ambiguous—every proper name names its customary Bedeutung in most contexts, but its customary Sinn in a variety of other contexts, including indirect speech. Dummett acknowledges that Frege views ambiguity as an imperfection of language and Dummett also thinks no coherent account can be given of a language with imperfections. Thus, if Frege’s explanation of indirect speech is correct, then as long as natural language has such constructions in it, it is not perfect, and cannot be given a coherent account. One might suspect, then, that the correct response would be to modify Dummett’s interpretation—to take Frege to be classifying some, but not all, ambiguities as defects of natural language. But this interpretive strategy does not work. Frege explicitly and repeatedly states, throughout his career—from his 1882 article “On the Scientific Justification of a Begriffsschrift” to his last publication in 1923, “Compound Thoughts”—that any ambiguity is a defect of natural language.⁶³ He writes, for instance, “In no way is it necessary to have ambiguous signs, and consequently such ambiguity is quite unacceptable. What can be proved only by means of ambiguous signs cannot be proved at all”.⁶⁴ He also describes “the rule of unambiguousness” as, “the most important rule that logic must impose on written or spoken language”.⁶⁵ Nor can we suppose that Frege did not notice that his view ⁶¹ Dummett (1981a), pp. 179–80. ⁶² Dummett (1981b), p. 32. ⁶³ He writes, Language [Die Sprache] proves to be deficient, however, when it comes to protecting thought from error. It does not even meet the first requirement which we must place upon it in this respect; namely, being unambiguous (OSJ p. 50) and We need a system of symbols [Ganzes von Zeichen] from which every ambiguity is banned, which has a strict logical form from which the content cannot escape. (OSJ, p. 52) Similar statements appear in BLA vol. i, p.19, pp. 306–7, 309. And they appear in his 1914 course notes LM NS pp. 228, 245/PW, pp. 213, 227. ⁶⁴ OFG I, p. 309.

⁶⁵ OFG I, p.385.

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of indirect speech is one on which natural language is systematically ambiguous. For he writes, in a letter to Russell, “To avoid ambiguity, we ought really to have special signs in indirect speech, though their connection with the corresponding signs in direct speech should be easy to recognize”.⁶⁶ There can be no question that Frege regards the indirect speech constructions of natural language as constructions involving one of the logical defects of natural language. The evidence from the letter to Russell suggests that Frege envisions the possibility of expanding Begriffsschrift to allow the expression and evaluations of inferences that are expressed in natural language by using indirect speech. But this is only a suggestion—it is not something Frege says explicitly. Indeed, he uses the suggested modification in his (natural language) letter in the interests of clarifying a disagreement he has with Russell. The modification in question is to underline the expressions appearing in indirect speech. Thus, one might suspect that he regards these underlined symbols as complex symbols that have the original symbols as constituents. But it is important to note that, even if he envisions using a similar modification in later expansions of Begriffsschrift, this is not to say that he would sanction any kind of ambiguity in Begriffsschrift. For, he writes of using “special signs” whose connection to the original signs should be easily recognizable. It does not follow that the original signs should be constituents of the special signs and it seems unlikely that he intended the original signs to be constituents, in any nontypographic sense, of the new signs. Perhaps the best comparison here is with the different signs Frege uses in Basic Laws for cardinal and real numbers. A real number is to be represented in the usual way with numerals. But a cardinal number is to be represented, instead, by a numeral with a slash through it. Given this notation, we can easily see the relation between the real number 1 and the cardinal number 1, but that is not to say that the two symbols are the same or that they have any constituent in common. There is no ambiguity in the use of numerals. And the evidence from Frege’s letter also suggests that a correct expansion of this sort would not introduce any systematic ambiguity into Begriffsschrift.⁶⁷ That is, even an expanded Begriffsschrift will not have expressions that function in the way natural language statements of indirect speech do. Hence it cannot be that, as Dummett writes “[W]e can apply Frege’s doctrines to natural language only in so far as we view its sentences as constructed in the same way as those of Frege’s formal language”.⁶⁸ ⁶⁶ December 28, 1902 WB, p. 236/PMC, p. 152. ⁶⁷ It may be worth noting that there is, in fact, a systematic ambiguity in the first version of Frege’s logic. The symbols that appear flanking the identity sign are supposed to name themselves in that context and what they customarily name in other contexts. Thus it may not be surprising that Frege doesn’t begin raising objections to ambiguity until a few years after the publication of Begriffsschrift. As I argue in Chapter 2, there is reason to think that Frege’s dissatisfaction with the Begriffsschrift account of identity is due at least in part to the requirement that the logical language be ambiguous in just this way. ⁶⁸ Dummett (1981b), pp. 151–2.

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The upshot is that Dummett’s characterization of Frege’s understanding of imperfection cannot be right. It cannot be that, as Dummett writes, defects of natural language are defects because no fully coherent account of a language exhibiting such features is possible.⁶⁹

and it cannot be that, it is because he thought he saw that no coherent account is possible that he regarded them as imperfections which had to be remedied when we devised a language fully apt for the expression of thoughts and the unassailable execution of deductive argument.⁷⁰

Ambiguity is an imperfection from Frege’s point of view. And Frege himself knows that his account of indirect speech describes some of the workings of a language in which all words are systematically ambiguous. But there is no indication that Frege thinks this prevents us from giving a coherent explanation of indirect speech. It may be useful, now, to take a moment to reconsider Dummett’s claim that, when Frege uses the term “logic,” he is talking about theory of meaning or an account of the workings of language. We have just seen that the conflation of “logic” and “theory of meaning” makes it seem that an imperfection in language must be something that creates an obstacle for developing, not just a scheme for evaluating inference, but a coherent account of language. And while the existence of ambiguous expressions in a language is a logical imperfection, as Frege’s discussions show, there is every reason to think that it is possible to give a coherent account of some languages that are imperfect in this way. There are reasons, then, to think that it is appropriate to separate Frege’s discussion of natural language from his project of providing a logical system for the evaluation of inference and a logically perfect language. First, this does not require us to go against Frege’s explicit statements. Typically, Frege uses the term “logic,” not for a general account of language, but for a scheme for the evaluation of inference. And while there is certainly a relation between the workings of language and a scheme for the evaluation of inference, these are, nonetheless, distinct subjects. Indeed, in a letter to Husserl he writes, “It cannot be the task of logic to investigate language and determine what is contained in a linguistic expression.”⁷¹ Second, Dummett’s criticism of Frege’s assimilation of sentences to proper names highlights another advantage of making this separation. For, as ⁶⁹ Dummett (1981b), pp. 32–3 emphasis added. ⁷⁰ Dummett (1981b), p. 33 emphasis added. ⁷¹ October 30, to November 1, 1906. WB, p. 102/PMC, p. 67–8.

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Dummett has claimed, we do not use sentences as a kind of proper name. And yet there is nothing wrong, for the purpose of developing a scheme for evaluating inference, with taking sentences to be a kind of proper name.

IVc. Could Frege Want a Theory of Natural Language as it Is? Let us now consider the possibility that, at least in “On Sinn and Bedeutung,” Frege means to be talking about natural language as it is. One obstacle to such a reading is that Frege is nearly always dismissive of natural language when the topic comes up. But for all that, as the microscope analogy shows, he does not think that we should stop using natural language as it is. Frege’s objection is to the use of natural language for a particular, scientific project. For other purposes, natural language is superior to logical language. And a look at his remarks about natural language supports the idea that it is only for purposes of logic that Frege dismisses natural language and its study. A typical statement, from a letter to Husserl, is Someone who wants to learn logic from language is like an adult who wants to learn how to think from a child. . . . The main task of the logician is to free himself from language and to simplify it.⁷²

Given that natural language (like the human eye or hand) is good for so many purposes, why should Frege not investigate natural language? And isn’t virtually all of “On Sinn and Bedeutung” one long discussion of the workings of natural language? If we can read “On Sinn and Bedeutung” as having natural language (as it is) as its topic, it need not be particularly puzzling that Frege should talk about indirect speech. But there are problems with taking Frege’s analyses of sentences involving indirect speech to be part of a study of the workings of a logically imperfect natural language. The discussion of indirect speech is also a discussion, at least in part, of truth-conditions. Thus, one might be inclined to think that this is part of a theory of the truth conditions for sentences of natural language as it is. But such an interpretation does not fit with what Frege actually does. Consider his discussion of expressions that have the grammatical form of object names yet do not designate objects “because the truth of some sentence is a prerequisite.”⁷³ His example is the expression “Whoever discovered the elliptic form of the planetary orbits.” But he does not suggest that we might give an account of how the truth or falsity is determined for sentences in which such an expression occurs. Rather, he gives a suggestion for eliminating this logical imperfection of natural language: the existence of such names can be avoided by designating a particular object to be what is named whenever the sentence in question is not true. ⁷² October 30 to November 1, 1906. WB, p. 102/PMC, p. 68.

⁷³ SB, p. 40.

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My arguments that Frege does not mean to be giving a theory, either for a logically perfect language or for natural language as it is, are not based on incidental remarks. Frege is emphatic in his dismissal of certain features of language (that is, he is emphatic about having no interest in giving an account of language in which such features occur). What distinguishes indirect speech from those features? Why should Frege do anything that remotely resembles giving an account of how indirect speech works? To answer this, it may help to consider the logical imperfection inherent in indirect speech, ambiguity.

IVd. Why Should Ambiguity be a Problem? As we have seen, when Frege identifies a feature of language as a defect, he does not suggest that this feature should be forbidden for all purposes. And there is no suggestion that the use of ambiguous expressions should always be forbidden. Rather, ambiguity cannot be tolerated in a language that is to be used for his particular scientific purposes. Why should ambiguity present a problem for Frege’s project? As we saw earlier, one of the most important roles of his logical language is to keep the chain of inference free of gaps and to prevent any presupposition from sneaking in unnoticed in our proofs.⁷⁴ Or, as he advertises in Basic Laws, each formula of Begriffsschrift displays all conditions on which it depends. He continues, This completeness, which does not tolerate any tacit addition of assumptions in thought, seems to me indispensable for the rigorous conduct of proof.⁷⁵

He writes, in a letter to Peano, that where inferences are to be drawn it is essential that the same expression should occur in two propositions and should have exactly the same meaning in both cases. It must therefore have a meaning of its own, independent of the other parts of the proposition.⁷⁶

Supposing that this requirement is always met—as Frege believes it is in Begriffsschrift—inference can be carried out as a calculation.⁷⁷ That is, there is a totality of rules which govern the transition from one sentence or from two sentences to a new one in such a way that nothing happens except in conformity with these rules.⁷⁸

But in a language that allows ambiguous expressions, there is a violation of Frege’s requirement that, whenever the same expression occurs in two propositions, it has ⁷⁴ BS, pp. 5–6. ⁷⁵ BLA vol. i, p. VI. ⁷⁶ Frege to Peano September 29, 1896. WB, p. 183/PMC, p. 115. ⁷⁷ PCN, p. 364–5. ⁷⁸ PCN, p. 365.

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exactly the same meaning. And, if such a language is used in the expression of proofs, these proofs will depend on a tacit assumption: that all occurrences of the expression in this proof have the same meaning. It is easy to see, then, exactly how the existence of indirect speech in natural language violates this requirement. To see this, let us begin with the following inference: (A)

The Morning Star = the Evening Star. The Morning Star is a planet Therefore, The Evening Star is a planet.

This inference looks to be a perfectly good inference that is a straightforward consequence of Leibniz’ law (or Law 52 of Begriffsschrift). Were the language in which it is stated devoid of ambiguity, it would be fine to say just this. However, the existence of indirect speech contexts makes our language systematically ambiguous. For we can express an apparently similar inference that looks to be a straightforward consequence of Leibniz’ law, but is not. The inference is: (B)

The Morning Star = the Evening Star. Alice believes that the Morning Star is a planet Therefore, Alice believes that the Evening Star is a planet.

In spite of the apparent similarities of these two inferences, the second is not a good inference. On Frege’s analysis of indirect speech, the meaning of “the Morning Star” in the first premise is different from its meaning in the second premise. Thus Frege’s analysis of indirect speech shows us that the problem lies, not with his logic but with the ambiguity of natural language. Since natural language is ambiguous, every statement of an inference requires a tacit presupposition: that all occurrences of an expression in the statement of the inference have the same meaning. In the first inference the tacit presupposition is correct; in the second it is not. Frege gives accounts, then, both of indirect speech and of a systematic ambiguity in natural language. Why is it important for his project to give these accounts?

IVe. Why Does Frege Analyze (some) Imperfect Parts of Natural Language? Frege’s discussions of language are in service of his development of a logically perfect language that can be used for his overall project. If he does not also have an independent interest in the workings of (imperfect) language, why would he want

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to understand a linguistic construction that will not be part of his logically perfect language? The key here is to realize that the very task of developing a logically perfect language presupposes that we are able, already, to use natural language in our expression and evaluation of inference. As he writes in one of his late diary entries, Indeed one might think that language would first have to be freed from all logical imperfections before it was employed in such investigations. But of course the work necessary to do this can itself only be done by using this tool, for all its imperfections.⁷⁹

It is not difficult to see why he should say this. We do not need Frege’s logical language or his system of evaluation to enable us to see, for example, that from (1)

The Morning Star is a planet

it follows that (2)

There is at least one planet.

Similarly, we do not need Frege’s logic to tell us that (1) does not follow from (2). To the contrary. It is these and similar observations that allow us to test a logical language and system of logical evaluation. Should Frege give us a logic on which (1) follows from (2) or (2) does not follow from (1), we will know that something has gone wrong. Without such tests, there would be no way to develop and test a new logic.⁸⁰ Moreover, provided we have an acceptable translation of (1) and (2) into Frege’s logical language, his logical system will pass this test. The same could

⁷⁹ NS, p. 285/PW, p. 266. ⁸⁰ Of course, there is a caveat here. On Frege’s view the claim that (2) follows from (1) depends on the presupposition that the terms in use don’t have certain logical defects. In particular, we presuppose that “the Morning Star” really names something and that the predicate “is a planet” holds or not of each object. What if the presupposition is not met? It is no part of Frege’s project to analyze how a proper name that does not name anything (or a predicate that does not determinately hold or not of each object) contributes either to the evaluation of inferences in which it appears or to determining the truth-value of sentences in which it appears. If the presupposition is not met, there is no evaluation at all. From the point of view of evaluating inferences, we simply presuppose that this defect does not exist. This suggests, of course, that there can be no inference that is without (unstated) presuppositions. For, after all, every inference is based on the presupposition that the names and predicates used in stating it really designate objects and concepts. We can, of course, add these presuppositions to the premises of the inference, but that will not solve anything. First, there is something suspect about these added premises—for they will not actually be used in the proof. Second, the added premises will themselves have presuppositions—that the names and predicates used in stating these premises designate objects and concepts. That is, it seems that we could never have any premise in any proof that is without presupposition. This issue is explored in Chapter 4.

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be said about more recent logical systems. What about the inferences labeled “(A)” and “(B)” above? These inferences certainly seem to fall under the purview of Frege’s project of evaluation. Indeed, both (A) and (B) look as if they should be straightforward consequences of Leibniz’ law (or Law 52 of Begriffsschrift). But if these are consequences of Frege’s logic, something must be wrong. For we know that the premises of (B) could be true and its conclusion false. Frege’s analysis of indirect speech shows us that his logic does not commit us to this incorrect evaluation. The appearance that the inference in question is correct is a result of a defect of natural language. Once we have Frege’s analysis of indirect speech this is easy to explain. In the first premise the expression “the Morning Star” appears in direct speech and names a particular planet. In the second premise, however, the same expression appears in indirect speech and names something else, its customary Sinn. That is, contrary to appearance, the second premise tells us nothing about the Morning Star. Thus Frege’s account of indirect speech tells us that what may look to be a problem with logic is no such thing—it is an artifact of one of the imperfections of natural language. It tells us more as well. For there are good arguments in which indirect speech contexts appear. Frege’s analysis of indirect speech identifies some constraints on how to expand Begriffsschrift so that it can be of use in evaluating such arguments, as well as what it would be like to translate such arguments into Begriffsschrift. And since, on Frege’s view, a word in indirect speech designates its customary Sinn, this also bolsters his claim to have identified a part of the content of linguistic expressions. His term “Sinn” is, after all, not just a term of art but an undefined term of art. If he is to convince his readers that there really is a Sinn associated with a linguistic expression, what better strategy than to show us that Sinn is something that we already talk about? It is now easy to identify the fragment of natural language that is of interest to Frege. It is the fragment in which we express apparently evaluable inferences. The linguistic constructions that are of interest to Frege are those that have significance for inference. The accounts he gives, of the workings of fragments of natural language, are those that he needs if he is to convince us that his logical system produces the correct evaluations of the arguments we already do understand.

V. So Where Are We Now? In this chapter, I have offered several arguments against one part of the Standard Interpretation: its general outline of Frege’s views about language. My target has been, not simply the Standard Interpretation, but also the view that the Standard Interpretation requires little by way of support; the idea that simply reading the words on Frege’s pages makes it dead obvious what he is doing. As I mentioned

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earlier in this chapter, Dummett writes that because Frege “is one of the clearest of all philosophical writers,” there is no need for “divergence over fundamental questions concerning what Frege was about.”⁸¹ A large part of what I have argued in this chapter is that Dummett and other supporters of the Standard Interpretation have not been willing to give Frege and the clarity of his writing their due. For the “obviousness” of the Standard Interpretation rests, in large part, on the assumption that Frege’s writings are either unclear or mistaken: on the insistence that Frege’s real projects are not described in his works; on the insistence that what he is doing is not what he says he is doing. In particular, I have emphasized that Frege uses the term “logic” for a specific purpose: logic comprises a language and system of evaluation that allows us to identify gapless proofs. The term “logic,” as Frege uses it, cannot be understood as another term for “theory of meaning.” One might argue that, in order to develop and/or defend a system of logic, one needs a theory of meaning. But Frege does not make such an argument. Nor, as I have argued, is there a straightforward reading on which Frege actually gives such a theory. I have also mentioned some well-known puzzles. Why did Frege assimilate sentences to proper names? Why did he not accord sentences a special status? And why did Frege describe the purpose of “On Sinn and Bedeutung” as that of solving a problem about identity? Although I have suggested that these questions present special problems for the Standard Interpretation, they are questions that might reasonably be asked of any interpreter of Frege’s work. In Chapter 2, I turn to the first version of Frege’s new logic; that is, the version set out in his 1879 monograph Begriffsschrift. Frege characterizes his departure from traditional logic as the replacement of the subject/predicate analysis of statements with a function/argument analysis. By working through Frege’s attempt to implement this new idea, we can see why there is a special problem about identity—a problem that Frege solves in “On Sinn and Bedeutung.” And we can see why the assimilation of sentences to proper names was far from a ludicrous deviation but, rather, a move that is essential to Frege’s conception of the new logic.

⁸¹ Dummett (1981b), p. xi.

2 Frege’s New Logic and the Function/ Argument Regimentation As we have seen, if we assume that “On Sinn and Bedeutung” is meant as a contribution to a theory of meaning—either for natural language or an improved version of natural language—then both Frege’s claim to be addressing a puzzle about identity and his view that sentences are names of objects seem obviously wrong. But, as we have also seen, there is scant evidence that Frege does mean to be developing such a theory of meaning. In particular, he does not use the term “logic” to talk about a theory of meaning or a theory of the workings of language (either natural language or an improved version of natural language). Rather, he uses the term to talk about a system of evaluation of inference, for which he uses a special kind of logically perfect language. And both the puzzle about identity and the view that sentences are names of truth-values, I have suggested, stem from the demands of developing a logic, in this sense. More specifically, both arise from problems with Frege’s original version of his logical system, the version described in his 1879 monograph Begriffsschrift. In this chapter, we will see how this works. In order to understand this, we need to start with the first version of Frege’s logic—the version he sets out in Begriffsschrift.

I. Why Does Frege Need a Microscope? As we have seen, Frege compares his logical language to a microscope, a tool useful for particular scientific purposes. What are these scientific purposes? The preface to Begriffsschrift gives us a brief sketch of a project whose pursuit, Frege tells us, led to his realization that he needed a new language; that natural language was inadequate for his purposes. The project he describes is a classificatory project. As we saw in Chapter 1, he writes that truths can be divided into two kinds, depending on what is needed for their proof. Some truths can be proved purely by means of logic. That is, they can be given a proof that “disregarding the particular characteristics of objects, depends solely on those laws upon which all knowledge rests”.¹ Truths of the other kind require a proof that is “supported by

¹ BS, preface.

Taking Frege at his Word. Joan Weiner, Oxford University Press (2020). © Joan Weiner. DOI: 10.1093/oso/9780198865476.003.0002

 ’       /  

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facts of experience.”² To which of these groups, he asks, do the truths of arithmetic belong? One might be inclined to think that Frege has been careless here. There cannot be a distinction, one might think, between logical truths and truths that are supported by facts of experience. For surely all knowledge—and therefore also the laws upon which all knowledge rests—must be supported by facts of experience. Frege would certainly agree with this much: experience is required in order for any particular person to have knowledge of any particular truth. As he says elsewhere, without evidence of the senses we would all be stupid as stones.³ But it is no part of Frege’s project to give an account of when an individual can be said to know a truth of arithmetic, an enterprise that I will be calling “personal epistemology.” Rather, his interest is in showing what is required for the justification, or proof, of the truths of arithmetic—justification that, as Frege understands it, has nothing to do with why or how any particular individual comes to believe a truth. All proofs require the support of laws on which all knowledge rests. But not all proofs require appeals to facts of experience, or particular facts about particular objects. And it is the project of distinguishing these two classes of truth that requires his new logical language. If a proof is to show us that a truth depends solely on the laws upon which all knowledge rests, one requirement is that there be no unstated presuppositions. For if an apparently purely logical proof requires an unstated presupposition, it may turn out that the unstated presupposition requires experience for its support. As a consequence, we need a means of determining whether an inference is gapless. There is, however, no mechanism for insuring that a proof stated in natural language is gapless. Frege writes about his new logical language that, Its first purpose, therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed, so that its origin can be investigated.⁴

Frege’s logic—his language, along with his laws and rules—is designed to be operated as a logical calculus. Once a proof is expressed in his new logical language, Begriffsschrift, it is supposed to be a mechanical task to determine whether or not it is gapless. Frege writes that his Begriffsschrift, is designed to produce expressions which are shorter and easier to take in, and to be operated like a calculus by means of a small number of standard moves, so that no step is permitted which does not conform to the rules which are laid down

² BS, preface. See also, FA, §91 and PCN, pp. 364–5.

³ FA, §106.

⁴ BS, preface.

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     once and for all. It is impossible, therefore, for any premiss to creep into a proof without being noticed.⁵

That is, for each line of a proof, we should be able to see by observation whether it follows by a rule of Begriffsschrift from earlier lines. The classification of the proposition that is proved will be inherited from the classification of the premises.⁶ But a gapless proof, on its own, may not be enough to tell us how to classify a truth. Suppose we have a gapless proof of a truth of arithmetic from other truths of arithmetic. This proof shows us something about the dependence of propositions on one another. But if the proof is to show us whether the proposition that has been proved belongs to logic, then we must look at the premises from which it has been proved. And the premises themselves might require proof. In that case, it may be that the next step is to prove them. At some point, however, proof must come to an end. And in order for it to be possible to carry out Frege’s project, it must be possible for our proofs to end with premises that do not require proof and that can be recognized (without proof) as belonging to logic. As he later writes, The problem becomes, in fact, that of finding the proof of the proposition, and of following it up right back to the primitive truths.⁷

Thus Frege’s logic cannot simply be a tool for evaluating inference. It must also have truths of its own. What are logical truths like? To answer this question, it will help to consider some of Frege’s other formulations of his classification scheme.

Ia. What Are Logical Truths? The Begriffsschrift Answer There is very little talk, in Begriffsschrift, about the nature of logic or what counts as a logical truth. Frege takes it for granted that his readers will not have questions about these issues. What he does say about logic and logical laws comes from his discussions of how logic is to be used in his project of classifying the truths of arithmetic. As we have seen, one of Frege’s characterizations of the classification scheme is that there are two sorts of truths: those that are provable from the laws upon which all knowledge rests and those that require the support of more specific truths for their proof. Truths in the former class are also characterized as those that can be proved “purely by means of logic.” Thus one characteristic of logical

⁵ FA, §91. See also, PCN, pp. 364–5. ⁶ For example, Frege writes later in BS, “If now (69) were a synthetic judgment, so would be the propositions derived from it.” (BS §23). ⁷ FA, §3.

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laws is that they are among the laws upon which all knowledge rests (laws that disregard the particular characteristics of objects).⁸ Some of Frege’s other remarks suggest that the classification scheme should be understood in terms of our faculties or sources of knowledge. One such remark appears in the beginning of Part III of Begriffsschrift. In Part III he offers proofs of some truths of arithmetic from general laws about sequences—proofs that, he thinks, will show us that (at least some) truths of arithmetic belong to logic. And he writes, Through the present example, moreover, we see how pure thought, irrespective of any content given by the senses or even by an intuition a priori, can, solely from the content that results from its own constitution, bring forth judgments that at first sight appear to be possible only on the basis of some intuition.⁹

One way of thinking about the distinction between logical and non-logical truths, this passage suggests, is to think of logical truths as truths that are given to us by pure thought, independent of the senses or intuition a priori. If we are to think of the distinction as tied to sources of knowledge in this way, however, we get three distinct classifications. For Frege indicates that there are three sources of knowledge: in addition to pure thought, there are also the senses and intuition a priori. This view of the importance of a three-part division is stated explicitly later (in Foundations), where he offers additional criteria for classifying truths. Those that can be derived from logical laws alone are logical or analytic truths; those that require the support of intuitions a priori (for example, the laws of Euclidean geometry) are synthetic a priori truths; and those that require appeals to particular facts about particular objects are synthetic a posteriori truths. This talk of faculties and sources of knowledge is also closely tied both to a conception of generality and, in particular, generality of domain. In the first section of Begriffsschrift, Frege mentions “the unrestricted domain of pure thought in general” [das umfassendere Gebiet des reinen Denkens].¹⁰ This domain is larger than, for instance, the domain of intuition. He writes that the propositions about sequences that he proves in Part III, far surpass in generality all those that can be derived from any intuition of sequences. If, therefore, one were to consider it more appropriate to use an intuitive idea of sequence as a basis, he should not forget that the propositions thus obtained, which might perhaps have the same wording as those given here, would still state far less than these, since they would hold only in the domain of precisely that intuition upon which they were based.¹¹

⁸ BS, preface.

⁹ BS, §23.

¹⁰ BS, §1.

¹¹ BS, §23.

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Indeed, one of the reasons Frege offers us later for taking the truths of arithmetic to be analytic or logical truths is that they are used in all sciences, that they appear to be applicable throughout all our thought. He writes, The truths of arithmetic govern all that is numerable. This is the widest domain of all [das umfassendste]; for to it belongs not only the actual, not only the intuitable, but everything thinkable.¹²

Analytic, or logical, truths—unlike other truths—cannot be denied in conceptual thought without resulting in contradictions. We have, then, three distinct characterizations of logical laws. First, logical laws are laws on which all knowledge rests, laws that transcend all particular characterizations of objects. Second, logical laws are the most general laws—that is, they apply everywhere. Third, logical laws are laws of pure thought: they are independent of the senses and intuition a priori. But, if these characterizations are distinct, they are also closely connected. We will turn to the issue of the significance of these characterizations, as well as just how they are related in Chapter 4, section IIIb. For now, however, it is worth noting that these characterizations are related. For example, the three sources of knowledge (the senses, intuition a priori, and pure thought) mark out different domains in a hierarchy of generality. Truths that require support from the senses cannot be generalized beyond the domain of things that can be sensed. We cannot, for example, derive general truths about spatial relations from the senses. The domain of geometry (the domain whose support is intuition a priori), then, is wider than that of what requires support of the senses. And the domain of logic—or that that requires only the support of pure thought—is wider still. It is also worth noting that the conception of logic and logical language in Begriffsschrift is very different from the contemporary conception. In particular, while Frege thinks of his logic as providing a means of evaluating inference, there is no attempt to explain correct inference in terms of truth under an interpretation of a formal language—or, indeed, in terms of any notion of truth.¹³ In fact, at points at which we might talk of truth or falsity (e.g., when the conditional is

¹² Frege FA, §14, p. 21. This is not to contrast the variables of arithmetic with variables of other sciences. For Frege, all variables range over an unrestricted domain. See Chapter 4, section IIIc. Rather, to say that the truths of arithmetic govern the widest domain of all is to say that they do not express, for example, “the peculiarities of what is spatial” as Frege says in FTA, p. 95. In his later writings, Frege describes this maximal generality somewhat differently. He describes laws of logic as, “prescribing how to think wherever there is thinking at all” (BLA vol. i, p. xv), as opposed to laws of geometry or physics, which provide a guide to thought only in restricted fields. See also, “Logik” (1897) NS, pp. 157–8/PW, pp. 145–6. ¹³ This claim about Begriffsschrift is uncontroversial. What is controversial is whether or not there are such attempts in Basic Laws. This issue will be discussed in Chapter 4.

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explained) Frege uses the expressions “affirmed” [bejaht] and “denied” [verneint]. How, then, is the logic Frege introduces in Begriffsschrift to be understood?

II. The First Version of Frege’s Microscope On Frege’s view, traditional Aristotelian logic is not adequate to play the role of logic in his classificatory project. In particular, there is a problem for any logic that relies on a subject/predicate analysis of sentences. Consider, for example, the statement that every natural number is odd. It is certainly a grammatical truth that this sentence has a subject and predicate. The subject of this sentence is “every number” and “odd” is its predicate. But does this grammatical analysis suffice for logical purposes? One of Frege’s objections to taking the grammatical analysis as a logical analysis is that such an analysis cannot recognize the distinction, in significance for inference, between “every natural number is odd” and “3 is odd.” Another is that there are many general truths of arithmetic that cannot be analyzed into subject and predicate. The claim that, for every number, there is some number that is greater appears to have two distinct subjects, “every number” and “some number.” And while some traditional logics would have resources to deal with this were there a separate predicate for each of these “subjects,” the linguistic predicate of our statement is a relation that, it seems, holds between the two subjects. There is no subject/predicate analysis that can work for such a sentence. And traditional Aristotelian logic cannot correctly evaluate the inferences that depend on statements that involve both relations and nested quantifications. Thus one of Frege’s key departures from traditional logics is to abandon the subject/predicate analysis of statements in favor of function/argument analysis. We can think of the sentence “Cato killed Cato,” he says, as a function of the argument “Cato” in various ways, corresponding to three different functions. If we here think of “Cato” as replaceable at its first occurrence, “to kill Cato” is the function; if we think of “Cato” as replaceable at its second occurrence, “to be killed by Cato” is the function; if, finally, we think of “Cato” as replaceable at both occurrences, “to kill oneself” is the function”.¹⁴

It may not be obvious from this example, however, that there is any advantage to Frege’s function/argument analysis. For, each of the analyses described in the above passage might reasonably be regarded as an analysis into subject and predicate. But Frege also gives us an example of an analysis that cannot be restated

¹⁴ BS, §9.

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as a subject/predicate analysis. Consider the statement that hydrogen is lighter than carbon dioxide. This statement is no more a statement about hydrogen than it is a statement about carbon dioxide. Here is it important that what is stated is the holding of a relation. This statement, Frege tells us, “can be regarded as [a] function of the two arguments ‘hydrogen’ and ‘carbon dioxide’.”¹⁵

IIa. The Begriffsschrift Notion of Function The notion of function Frege uses here, of course, is not the familiar mathematical notion of function. If the function/argument analysis is going to be of help, we need to know what this notion of function is. Frege says the matter can be expressed generally as follows, If in any expression, whose content need not be capable of becoming a judgment, a simple or a compound sign has one or more occurrences and if we regard that sign as replaceable in all or some of these occurrences by something else (but everywhere by the same thing), then we call the part that remains invariant in the expression a function, and the replaceable part the argument of the function.¹⁶

I will call the above passage, which is highlighted in the original, the “official account” of functions. Two features of the view described in the official account are both striking and surprising. First, there is no objective distinction between functions and arguments. Whether some part of an expression is a function or argument is determined by how we regard the expression. Second, functions (and arguments) are expressions. A function is the part of the expression that remains constant when we view a sign as replaceable (or replace it) with a different sign. An argument is an expression that is viewed as replaceable (or replaced). Before continuing, it will be helpful to think about the kind of examples Frege gives. Consider “Cato killed Cato,” the example we looked at a moment ago. Frege claims that one acceptable analysis of this sentence is into the function “to kill Cato” and the argument “Cato”. But it may seem that this must be wrong. For, according to the official account, any function we find in the sentence must be an expression that appears in the sentence. And the expression “to kill Cato” does not actually appear in the sentence “Cato killed Cato”. This objection, however, rests on the assumption that the function and argument can be natural language expressions. As we have already seen, Frege thinks that natural language is inadequate for his purposes. Thus one would expect the functions and arguments to be expressions, not of natural language, but of his logically perfect language.

¹⁵ BS, §9.

¹⁶ BS, §9.

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And, indeed, Frege begins the section in which he introduces the notion of function by asking us to suppose “that the circumstance that hydrogen is lighter than carbon dioxide is expressed in our formula language.”¹⁷ He then talks about regarding signs in this formula language expression as either replaceable or constant. Frege’s examples, however, are almost invariably natural language examples. It is not difficult to see why this is: Frege’s logical language, as it is introduced in Begriffsschrift, has very limited expressive power. In what follows, I will be using natural language examples, as Frege does, to talk about function/ argument analyses that, strictly speaking, only work for expressions in a logical language. Let us turn, now, to some of the surprising details of the official account.

IIb. Functions and How We Regard an Expression Frege’s replacement of the subject/predicate analyses of statements with his new function/argument analysis is one of the central advances of his logic over traditional logics. But, given his renowned anti-psychologism, it is surprising that he should say that how we regard the expression determines what is its function or argument. Thus, one might be inclined to suspect that the mention of how we regard the expression in his official account is an inadvertent slip. But it is not. Frege introduces the term “conceptual content” for the content of an expression that has significance for the inferential sequence. And he writes, about the distinction between function and argument, The distinction has nothing to do with conceptual content; it comes about only because we view the expression in a particular way.¹⁸

This is especially puzzling, since the function/argument analysis of a statement is brought in precisely because of its significance for the inferential sequence. In order to understand what Frege means here it will be helpful to think about how his logical language could be used in analyses of everyday natural language statements and arguments. Although Frege does not actually give us examples of translation from natural language to Begriffsschrift, his explanations of Begriffsschrift expressions include information that will allow us to construct such translations. For example, he writes, ⊢Ψ(A,B) can be translated by “Β stands in the Ψ relation to A” or “Β is a result of the application of the procedure Ψ to A.”¹⁹

¹⁷ BS, §9.

¹⁸ BS, §9.

¹⁹ BS, §10.

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Let us use this to consider how we could express something like “Brutus stabbed Caesar” in a language like Begriffsschrift.²⁰ This natural language statement tells us that Brutus stands in a particular relation to Caesar. It is reasonable to suppose, then, that “Brutus stabbed Caesar” would be translated by something like: s(B,C). Now let us think about how to identify the function and argument in that expression. We can regard “B” as replaceable in “s(B,C).” Replacement of “B” gives us expressions like: s(A,C), s(C,C), s(D,C), etc. If we think of the expression in this way, the function (what is common to all the expressions we get by replacing “B”) is, by analogy to Frege’s Cato example, “to stab Caesar.”.²¹ But that is not to say that “to stab Caesar” is the function in our original statement. For, just as we can regard the second “Cato” in Frege’s example as being replaceable, we can regard “Caesar” (or “C”) as replaceable in our example, in which case we get: s(B,A), s(B,B), s(B,D), etc. By thinking of replacing “C,” we are regarding the original proposition as having the function “to be stabbed by Brutus.” What this shows us that there is no fact of the matter about what is the function and what is the argument in “Brutus stabbed Caesar.” Or, in other words, it depends on how we regard the sentence. But while changes in our way of regarding the expression can change what is the function and what is the argument in our analysis of the statement, it does not in any way change the expression or its content. The same Begriffsschrift-style expression, “s(B,C),” can be regarded as having different functions and different arguments. Moreover, the Begriffsschrift-style expression is clearly both about Brutus and about Caesar; inferences can be drawn from it about either of these individuals, depending on what other premises are used. Indeed, it is a mechanical task to derive, from this, using a contemporary version of Frege’s logic, both (Ǝx)s(x,C) (i.e., that someone stabbed Caesar) and (Ǝx)s(B,x) (i.e., that Brutus stabbed someone). This exhibits the greater flexibility of the Begriffsschrift function/argument expressions over, for example, Boole’s subject/predicate expressions. Unlike a Boolean expression of the above statement, a Begriffsschrift style expression of this statement will have three letters. Begriffsschrift style expressions of other statements may have even more signs, since a function can have many arguments. In contrast, using Boole’s notation, the expression of the proposition will have only two terms, a subject and a predicate. Thus, in order to express the proposition in question in Boole’s notation, a subject and predicate must be chosen. The two terms could be “Brutus” and “stabbed Caesar” or “Caesar” and “was stabbed by Brutus.” And once one of these predicates is chosen and used to give a Boolean

²⁰ I talk about a language that is like Begriffsschrift rather than Begriffsschrift itself because in the actual language Frege sets out in Begriffsschrift, unlike contemporary notations, there are no nonlogical constants. Thus it is not possible to give actual Begriffsschrift translations of his examples. ²¹ Note that, strictly speaking, “to stab Caesar” is not the function but, rather, the translation of the function into English. For convenience I follow Frege in writing as if the function belongs, not to the logical language, but to natural language.

 ’       /  

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symbolization of the proposition, the other predicate cannot be recovered. Were we to choose “Caesar” as the subject, we would need a simple predicate (a single letter) for “was stabbed by Brutus.” There would be, then, no name for Brutus in the symbolization and, hence, no way to infer, say, from the additional information that Brutus was a Roman, that a Roman stabbed Caesar.

IIc. Do Functions and Arguments Differ in Kind? We have seen that the conceptual content of a sentence does not determine a unique function/argument decomposition. But that is not to say that the distinction between function and argument has nothing to do with conceptual content. And surely, one might think, there are constraints on how we regard the expression that are dictated, not by us or our psychology, but by features of the expression itself—features that do have to do with conceptual content. Granted, we can regard the content of “s(B,C)” as being decomposable into a two-place function applied to two arguments, B and C, or a one-place function applied to B, or a one-place function applied to C. But these seem to be the only options. And these seem to be the only options for an important reason: there appears to be an obvious difference between function expressions (whether simple or complex) and expressions like “B” and “C,” that seem to be used as names of objects. But on Frege’s official account no such distinction is made. What can be taken as a function, on this account, is dependent simply on regarding part of the expression as replaceable and another part as invariant. Thus, while we can regard “s(B, )” as invariant and “C” as replaceable, we can just as easily regard “C” as invariant and “s(B, )” as replaceable. For example, consider the following sentences: Hydrogen is lighter than carbon dioxide Hydrogen is an element with atomic number 1 Hydrogen is a gas at room temperature The invariant part of all these sentences is the expression “Hydrogen.” Hence, given Frege’s official characterization of the notion of function, “Hydrogen” is a function. One might suspect that this must have been something Frege overlooked. However, it is not. For he seems to consider this consequence of his official account, when he writes, Since the sign Φ occurs in the expression Φ(A) and since we can imagine that it is replaced by other signs, Ψ or Χ, which would then express other functions of the argument A, we can also regard Φ(A) as a function of the argument Φ.²²

²² BS, §10, emphasis in the original.

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On Frege’s account of the notion of function there is, then, no explicit distinction between a function-letter and an argument-letter. Still, one might think, this must be wrong and Frege must have seen that it was. For there is an important distinction between “Φ” and “A.” Suppose, for the moment that “Φ(A)” expresses a content that can become a judgment (i.e., is a sentential expression). When we regard “A” as replaceable in “Φ(A),” we can distinguish those letters that, when substituted for “A,” yield a content that can become a judgment and those that do not. We might, for example, get other judgeable contents by substituting “B” and “C” for “A.” However, if we substitute “Φ” for “A” the resulting expression does not have a content that can become a judgment. Similarly, if we replace “Φ” with “B” the resulting expression not only fails to have judgeable content, it would seem to be gibberish. What looks to be needed here is a list of rules of formation for the language of Begriffsschrift.²³ And one might be tempted to infer that, had Frege noticed this, he would have abandoned the view that what parts of an expression can be counted as a function and what parts as arguments is a feature of how we regard the expression. It is important to see, however, that the above argument depends on the understanding of Frege’s Begriffsschrift contrast between function and argument as an early version of his later contrast of function and object. And this understanding is not quite right. Indeed, the notion of argument, which continues to be present in Frege’s later writings, is not the same as that of object. In particular, since his logic is not a first-order logic, he needs to be able to view first-level functions not only as functions but also as arguments. This comes into play in Begriffsschrift immediately after he tells us that either “Φ” or “A” can be taken as the argument of “Φ(A)”. For his next step is to introduce the concavity symbol (his symbol for the universal quantifier). To use the concavity symbol as a quantifier, one puts German letters both in the concavity space (that is, in the quantifier) and in the part of the formula that is governed by the quantifier. The first example of the use of such a quantifier is as governing the slot occupied by “A” in “Φ(A).” He then continues, Since a letter used as a sign for a function, such as Φ in Φ(A), can itself be regarded as the argument of a function, its place can be taken, in the manner just specified, by a German letter.²⁴

Quantification over first-level functions appears throughout Begriffsschrift and is especially important in the Part III proofs about sequences, including his proof of

²³ In fact, Frege offers us no such list and, as we have just seen, the account of function and argument gives us no tools to rule out such an expression as ill-formed. ²⁴ BS, §11.

 ’       /  

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mathematical induction. Just as first-level function slots can be quantified over, there can be second-level functions that take first-level functions as arguments. For example, the function F is hereditary in the f sequence takes two arguments, F and f, both of which are first-level functions. Thus it is important for the use of the logical language that a single expression can be regarded either as a function or as an argument. Frege’s talk of function and argument, then, is not meant to be understood as classifying things either as functions or as arguments. Rather, it is talk that is helpful for enabling us to analyze an expression, particularly a sentential expression, in a way that allows us to formulate and evaluate inferences. It is important to realize, however, that—while Frege does not yet explicitly distinguish objects from functions and first- from second-level functions in Begriffsschrift—his understanding of the new logic already relies on a recognition of these differences. For in the actual definitions and proofs there is never any confusion about whether a particular slot is an object slot or a function slot, or whether a function slot is a first- or second-level slot. Frege also briefly discusses the difference between the two propositions “The number 20 can be represented as the sum of four squares” and “Every positive integer can be represented as the sum of four squares.” And he talks about requirements on the use of German letters in the quantifier so that what follows is always a content that can become a judgment. All of this is eventually made explicit in later works. It is an overstatement to say that the identification of function and argument has nothing to do with conceptual content. But, as we have seen, while there are constraints on how an expression can be divided into function- and argumentexpressions, there is also an important respect in which Frege is right to say that this is determined by our way of regarding the expression. For all that, there are difficulties with the official account of function in Begriffsschrift. Let us turn next to the second surprising feature of Frege’s official account of function and argument: his claim that functions and arguments are linguistic expressions.

IId. Some Problems with the Linguistic-expression View of Function The claim that the functions and arguments involved in our logical regimentations are linguistic expressions is surprising because it does not fit with something that is central to Frege’s project: the conception of the logical language as a language designed for expressing content. In particular, he describes his replacement of the notions of subject and predicate with those of function and argument

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as “regarding a content as a function of an argument.”²⁵ Moreover, this is not a one-time remark. For example, after writing, “The circumstance that carbon dioxide is heavier than hydrogen” and “The circumstance that carbon dioxide is heavier than oxygen” are the same function with different arguments . . . ²⁶

He says, of this example, [W]e can also conceive of the same conceptual content in such a way that “carbon dioxide” becomes the argument and “being heavier than hydrogen” the function.²⁷

And, later in the same section he writes, For us the fact that there are various ways in which the same conceptual content can be regarded as a function of this or that argument . . . ²⁸

To say that a conceptual content can be regarded as a function of an argument is, it seems, to say that the value of a function on an argument is a conceptual content. But how does this fit with the official account of function and argument? According to the official account, the function in question is the part of the two expressions that is the same. That is, it is the expression “the circumstance that carbon dioxide is heavier than . . . ” Moreover, according to the official account, arguments as well as functions must be linguistic expressions. Thus, the different arguments that Frege mentions in the above example must be the linguistic expressions “hydrogen” and “oxygen.” What about the values of these functions on these arguments? If functions and arguments are simply linguistic expressions, it is difficult to see what the value of a function on an argument can be, if not the concatenations of the expressions. But this conflicts with Frege’s statement that, in analyzing those two sentences, he is regarding the content (not the linguistic expression) as a function on an argument. For the contents of the sentences in the above example are not linguistic expressions.²⁹ This does not show that functions and arguments are not linguistic expressions. But it does exhibit a tension between Frege’s idea that functions and arguments are linguistic expressions and his idea that we can give a function/argument analysis of a (nonlinguistic) content.

²⁵ BS, preface, emphasis added. ²⁶ BS, §9. ²⁷ BS, §9, emphasis added. ²⁸ BS, §9, emphasis added. ²⁹ The idea that the value of the function on the argument is a linguistic expression also conflicts with Frege’s view of other expressions that appear in these statements.

 ’       /  

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There are also other problems with reconciling Frege’s view of functions and arguments as linguistic expressions with his general views about how the logical language works. In particular, the linguistic-expression view of functions and arguments does not fit with his claim that “signs are merely representatives of their content.”³⁰ For consider, again, the expression carbon dioxide is heavier than hydrogen It seems entirely reasonable, as well as in line with the idea that signs are representations of their content, to say that, supposing the above expression to express a judgment, the judgment in question is about hydrogen, not about the word “hydrogen.” It ought, then, to be equally correct to express the same judgment by using a different name for hydrogen, say, the chemical symbol, “H.” That is, the above expression and carbon dioxide is heavier than H give us, not the same function on different arguments but, rather, the same function on the same argument. But, if functions and arguments are linguistic expressions, as they are on Frege’s official account, the arguments are different. Moreover, the official account of functions and arguments as linguistic expressions also seems to conflict with the way in which the logical system of Begriffsschrift actually works. For example, were sameness of argument simply sameness of expression, then there would be no role to be played by definitions. Nor would there be any way to use Frege’s version of Leibniz’ Law—that is, Law 52: c ≡ d ! (f(c) ! f(d)). It is, thus, in line with the way Frege’s logical system works, to take the argument of a function to be, not a linguistic expression that is viewed as replaceable in a larger expression but, rather the conceptual content of such an expression. And, since we have seen that any expression that can be taken as a function can also be taken as an argument, the same problem arises for the function part of the official account. It should be no surprise, then, that when Frege returns to the issue of what a function is, in his 1891 Function and Concept, he claims that it will not do to take functions to be linguistic expressions because no distinction is made between form and content (Form und Inhalt), sign and thing signified (Zeichen und Bezeichnetes) . . . a mere expression, the form for a content, cannot be the heart of the matter; only the content itself can be that.³¹

³⁰ BS, §8.

³¹ FC, pp. 2–3.

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We will turn to Frege’s new view of function and argument later in this chapter. For now, however, it might be worth noting something interesting for the interpretation of Begriffsschrift. When Frege explicitly rejects the idea of taking a function to be a linguistic expression, he does not identify it as a view he had held in Begriffsschrift. Rather, he suggests that it is the kind of answer we are likely to see in books of mathematics. Frege may well have simply adopted this familiar way of talking about functions without much thought and, in particular, without realizing that it does not fit with his actual use of the function/argument analysis of statements. That would explain the fact that an account that better fits the way the logic of Begriffsschrift actually functions, would be something like the following, If in an expression, whose content need not be capable of becoming a judgment, a simple or a compound sign has one or more occurrences and if we regard that sign as replaceable in all or some of these occurrences by something else (but everywhere by the same thing), we call the content of the part that remains invariant in the expression a function, the content of the replaceable part the argument of the function.³²

Although Frege never explicitly introduces this modified version of the official account, it is suggested by his treatment of the relations among concepts, objects, and judgments in “Boole’s logical Calculus and the Concept-script.”³³ In what follows, I will refer to this modified version of the official account as the “conceptual content view.”³⁴

III. Two Problems with the Begriffsschrift Account of Identity Why does Begriffsschrift require an identity symbol? It is obvious that, insofar as Begriffsschrift is a tool for expressing truths and inferences of arithmetic, there needs to be a way of expressing identity statements about numbers, e.g., “1+1=2.” But the identity sign in Frege’s Begriffsschrift can also be used between expressions whose contents can become judgments, i.e., sentences. Why should Begriffsschrift need an identity sign that can be used between sentences? The ³² This is a modification of Frege’s official account of function in BS, §9. The modification consists of inserting the two bolded phrases. ³³ BLC. NS pp.18–19/PW, pp. 17. ³⁴ Michael Beaney argues that the conceptual content view is actually what Frege has in mind in Begriffsschrift, although Beaney acknowledges that Frege does not say this explicitly. See Beaney (2007). I do not find this entirely convincing, given the large number of statements Frege makes that conflict with the conceptual content view. Beaney also argues that Frege’s recognition of truth-values as objects arises from problems with identity.

 ’       /  

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answer is that this need is a consequence of Frege’s function/argument analysis of sentences. A function must have a value for each argument. If we take the predicate of a sentence to be a function (or a function-sign), what kind of value can it have on an argument? As we have seen, Frege’s statements in Begriffsschrift appear to give us two options: either the value is the sentence itself or the value is its conceptual content. The official account of functions may suggest that the value of a function must be a linguistic expression. It should be obvious that, on this view, the functions in question are single-valued. And, since identity between values requires the exact same linguistic expression, there will be no need for an identity sign to be used between sentences. However, as we have just seen, the view that fits better with many of Frege’s statements is that the value is the conceptual content of the sentential expression.³⁵ Since it is possible to have distinct expressions (including sentences) with the same conceptual content, there is a need for an identity sign that can be flanked by sentences. This view also fits with Frege’s introduction of the identity sign, “≡”. He writes, Now let

⊢(A ≡ B) mean that the sign A and the sign B have the same conceptual content, so that we can everywhere put B for A and conversely.³⁶

The significance that the identity sign has for inference is encoded in Begriffsschrift as Law 52: c ≡ d ! ( f(c) ! f(d)). And this is also a respect in which the understanding of function and argument in Begriffsschrift seems to rely implicitly on the conceptual content view—the view that functions and arguments are not linguistic expressions but, rather, conceptual contents of linguistic expressions. For the antecedent requires the argument-expressions not to be the same but to have the same conceptual content. There are problems with this explanation of the identity sign, however.

IIIa. The First Identity Problem How are we to recognize sameness of conceptual content? Frege’s answer, for contents of sentential expressions, is:

³⁵ It may be worth mentioning that, if we try to fit together all Frege’s statements—that is, if we take functions and arguments to be linguistic expressions but the values of these functions to be conceptual contents, there will be a problem with composing functions. ³⁶ BS, §8.

    

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the consequences derivable from the first, when it is combined with certain other judgments, always follow also from the second, when it is combined with these same judgments.³⁷

This is a very fine-grained notion. To appreciate how fine-grained it is, let us consider two similar inferences that differ only in one premise. (1) (2) (3)

8 is even All even numbers are divisible by 2 Therefore, 8 is divisible by 2

and (4)

The number of planets is even All even numbers are divisible by 2 Therefore, 8 is divisible by 2

Obviously, the substitution of (4) for (1) makes a difference to the status of the inference. For, in the first, the conclusion is a consequence of its premises ((1) and (2)) and logical laws. To infer the same conclusion from premises (2) and (4), however, requires an additional appeal to facts about the external world. That is, the inferential consequences of (1), when it is combined with (2) are not the same as the consequences of (4) when combined with (2). Thus (1) and (4) do not have the same conceptual content. How does this fit with a Fregean function/argument analysis? According to the official account, the expression that is shared by (1) and (4), i.e., “is even,” is a function. Since the conceptual contents of (1) and (4) are different, and since functions must be single-valued, we can infer that (1) and (4) have different arguments. On the official account, there is no question that this is right. For the arguments are simply linguistic expressions, and “8” and “the number of planets” are distinct linguistic expressions. But, as we have already seen, the official account is problematic: Frege also claims that signs are merely representatives of their content. What about the conceptual content account view? This view is also consistent with our distinguishing (1) and (4). After all, the expressions “8” and “the number of planets” do have distinct conceptual contents (as the examination of the above two inferences indicates). But there is still a problem. Frege’s Begriffsschrift is supposed to be usable, not only for arithmetic, but for other sciences, including astronomy. Thus it must be possible to make true and informative identity

³⁷ BS, §3.

 ’       /  

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statements about, say, the number of planets. To say that the number of planets is 8 is to make an identity statement that we currently accept as a truth. But, given that the expressions “the number of planets” and “8” do not have the same conceptual content, according to Frege’s explanation of identity, the identity statement, “the number of planets ≡ 8” is false—as is “the number of planets ≡ 9”. The upshot is that there seems to be an inescapable problem with the Begriffsschrift conception of identity.³⁸ For, as a matter of fact, if we want Begriffsschrift to be useful in the expression of scientific statements and inferences, sameness of conceptual content is too strong a requirement for identity. We need a notion of identity on which it can be true that the number of planets is identical to 8. Moreover, supposing the number of planets is identical to 8, and given that functions must be single-valued, the value of a function on 8 as argument must be the same as its value with the number of planets as argument. Thus, (1) and (4) above should be the result of applying the same function on the same argument. That is, there should be some respect in which (1) and (4) are the same. But what could this be? In Begriffsschrift, we can recognize only expressions and their conceptual contents. (1) and (4) are different linguistic expressions and they have different conceptual contents.

IIIb. A Second Identity Problem There is also another problem with the Begriffsschrift account of identity. To see this problem, let us begin with the opening remark from the section on identity. Frege writes, Identity of content differs from conditionality and negation in that it applies to names and not to contents. Whereas in other contexts signs are merely representatives of their content, so that every combination into which they enter expresses only a relation between their respective contents, they suddenly display their own selves when they are combined by means of the sign for identity of content; for it expresses the circumstance that two names have the same content. Hence the introduction of a sign for identity of content necessarily produces a bifurcation in the meaning of all signs: they stand at times for their content, at times for themselves.³⁹

What is special about identity, Frege tells us, is that it is about the names that appear in identity statements, rather than their content. How is it that Frege thinks

³⁸ Robert May (2012) makes a similar point.

³⁹ BS, §8.

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identity statements are about names? And what significance does this have for his logical system?⁴⁰ An identity statement is meant to be a license for the intersubstitution of names in inferences. More specifically, an identity, say “c ≡ d”, is supposed to license the inference, from any sentence in which “c” appears, to any sentence that results from substituting “d” for one or more of the occurrences of “c.” Described in this way, Frege’s identity symbol does seem to concern names rather than their contents.⁴¹ But there is a problem with viewing identity in this way. In order to make an inference from an identity statement, we need to employ Law 52: c ≡ d ! (f(c) ! f(d)). One result is that most instances of Law 52 contain ambiguous signs.⁴² Whatever names replace “c” and “d” appear, in the first part of the sentence, as names for themselves and, in the second part, as representatives of their content. Frege may not have seen any difficulty with this ambiguity when he wrote Begriffsschrift but he came to view ambiguity as a serious logical defect shortly thereafter and he continued to hold this view of ambiguity for most of his career.⁴³ In particular, he repeatedly voices this view in the years between the publication of Begriffsschrift and Foundations. In one of his papers about his logical language written during this period he writes that the “first requirement” is that such a language not be ambiguous.⁴⁴ “We need a system of symbols from which every ambiguity is banned.”⁴⁵ Ambiguity is a defect that, according to Frege, can be found not only in natural languages, but also in a number of logical

⁴⁰ This is a topic that has been much discussed in the literature, but most of that discussion is irrelevant for the issues under consideration here. For example, Pardey and Wehmeier (2019) argue that there is no problem from the point of view of the logical system. But Pardey and Wehmeier do not consider the significance of ambiguity for the epistemological project. As we saw in Chapter 1, Frege’s objection to ambiguity in logical languages is that some inferences, as stated in such a language, will not be gapless. Thus, the correctness of an inference that contains an ambiguous expression might depend on a presupposition—a presupposition about how the ambiguity is to be resolved—that is not explicitly stated in the inference itself. The arguments Pardey and Wehmeier give do not address this issue. It is also worth noting that Pardey and Wehmeier, as well as most of the other people who have written on this issue, simply assume that the Begriffsschrift view of identity statements is that they tell us that the terms flanking the identity sign are co-referential. However, as we have seen in this chapter, this is mistaken. Frege’s claim, in Begriffsschrift, is that identity statements tell us that the terms flanking the identity sign have the same conceptual content. Sameness of conceptual content is a much more finegrained relation than sameness of reference. In what follows, we will see how Frege’s objection to ambiguity, along with some of his other views, may have pushed him to draw the Sinn/Bedeutung distinction in “On Sinn and Bedeutung.” For all that, as we have already seen, Frege does make a few remarks in which something close to the view of identity statements as stating that expressions are coreferential is implicit. ⁴¹ Of course, this difference between the identity sign and, say, the conditional sign is only apparent. The description of the inferences licensed by a conditional can also be expressed by talking about names. ⁴² The exceptions are instances in which “f(c)” is itself an identity statement. ⁴³ And Frege continued to hold this view throughout his career. See, for example, PCN 362, 367; OFG II, pp.307–9; BLA vol. i, p. 51, BLA vol. ii, p. 82, LM NS pp. 228, 245/PW, pp. 213, 227. ⁴⁴ OSJ, p. 50. CN (1972), p. 84. ⁴⁵ OSJ, p. 52 CN (1972), p. 86.

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languages proposed by others.⁴⁶ In another of these papers he writes “we may not use the same symbols with a double meaning in the same context.”⁴⁷ Unfortunately, as we have just seen, on Frege’s account of identity, the same symbol is used with a double meaning in an instance of Law 52. The system of symbols Frege gives us in Begriffsschrift is, by his own account, ambiguous.

IV. Frege’s Introduction of Sinn and Bedeutung as a Solution to the Identity Problems Where are we now? We have seen that one of Frege’s fundamental innovations is his substitution of function/argument analysis for subject/predicate analysis of statements. The notion of function that is used, of course, is not a traditional mathematical notion. And, as we have seen, Frege does not give us a consistent story in Begriffsschrift about what this new notion of function is. Moreover, as we have also seen, there is no easy fix here. As long as Frege sticks to the view that the only content that can be of concern to logic is conceptual content, there is no way to give a good account of the new notion of function and no way to solve the first problem with identity. In particular, sameness of the conceptual content of the expressions flanking the identity sign is simply too strong a condition for the truth of an identity statement. At this point, it should be obvious to most contemporary readers that at least part of Frege’s solution will be to introduce the notions of Sinn and Bedeutung. Let us turn, then, to “On Sinn and Bedeutung.” Frege begins “On Sinn and Bedeutung” with a discussion of a problem about identity and he claims to be rejecting the Begriffsschrift view. However, there is no discussion of the Begriffsschrift definition of identity, nor is there any mention of conceptual content in “On Sinn and Bedeutung.” The view Frege criticizes is not actually the Begriffsschrift view, but a modified version of that view. What the modified version has in common with the original version is that, on both views, identity statements are statements about linguistic expressions. In order to understand the objection and Frege’s ultimate view, it will help to begin by seeing how Frege goes from the view of identity in Begriffsschrift to the view that he criticizes in “On Sinn and Bedeutung.”

⁴⁶ He writes, Exactly the opposite holds for the symbolism for logical relations originating with Leibniz and revived in modern times by Boole, R. Grassmann, S. Jevons, E. Schröder, and others. Here we do have the logical forms, though not entirely complete; but content is lacking. In these cases, any attempt to replace the single letters with expressions of contents, such as analytic equations, would demonstrate with the resulting imperspicuity, clumsiness—even ambiguity—of the formulas how little suited this kind of symbolism is for the construction of a true conceptual notation. (OSJ, p. 54/CN, p. 88.) ⁴⁷ AIMCN, p. 4; CN, p. 93.

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IVa. The Modified Begriffsschrift View of Identity Let us start with a closer look at the Begriffsschrift view. As we saw earlier, Frege writes that an identity statement of the form “A ≡ B” means the sign A and the sign B have the same conceptual content, so that we can everywhere put B for A and conversely.⁴⁸

There are two distinct parts to this account of identity. The first part purports to tell us about the content of identity statements. And this, as we have seen, is wrong: sameness of conceptual content is too strong a requirement for the truth of the identity statement. What about the second part? This part is not simply a rephrasing of the first. Rather, the second part tells us how the appearance of an identity statement in a proof should affect our evaluation of the proof. The identity statement licenses certain inferences: we are entitled to infer any statement that results from taking a statement on an earlier line and replacing one of A and B with the other. This is a correct account of the significance the appearance of an identity statement in a proof has for our evaluation of that proof. Indeed, at various points, Frege toys with the idea of using this as a definition of identity. For example, he writes, in Foundations, Now LEIBNIZ’S definition is as follows: “Things are the same as each other, of which one can be substituted for the other without loss of truth”. This I propose to adopt as my own definition of identity.⁴⁹

And this view, as we saw earlier, is encoded in Begriffsschrift as Law 52: c ≡ d ! ( f(c) ! f(d)). Suppose that we understand an identity statement as whatever statement is needed to license us to infer the consequent of Law 52. What needs to hold, then, of the two expressions flanking the identity sign is that they name the same thing. But as we just saw, when we considered the sentence “the number of planets = 8,” we cannot conflate what it is that an expression names with the conceptual content of that expression. It is not difficult to see how Frege might move from the official Begriffsschrift account of identity (that “A ≡ B” means that “A” and “B” have the same conceptual content) to the modified version that he criticizes in “On Sinn and Bedeutung” (that “A ≡ B” means that “A” and “B” name the same thing). Indeed, this modified version of the account seems to be implicit in the Begriffsschrift

⁴⁸ BS, p. 21.

⁴⁹ FA, §65.

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discussion of identity. For, in the example Frege uses in this discussion, he characterizes two names of the same point as having “the same content” [denselben Inhalt].⁵⁰ And yet (although he does not explicitly acknowledge this) it is evident from the ensuing discussion that these two names do not have the same conceptual content. “On Sinn and Bedeutung” begins with an explanation of why it is plausible to take an identity statement to tell us that the names flanking the identity sign name the same thing—an explanation very similar to the one he gives in Begriffsschrift. There appear to be two possibilities. Either an identity statement is about objects or it is about names of objects. The first of these is, for us, the more familiar view. But there is an apparent problem with this view, a problem arising from considerations that are epistemological in Frege’s sense. To see this, and to see its significance, we need to remember that the term “epistemology” as Frege uses it, has to do, not with individuals and their claims to knowledge, but, rather, with what constitutes the best proof of certain truths. As he points out both in Begriffsschrift and in “On Sinn and Bedeutung,” there are true identity statements that cannot be given purely logical proofs. Some identity statements require laws of Euclidean geometry for their proof—they are, he writes, “synthetic in the Kantian sense.”⁵¹ In “On Sinn and Bedeutung,” he writes that there are true identity statements that cannot be established a priori—among these he mentions the discovery that the rising sun is not new every morning or the re-identification of a planet.⁵² Such discoveries require empirical support. And these observations seem to undermine the view of identity as a relation between objects. For, after all, if identity statements are about objects, it would seem that a true identity statement simply says, of a particular object, that it is identical to itself. Since it is a law of logic that, for any a, a = a, should it not be possible to establish any true identity statement using logical laws alone? But what, then, of the true identity statements in which different names appear on the two sides of the identity sign? There is no reason to think that, whenever two different symbols name the same thing, this should be provable by logic alone.⁵³ The natural solution seems to be to take ⁵⁰ BS, §8. ⁵¹ BS, §8. What actual influence Kant had on the development of Frege’s Sinn/Bedeutung distinction is not directly relevant to the story I want to tell here. However, Michael Kremer (2010) makes an interesting case for Kantian influence on this development. ⁵² SB, p. 25. ⁵³ It may seem that there is something odd about this description of the motivation for Frege’s original explanation of identity statements. For in the above explanation (as well as in Frege’s actual words) there is an indication that the signs flanking the identity sign are names of particular objects or individuals. But in Begriffsschrift the only content associated with a symbol is its conceptual content and, as we have seen above, the conceptual content of a symbol cannot be the object or individual it names. However, this is less mystifying if we bear in mind the many conflicting explanations that appear in Begriffsschrift. For example, as we saw earlier, Frege vacillates between taking arguments and functions to be expressions and taking them to be conceptual contents of expressions. Throughout most of the book his notion of content is conceptual content. But in both §8, his discussion of identity, and §9, his introduction of the notion of function, he talks of the object that a name names as its

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identity statements to be about expressions: that is, an identity statement tells us that the two names flanking the identity sign are names for the same object. In “On Sinn and Bedeutung,” Frege turns next to the explanation of what is wrong with taking identity statements to be about names. Consider, again, the reidentification of a planet (e.g., the discovery that the Morning Star is identical to the Evening Star). These are substantive astronomical discoveries. Yet, on this view, the statement that “the Morning Star” and “the Evening Star” name the same object is not a statement about astronomy but about words and how they are used. Something has gone wrong. The discovery that such a statement is true is supposed to be the kind of thing that can be a substantial astronomical discovery about planets, not a discovery about an arbitrary decision about how to name things. Frege concludes that it is a mistake to say that identity statements are about the expressions flanking the identity sign.

IVb. The New View of Identity Frege’s solution is introduce a version of the view that an identity statement is about objects, a version that involves the recognition of another sort of content that is associated with a linguistic expression: its Sinn. The introduction of the notion of Sinn is supposed to show us that the content of an identity statement need not be trivially true; that the recognition of the truth of an identity statement can be a substantial scientific discovery. To see how this works, let us begin with one of Frege’s examples. Suppose that we have a triangle and three lines, a, b, and c, each of which connects one of the vertices of the triangle with the midpoint of the opposite side. The following two statements are true: (A)

The point of intersection of a and b = the point of intersection of b and c

and (B)

The point of intersection of a and b = the point of intersection of a and b

But from the point of view of the epistemological questions that concern Frege, we need to recognize an important distinction between the classifications of

content. In fact, one of the interesting features of the Begriffsschrift discussion of identity is that there is a sense in which both the notions of Sinn and of Bedeutung are already recognized. For Frege indicates that expressions like “the point of intersection of a and b” are associated with both a point (the object that will later be called its Bedeutung) and a way of determining the point (something very like what will later be termed its Sinn). But the official restriction to recognizing only one kind of content associated with an expression (its conceptual content) precludes these notions from playing any role in the official explanation of identity that appears at the end of the section.

 ’       /  

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(A) and (B). The proof of (A) is non-trivial and requires laws of Euclidean geometry. In contrast, (B) can be given a very short proof using logical laws alone. Since both of the above sentences are about a particular point, if the only content associated with each name is what it names, there appears to be no way of distinguishing the content of the two sentences. But the difference in what is required for the proofs of these statements shows us that there must be some difference in content that has significance for inference—that there is something in Frege’s original notion of conceptual content that needs to be captured. Frege’s new notion of Sinn is meant to play this role. What is Sinn and how does the introduction of Sinn solve the problem? Frege says surprisingly little about what Sinn is. As we saw in Chapter 1, Frege does not define Sinn as mode of presentation [Art des Gegebenseins] but says, rather, that the mode of presentation is contained in the Sinn. It should be clear that such a remark cannot be meant as a definition. For it leaves open the possibility that Sinn may contain other things as well.⁵⁴ Nor does Frege think it is entirely obvious exactly what mode of presentation is, as his subsequent attempts to explain Sinn make clear. But even if we do not know exactly what mode of presentation is, the phrase directs us to a difference in content between the expressions on the two sides of the identity sign in (A) above. For while both expressions give us a way of identifying the same point, the procedure for identifying the point given by the expression on the left side (find the point of intersection of two lines, a and b) is different from the procedure given by the expression on the right side (find the point of intersection of b and c). For (A) to be true, the two procedures must identify the same object. Thus it should be no surprise that (A) must be established by a substantive geometric proof. In contrast, (B) follows almost immediately from the laws of logic. This should not be particularly surprising, given that (B) offers us only one way to identify the object. (B) just tells us that the point of intersection of two lines is the same as the point of intersection of the same two lines. I have characterized Frege’s problem with identity as epistemological, in his sense. That is, that what is of concern is not the status of an individual’s claims to knowledge but the sort of support required to justify a truth. This is somewhat at odds with the widely accepted views that the difference between (A) and (B) is to be understood as a difference in information content⁵⁵ or a difference in “cognitive value.” The latter locution, in particular, suggests that Frege is talking about mental processes or about what significance a truth has for an individual. However, a closer look at the text should make it clear that his concern is epistemological in

⁵⁴ It is also important to note that the expression “mode of presentation” [Art des Gegebenseins] appears only rarely in Frege’s writings. In particular, it does not appear in other places in which he introduces the notion of Sinn, e.g., in Function and Concept, Basic Laws, or Logical Investigations. ⁵⁵ See Salmon (1986), p. 12.

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the sense described earlier and, indeed, that the translation of Erkenntniswert by “cognitive value” is misleading. For, Frege cashes out Erkenntniswert in terms of the analytic/synthetic and a priori/a posteriori distinctions, distinctions that concern what is required for the proof or justification of a truth.⁵⁶ For example, in explaining why the view that identity statements are about names seems plausible he writes, The reasons which seem to favour this are the following: a = a and a = b are obviously statements of differing cognitive value [Erkenntniswerte]; a=a holds a priori and, according to Kant, is to be labelled analytic, while statements of the form a = b often contain very valuable extensions of our knowledge and cannot always be established a priori.⁵⁷

This is in line with Frege’s other uses of Erkenntnis and its cognates. In Basic Laws, for example, he writes that gapless proofs are necessary for us to get insight into the epistemological nature of a law [erkenntnisstheoretische Natur eines Gesetzes]. And this connection of gapless proof with Erkenntnis or Erkenntnistheorie appears in writings throughout most of his career—from Begriffsschrift to the notes titled “Sources of Knowledge [Erkenntnisquellen] of mathematics and the mathematical natural sciences” and “A new attempt at a foundation for Arithmetic,” written shortly before his death.⁵⁸ Thus the value in question is value that is of interest for epistemology in Frege’s sense; that is, of what sort of support is required to establish a truth. A more accurate, if less colloquial, translation would be “value for knowledge.” Indeed, with the exception of sentences appearing in “On Sinn and Bedeutung,” sentences of Frege’s in which both the terms Erkenntnis and Wert appear are typically translated by sentences in which expressions like “value for knowledge” appear. This is the translation I will use in what follows. Let us return now to the way in which “On Sinn and Bedeutung” offers a solution to the identity problems. First, as we noted in section IIIa there is a problem with the Begriffsschrift view, on which an identity statement tells us that two expressions have the same conceptual content. The introduction, in “On Sinn and Bedeutung” of two distinct kinds of content associated with an expression solves this problem. It is now possible to say that the truth of the identity statement requires the expressions flanking the identity sign to name the same thing, but it does not require the expressions to have the same Sinn. For example, while the expressions “the Morning Star” and “the Evening Star” do not have the same Sinn they do name the same thing. Hence (1)

The Morning Star is identical to the Evening Star

⁵⁶ See, e.g., FA, §§3, 17, 88. ⁵⁷ SB, p. 25. ⁵⁸ See, e.g., BS preface (Erkenntnis), FA, p. xi (Erkenntnistheorie).

 ’       /  

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is true. Given that (1) is true if and only if (2)

“the Morning Star” and “the Evening Star” name the same thing,

one might be tempted to take (2) as a paraphrase or analysis of the content of (1). This is the view, however, that Frege criticizes at the outset of “On Sinn and Bedeutung.” What replaces this view? Frege’s new position is simply the straightforward view that identity statements are about objects—it is, in fact, about the objects named on either side of the identity sign. Consequently, there is no ambiguity problem. The ambiguity problem arises from the view that, when names appear flanking the identity sign they name themselves rather than whatever they customarily name. But on Frege’s new view, when an expression appears in an identity statement there is no change in what it designates. For example, “the Morning Star” designates the same thing, when it appears the sentence “the Morning Star is identical to the Evening Star,” that it designates in the sentence “the Morning Star is a planet.” It is also worth noting that Frege’s new view is not that the identity statement tells us about the Sinn of the expressions flanking the identity sign. To say that the Morning Star is identical to the Evening Star is no more to make a claim about what certain expressions express than it is to make a claim about the expressions themselves. The content of this statement belongs to astronomy—not to the topic of the Sinn of a linguistic expression. But the Sinn is what is expressed. It is via the recognition of Sinn as a part of the content of linguistic expressions that we can see that what is expressed by “The Morning Star is identical to the Evening Star” differs from what is expressed by “The Morning Star is identical to the Morning Star.” Given that “On Sinn and Bedeutung” does provide a solution to a problem about the views of identity in Begriffsschrift, pace Salmon,⁵⁹ it seems perfectly reasonable to take Frege at his word when he tells us that the purpose of “On Sinn and Bedeutung” is to solve a problem with identity. Moreover, the problem with identity is a problem from the point of view of the role Frege’s logic is to play in his epistemological project. It is not a problem from the point of view of what is required for a theory of meaning for the simple reason that, as we saw in Chapter 1, there is no evidence that Frege was interested in the theory of meaning. This is the first of the puzzles I promised to address in this chapter. The second puzzle is why Frege assimilates sentences to proper names.

⁵⁹ See Chapter 1, section Ib.

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V. Functions, Arguments, and Objects Before we turn to the assimilation of sentences to proper names, it will help to think a bit about Frege’s notion of a proper name. It should not be surprising that Frege does not talk about proper names in Begriffsschrift. For a proper name, as Frege understands this notion, is an expression that names an object. In Begriffsschrift, the notion of objecthood is not explicitly introduced; he writes about functions and arguments, not functions and objects. And, as we saw earlier, the terms “argument” and “object” are not interchangeable for Frege. For his logic (both early and late) is second-order. Thus, not all arguments are objects. An object is the kind of thing that can be the argument of a first-level function, arguments of second-level functions are not objects; they are, rather, first-level functions. One might suspect that Frege simply did not yet have a notion of objecthood when he was writing Begriffsschrift. After all, according to Frege’s official definition of functions as the invariable part of an expression, any expression that can be a constituent of a larger expression can be viewed as a function. Thus, in the sentence “Hydrogen is lighter than carbon dioxide,” we can take “Hydrogen” to be, not the argument, but the function. But it is important to remember that while, at this point, he does not explicitly distinguish object-names from functionnames, the first version of his new logic nonetheless depends on our recognition of these different levels. Although we can take “hydrogen” to be a function, it does not follow that, say, “carbon dioxide” can be used as the argument of this function. The argument of the function “hydrogen” must, in fact, be a first-level function. It is clear (and not surprising) that Frege recognized the distinction between objects and functions—as one would expect, he never replaces a first-level function sign with an object-expression or vice versa. Even if this is not made explicit in Begriffsschrift, the workings of the new logic require us to distinguish between object-expressions, first-level function expressions, and second-level function expressions. It is also important to note that our understanding of Frege’s logic depends on our not conflating grammatical and logical classifications. Frege is explicit about this, when he writes, If we compare the two propositions “The number 20 can be represented as the sum of four squares” and “Every positive integer can be represented as the sum of four squares”, it seems to be possible to regard “being representable as the sum of four squares” as a function that in one case has the argument “the number 20” and in the other “every positive integer”. We see that this view is mistaken if we observe that “the number 20” and “every positive integer” are not concepts of the same rank. What is asserted of the number 20 cannot be asserted in the same sense of “every positive integer”, though under some circumstances it can be

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asserted of every positive integer. The expression “every positive integer” does not, as does “the number 20, by itself yield an independent idea but acquires a meaning only from the context of the sentence.⁶⁰

If we analyze the two statements mentioned in the above into subject and predicate, the only difference we can recognize is that they have different subjects. But in Frege’s logical language, as in contemporary logical languages, the analyses of these two sentences are very different. Leaving aside whatever complexity the predicate has, we have, in one case, an analysis that would be represented, today, by something like “St” and, in the other, something like “(x)(Px ! Sx).” It is not just that the sentences have different subjects. The contribution that “every positive integer” makes to the content is more complex than its simply playing the role of a subject. And this complexity has significance for the roles the two sentences can play in inferences. In particular, then, although the subjects of the two sentences can occupy the same slot in a natural language sentence, they could not occupy the same slot in a logical language that satisfies Frege’s demands. This is why Frege objects to the subject/ predicate analysis of “every number is odd.” Although the notion of objecthood is not mentioned in Begriffsschrift, it is mentioned explicitly very shortly after the publication of Begriffsschrift. One mention appears in his 1882 article, “On the Scientific justification of the Begriffsschrift,” where he writes, one frequently recurring phenomenon may be mentioned here: the same word may serve to designate a concept and a single object which falls under that concept. Generally, no strong distinction is made between concept and individual. “The horse” can denote a single creature; it can also denote the species, as in the sentence: “The horse is an herbivorous animal.” Finally, horse can denote a concept, as in the sentence: “This is a horse.”⁶¹

This passage also contains another term that—oddly, given the title of Begriffsschrift,—appears only rarely in that work. This is the term “concept” (Begriff) which, Frege indicates in the above passage, is a term for what is denoted by a predicate and which he uses for this purpose from 1881 on. During the period from 1881 to 1891, there is little discussion of how the notion of concept is to be fit in with his function/argument analyses. Frege deals with this explicitly and in detail in Function and Concept (1891), where he first states the view that a concept is a function all of whose values are truth-values. In the remainder of my discussion of Begriffsschrift, I will follow Frege’s practice of avoiding the use of the term “concept.” ⁶⁰ BS, §9.

⁶¹ OSJ, p. 50/CN, p. 84.

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VI. Frege’s View of Sentences as Truth-value Names VIa. Sentences in Begriffsschrift Bearing all this in mind, let us consider the role of sentences (or expressions with judgeable content) in Begriffsschrift. Frege is explicit about one difference in role between sentences and other expressions in his logical language. In order to understand this difference, we need to look at some of the differences between Frege’s logical language and contemporary logical notations. Two of Frege’s basic logical symbols, his judgment stroke and his content stroke, have no analogue in our contemporary logical notations. The judgment stroke is a small vertical line and the content stroke is a horizontal line (which can be of various lengths, depending on context). Both symbols must be used in order to express a judgment in Begriffsschrift. Supposing, for instance, that “A” is a Begriffsschrift sentence, then if we wish to assert it in the context of an inference, we would write,

⊢A This symbol can be prefixed only to an expression that has judgeable content. Part of Frege’s point in introducing this notation is to draw our attention to the fact that not every sentence that appears in a Begriffsschrift proof is asserted in the proof. For example, were Frege to give a logical proof that 0 is not equal to 1 in his language, the statement “0≡1” would appear in that proof, but it would not be asserted. The statements that are asserted are either premises (the initial lines in the proof) or statements that are inferred from earlier lines in the proof. As Frege also points out, not every expression expresses a content that can be judged. To write

⊢ house would seem to make very little sense. Thus Frege restricts the use of the content stroke: it can be prefixed only to an expression whose content is judgeable (that is, a sentence). Officially, then, sentences and object-names play different roles in this first version of Frege’s logical language. And, if we consider only the features of the language just mentioned, it may seem that the difference is the same as the difference between sentences and object-names in natural language or in today’s standard logical notations for first-order logic. But this is not true. For there are no contexts in either natural language or standard logical notation that can be occupied either by an object-name or by a sentence. In contrast, the Begriffsschrift identity symbol (the triple bar) can be flanked either by sentential

 ’       /  

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expressions or by object-names.⁶² Thus there is a sense in which the feature of the assimilation of sentences to proper names that Dummett takes to be so objectionable—that, on this view, sentences can be put in proper name slots— already appears in the first version of Begriffsschrift. It is not difficult to see why Frege would be pushed to take sentences to be object-names. This follows almost immediately from the ideas that sentences should be given function/argument analyses and that the application of a function to an argument must give us a value. How does this work in Begriffsschrift? The official view of Begriffsschrift is that both function and argument are linguistic expressions. His analyses are analyses of a sentence into constituent linguistic expressions. One of the analyses of “Cato killed Cato” that he offers in Begriffsschrift is into two constituents: the function “to kill Cato” and the argument “Cato.”⁶³ Given that function and argument are linguistic expressions, on the Begriffsschrift view, it would seem that the only option is to taken the value of the function on the argument to be a linguistic expression. Which linguistic expression? The natural answer is that it is the complete sentence formed by putting the argument in the function slot. In this case, the value would be the sentence, “Cato killed Cato,” an answer that, as we have seen, is problematic. The new view of functions is introduced in Function and Concept, a lecture that Frege presented to the Jena Medical and Scientific Society and subsequently published. He begins by mentioning an earlier lecture to the same society, in which he introduced “the symbolic system I entitled Begriffsschrift.”⁶⁴ The purpose of Function and Concept he goes on to say, is to tell his audience about “some supplementations and new conceptions, whose necessity has occurred to me since then.”⁶⁵ In this lecture, Frege officially abandons the idea that function and argument are linguistic expressions. We can still divide a sentence into the constituents “to kill Cato” and “Cato.” But we are talking, not about the linguistic expressions, but about a function and argument that are named by the expressions. Once Frege abandons the view that functions are linguistic expressions, there is no longer any reason to take the value of a concept to be a linguistic expression (i.e., a sentence). The sentence should be, not the value, of the function on the argument, but a name of that value. This already makes sentences into object-names. For example, functions must be single-valued and it must be possible to say that a particular function is single-valued. But, if we are to be ⁶² For example, in the applications of Law 52 [c ≡ d ! (f(c) ! f(d)] in the proofs of propositions 75, 89, and 105, the object-letters “c” and “d” are replaced by sentences. On the other hand, proposition 100 tells us that if z is identical with or follows x in the f-sequence, and z does not follow x in the f sequence, then z ≡ x. The sequences for which these theorems are meant to hold include the natural number sequence. Hence, the applications of this proposition will include propositions in which the identity sign is flanked by object names. ⁶³ Although the words “to kill Cato” do not appear in the actual sentence, it is likely that Frege has in mind a Begriffsschrift rendering of the sentence. See section IIb above. ⁶⁴ FC, p. 1. ⁶⁵ FC, p. 1.

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able to say that a particular concept is single-valued, we must be able to put sentences on either side of the identity sign. Immediately after introducing the new notion of function, in Function and Concept, and using this to explain why predicates are names of functions, Frege introduces the truth-values (the True and the False) as the values of these functions. There are several reasons for taking truth-values to be the values of these functions, some of which will be easier to see when we look at Frege’s new notion of function, in the next chapter. But one important reason for taking sentences to name truth-values should be clear, given the demands of Frege’s logic. For Frege does not change his view about the significance that an assertion of an identity has in a proof. As he has told us in Begriffsschrift, the significance of “A = B” is “that we can everywhere put B for A and conversely”.⁶⁶ And it is sameness of truth-value that licenses such an intersubstitution of sentences. On the new view, truth-values are objects and sentences are object-names that name truth-values. Although there is some defense of this view in Function and Concept, he does raise a worry in a footnote that “this way of putting it may at first seem arbitrary and artificial” and thus, he continues, “it would be desirable to establish my view by going further into the matter.”⁶⁷ This he proposes to do in his essay “On Sinn and Bedeutung.” Let us see how this works.

VIb. Sentences in “On Sinn and Bedeutung” Early in “On Sinn and Bedeutung” Frege offers a brief argument designed to counter the charge that the view of sentences as names of truth-values is arbitrary. There is no difficulty in seeing that sentences contain thoughts. But if a sentence has Bedeutung (names something), he argues, it cannot be the sense of the sentence. Do sentences have Bedeutung? Frege’s initial argument that (some) sentences do comes from his examination of the Bedeutung of subsentential constituents of a sentence. Why, he asks, is the thought not enough for us? And his answer is, “Because, and to the extent that, we are concerned with its truthvalue.”⁶⁸ One might expect Frege to stop there or to give additional, straightforward, arguments that the Bedeutung of a sentence must be its truth-value. But this is not how he proceeds. Rather, he introduces two tests of the view that the truth-value of a sentence is its Bedeutung. Indeed, these two tests occupy nearly thirteen of the twenty-five pages of this essay. The first test is to consider whether the truth-value of a sentence remains unchanged when a constituent of the sentence is replaced by another expression with the same Bedeutung. The

⁶⁶ BS, §8.

⁶⁷ FC, p. 14.

⁶⁸ SB, p. 33.

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second is to consider whether this holds if the expression that is replaced is itself a sentence and it is replaced by another sentence with the same truth-value. The thirteen pages in question are, of course, the pages in which Frege gives analyses of a variety of natural language sentences. What is important to see, however, is that all of these analyses are in service of the test for whether sentences are truth-value names. And, indeed, one result is that Frege retreats from his initial claim that all sentences are truth-value names. His ultimate position is that all sentences that have thoughts as their senses are truth-value names. One interesting features of these tests is that they do not address the issue that Dummett found so troubling: there is no attention at all to the issue of whether sentences do (or should) play the same role in a version of natural language as proper names. Is the explanation that, while Frege means to be giving a partial theory of natural language, this issue simply does not occur to him? There is, after all, an important difference between the discussions of “On Sinn and Bedeutung” and the discussions of Function and Concept—a difference that might lead us to think that the topic of “On Sinn and Bedeutung” is natural language (or some version of it). In Function and Concept there is no question that the language of interest is Begriffsschrift, his logical language. And, although he describes how the language works, that is not the only reason. Most of his examples are either actual expressions of Begriffsschrift or natural language statements for which he intends, very shortly, to introduce Begriffsschrift translations.⁶⁹ But in “On Sinn and Bedeutung” virtually all of the examples are of natural language sentences. And these examples are, for the most part, of natural language sentences whose translation has no role to play in his project of showing that the truths of arithmetic belong to logic. Moreover, they are, for the most part, sentences that—even after we have seen Frege’s analyses—do not have obvious translations into Begriffsschrift. Frege does not have any plan, for instance, to translate sentences involving indirect speech into Begriffsschrift.⁷⁰ Why, then, does he devote so much of “On Sinn and Bedeutung” to discussion of indirect speech? If the answer is not that he has an interest in natural language, what could it be? One clue is that indirect speech is mentioned in the preface to the first volume of Basic Laws. Frege describes his view that a sentence is associated both with a Sinn (the thought it expresses) and a Bedeutung (a truth-value) and he refers his readers to “On Sinn and Bedeutung” for a defense of this view. He then goes on to say,

⁶⁹ There are exceptions, however. He also writes about analyses of “Caesar conquered Gaul” and “the capital of the German Empire”, FC, pp. 17–18. ⁷⁰ The suggestion of underlining the expressions in question that he makes in the letter to Russell of December 28, 1902 (WB p. 236/PMC p. 152) is, after all, a suggestion for how to avoid confusion in natural language (and he uses it for precisely that purpose in the letter). But he not only fails to suggest how translations into Begriffsschrift will go, he does not have a plan for how to introduce such translations.

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Here, it might merely be mentioned that only in this way can indirect speech be accounted for correctly. For in indirect speech, the thought, which is normally the sense of the proposition, becomes its reference.⁷¹

This is surprising because Begriffsschrift, the only language of interest to Frege in Basic Laws, has no indirect speech locutions. The few discussions of ordinary language in Basic Laws are comments about how expressions of ordinary language are best rendered in Begriffsschrift. Nor does Frege offer any strategy for expressing in Begriffsschrift what is expressed by using indirect speech in natural language. Why, then, should Frege mention indirect speech in the preface to Basic Laws? It is important, here, to recall the purpose of Begriffsschrift. Frege wants to convince his readers that it is possible to use his logical system to give a correct evaluation of any inference on any topic. Thus his logic must answer to the logical standards of evaluation that we already recognize as correct; to our antecedent evaluations of arguments. And our antecedent evaluations are, of course, carried out in natural language. The problem with indirect speech is that, as we saw in Chapter 1, Frege’s standards of evaluation appear to give us the wrong verdicts.⁷² Consider, again, the significance identity statements have for inference: “that we can everywhere put B for A and conversely.”⁷³ If we take sentences to be names of truth-values, then all true (false) sentences name the same thing. We have, among others, the following identity: (1)

The Morning Star is a star ≡ the Evening Star is a star

This identity licenses us to infer, for example, from (2)

It is not the case that the Morning Star is a star

that (3)

It is not the case that the Evening Star is a star.

This is the correct verdict. But the very same law along with the very same identity appears to license us to infer from (4)

Alice believes that the Morning Star is a star

that (5)

Alice believes that the Evening Star is a star.

⁷¹ BLA vol. i, p. x.

⁷² Chapter 1, section IVe.

⁷³ BS, 8.

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And this, of course, is the wrong verdict. It appears, then, that we have a counter-example to Law 52, Frege’s version of Leibniz’ law. What has gone wrong? It could be that sentences really do not name truth-values. Alternatively, it could be that this illustrates one of the logical defects of natural language. Frege tries to show us that the problem stems from a logical defect of natural language, a defect that is manifest when we consider sentences involving indirect speech. The sentence (M) “The Morning Star is a star,” for instance, is a constituent of both (1) and (2) above. In these (direct speech) contexts (M) names a truth-value (the False). But, while (M) is also a constituent of (4), it appears there in indirect speech. Thus, on Frege’s analysis of (4), (M) names, not a truth-value but its customary Sinn, which is to say the thought it customarily expresses. That is, (M) names different things in different contexts. This is an example of the ambiguity that runs through natural language and ambiguity is a logical defect. Moreover, this ambiguity shows us why we seem to get the wrong verdict in the second inference. (M) names something in (1) that it does not name in (4). Thus, in spite of the common linguistic constituent of (1) and (4), (1) does not license us to infer anything from (4). This is an example of why it is important for Frege to discuss natural language. For we do make and evaluate inferences from statements that involve indirect speech. Were it impossible, in principle, for Frege’s logical system to give us the correct evaluations of these inferences, then this would be a defect, not of natural language, but of the logical system. What we have seen is that the indirect speech analysis of (4) and (5) above explains what would otherwise seem to be a counterexample to the possibility of using Frege’s logical system for the evaluation of inferences on any topic. Should we wish to expand Begriffsschrift in order to evaluate inferences involving indirect speech (for example, should we wish to expand Begriffsschrift in order to evaluate inferences belonging to psychology), this analysis illustrates some of the constraints that must be satisfied by the expanded language. One might assume that, were this Frege’s motivation, he would support his analysis of indirect speech with a discussion of its use in the evaluation of inferences. And he does not do this. But this assumption is based on a mistaken idea about Frege’s practice. Consider his discussions of the analyses that appear in Begriffsschrift. In this work he explicitly claims that the only content that he is interested in is content that has significance for inference. Yet he does not support the analyses that he offers in Begriffsschrift with discussions of the evaluation of inferences. For example, when he introduces his function/argument analysis in §9 of Begriffsschrift he does not justify it by talking about how sentences function in inference. Similarly, there is no talk of inference when he discusses the content of: If the moon is in quadrature with the sun, the moon appears as a semicircle.

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He writes that this natural language statement cannot be expressed in Begriffsschrift by a statement of the form “A ! B”, for the natural language statement expresses a causal connection.⁷⁴ Thus, while the content that is of import here is content that has significance for inference, Frege does not undertake to defend his analyses of the content of a statement or group of statements with a discussion of how it actually figures in inferences. I have been arguing that one purpose of “On Sinn and Bedeutung” is to give analyses of sentences belonging to a significant part of natural language: the part of natural language in which we express and evaluate inferences. Suppose we agree that Frege’s interest in natural language is limited to the fragment of natural language in which we express and evaluate inference. How is this different from the Standard Interpretation? Why is this not just to say that Frege is giving a theory of the workings of this particular fragment of natural language? The answer is simple. On the interpretation I have offered, as opposed to the Standard Interpretation, the workings of natural language, on their own, provide no grounds for criticizing Frege’s claim that sentences are names of objects. What is at stake here is nothing more nor less than the correct evaluation of inferences. An account that yields the correct evaluations of inferences is a perfectly good account—regardless of whether it accords with other aspects of our actual (or proposed) use of natural language expressions. We have seen in this chapter some of the problems that arise on the official Begriffsschrift view of functions: the view on which a function is simply an expression that can be a component of other expressions. As we have seen, on the new understanding of the notion of function, which Frege introduces in Function and Object, a function is not an expression but something that the expression names. And a function needs to have a value for each argument. If predicates are to be function-names, the values of the functions in question must be truth-values. And thus sentences must be names of truth-values. This also requires a new account of identity and the recognition of a distinction between two sorts of content an expression can have (Sinn and Bedeutung). As I have also mentioned in this chapter, on Frege’s view truthvalues must be objects. Why? As we have seen, in Frege’s logical system sentences can appear flanking the identity sign and identity statements are statements about objects. But Frege says very little about this in “On Sinn and Bedeutung.” He writes only, These two objects are recognized, if only implicitly, by everybody who judges something to be true—and so even by a sceptic. The designation of

⁷⁴ BS, §5.

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the truth-values as objects may appear to be an arbitrary fancy or perhaps a mere play upon words, from which no profound consequences could be drawn. What I am calling an object can be more exactly discussed only in connexion with concept and relation. I will reserve this for another article.⁷⁵

We will turn, in Chapter 3, to the new notions of function, concept, and object.

⁷⁵ SB, p. 34. The article to which Frege refers us is, of course, “On Concept and Object”.

PART II

METAPHYSICS AND THE STANDARD INTERPRETATION

3 Metaphysics and the Standard Interpretation As we saw in Chapter 2, Frege’s function/argument analysis of statements requires us to recognize predicates as names for concepts—functions that have truthvalues as values. We can see from this assimilation of concepts to functions that Frege’s notion of function is not a familiar mathematical notion. But what, exactly, is this notion? It may seem that once the view of functions as linguistic expressions is abandoned, functions must occupy not only a special logical niche but also a special metaphysical niche. Tyler Burge writes “Logic and ontology are mutually entangled in Frege.”¹ And “Frege’s most fundamental ontological categories (function and object) are logical categories.”² In what way are the categories of function and object ontological categories? The answer may seem simple: we can see, for instance, that functions are not spatiotemporal entities and that they are not objects. On the Standard Interpretation, Frege’s account of the truths of arithmetic are undergirded by metaphysical views: he is a Platonist. The support for this position is to be found in a series of his statements: Truths of arithmetic, he writes, are eternally true; they are independent of us and our judgments and thoughts. Although numbers are not spatio-temporal objects, not every object has a place. He writes that there is a “third realm” containing objects that cannot be perceived by the senses but are not ideas.³ But does this suffice to show that Frege’s view is metaphysical? Burge acknowledges that it does not. He writes, Many of these things might be maintained by someone who was not a Platonist. One might make the remarks about imperceptibility, non-spatiality, atemporality, and causal inertness, if one glossed them as part of a practical recommendation or stipulation for a theoretical framework, having no cognitive import—or as otherwise not being theoretical claims or claims of reason. . . . Or one might have some other basis for qualifying these remarks, reading them as “nonmetaphysical” or as lacking their apparent ontological import. . . . Platonism has

¹ Burge (1992), p. 644n. ² Burge (1992), p. 644n. ³ It is worth noting, however, that the claim about the third realm appears only once in Frege’s writings—in a paper that was meant as the first chapter of a logic textbook.

Taking Frege at his Word. Joan Weiner, Oxford University Press (2020). © Joan Weiner. DOI: 10.1093/oso/9780198865476.003.0003

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How metaphysical, then, is Frege’s understanding of the notion of function? In order to answer this question, let us begin with Frege’s introduction of his new notion of function and his explanation of the notion of concept as a special kind of function in Function and Concept. We will begin with the official, “logical” account and then consider whether Frege’s notions of function and object are ontological, as well as logical categories.

I. Frege’s Logical Notion of Function Ia. Familiar Function-expressions and Functions As we saw in Chapter 2, the account of function in Begriffsschrift does not work—in particular, functions cannot be linguistic expressions. This is something Frege himself came to see and, consequently, the account of the notion of function in the second version of his logic is very different from the Begriffsschrift account. The new account of functions is first set out, along with the new version of his logic, in Function and Concept. A function will have to be, not an expression but, rather, what we are talking about when we use a functionexpression. But what is a function-expression? Frege begins by asking us to consider the three expressions, “2.1³ + 1,” “2.4³ + 4,” and “2.5³ + 5.” These expressions seem to have a constituent in common—a constituent expression that refers to a particular function of arithmetic. In fact, each expression on the list appears to be a name of the value that function has for a particular argument. What is this function? Our usual method for naming a function, is to use a variable. That is, we might say that the function in question is: 2.x³ + x. But Frege disagrees. He writes, From this we may discern that it is the common element of these expressions that contains the essential peculiarity of function; i.e. what is present in

‘2:x3 þ x’ over and above the letter ‘x’. We could write this somewhat as follows: ‘2( )³ + ( )’.⁵

Why should “x” not be a part of the name of the function in question?

⁴ Burge (1992), p. 637.

⁵ FC, p. 6, emphasis added.

    

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The first thing to note is that, even if we were to agree that “2.x³ + x” is a name of the function in question, it is not a name that appears in the expressions listed above. For none of these expressions contains the letter “x”. Yet, if we agree that the same function is mentioned in all these expressions, it seems reasonable to suppose that a name for this function is a constituent of each of these expressions. The expressions on the list, then, contain a name of the function and this name does not contain the letter “x.” Frege’s position, however, is stronger than this. On his view, there is no name of this function in which the letter “x” appears. This is not to say that the letter “x” can never appear in a discussion of the function in question. Indeed, it frequently is used to indicate the location of the gaps in the real function-name, the places that require supplementation with an argument expression. But in such uses the letter is not part of the function-name. That is, while expressions that result from putting some variable into the gaps in the function-name—such expressions as “2.x³ + x”—do have a use, their use is not to name a function. Rather than naming a function, the expression “2.x³ + x” indefinitely indicates a number or, to be more specific, it indicates the value of this function for x as argument. It can be used to make general remarks about the values of the function. To see what this comes to, it may help to begin with a sentence in which a function-name, but no variable appears. Consider, ðAÞ

2:13 þ 1 ¼ 3:

There are, of course, several ways of understanding this; there are several distinct function-names in (A). But let us concentrate on the function that is common to the list with which we began. (A) tells us that the value of that function, for 1 as argument, is 3. If we replace “1” with “x”, we get ðBÞ

2:x3 þ x ¼ 3:

We do not mean, thereby to say that the function in question is identical to the number 3. Rather, we mean to say that the value of that function, for x as argument, is the number 3. Should this appear in a proof, we could deduce that x = 1. That is, (B) does not tell us just about a function, “x” plays a role other than that of marking the empty places in the function-expression. In this case the role is very like that of a number name. Of course, there are sentences in which “2.x³ + x” appears and in which “x” appears, not as a number name, but as a variable designed to express generality. For example, ðCÞ

2:x3 þ x ¼ ð2:x2 þ 1Þx:

One might think that, in the context of this discussion, (C) is meant to express an identity between functions and that “x” is merely indicating where the argument

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places of the function lie. But there are two problems with this. First, (C) is just shorthand for a universally quantified statement, that is, ðDÞ

ðxÞð2:x3 þ x ¼ ð2:x2 þ 1ÞxÞ

A variable that expresses generality, as “x” does in this case, is not a mere placeholder that is used to form a function-name. Second, we can see that “x” does more than indicate argument places because the following statement does not express identity between functions: ðEÞ 2:x3 þ x ¼ ð2:y2 þ 1Þy: This is not to say that we can’t, with appropriate commentary, use (E) in a statement about sameness of functions. But the most appropriate commentary is to say that what is named on the left side of the equation “2.x³ + x = (2.² + 1)y” is a function with x as argument and on the right side what is named is the same function with y as argument. That is, the most appropriate commentary apparently involves taking “x” and “y” not to be parts of function-names. It may seem, then, that there is a way of naming functions without using variables standing in the argument spaces. After all, Frege says that a function expression has gaps. Why not simply leave gaps in the name as in the second expression that Frege displays? That is, why not say—as Frege suggests in the passage set off above—that the real function-name is “2( )³ + ( )”? The problem with this strategy is that the expression in question is not really an expression with gaps. It is not, in particular, the expression that results when the argument names are removed from “2.x³ + x”. After all, “2.x³ + x” contains no parentheses. The parentheses and even the blank spaces that appear in the second expression are simply different typographical symbols that are used to indicate, just as the use of the letter “x” does in the first expression, what sort of supplementation is required. Parentheses, and even blank spaces, are characters, not gaps. Furthermore, this way of indicating what supplementation is required is not as good as the use of variables. If we use the expression “2( )³ + ( )” we cannot distinguish the function in which both sets of parentheses must be filled with the same argument from the function in which the two sets can be filled with different arguments. That is, this strategy for naming functions does not allow us to distinguish functions of one argument from functions of two arguments. So how can we use names to talk about functions? Frege introduces another type of expression in Basic Laws. Instead of using a variable or parentheses with a blank space to mark the supplementation that is required, he fills the gap with a lower-case Greek letter. For example, instead of using “2.x³ + x” or “2( )³ +( )” as a functionname, he uses “2ξ³+ξ.” This has the advantage that we can indicate which slots need to be filled with the same argument (since different lower-case Greek letters can fill different slots). It also has the advantage that the symbols that are used to fill the

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slots cannot be used to express generality. Thus, we do not have the kind of problem with, say, “2.ξ³+ξ = (2.ζ²+1).ζ” that we have with “2.x³ + x = 2.y³ + y”. Frege uses such expressions extensively in Part I (Exposition of the Concept-script) of volume i of Basic Laws.⁶ But, as we saw earlier, on Frege’s view there is something wrong with calling such an expression a function-name. For to paraphrase his earlier statement, “2.ξ³+ ξ” is not a function-name. Rather, the function-name is what is present in “2.ξ³+ ξ” over and above the letter “ξ.” He writes, The nature of function lies therefore in that part of the expression that is present without the ‘x’. The expression of a function is in need of completion, unsaturated. The letter ‘x’ serves only to hold open places for a number-sign that is to complete the expression, and so marks the special kind of need for completion that constitutes the peculiar nature of the function just designated. In the sequel, instead of ‘x’ the letter ‘ξ’ will be used for this purpose.⁷

He goes to claim that, just as the letter “x” is not part of the function-name, the expression “ξ” is not part of the function-name. Indeed, he writes that ‘ξ’ itself will never occur in the concept-script developments; I will only use it in the exposition of the concept-script and in elucidation.⁸

And this holds for his later writings as well. This is important because real function-names must be used in the logical theory—a theory that, on Frege’s view, must be expressed in the concept-script (Begriffsschrift). We will see shortly the significance of Frege’s view that the exposition of the concept-script and elucidation belong, not to the logical theory, but to its propaedeutic. There is something else that should be evident from the above discussion: any time we try to use an expression on its own as a function-name we need to add some sort of commentary. What gives us the function is not the expression alone, but the expression plus the commentary. On Frege’s view there is no way simply to write out a function-expression on its own. Only in combination with argument expressions (either variables, lower-case Greek letters, or object-names) can function-expressions actually be written out. This is a sense in which a function-expression is incomplete (or, as he also says, in need of supplementation, or unsaturated). And these incomplete expressions are used to talk of functions, which are incomplete. When a function-expression is completed with a name or names—for example, when we get, “2.1³ + 1” by completing the original function-

⁶ This notation is also used a few times in BLA vol. ii. See, for instance, the heading of §§193–5, in which the upper limit in a positival class is defined. But what is interesting to note is that the notation is used only in the heading of the sections. It is not used in the actual definitions. ⁷ BLA vol. i, pp. 5–6. ⁸ BLA vol. i, p. 6.

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name with the numeral “1”—the result is a name of something complete, an object. Unsaturatedness, Frege says, is the “essence of functions.”⁹ Thus, it is also important to note that, in the exposition of the concept-script, Frege frequently uses such expressions as “the function ξ² = 4.” On Frege’s view, such an expression, as the definite article indicates, is not unsaturated: it is an object-name, not a function-name. And Frege’s objections to taking such expressions as “the concept horse” to be concept-expressions, which are discussed in section IIa below, also apply to taking such expression as “the function ξ² = 4” to be function-names.

Ib. Unsaturatedness and some Unfamiliar Function-names and Functions We saw in Chapter 2 that Frege’s use of function/argument analysis as a basis for his logical system requires us to think of predicates as function-names. But, as we also saw, Frege’s understanding of functions in Begriffsschrift was problematic. As we have just seen, the new conception of function, which Frege introduces in Function and Concept, is tied to the notion of unsaturatedness. But how is this to be connected with the notion of function that underlies Frege’s logical system? Frege tells us that unsaturatedness can be found, not only in the complex number names formed using familiar function-expressions, but in many other expressions as well. In addition to his list of complex number names (that is, “2.1³ + 1”, “2.4³ + 4”, and “2.5³ + 5”), he gives us another list of expressions with a common, unsaturated, constituent. The second list is: (–1)² = 1 0² = 1 1² = 1 2² = 1. What is it that is common to the expressions on the above list? If we remove from these lines, respectively, “–1,” “0,” the first “1,” and “2”, we get, it seems, the same predicate. As with the more familiar functions of arithmetic, the easiest way to indicate the common constituent is to replace the relevant numerals by a variable: that is, “x² = 1”. And, as before, if we do this, we have to recognize that the variable “x” is not part of the unsaturated expression. The same thing could be indicated by using a different variable or by writing, “( )² = 1”. If, as Frege claims, ⁹ In fact since, as we have seen, the unsaturatedness is found in the linguistic expressions themselves, strictly speaking, it is not just that certain expressions name functions, these expressions also are functions. But this is not to say that when we use the expressions in question, the expressions are the things we are talking about.

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unsaturatedness is the essence of functions, then sentences as well as composite number names have functions as constituents. But is this sufficient to classify predicates as a kind of function-name? In order to define a function, we need to indicate what values the function has for each argument. When we complete one of the familiar function-names of arithmetic the resulting expression is a composite number name—that is the name of the value of the function in question on the argument whose name appears in the argument slot. But when a predicate is completed by an argument-name we get, not a composite name, but a sentence. Moreover, while predicates (that is, concept-names) can be recognized as incomplete in the way that ordinary function-names can, defining a concept is different from defining a function. To define a concept is not to give a value for each object but, rather, to indicate for each object whether or not it falls under the concept. If, however, we want a concept to give us a value for each object, as we saw in the previous chapter, it is far from clear what these values could be. However, there is something that a definition of a concept will give us for each object: either the answer “true” (for the objects that fall under it) or the answer “false” (for those that do not). Frege writes, about the expressions on his second list, Of these equations the first and third are true, the others false. I now say: ‘the value of our function is a truth-value’, and distinguish between the truth-values of what is true and what is false. I call the first, for short, the True; and the second, the False.¹⁰

This is Frege’s first introduction of the idea that a sentence is just a kind of name and it is his first introduction of the kind of thing sentences name, that is, truthvalues. A concept (or what is named by a predicate) is a function all of whose values are truth-values. Frege continues, in this way, to widen the notion of function further. Just as the predicates of arithmetic can be seen as unsaturated, any other predicate can be seen as unsaturated. Just as the sentences of arithmetic that appear on Frege’s second list share a common predicate, so do the following sentences: “The Earth is a planet,” “The Sun is a planet,” “Venus is a planet.” And now that we have a way to identify sentences as names, we can identify the predicate of these sentences as a function that has a value for each argument. On Frege’s analysis, we should take the first and third of these sentences to name the True, and the second to name the False. This is a development of the function/argument analysis of simple predication that is part of the foundation of Frege’s introduction of his new logical language. Having finished his introduction of this notion, Frege notes that his wider version of the mathematical notion of function amounts to admitting “objects without ¹⁰ FC, p. 13.

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restriction as arguments and values of functions” and raises the issue of what an object is. But, he tells us, no definition of objecthood is possible and he writes, Here I can only say briefly: An object is anything that is not a function, so that an expression for it does not contain any empty place.¹¹

Given this explanation of the notion of objecthood, it should not be surprising that he continues, A statement contains no empty place, and therefore we must take what it means to be an object. But what a statement means is a truth-value. Thus the two truthvalues are objects.¹²

It is worth noting that, if the explanation in terms of expressions is all there is to be said about what it is to be an object, then this statement about why truth-values are objects is exactly right. But it may seem that there is something wrong with the idea that this could be all there is to be said about what it is to be an object. Suppose, for example, I were to say that to be a whiffle is to be something whose name begins with ‘a’. One might think that something has gone wrong here. For one thing, we can give the name “Ava” to anything we like—thereby making it into a whiffle. The distinction between whiffles and non-whiffles is entirely unsuitable to play any interesting role in logic, metaphysics, or any other field. But that is no real problem unless I have advertised the significance of this distinction for some field. After all, there is nothing wrong with the predicate “has a name that begins with the letter ‘a’.” If all I want to do is to introduce the term “whiffle”—for whatever strange reason—then there is nothing wrong with what I have just said. What about Frege’s notions of function, concept, and object? These notions, unlike my notion of whiffle, are supposed to be significant. But, before we insist that there is something wrong with simply saying that any complete expression that designates something designates an object, we need to think about what significance Frege takes the function/object distinction to have. For this purpose, it will help to turn to Frege’s response, in “On Concept and Object,” to an objection to his concept/object distinction.

Ic. Concept and Object Although it may seem that Frege first describes his notion of concept in his 1891 publication of Function and Concept, this notion plays an important role in the earlier discussions in Foundations. He is explicit, in Foundations, that no objects ¹¹ FC, p. 18.

¹² FC, p. 18.

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are concepts and he argues, at length, that numbers are objects. In a paper published several years after Foundations appeared, Benno Kerry criticized the Foundations view of concepts.¹³ Frege responds to Kerry in “On Concept and Object.” Frege first tells us that his use of these terms is not Kerry’s and that his use is a logical use, before declining to offer a definition. The notions of concept and object, he tells us, cannot be defined because they are logically simple. All that can be done is to lead the reader, by means of hints, to an understanding of the terms in question. As with the introduction of his notion of function, the hints rely almost exclusively on a discussion of different categories of expression. His first try, on the second page of the article is, A concept (as I understand the word) is predicative.¹ On the other hand, a name of an object, a proper name, is quite incapable of being used as a grammatical predicate. ¹It is, in fact, the meaning [Bedeutung] of a grammatical predicate.¹⁴

This talk of linguistic expressions is not simply an introductory device—a way of speaking that he means to abandon in favor of a better formulation—for he continues to write about concepts in terms of linguistic predicates throughout the article. A number of pages later, he writes, We may say in brief, taking ‘subject’ and ‘predicate’ in the linguistic sense: a concept is what is meant by a predicate [Bedeutung eines Prädikates]; an object is something that can never be the total meaning [Bedeutung] of a predicate, but can be what a subject means [Bedeutung eines Subjekt].¹⁵

These characterizations of concepts in terms of predicates (or, conceptexpressions) provide a response to one of Kerry’s arguments. According to Kerry, Frege must recognize that there are concepts that are also objects. The argument for this is based on the assumption that the following sentence is true: The concept horse is a concept easily attained. It follows from this statement, that the concept horse is a concept. Yet, because the expression “the concept horse” functions as a proper name in that sentence, it is also, given Frege’s criteria, an object. Frege’s response should not be particularly surprising: he agrees that the concept horse is an object but denies that it is a concept. It is not simply that

¹³ Kerry (1887).

¹⁴ CO, p. 193.

¹⁵ CO, p. 198.

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the expression “the concept horse” is the grammatical subject rather than the predicate of the sentence in question. It is that it cannot be the predicate of any sentence. Frege offers a simple criterion for distinguishing complex objectexpressions from concept-expressions: object-expressions make use of definite articles where concept expressions make use of indefinite articles. Thus “the Morning Star” is an object-expression, while “a planet” is a concept-expression. In the sentence (1)

The Morning Star is a planet

a concept, that of being a planet, is predicated of an object, the Morning Star. One might suspect, however, that this simple criterion will not quite work as a criterion for distinguishing objects and concepts. For it may seem that expressions that count, on this criterion, as object-names can also be used as predicates. For example, “the Morning Star” designates the same thing as “Venus.” Thus, if “the Morning Star” is an object-expression, so is “Venus.” And it may seem that, in the sentence (2)

The Morning Star is Venus,

Venus is predicated of the Morning Star. Thus Venus would be both an objectexpression and concept-expression. But this suspicion, Frege tells us, is mistaken. The reason is that the word “is” plays different roles in the two sentences set off above. In the first sentence the word “is” is used merely as a copula—that is, a grammatical device to connect subject and predicate. But in the second, “is” is used to express identity. That is, it is not inappropriate to regard “is” as an abbreviation of “is identical to”: the thought can also be expressed by the sentence “The Morning Star is identical to Venus.” The predicate here is not “Venus” but a complex predicate “identical to Venus.” It is important to see that the point here is a logical rather than a grammatical point. Were we to express (1) in a logical language like Begriffsschrift, we would need a Begriffsschrift expression to play the role of “the Morning Star” and a Begriffsschrift expression to play the role of “a planet.” But “is” plays no logical role and hence there would be no Begriffsschrift expression corresponding to it. In contrast, a Begriffsschrift expression of (2) would have, in addition to expressions playing the roles of “the Morning Star” and “Venus” an occurrence of the identity sign. Another way to think of this is that (1) can be regarded as stating that a particular object falls under the planethood concept. But in (2) what is predicated of the Morning Star is not a single concept that we might call “Venushood.”Rather, what is predicated in (2) is a concept that has two distinct constituents, Venus and the identical to relation. That is, Venus is the meaning

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of only part of the predicate. It is not (and can never be) the total meaning of a predicate. “Venus” and “the Morning Star” are both object-expressions. Now let us return to the sentence with which we began, “the concept horse is a concept easily attained.” Our simple criterion tells us that “the concept horse” is an object-expression, for it begins with a definite article. But what of the other criterion? Can the concept horse be the meaning of a grammatical predicate? We certainly can say, Venus is the concept horse. But, as we saw in the discussion of “Venus,” this is not sufficient to show that the concept horse can be the meaning of a grammatical predicate. In order to determine whether the expression “the concept horse” is the predicate of the sentence in question, we need to determine whether the word “is” functions as a copula or as the “is” of identity. And, on Frege’s view, what is being said is that Venus is identical to the concept horse. That is, in this sentence the concept horse is not being predicated of Venus. To see why Frege holds this view, it will be helpful to consider an alternative view. One might think that this is the wrong way to go about trying to predicate the concept horse of something. The problem, one might think, is that Frege is not appreciating the flexibility of language. For surely, one might think, any expression can be used as a name for anything. How might this be used as a response to Frege? Consider the following alternative view. If we do not insist that “the concept horse” and “a horse” must designate different sorts of things, we can see (one might think) that the expression “the concept horse” designates what is designated by “a horse” in (3)

Venus is a horse.

On this alternative view Kerry is right—it is true that the concept horse is a concept easily attained. This alternative view, however, will not work. For expressions that designate the same thing must be intersubstitutable in inferences. That is, supposing we adopt this view and thus conclude that it is true that (4)

The concept horse is a concept easily attained.

We should then be entitled to substitute, “a horse” for “the concept horse” and infer that, (5)

A horse is a concept easily attained.

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The juxtaposition of (4) and (5) exhibits the flexibility of natural language, for both are meaningful, well-formed sentences. But it also exhibits Frege’s point in distinguishing concept- and object-expressions. For, even if we claim to accept the alternative view, we can see, not only that (4) and (5) are different statements, but also that (5) does not follow from (4). The problem is that to begin a sentence with the expression “a horse” is to say something about horses—either, depending on context, about all horses (as in, “A horse is a mammal”) or about a particular horse (as in, “A horse is in my backyard”). Thus to say that a horse is a concept easily attained is to make a transparently false statement about horses. Indeed, this makes it easier to see that the problem is not grammatical at all. There is no difficulty in using the expression “a horse” as the grammatical subject of a sentence. But this is one of the important reasons for abandoning the subject/ predicate analysis of sentences for the function/argument analysis. The content of a sentence whose subject is “a horse” will be very different from that of the sentence that results by replacing “a horse” with any proper name, including “the concept horse.” The distinction is not grammatical, but logical. Object-names play a different logical role from function-names. In the sentence, The concept horse is a concept easily attained. “the concept horse” functions as an object name. The sentence purports to tell us something about a particular thing, the concept horse; it purports to tell us that this particular thing is a concept easily attained. But if “the concept horse” ’ is replaced by “a horse,” the resulting sentence, A horse is a concept easily attained. does not purport to tell us about a particular thing. Rather, it appears to tell us something general about horses—that they are concepts easily attained. As we saw in Chapter 2, this difference would be evident were we to translate these two natural language sentences into the kind of logical language that Frege introduces. The rules for constructing expressions in Frege’s logically perfect language prohibit us from substituting a concept-expression for an object expression. If we try to make such a substitution the result will not be a legitimate Begriffsschrift expression at all. The first natural language sentence has the form of a predication. Thus, if we want to translate the first sentence into Begriffsschrift, the Begriffsschrift expression that serves as the translation of “the concept horse” will be an object-expression. In contrast, the translation of the second natural language sentence will be a generalization. The expression that serves as the translation of the subject (“a horse”) of the natural language sentence will be a complex expression that includes, not only a concept expression for “a horse” but also a quantifier whose scope is the entire sentence. In his early paper “On the

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Scientific Justification of the Begriffsschrift,” Frege writes, about the flexibility of natural language, that it can be compared to the hand, which despite its adaptability to the most diverse tasks is still inadequate. We build for ourselves artificial hands, tools for particular purposes, which work with more accuracy than the hand can provide. And how is this accuracy possible? Through the very stiffness and inflexibility of parts the lack of which makes the hand so dextrous.¹⁶

The flexibility of natural language leads to an awkwardness when we try to talk about concepts. But there is no similar awkwardness in Frege’s logically perfect language.

II. Function and Object as Metaphysical or Ontological Categories Let us return now to Burge’s claim that Frege’s notions of function and object are ontological categories, as well as his claim that the third realm contains “some objects and all functions.” Is Burge right? In particular, are Frege’s notions of function and object both logical and ontological categories?¹⁷ It is worth noting at the outset of this discussion that there is something very odd about thinking of Frege’s function/object and concept/object distinctions as ontological distinctions, as distinctions that categorize entities in the universe. On this view some entities are functions, some are objects. That is, belonging to one of these categories (e.g., being a function, being unsaturated) is a property of some, but not all, entities in the universe—much as being red is a property of some, but not all, entities in the universe. But it is a very different matter to say what it is to be a function than it is to say what it is to be red. And this is not simply because redness is an observable property. If asked whether a particular entity is red, one might examine it to see if it is sufficiently close to the color of blood (see Merriam Webster’s definition). One might consult a color chart. Or one might perform some sort of test to measure the frequency or wavelength of the light reflected. But one thing that would not come up is the kind of expression used to talk about or name that entity. In contrast, when Frege brings up the issue of what it is to be a function or concept, he invariably talks about linguistic expressions and how they work. He introduces the notion of function in Basic Laws, for example, by introducing functionexpressions and then writing “The expression of a function is in need of ¹⁶ OSJ, p. 52. ¹⁷ The support for the Standard Interpretation that follows is not the support that Burge gives. For my objections to Burge’s account, see Weiner (1995a), pp. 363–82 and (1995b), pp. 585–97.

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completion, unsaturated.”¹⁸ In “On Concept and Object”, as we have seen, he tells us that concepts are predicative and, by way of explaining this, adds in a footnote, that a concept is the Bedeutung of a grammatical predicate. It is not obvious that Frege thinks there is anything to be said about the distinction without talking about our use of language. Is there a sense in which, nonetheless, these categories might be counted as ontological categories? Burge suggests a test for whether an apparently metaphysical statement should be viewed as belonging to metaphysics—determine whether the statements in question are meant to play some theoretical role in the explanation of some phenomenon. Is there a theoretical role to be played by Frege’s explanations of the notions of function, concept, and object? And, if so, does the theory in question belong to metaphysics? There is certainly a trivial sense in which Burge is correct to claim that logic and ontology are “mutually entangled in Frege.” On Frege’s view, the laws of logic are, after all, the laws on which all knowledge rests. Such laws are used in arguments and reasoning throughout the sciences. Thus, no matter what science we choose, there is a sense in which logic and that science could be said to be mutually entangled. In this sense, the notions of function and object belong to every science—for, on Frege’s view, no matter what our topic, our statements can be analyzed into functions and arguments. But for all that, if we think of these explanations as statements of truths belonging to a science, then they clearly do not belong to most sciences. For example, no truth about the nature of functions and objects plays a role in the proofs of Euclidean geometry. Nor would it be right to say that function and object are categories belonging to geometry. What would it be, then, for logic and ontology (or metaphysics) to be more closely entangled than logic and geometry—what would it be for them to be so closely entangled that some logical categories are also ontological categories? Frege is very explicit about the kinds of relations that hold between sciences. In particular, arithmetic, he believes, is more closely entangled in logic than geometry. More specifically, arithmetic belongs to logic. And Frege is specific about what it means for arithmetic to belong to logic: all truths of arithmetic must be provable from truths that are identifiably primitive logical laws. This is in contrast to geometry which, he says, requires certain axioms peculiar to it where the contrary of these axioms— considered from a purely logical point of view—is just as possible, i.e., is without contradiction.¹⁹

¹⁸ BLA vol., i, p. 5.

¹⁹ FTA, p. 94.

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What about metaphysics? First, it should be clear that logic cannot be viewed as a part of metaphysics. For, it is logic, not metaphysics, that is the more fundamental science. Frege writes, I take it to be a sure sign of error should logic have to rely on metaphysics and psychology, sciences which themselves require logical principles.²⁰

Second, metaphysics, like any science, either belongs to logic or does not. If not, the truths of metaphysics must be synthetic a priori or synthetic a posteriori—in which case, it is difficult to see how the categories of function and object can be metaphysical categories. Thus, if Frege does have a view on which there is a special relation between metaphysics and logic, it is difficult to see how his view could be anything other than that metaphysics is a part of logic. Of course, Frege never says anything like this—nor does he say that metaphysics is a distinct, synthetic, science. That is, there is no explicit evidence, in his writings, for taking Frege to have the view that there is any special relation between logic and metaphysics. One might think, however, that there is a different reason for the claim that metaphysics has a special relation to logic for Frege. For one might think that Frege’s notion of logic already encompasses what we mean by “metaphysics” or “ontology.” How can we determine whether Frege’s logic includes a metaphysical theory? One way to do this would be to find statements that Frege identifies as belonging to logic and that, on contemporary views, belong to metaphysics. Among the possibilities are Frege’s statements about functions and objects. He writes, for instance, that unsaturatedness is essential to the nature of functions; that functions cannot be objects and that objects cannot be functions. Does Frege mean these statements to be understood as statements belonging to a theory? And, if so, is there a metaphysical phenomenon that Frege means to use his distinction to explain?

IIa. Functions and the Problem of Predication For the most part, the discussions of “On Concept and Object” give us no reason to regard the concept/object distinction as a metaphysical distinction. There is, however, one paragraph of “On Concept and Object” in which Frege seems to use his concept/object distinction to address a traditional metaphysical issue. This is the discussion that follows his acknowledgment of a problem with his notion of function, which he calls an “awkwardness of language.” He writes, “I mention an object, when what I intend is a concept.” Given his conception of concepts as unsaturated, a concept-expression must be incomplete. Thus while it is tempting ²⁰ BLA vol. i, p. xix.

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to think we can use such complete expressions as “the concept horse” as names of concepts, these expressions must be object names. Having admitted this difficulty with his own use of “concept,” Frege turns, in the penultimate paragraph of the paper, to the issue of whether this difficulty is “artificially created,” whether it might be avoided by someone who understands the notion of concept as Kerry does. In arguing that even Kerry cannot avoid this difficulty, Frege may seem, for the first time in the paper, to be raising a metaphysical issue. The issue in question is the traditional issue of the unity of the proposition. What is it that distinguishes a sentence—a string of words that expresses a thought—from a string of words that does not express a thought? Frege argues that the only way to answer this question is to recognize unsaturatedness. On his view, the answer comes from recognizing the unsaturatedness of predicates. But even if we do not recognize predicates as unsaturated, we must, Frege argues, recognize unsaturatedness if we are to solve the problem. In order to understand this argument, let us consider Kerry’s view that “the concept horse” is a name of a concept. The price (or advantage—depending on your point of view) of Kerry’s view is that concepts and objects are on a level. To say that 2 is a prime number is to assert a relation between the number 2 and the concept prime number: 2 falls under the concept prime number. On this view, there is something misleading about the sentence “2 is a prime number.” For it seems that the only significant expressions in this sentence are “2” and “a prime number.” And if the only significance these expressions have for expressing a thought lies in their expressing senses that pick out the number 2 and the concept prime number, then we ought to be able to express a thought via the use of other expressions that pick the same things out. That is, 2 the concept prime number should express a thought. But it does not. What, then, explains the fact that 2 is a prime number actually expresses a relation between 2 and the concept prime number? The answer is that one of the constituents of what is said by this sentence—the relation that is said to hold—is not explicitly mentioned in the sentence itself; its presence is implicit in the juxtaposition of the subject and predicate. This can be fixed, of course, by mentioning the relation explicitly. We can say 2 falls under the concept prime number. We now have a sentence that explicitly contains expressions representing all constituents of the thought.

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But on this new account, Frege claims, the difficulty has simply reappeared in another guise—the expression “falls under” is unsaturated. Of course, someone might respond that there need be nothing unsaturated here. For the expressions “falls under” and “the falling under relation” surely name the same thing. But then it cannot be that what is required to express a thought is simply to mention an object, a concept, and the falling under relation. For the words “2 the falling under relation the concept prime number” do not express a thought. One could, of course, recognize something else as missing from this string of words. That is, one could say, not only that “2 is a prime number” is an abbreviation of “2 falls under the concept prime number” but that this, too, is an abbreviation. It is an abbreviation of: The falling under relation holds between 2 and the concept prime number. But now we have a new, potentially unsaturated, constituent. Is holding between unsaturated, or do we need to recognize yet another relation that is left out of our current rendition of the real content of “2 is a prime number”? It should be clear, at this point, that either we must accept that the “artificially produced” difficulty is not artificially produced at all or we are committed to an infinite regress—that is, there is no way to identify all the constituents of the statement. A look at this argument may seem, then, to show that Frege’s notion of unsaturatedness is a metaphysical view, designed to solve the problem of the unity of the proposition. This is how Davidson reads it. His response is, So there are entities that are not objects. The best Frege could do was to relapse into metaphor: the entities were “incomplete,” “unsaturated.” Frege remarks, “It must indeed be recognized that here we are confronted by an awkwardness of language, which I admit cannot be avoided, if we say that the concept horse is not a concept.” This is more than a superficial difficulty: if, as Frege maintained, predicates refer to entities, and this fact exhausts their semantic role, it does not matter how odd or permeable some of the entities are, for we can still raise the question of how these entities are related to those other entities, objects.²¹

Indeed, for this reason Davidson believes that Frege’s view, itself, invites a regress. But what, exactly, is Davidson talking about here? In particular, how does he envision raising the question of how concepts are related to objects? The suggestion is that what gets the regress going is that, having identified the entities to which each constituent refers, we can ask whether the one that is a concept holds of the one that is the object. But Davidson does not make it clear how we can ask

²¹ Davidson (2008), pp. 144–5.

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this. And, given Frege’s views, it is far from clear that we can ask any such thing. Consider, again, 2 is a prime number. The subject “2,” on Frege’s view, refers to an object, the predicate “a prime number” to a concept. How do we ask whether or not that concept holds of the object? We cannot simply use the constituents of our sentence, for the question Does a prime number hold of 2? is the wrong question. It is not a question about the relation of two particular entities but, rather a general question about prime numbers. It is also a question that is correctly answered in the negative—prime numbers are not the sort of thing that can hold or not of objects. Nor will it help to change the order and write, Does 2 fall under a prime number? Davidson’s claim, then, that we can raise the question presupposes that there are ways to name concepts other than by using predicates. But how can we do this? As we have already seen, Frege would deny that the following does the job, Does the concept prime number hold of 2? For Frege’s view that the expression “the concept horse” cannot name a concept applies equally to the expression “the concept prime number.” Is Davidson’s claim that we can always ask such a question correct? It is, in any case, a claim with which Frege disagrees. To see why, it may help to consider some examples of what goes wrong when we try to construct a name of a concept that is not a predicate. We might start with the idea that the concept named by “a prime number” names the same thing as the predicate of “2 is a prime number.” But there are problems with this idea. One reason is that, as we saw earlier, “a prime number” is not actually the predicate of “2 is a prime number” (it has no gap). Nor, as we also saw earlier, can we easily correct this by replacing “a prime number” by “x is a prime number” or “( ) is a prime number.” Still, one might think that this kind of problem is easily circumvented. For we can use—instead of the expression “the concept named by ‘a prime number’ ”—“the concept named by the predicate of ‘2 is a prime number’.” That is, we might try:

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The concept named by the predicate of “2 is a prime number” holds of 2. But a problem remains. The expression “the concept named by the predicate of ‘2 is a prime number’ ” is a complete (object-) expression. It is no more available as a name of something incomplete than “the concept horse” is. Davidson, as we have seen, argues that Frege has not succeeded in solving the problem of predication. But Davidson’s argument is based on a reading of Frege that directly contradicts many of Frege’s explicit statements. For Frege’s view only gives rise to a regress if, given a predicate, we can construct an object-name that names whatever the predicate names. For example, to get a regress going using the sentence “2 is a prime number,” it must be possible to construct an object-name for whatever is named by “a prime number.” Davidson simply presupposes that, no matter what one’s metaphysical theory, any entity can be given an objectname. As he writes in Truth and Predication, “entities are entities.” Yet Frege does not agree. It is not just that Frege never writes that functions and concepts are entities. Frege also explicitly denies that functions and concepts can be named by complete expressions (that is, object-names). The view that Davidson attributes to Frege—the view that, Davidson believes, is meant to solve the problem of predication—is not Frege’s view. If metaphysics is understood in Davidson’s sense, Frege’s discussions of functions and concepts are not meant to belong to metaphysics. This is one reason for reconsidering the idea that Frege’s statements are meant to be part of a metaphysical theory that solves the problem of predication. There are other reasons as well. Much has been made of the paragraph from “On Concept and Object” in which Frege seems to address the problem of predication—probably because of its relation to a traditional philosophical problem. But it is worth emphasizing that Frege does not make much of this particular issue himself. His entire discussion of the problem of predication appears in a single paragraph of “On Concept and Object.” Nor does Frege actually claim, in this paragraph, to be solving the problem of predication. Rather, the paragraph provides a response to a particular objection to his notion of concept. Frege argues that an understanding of predication requires the recognition of unsaturatedness at some level of our analysis of sentences. But there is no indication that he is advocating a metaphysical theory one of whose truths is that unsaturated entities are among the entities in the universe. Indeed, he ends the paragraph by denying this. He writes, “Complete” and “unsaturated” are of course only figures of speech, but all that I wish or am able to do here it to give hints.²²

²² CO, p. 205.

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Were Frege’s claims about unsaturatedness meant as part of a metaphysical theory that classifies different entities as saturated or not, then “unsaturated” would not be a figure of speech but, rather, a technical term. Were these claims meant as an explicit solution to the problem of predication, then it would be wrong to say that all he wishes or is able to do is to give hints. What all of this suggests is that Frege does not view the statements in this paragraph either as a part of a metaphysical theory or as a solution to the problem of predication.

IIb. Platonism and Frege: Burge’s “Literal” Interpretation But if Frege does not mean to be offering a solution to the problem of predication, it does not follow that he offers us no metaphysical theory at all. Is there another sense in which Frege is advocating a metaphysical view—a sense, in particular, in which he is a Platonist? Burge thinks that on the obvious, literal interpretation of Frege’s words, he is a Platonist. And he offers several characterizations of what Frege’s Platonism comes to. Burge writes, As is well-known, Frege thought that extensions—including numbers— functions—including concepts—and thought contents are imperceptible, nonspatial, atemporal, and causally inert.²³

Burge also writes, Platonism, as I understand the doctrine, regards some entities (for Frege, some objects and all functions) as existing non-spatially and atemporally.²⁴

And he writes that had Frege maintained that extensions, functions, or thought contents were dependent on human conceptualization or human language, judgment, or inference (actual or possible), he would have said so, and thereby qualified the numerous remarks that have traditionally invited the Platonic interpretation of his work.²⁵

This is a point he emphasizes again, writing: Independence is independence. Frege’s repeated remarks about mindindependence of non-spatio-temporal entities would not have been literally true, if they had been backed by a set of unstated qualifications of the sort that

²³ Burge (1992), p. 636.

²⁴ Burge (1992), p. 637.

²⁵ Burge (1992), p. 638.

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an idealist (or deflationary) interpretation of them would require. . . . Frege enters no such qualifications.²⁶

As the above passages indicate, it is a central part of the Platonism that Burge attributes to Frege that the “third realm” of non-spatio-temporal mindindependent entities, contains not only objects, but functions and concepts; that it is a literal truth that functions and concepts are non-spatio-temporal and mindindependent. Burge’s evidence is that Frege does not qualify his “repeated remarks about mind-independence of non-spatio-temporal entities”. But is this claim correct?

IIb.i. What Does Frege Say about Non-spatio-temporal Entities? As is common with supporters of the Standard Interpretation, Burge presents this part of his argument as mere reportage. But as is also common with supporters of the Standard Interpretation, the reportage is not entirely accurate. According to Burge, Frege meant his Platonism to be a theory, one of whose central doctrines is that all functions are non-spatio-temporal objective entities. In the above passages, Burge mentions Frege’s “numerous remarks” and “repeated remarks” but Burge is stingy when it comes to providing citations. In particular, Burge provides no citation to a passage in which Frege claims that all functions are non-spatiotemporal objective entities. If Burge is right, one would expect Frege to have explicitly stated this central doctrine at least somewhere in his writings. One would certainly expect this view to be stated in his writings titled Function and Concept and “What is a Function?” But these writings contain no such statement. One would also expect Frege to characterize functions as non-spatio-temporal objective entities in his explicit discussions of the “essence of the function.” He does not.²⁷ This is not to say that Frege would deny that functions are non-spatio-temporal and mind-independent. I suspect that, if asked, Frege would have said that functions are non-spatio-temporal and mind-independent. But the fact that he does not say so explicitly suggests that he does not regard this as a central tenet of a theory. Moreover, even if he had said so, as Burge himself recognizes, that is far from what is required for showing that this is a central tenet of Frege’s doctrine or that it bears the metaphysical weight Burge says it does. For, as we saw earlier, Burge writes, Many of these things might be maintained by someone who was not a Platonist. One might make the remarks about imperceptibility, non-spatiality, atemporality, and causal inertness, if one glossed them as part of a practical recommendation or

²⁶ Burge (1992), p. 641.

²⁷ See, e.g., BLA p. 5 and FC, p. 6.

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As I shall argue shortly, there is good evidence that, had Frege been asked whether it was a literal truth belonging to a metaphysical theory that functions are nonspatio-temporal and mind-independent, he would have said that it was not. Of course, the functions that Frege is most interested in are concepts. And concepts are mentioned in the passages that, Burge claims, tell us that “Concepts or other functions are counted atemporal and by implication imperceptible, nonspatial, and causally inert.”²⁹ What are the claims in question and what is the evidence that they are meant as statements belonging to a metaphysical theory? The passages come from two works. Frege writes in the introduction to Foundations, We suppose, it would seem, that concepts sprout in the individual mind like leaves on a tree, and we think to discover their nature by studying their birth; we seek to define them psychologically, in terms of the nature of the human mind. But this account makes everything subjective, and if we follow it through to the end, does away with truth. What is known as the history of concepts is really a history either of our knowledge of concepts or of the meanings of words.³⁰

To say that concepts have no history is, arguably, to say that they are atemporal. If we presuppose that Frege has a metaphysical theory in which concepts are counted among the entities of the universe, and we want to know whether or not concepts are temporal entities, then this passage gives us a clear answer. But if we do not make this presupposition, there is scant support in this passage for the view that Frege has a metaphysical theory in which concepts are counted among the entities. It is not just that the introduction to Foundations would be an odd place to introduce this view. Nothing Frege says in this passage or the surrounding pages of the introduction, indicates that he means to be making a statement belonging to a metaphysical theory. Rather, he wants to distinguish his use of the term “concept” from another use: a use on which a concept is an idea, something belonging to a human mind. The project that Frege describes in Foundations is the project of defining the concept number and the numbers and of proving the basic truths of arithmetic from logical laws. When he asks for a definition of the concept number, then, he is asking for a definition that can be

²⁸ Burge (1992) p. 637, emphasis added. ²⁹ Burge (1992) p. 636n9. In the above discussion I do not discuss the additional two passages in which, he claims, Frege “suggests these points about concepts indirectly.” The matter at hand is whether explicit remarks Frege makes should be construed other than as statements belonging to a theory. Indirect suggestions are beside the point. ³⁰ FA, p. vii.

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used to accomplish this. The purpose of the comments about concepts in the above passage is to make clear to the reader what is required of a definition of the concept number. In particular, no facts about the history of our knowledge of number or of any individual’s knowledge of number will be relevant to the evaluation of a proposed definition. The theory, which is to be given in Basic Laws, will consist of definitions, laws, and proofs, all expressed in Begriffsschrift. Another of the three passages Burge cites comes from an article titled “On the Law of Inertia”. This article is a discussion of Ludwig Lange’s use of his notion of an inertial frame of reference as a replacement for Newton’s absolute space and time. Both Frege’s remarks and the context are very similar to the remarks in the passage set off above. Finally, the third of these passages is to a footnote in Foundations. The word “concept” appears only once in that footnote, where Frege writes, Objective ideas can be divided into objects and concepts.³¹

None of the passages that Burge cites, then, appears to contain a statement about concepts that is meant to belong to a metaphysical theory. If Frege does mean to be presenting a metaphysical theory on which there is a third realm containing, to quote Burge “some objects and all functions,” it is surprising that Frege is nowhere explicit about the metaphysical status of functions and concepts. Burge’s attribution of a metaphysical theory to Frege is based, not on what Frege says he is doing, but on what, according to Burge, Frege does. But are Burge’s reports of what Frege does accurate?

IIb.ii. Does Frege Qualify these Remarks? Burge claims that Frege does not qualify his “repeated remarks about mindindependence of non-spatio-temporal entities.” But this is false. Frege, in fact, repeatedly qualifies all claims in which the expression “concept” appears. Some of these qualifications are exactly the sort of qualifications that we would expect to find, were a deflationary interpretation—an interpretation on which the statements about functions and concepts are not meant to be taken as literal truths of a theory—correct. Let us consider some of these qualifications. It is worth noting, first, that it is probably no accident that the term “entity,” which is so prominent in Burge’s discussion, does not appear in any of the passages from Frege that Burge cites and, indeed, rarely appears in any of Frege’s discussions of functions, concepts and objects. “Entity” is a useful term if one wants to talk about features common to functions, concepts and objects. But one of Frege’s explicit claims is that there are no features common to functions, concepts, and objects. He writes

³¹ FA §27.

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that what can be said about objects cannot be said about concepts or functions. This, he says, can be expressed metaphorically: “There are different logical places; in some only objects can stand and not concepts, in others only concepts and not objects.”³² Frege is explicit that numbers are not spatio-temporal objects. He writes, in Foundations, “not every object has a place” and “not every objective object has a place.”³³ If it can be said of an object that it is objective but not spatiotemporal, any attempt to assert the same thing about a concept (or function) will fail. He also writes, I do not want to say it is false to say concerning an object what is said here concerning a concept; I want to say it is impossible, senseless (sinnlos) to do so.³⁴

Thus it follows from Frege’s explicit remarks that either the statements about concepts or those about objects must be abandoned as literal truths of a theory. In other words, it follows from Frege’s explicit statements that the view that, according to Burge, is well known to be Frege’s, that extensions—including numbers—functions—including concepts—and thought contents are imperceptible, non-spatial, atemporal, and causally inert.³⁵

cannot be stated. This can certainly not, then, be a statement belonging to a theory, whether metaphysical or not. Frege does, however, indicate that, while relations that hold between objects cannot hold between concepts, there are corresponding relations that hold between concepts.³⁶ Burge might reply that even if the same properties of being non-spatio-temporal and objective cannot hold of objects and concepts, corresponding properties hold. But this will not help. It remains to explain what statements about concepts and functions correspond to the claims that objects are objective but not spatio-temporal. Frege offers no such explanation. And, given some of Frege’s remarks about the expressions “concept” and “function” this should not be very surprising. He writes, the word “concept” itself is, taken strictly, already defective, since the phrase “is a concept” requires a proper name as grammatical subject; and so, strictly speaking, it requires something contradictory, since no proper name can designate a concept; or perhaps better still, something nonsensical [einen Unsinn].³⁷

³² ³⁴ ³⁵ ³⁷

OFG I, p. 372. ³³ FA §61. CO, p. 200; see also CO, p. 198; and “Logik” NS, p. 130/PW, p. 120. Burge (1992) p. 636. ³⁶ Review of Husserl, p. 320. “On Schoenflies” NS, p. 192/PW, pp. 177–8.

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Given that the replacement of “x” in “x is an object” with any object-name will give us a true sentence, we seem to be committed to saying that everything is an object. And, similarly that nothing is a concept. The problem with “concept” is also a problem for the term “function.” For the problem results from the predicative nature of concepts which, Frege says, “is just a special case of the need of supplementation, the ‘unsaturatedness’ that I gave as the essential feature of a function in my work Funktion und Begriff.”³⁸ The expression “the function f(x),” Frege says, does not designate a function.³⁹ But it is not surprising—given that every primitive symbol in Frege’s logical language, Begriffsschrift, is a function-name— that, as Frege also says, it was “scarcely possible to avoid the expression ‘the function f(x)’” in the introduction of his logic. Indeed, in a letter to Bertrand Russell, he writes, “the words ‘function’ and ‘concept’ should properly speaking be rejected.”⁴⁰ What these remarks show us is that Frege explicitly and repeatedly qualifies all his remarks in which the terms “function” and “concept” appear. On his view, any other attempt to say something about functions or concepts using such defective terms as “function” or “concept” will likewise founder. Such statements include any attempt to make a claim about functions or concepts corresponding to those about the non-spatio-temporal objectivity of numbers and thoughts. Frege does not simply abstain from saying that all functions are non-spatio-temporal entities, he explicitly denies that his remarks about the nature of functions and concepts are literally true. This is exactly the sort of denial one would expect from someone who did not mean these remarks to be taken as metaphysical doctrines. Moreover, to say that functions—in the guise of unsaturated entities—are non-spatiotemporal, mind-independent denizens of a third realm is a large, controversial doctrine. Indeed, it is a doctrine that few of Frege’s readers (few, even, of his fervent admirers) find plausible. It is also a doctrine that is nowhere clearly stated in his writings. Worse still, it is a doctrine that contradicts many of Frege’s explicit statements. Burge writes, It is dubious historical methodology to attribute to a philosopher with writings that stretch over decades, a large, controversial doctrine, if he nowhere clearly states it in his writings.⁴¹

By Burge’s own lights, then, it is bad historical methodology to attribute this view to Frege.

IIb.iii. Can There be a Literal Statement that Functions Are Atemporal? Frege says, as we have seen, that numbers are non-spatio-temporal. Before continuing, it may be useful to look more carefully at why, on his view, we cannot ³⁸ CO, p. 198n. ³⁹ CO, p. 198. ⁴⁰ Letter to Russell, July 28, 1902, WB, p. 224/PMC, p. 141.

⁴¹ Burge (1992) pp. 639–40.

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say that functions are, likewise, non-spatio-temporal entities. For clarity, I will consider only what it would be to say that functions are atemporal. How would we express this view in Frege’s logical language? It looks like a universally generalized conditional, something of the form: (x)(if x is a function, then x is atemporal). But such a strategy for beginning the translation of this claim into Begriffsschrift will not do. It is not simply that Frege says the expression “is a function” is defective. The variable in the partially regimented statement above is a first-level variable. The “x” in “if x is a function, then x is atemporal” can be replaced only by an object-expression. Thus, if we understand “x is a function” as Frege would like, it must be universally false. The generalized conditional, then, is trivially true and cannot constitute a substantive metaphysical claim. Indeed, it is also trivially true that functions are spatio-temporal objects. Of course, the problem with this strategy for representing a generalization about functions should be no surprise, since Frege’s logic is a second-order logic. Any generalization about functions should have a second-level quantifier. How might the claim be expressed, then, using a second-level quantifier? Frege tells us that we need to distinguish object-letters from function-letters.⁴² So, perhaps what is needed is something like: (g)(g is atemporal). But this will not work either. In Frege’s logical language, the use of a function letter is not sufficient to give us a second-level quantifier. Frege tells us that a function-variable not only needs to be a different sort of letter, it also needs to carry brackets with it and either one or two places for an argument. It may seem, then, that we should write something like: (g)(g( ) is atemporal). But this cannot be transformed into a Begriffsschrift expression that can appear on its own. In a Begriffsschrift expression the empty place following a function-expression must be filled by an objectname or a first-level variable. How can our claim be expressed if this condition is satisfied? Since our claim appears not to involve any particular object, it will not do to insert an object-name into the empty space. The alternative is to put in a variable. But, in Frege’s notation the expression “(g)(g(x) is atemporal)” is simply shorthand for a universal generalization, “(x)(g)(g(x) is atemporal).” And this can not express the claim in question. It does not say that every function is atemporal but, rather, that for any argument, the value of a function at that argument is atemporal. This is not only the wrong sort of claim; it appears, on Frege’s view, to be false. For example, the expression “the mother of x” seems to be a perfectly good function whose values are never atemporal.⁴³

⁴² BLA vol. i, pp. 34–5. ⁴³ Strictly speaking, of course, “the mother of x” is not a function in Frege’s sense because a function must have a value for every object. However, that does not affect the argument here, since Frege also assumes that in cases like this a designated value (e.g., 0) can be stipulated for the objects that do not have mothers. His proposal that we do something like this appears in SB, p. 42.

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As we saw earlier, Frege introduces a solution to the problem of how to talk about functions without putting an object-name or variable in the argument slot of the function-name. The solution is to use lower-case Greek letters. It might seem that we could use an expression like “(g)(g(ξ) is atemporal)” to express this. But, as we saw earlier, the lower-case Greek letters do not belong to the language of Begriffsschrift—they are introduced to be used in elucidations. And the expressions in which they are used (e.g., “the function ξ + ξ”) are intrinsically defective. Thus, were this strategy required for stating that functions are atemporal, the statement in question would not be expressible in Begriffsschrift. Frege’s logical language is supposed to be usable to evaluate inferences of any science on any topic. If the statement in question belongs to metaphysics, then metaphysics would not count as a science. There is a way in which the above problems should not be surprising. For one feature of all the strategies discussed above is that all employ a first-level atemporality predicate. And it should be no surprise that something goes wrong every time we try to use a first-level predicate to predicate something of functions. If we are going to make a statement in which something is predicated of, say, first-level functions, the predicate should not be a first-level predicate but a second-level predicate. And Frege does give us strategies, in volume i of Basic Laws, for constructing Begriffsschrift expressions that are second-level predications. Why would he not make use of such a predication to construct a statement that all firstlevel functions are atemporal? Using the notation Frege introduces, the statement that all first-level functions are atemporal, might be written: (f)(atemporalx(fx)).⁴⁴ The problem with this strategy, however, is that there is no obvious way to make a connection between the first-level atemporality predicate and the proposed second-level atemporality predicate. As we saw earlier, Frege writes that it is impossible to say about an object what can be said about a concept. Thus what is predicated of the number 1 if we say that the number 1 is atemporal, cannot be predicated of a function or concept. But one might think that there can be some important relation between the first-level atemporality predicate and this kind of second-level atemporality predicate. After all, while Frege says that identity is not a relation that can hold between functions, he also says that there is a corresponding relation that can. Can something similar be said about different atemporality relations? To answer this question, let us consider first, the identity relation and the corresponding relation that can hold between functions. The corresponding relation is one that holds between functions when they always give the same value for the same argument, (x)(f(x) = g(x)). There is an obvious connection between identity and the corresponding relation: the corresponding relation is a

⁴⁴ See BLA vol. i, §§23–5.

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generalized identity. There is also another interesting way of seeing the connection. Frege relies on Leibniz’ explanation of identity: two things are the same when one can be substituted for the other without loss of truth.⁴⁵

The licensing of this substitution is first encapsulated in Law 52 of Begriffsschrift and, later, in Basic Law III of Basic Laws. These laws, along with Frege’s other laws and rules, allow us to infer f(b) from f(a) and a = b—that is, to substitute b for a. And the corresponding statement about functions also licenses intersubstitution. For example, from f(a) and (x)(f(x)=g(x)), we can infer g(b)—that is, to substitute g for f. Thus, we can also see the correspondence of these distinct relations in terms of significance for inference. But there is no similar claim that we can make about the use of “atemporal” as a predicate that can hold or not of objects and our imagined predicate “atemporalx” that can hold or not of first-level functions. One might respond that the reason is that the atemporality properties are not logical properties but metaphysical properties. But this does not help. For what we need is some explanation of what ties first-level atemporality to second-level atemporality. The sense that there is a tie comes from the presupposition that there is something we can say, both about objects and about functions of various levels: that they occupy an extra-linguistic universe. But notice that this is exactly the kind of thing that, on Frege’s view, is unstatable. Moreover, it is important to note that there is a big difference between the first-level atemporality predicate and higher-level atemporality predicates. For the first-level atemporality predicate holds of some objects (e.g., the number 1) but not others (e.g., a tree or a house). In contrast, there is no content to the second-level atemporality predicate. Anything of which second-level atemporality can be predicated is second-level atemporal. This makes the second-level atemporality predicate similar, not to the first-level atemporality predicate, but to the objecthood predicate. For anything of which objecthood can be predicated is an object. There is no Begriffsschrift expression for predicating objecthood and it would serve no purpose to add such a predicate. Similarly, it would serve no purpose to add a second-level atemporality predicate. That is, it is not a contentful statement of any science, whether metaphysics or not, to say that first-level functions are atemporal.

IIc. What Work Does Frege’s Function/Object Distinction Do? One might object that the problem here is that the above story still implicitly relies on Frege’s notion of unsaturatedness. For, after all, if there can be object-names of ⁴⁵ This statement appears both in his 1894 review of Husserl’s Philosophy of Arithmetic (p. 320) and in FA, §8.

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functions, then there should be nothing wrong with the original strategy for formulating the claim that all functions are atemporal, that is “(x)(if x is a function, then x is atemporal).” But it is important to remember why Frege is so insistent that, for example, the concept horse is not a concept. The terms “function,” “concept,” and “object,” as he uses them, are all introduced as a means for explaining his new logic and, in particular, the difference between the traditional subject/predicate analysis of statements and the function/argument analysis that, he argues, is superior. Predicates, Frege acknowledges, can appear either in the subject or predicate slot of a sentence. Thus we can say: (A) A horse is a mammal. (B) Venus is a horse. From the grammatical point of view, both (A) and (B) are subject/predicate statements. And, while (A) and (B) share a predicate, the predicate they share appears in the grammatical subject slot in (A) but in the grammatical predicate slot in (B). It is because of this that, from Frege’s point of view, the correct logical analyses of these sentences are very different. (B) expresses a simple predication— an object is said to fall under a concept—and an expression of this kind of content in a logically perfect language requires only two (possibly complex) names. In contrast, what (A) expresses is not a simple predication but, rather, a quantificational statement; a statement that (as Frege sometimes characterizes it) one concept is subordinate to another. The expression of this kind of content in a logical language requires not just two (possibly complex) concept-expressions but also some signs for logical functions, including a quantifier symbol and conditional stroke. What enables us to recognize that (A) is a quantified statement is that a predicate, or concept-expression, appears as grammatical subject. Now let us consider a third sentence, (C)

The concept horse is realized.

The correct analysis of (C), unlike that of (A), requires no quantifier. On Frege’s view, (C) is to be analyzed as a simple predication. But if we reject Frege’s claim that the concept horse is not a concept and insist that “the concept horse” must name the same thing as “a horse”—whatever that is—then the concept/object (function/object) distinction can tell us nothing about the way in which the correct analyses of (A) and (C) differ. Both (A) and (C) are, from a grammatical point of view, subject/predicate sentences and both sentences have, as subjects, concept-expressions. Thus, if we insist that “a horse” and “the concept horse” are both concept-expressions, the occurrence of a concept-expression as the subject of the sentence gives us no reason to infer that the sentence is to be analyzed as a

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quantified statement. And this undermines the point of Frege’s introduction of the understanding of concepts as functions. Frege’s claim that what can be said of objects cannot be said of concepts is one of those that is typically disregarded. For one thing, it is one of the many things that Frege both says and takes back—it cannot be a statement of a literal truth. But the above discussion shows that we cannot simply disregard the explicit claim. For, if we pay attention to the work that Frege’s function/object and concept/ object distinctions are supposed to do, we can see that he is committed to the apparently paradoxical statements about functions, concepts, and objects. It is only against the background of such statements that we can understand, for example, what it is to say that numbers are objects. Moreover, as we have just seen, it is not Frege’s apparently paradoxical statements that preclude the statement of metaphysical theories about functions and concepts. It is, rather, his view of Begriffsschrift notation as a language adequate for the expression of all content with significance for inference. Any charitable interpretation of a philosopher’s work will involve some picking and choosing among that philosopher’s statements. But it is important to be careful about any choice to disregard some statement. In Frege’s case, if we choose to disregard his statements about functions and concepts, we must thereby reject some of his fundamental views—his views about the purpose of his logical language. Insofar as all theories must be statable in Begriffsschrift, there is no way to state a metaphysical theory of the sort that either Burge or Davidson attributes to Frege.

IId. “Function” and “Object” in Statements of Literal Truths I have argued that, if we pay attention to Frege’s statements, we will see that his notions of function, concept, and object do not belong to a metaphysical theory. By giving up the idea that these notions are meant to form part of a metaphysical theory, we avoid some of the apparent problems with Frege’s statements. But there are other apparent problems with these notions that have not been discussed. Let us turn to this issue next. One might suspect that the problem with Frege’s understanding of the terms “function” and “concept” is that their defective nature prevents us from talking about functions or concepts. We have seen that we cannot talk about the nature of functions by writing, for example, “all functions are atemporal” and that we cannot talk about a concept by using the expression “the concept horse.” This suggests that we can make neither specific claims about particular concepts or functions nor general claims about concepts or functions. But this is misleading. First, we can, and routinely do, make specific claims about particular concepts and functions. After all, predicates are concept-names and every sentence has at least one constituent that is a predicate. Thus every time we make a statement we are

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talking about concepts. “2 is a prime number” tells us, not only about 2, but also about a concept. Is there a problem, then, with making general statements about concepts? Again, the answer is that there is no problem. Frege’s logic, after all, is a second-order logic and it is possible to quantify over functions and concepts. Indeed, many of his general remarks about the nature of concepts can be expressed without difficulty in his logical language. Frege writes, for example, that a concept must hold or not of each object, something that might be expressed (in a contemporary version of Frege’s notation) as: (F)(x)(Fx v ~ Fx). The real problem has to do with Frege’s explicit statements. Frege is in no position to banish these intrinsically defective words from his writings. They are needed for his introduction of the new logic. It is no idle claim that concept and relation are the foundation-stones on which he builds his structure.⁴⁶ Moreover, much of the work Frege does in Foundations requires him to use defective terms. Consider, for example, two of the important statements from Foundations: that numbers are objects and that statements of number are assertions about concepts.⁴⁷ Both of these statements appear to be substantive literal (and nonparadoxical) truths that constitute part of the foundation of Frege’s theory of numbers. And yet it is a consequence of his understanding of the notions of concept and object that, if we construe these statements literally, neither can do the job Frege seems to require them to do. To see this, it will help to look at each of them individually. Let us begin with the statement that numbers are objects. Although the term “object” does not exhibit the problems associated with the terms “function” and “concept,” it seems to be a consequence of Frege’s views that there is no substantive issue here: nothing to debate or defend. For, as we have seen, for Frege, any name that can be used to fill in that space will be an object expression. Anything of which objecthood can be predicated is an object. That is, it appears to be a literal truth that everything is an object. And if everything is an object, it can hardly be informative to say that numbers are objects. What about the claim about statements, or ascriptions, of number? Since its expression contains the word “concepts” it is, on Frege’s view, defective. As we have seen, this is not a problem in itself. For there are natural language statements

⁴⁶ BLA vol. i, p. 3. ⁴⁷ Actually, as Michael Kremer points out, in his Review of the Jacquette translation of Foundations Kremer (2008), what Frege says explicitly is not that statements of number are assertions about concepts but, rather, that they contain assertions about concepts. This may seem to suggest that there is more to the content of statements of number than the assertions about concepts. In a sense, of course, there is. After all, one of Frege’s aims is to show us how such statements can also be understood as statements about an object (the number in question). However, this is not to say that the assertion about the number is something additional that is contained in the statement of number, for it is also Frege’s view that the two varieties of expressions should be logically equivalent. The difference in question, however, does not affect the point I make in what follows. For simplicity, I will continue to characterize Frege as saying that statements of number are assertions about concepts.

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that are likewise defective but present no particular problem for Frege. Consider, for example, the statement that a concept must hold or not of each object. This example presents no problem because there is a non-defective Begriffsschrift expression that expresses what we seem to be trying to express in the natural language statement. We can simply regard the natural language statement as a flawed attempt to state the correct logical law. However, this does not solve the problem with the statement that ascriptions of numbers are assertions about concepts. For this statement is not a flawed attempt to state a logical law. Nor does any Begriffsschrift correlate of this statement appear in Frege’s attempts to prove truths of arithmetic from logical laws. So, it seems to follow from Frege’s views about concepts and functions that one of the central statements he takes himself to have established in Foundations (that numbers are objects) is, in fact, uninformative and that another (that ascriptions of numbers are statements about concepts) is defective. If the statements in question are meant to be truths of a theory, this is a serious problem. But are they meant to be truths of a theory? We have already seen that they are not meant to be truths of a metaphysical theory. Are they meant to be truths of a logical theory? Insofar as the truths of a logical theory must be logical laws, statable in Begriffsschrift, they are not. One might suspect that some of Frege’s views about concepts and functions are meant to be, not logical truths stated in Begriffsschrift, but truths belonging to the metatheory. We will examine this claim in Chapter 4. However, the two claims in question—that numbers are objects and that ascriptions of number are assertions about concepts—do not look like candidates for metatheoretic truths. Indeed, neither statement comes up when Frege tries to show us the workings of his logic in Part I of Basic Laws. I will close this chapter with an argument that the statements in question, as well as many of Frege’s other statements about function and concept, are not meant to be literal truths of a theory but, rather, are meant to play another role, that of elucidation. And the success of statements that are meant to be elucidatory, as we shall see shortly, does not depend on their being taken as literal truths.

IIe. Elucidation, its Uses and Evaluation Let us begin with Frege’s attitude towards his explanations of the notions of function and concept. Frege tells us in “On Concept and Object” that the notion of concept cannot be defined because it is logically simple.⁴⁸ How, then, is he to communicate what he means by “concept” and “object”? He writes,

⁴⁸ CO, p. 193.

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On the introduction of a name for something logically simple, a definition is not possible; there is nothing for it but to lead the reader or hearer, by means of hints, to understand the word as is intended.⁴⁹

This remark is reiterated it a few pages later.⁵⁰ And the view in question is not one that he abandoned later. He writes, in a later paper on the foundations of geometry, My opinion is this: We must admit logically primitive elements that are indefinable. Even here there seems to be a need to make sure that we designate the same thing by the same sign (word). Once the investigators have come to an understanding about the primitive elements and their designations, agreement about what is logically composite is easily reached by means of definition. Since definitions are not possible for primitive elements, something else must enter in. I call it elucidation [Erläuterung]. It is this, therefore, that serves the purpose of mutual understanding among investigators, as well as of the communication of the science to others.⁵¹

And, in an unpublished piece of 1924/5, [W]hat is simple cannot be analysed and hence not defined. If, nevertheless, someone attempts a definition, the result is nonsense [kommt Unsinn heraus]. All definitions of function belong to this category.⁵²

How are we meant to understand this view? One might suspect that Frege is suggesting that there is a technique (elucidation), which is an alternative to definition and is confined to a particular use: that of introducing terms for something logically simple.⁵³ One might also suspect that this is a technique that can enable us to avoid nonsense in this particular situation. If so, this may seem to enable Frege to avoid Kerry’s problem: his remarks introducing the notion of concept are elucidatory. But this will not work. First, Kerry’s problem is not a problem about the attempt to define the notion of concept. The problem is framed without any talk of defining. Second, if some of Frege’s statements are nonsensical, this cannot be changed by labeling these statements “elucidations.” And, finally, there is no indication that Frege thought the “elucidation” label would have such an effect. ⁴⁹ CO, p. 193. ⁵⁰ CO, p. 195. ⁵¹ OFG II, p. 301. ⁵² Sources of Knowledge. NS, p. 290/PW, p. 271. ⁵³ It is a widely held view that Frege understood elucidation to be limited to the introduction of primitive terms and/or for the purpose of conveying concepts that are too simple to admit of further analysis. See, e.g., Blanchette (2012), p. 173n31. As we shall see below, however, this is demonstrably false.

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There is, however, a more plausible interpretation. The aim of statements that introduce terms for what is simple, Frege claims, is to reach a common understanding. The more plausible interpretation is that he thinks the only criteria for evaluating such statements—insofar as they can be evaluated—concern whether or not this aim is achieved. If this is so, then even nonsense may sometimes be valuable. After all, heuristics are often useful for communication even when they require one to entertain something that is absurd. It may be useful to repeat, in this context, the example I brought up in the preface. A flute player of my acquaintance, who was having difficulty getting to the end of a particularly long phrase, was told to store some air in her feet when she breathes in and to let the air up out of her feet when she comes to the end of the phrase. And, in fact, the heuristic was helpful—it enabled her to get to the end of the phrase. One might well say that what she was told is absurd—that it makes no sense because air can not be stored in the feet. But it would be misguided to object to the use of this heuristic on the grounds that it can not be true or does not make sense. In this particular case, the heuristic had value because it enabled a flute player to play a problematic phrase. Similarly, it is misguided to object to Frege’s non-literal hints about how to understand his logic and logical language on the grounds that they are not coherent or not literally true. Coherence and truth do not belong among standards of evaluation for such non-literal hints. Does this interpretation help explain Frege’s equanimity in the face of his problematic uses of “concept” and “function”? It does if the problematic uses are limited to attempts to get us to understand the terms “concept” and “function.” Since these are simple terms, even nonsense may be acceptable in their introduction, provided it is effective. Frege’s problematic statements, however, are not limited to the introductions of these terms. As we have seen, he thinks the terms themselves are intrinsically defective; every use of them is problematic. Even so, the above passages suggest that there will be no problem if these terms are used only to introduce other primitive logical elements. But while Frege does use these terms to introduce primitive terms of his logic, we have already seen that he also makes other uses of them. His statements that no concept is an object, that numbers are objects, that ascriptions of number are statements about concepts are not designed to introduce primitive terms. What is the status of these statements? All these statements belong to Frege’s project of providing logical foundations for arithmetic but none of them is an actual part of those foundations. At no point does any of them appear as a premise in a proof. And at no point does Frege offer a proof of any of them. Nor are there expressions for objecthood or concepthood in his logical language. One might think, however, not only that it would be of no help to describe these statements as elucidatory, but that such a description is not even an option. Let us begin by considering whether or not this is an option.

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We have seen passages in which the notion of elucidation comes up in the context of a discussion of the introduction of a primitive, undefinable term. If we understand elucidation as a procedure for introducing a primitive term, then the statements in question do not qualify. Frege never claims, however, that elucidation is confined exclusively to attempts to introduce primitive terms. And this is not his view. For there is at least one example of a discussion that Frege explicitly characterizes as elucidatory that has nothing to do with the introduction of primitive terms. This is his attempt in §34 of Basic Laws to help us understand, not the meaning of a primitive term, but the meaning of a term for which he offers a complex definition. He writes that, although all the terms used in the definition are already known, “a few elucidations” [einige Erläuterungen] may help us understand the definition. Thus the term “elucidation” is not restricted to remarks made in the introduction of a primitive term. But how can we recognize elucidation? When is it legitimate to describe a statement as elucidatory and how can this help with our problem? One might suspect that elucidation is recognizable because it appears to be nonsensical. But this is surely false. First, we can not simply include nonsensical statements in our theory and defend them on the ground that they are meant to be elucidatory. What would be the point of that? Second, few, if any, of Frege’s problematic statements about the notion of concept really appear to be nonsensical. A more typical characterization of these statements is that they miss his thought and, consequently require a reader who will not begrudge him a pinch of salt.⁵⁴ Moreover, there are elucidations that are clearly not nonsensical. Indeed, to a reader who does not notice that Frege uses the term “elucidations” for the discussion in §34, this discussion will seem virtually indistinguishable from an informal, natural language proof. Frege begins by dividing the contexts in which the defined term might appear into cases. Then he gives, for each case, an apparently good argument that the resulting object-expression will designate a particular object. There is nothing nonsensical in this discussion. What is it, then, that marks the discussion in §34 as elucidatory? One answer is that the argument is elucidatory because it belongs to the propaedeutic, not the theory. The §34 argument is largely an attempt to get the reader to understand how the definition works. The function in question, which is sometimes called the application function, is a two-place function that is designed to allow us to express a statement that a function holds of an object, something of the form Φ(Δ) as a statement about a relation’s holding between two objects, Δ and the value-range of Φ (i.e., ἀ Φ (α)).⁵⁵ The symbol for the application function is: ⌒. And, using this ⁵⁴ CO, p. 204. ⁵⁵ Just as the extension of a concept is an object with a special connection to the concept, the valuerange of a function is a kind of object associated with each function. The application function is introduced as a way of replacing the idea of a function holding of an object with the idea of something like a set theoretic membership relation between two objects. A Begriffsschrift name for the value-range

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symbol, we can express what is stated in the form “Φ(Δ),” instead as: Δ⌒ἀΦðαÞ. One purpose of Frege’s argument in §34 is to show us that the new function sign gives us this result in the cases we care about. Another purpose of the argument is to show us what happens in the don’t-care cases, that is, the cases in which the second argument is not a value-range. In these cases, Frege shows us, the value of the function is the False. Do the arguments of §34 belong to the logical theory? In §35, Frege makes it clear that they do not. He begins by noting that we can see from the arguments of §34 that the function in question designates a function. He then continues, This alone is fundamental for the conduct of the proofs to come; for since the definition itself is the foundation for their construction, our elucidation could be wrong in incidental ways without thereby calling into question the correctness of those proofs.⁵⁶

That is, should the arguments of §34 be flawed, this would not undermine the theory in any way. The only part of all this that belongs to the theory is the definition itself. As long as the function-sign does designate a function, there can be no problem. There is, then, no requirement that elucidation either be nonsensical or be restricted to the introduction of primitive terms. The §34 discussions give us a different way of thinking about Frege’s notion of elucidation: as a statement or argument that, while it does not actually belong to the logical theory, is useful for getting the reader to understand something that does belong to the logical theory. It is not only the arguments of §34 that qualify. Volume i of Basic Laws is divided into two parts. Part I, which is titled “Exposition of the Concept-script,” gives us his introduction to, and explanation of, the logical system. This includes, not only the logical language but also the basic laws and rules, as well as derivations of some of the consequences of the basic laws and most of the definitions that will be used in the proofs of Part II. Part II is titled “Proofs of the Basic Laws of Cardinal Number” and each section is concerned with the proof of a proposition. However, Frege does not claim to be giving a proof in every section of Part II. As he notes in the introduction, he has divided the sections into those headed “Analysis” (Zerlegung) and those headed “Construction” (Aufbau). He writes, The proofs are contained solely in the sections entitled “Construction”, while those headed “Analysis” are meant to facilitate understanding by providing a

of a function, Φ, looks like this: ἀΦðαÞ. For a more complete explanation of this notion, see Chapter 4, section IIe. ⁵⁶ BLA vol. i, §35.

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preliminary and rough sketch of the proof. The proofs themselves contain no words but are carried out solely in my symbolism.⁵⁷

The stated purpose of the Analysis sections—to assist the understanding—is exactly the kind of purpose that we identified as the purpose of the arguments that Frege labels “elucidations” in §34. Moreover, in a brief section of Part II titled Preliminaries, Frege reiterates the importance of the distinction between the Analysis and Construction sections. He refers to the Analysis sections as commentaries and he writes that they are merely intended to serve the convenience of the reader; they could be omitted without compromising the force of the proof, which is to be sought under the heading “construction” only.⁵⁸

This is almost exactly what he writes in §35 when he claims that the elucidations of §34 might be flawed without placing the correctness of proofs in question.

IIf. The Concept Horse Revisited Does this help with the problem we have been considering? That is, can we identify an elucidatory role that is (successfully) played by Frege’s explanations of the terms “function,” “concept,” and “object” and by his claims that numbers are objects and that an ascription of number is a statement about a concept? One way to think about this is to think about the purpose of Frege’s Begriffsschrift. It is meant to be a language that expresses all content with significance for inference. Thus any statements that are useful either in explaining the workings of Begriffsschrift or in explaining how to express a statement in the language can be viewed as successful elucidations. And surely all of the statements under consideration count. It is not difficult to see the role played by Frege’s terms “function,” “concept,” and “object” in introducing Begriffsschrift. There is a difference between Begriffsschrift object-expressions and Begriffsschrift function-expressions and a difference between the slots that can be occupied by the different categories of expressions. These differences have significance for our understanding and evaluation of inferences. The same holds for the claims that numbers are objects and ascriptions of number are statements about concepts. Both claims can be understood as claims about the correct Begriffsschrift expressions of statements about numbers.

⁵⁷ BLA vol. i, p. v. See also, p. 70.

⁵⁸ BLA vol. i, p. 70.

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To say that numbers are objects is to say something about the sort of definitions we want and about how statements about numbers function in correct inference. Suppose, for example, one has proved a universal generalization from Frege’s logical laws, (x)Φx. Given that the numeral “1” has been introduced into his notation by an appropriate definition, one can prove Φ(1). In contrast, given that a concept expression, say θ, has been introduced into his notation by an appropriate definition, one cannot prove Φ(θ). Indeed, the result of putting θ in the argument place (without adding an argument place for θ) will be ill-formed. Similarly, the statement about ascriptions of number is designed to show us how some things (in this case, certain statements about numbers) are to be expressed in Begriffsschrift. Consider, for example: there are exactly two moons of Mars. Frege’s claim about ascriptions of number tells us that this is a statement about a concept, moon of Mars. Described in terms of the logical language, what Frege’s claim about ascriptions of numbers tells us about this statement is that it should be translated in the way we teach logic students to translate it i.e., (using contemporary notation): (Ǝx)(Ǝy)(Mx & My & x ≠ y & (z)(Mz $ (z=x v z=y))). It should be evident that these claims can all be understood as belonging to the propaedeutic of Frege’s logic. It should also be evident that these claims can be seen as perfectly reasonable attempts to convey an understanding to the reader. Indeed, the claim that ascriptions of number are assertions about concepts is one that can be helpfully used in teaching logic today. One might think, however, that such claims should be viewed, not as belonging to a propaedeutic, but as belonging to a theory: a theory of how to translate natural language into Begriffsschrift. It is important to realize, however, that even if this is right—even if Frege means to be stating literal truths about correct translations of natural language statements into Begriffsschrift—these literal truths form no part of the foundations of Frege’s arithmetic or his logic. First, we can see, from what he actually does, that Frege does not regard such truths as part of the foundations of arithmetic. For Frege intends to provide gapless proofs of the truths of arithmetic from the primitive logical truths on which they depend. If, among the primitive logical truths, there are truths about the correct translations of natural language statements into Begriffsschrift, then his explicit methodology requires him to provide gapless proofs of the truths of arithmetic from these truths about language. Gapless proofs, of course, can be provided only in Begriffsschrift. Thus, if he is to accomplish his aim, the logical truths that underlie arithmetic must be, not merely expressible in Begriffsschrift, but expressed in Begriffsschrift. Second, it is not difficult to see why Frege should not have regarded such truths as logical truths. For truths about the correct translation of natural language statements are truths about natural language, not about logic. A study of natural language and its workings is a special science, not a foundation for logic. And, finally, as we saw in Chapter 1, it is not a part of Frege’s project to give a theory of the workings of natural language.

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Where does this leave us? Frege tells us that the concept horse is not a concept. The apparently paradoxical nature of this statement is due to the defects of the expression “concept” and he argues that, given his understanding of the notion of concepthood, there is no way to introduce a natural language expression for this notion that does not exhibit this kind of defect. Is this a problem? The mere fact that Frege’s account of the notion of concept commits him to apparently paradoxical remarks is not, of itself, an objection to what he is doing. There is, after all, no inconsistency in saying that the concept horse is not a concept, it is merely odd. I have argued that, if we want to evaluate Frege’s view and, in particular, his statements about the notions of function, concept, and object, we need to see what he wants to accomplish by making these statements. As we have seen, if his statements about functions and concepts are taken to be literal truths of a theory, then there is something wrong with the theory. But Frege himself denies that they are meant as statements of literal truths. And, as we have also seen, Frege appears to have used these statements successfully to accomplish a perfectly good task. I have argued that the role many of Frege’s statements about functions and concepts play in his overall project is elucidatory. And I have argued that, even if they are defective statements rather than statements of literal truths, this does not interfere with their successfully playing an elucidatory role. There may seem to be something missing, however, from the discussion of this strategy for resolving the problem. What is missing from this discussion, one might think, is the role that functions and statements about functions must play in a different theory: the metatheory for Frege’s logic. But does Frege have a metatheory for his logic and, if so, is it a metatheory that requires the statement of literal truths using such terms as “function”? We will turn to this issue in Chapter 4.

PART III

METATHEORY AND THE STANDARD INTERPRETATION

4 Soundness, Epistemology, and Frege’s Project As we have seen, Frege tells us that many of his explicit statements—including all those in which the terms “concept” and “function” appear—are defective. And, as we have also seen, these statements are meant to play an elucidatory role in his project. This is not to say that the remarks in question all introduce primitive terms—we saw in Chapter 3 that there are statements that Frege explicitly labels as elucidatory but that are not used to introduce primitive terms. Nor is it to say that the remarks in question are nonsensical—we saw in Chapter 3 that there are unproblematic statements of literal truths that Frege explicitly labels as elucidatory. Rather, what distinguishes elucidatory remarks from other remarks is the role they play. Elucidation belongs, not to a theory, but to its propaedeutic. And, as we saw in Chapter 3, there is no problem in seeing how certain statements, although they are defective, can nonetheless play an elucidatory role. I have argued that this is a solution to the puzzle of how Frege can have explicitly disavowed so many of his statements—including statements that appear to state central features of his view. In order for this to be a solution, however, the defective statements must not belong to the theory. And, on the Standard Interpretation, many of Frege’s natural language statements do belong to a theory, a metatheory for his logic. Insofar as these natural language statements include the kind of statements that Frege identifies as defective, the Standard Interpretation requires us to say that Frege has made a somewhat inexplicable mistake. Frege was explicit about the defective nature of the terms “function” and “concept” during the period in which he was formulating the new version of his logic. Indeed, he described these defects during this period. Had he viewed himself as providing a metatheory for his logic—a metatheory that must be stated using the terms “function” and “object”—it is difficult to see how he could have been so cavalier about the defective natures of these terms. That is, as with the issue of taking sentences to be names of truthvalues, this apparent problem is also an apparent problem with the Standard Interpretation. Does Frege mean to provide a metatheory for his logic? It should be evident that the answer to this question is dependent on what is meant by “metatheory.” Insofar as the enterprise of using natural language to introduce, discuss, or argue about features of a formal system is metatheoretic, there can be no debate:

Taking Frege at his Word. Joan Weiner, Oxford University Press (2020). © Joan Weiner. DOI: 10.1093/oso/9780198865476.003.0004

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Frege does have a metatheory. He introduces and discusses both versions of his new logic in natural language; he argues that his logic is superior to Boole’s by discussing formal features of both logical systems.¹ So there is an obvious and uncontroversial sense in which Frege has a metatheory. But there is something a bit strange about talking about the above activities as metatheory. For, typically, we think of theory as an enterprise that involves the statement of literal truths and the proof of some literal truths on the basis of others. There is no question that Frege is engaged in constructing a theory of this sort for logic and arithmetic. But, as I shall argue, the discussions mentioned above—and, in particular, the discussions by which he introduces his new logic—are not meant to belong to his theory in this sense: they are not meant to involve proofs of literal truths. The natural language statements that he identifies as defective create a problem for Frege only if he needs these statements—as on the Standard Interpretation he does—to be literal truths that are usable in proofs. I shall argue that the statements in question need not be statements of literal truths to play the role Frege needs them to play. If we understand these statements as Frege means us to do, there is no problem. What kind of metatheory does Frege have on the Standard Interpretation? It is widely acknowledged that he does not offer the kind of model theoretic metatheory with which we are familiar today. Frege objects to his logical language, Begriffsschrift, being considered to be subject to multiple interpretations.² But while his objections to viewing sentences as subject to interpretation give him a reason for not introducing a truth-under-interpretation relation, this is no reason for not introducing a truth predicate. It is widely believed that Frege meant to give his logical language a unique interpretation and to use this to offer a kind of protosoundness proof. And we need to use a truth predicate in order to demonstrate that, for example, if a conclusion follows by a particular rule of inference from true premises, then the conclusion will be true. Dummett writes, In Frege’s logical theory, there was for the first time offered an account of the determination of the sentences of a considerable fragment of language as true or as false, and therefore, also for the first time, the possibility, not merely of specifying certain rules of inference as valid, but of demonstrating their validity in the sense of yielding true conclusions from true premises.³

¹ My thanks to Ian Rumfitt for pointing out that some of the statements in my earlier writings suggest that Frege would not have engaged in such discussions. I did not mean to suggest this. ² Frege introduces his logical language, Begriffsschrift, not as a means for representing logical form, but as a means for expressing content (or, as he says in the preface to BS, begriffliche inhalt)). And Frege holds the view of Begriffsschrift as a language expressing content throughout his career. The title of §32 of BLA is “Every concept-script proposition expresses a thought.” He writes, “If something is supposed to express now this thought, now that, then in reality it expresses no thought at all” (OFG II, p. 424). He also writes, “The word ‘interpretation’ is objectionable, for when properly expressed, a thought leaves no room for different interpretations” (OFG II, p. 384). ³ (1981a) pp. 81–2.

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My focus in sections I and II of this chapter will be on whether Frege has a metatheory in the sense just described: that is, a theory that includes justifications of the basic laws and rules of inference, justifications in which a truth predicate—a predicate that holds of true sentences or thoughts—plays an ineliminable role and in which there is a need for variables that range over an infinite number of sentences.⁴ This is the view that I will be calling the “Standard Interpretation” in this chapter. And, for simplicity, I will be using “semantics” and “metatheory” as terms for this sort of theory. In what follows, the logic under examination will be the second version of Frege’s logic: the version set out in Function and Concept and in the first volume of Basic Laws. Why limit the discussion to this version of Frege’s logic? One reason is that there clearly is no truth predicate in the first version, the version set out in his monograph, Begriffsschrift.⁵ Another is that the plausibility of the Standard Interpretation depends on Frege’s having employed a truth-predicate in the discursive sections of Basic Laws. Had he employed a truth predicate in Begriffsschrift and then abandoned it in Basic Laws, this would be a very strong rejection of the metatheory view. As I shall show, the sort of metatheory in question is not part of Frege’s logic in Basic Laws. Frege uses no truth predicate in the justification of his logical rules or basic laws. For, as we shall see, the predicate “is the True”—which does appear in Basic Laws—is not a truth predicate. Nor is Frege’s horizontal a truth predicate.⁶

I. Logical Laws and Metatheory Ia. Quine’s Argument for Semantic Ascent Why has it seemed obvious to so many philosophers that Frege does use a truth predicate in his justification of logical rules? The answer lies, in part, with the contemporary understanding of what is required, not just for the justifications of

⁴ Both the claim that Frege makes ineliminable use of a truth predicate and that he needs to use variables that range over an infinite number of sentences come from Stanley (1996), p. 53. ⁵ Instead of using expressions like “is true” and “is false”, Frege uses the expressions affirms (bejaht) and denies (verneint). ⁶ David Bell has suggested to me in conversation that there is a sense in which the horizontal is certainly a truth predicate—the content of the horizontal concerns truth and only truth. This is certainly correct. However, in this sense, the negation stroke and conditional stroke are also truth predicates. But none of these are truth predicates in the sense of “truth predicate” that is under discussion in this chapter. A truth predicate, in the sense that is under discussion here, is a predicate that holds either of all and only true sentences or of all and only true thoughts. That is, a truth predicate for sentences should be a phrase that can be translated either “is true” or “refers to the True.” The sections in which Frege explains why his basic laws and rules are good laws and rules are §§14–18, 20, and 25. There is no phrase that can be translated as “is true” or “refers to the True” in these sections. Below, I explain why “is the True” is not a truth predicate.

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the rules of logic, but also for its laws and even for expounding the logical theory.⁷ Dummett writes, “Logic can begin only when the idea is introduced of a schematic representation of a form of argument.”⁸ Why is this? Quine gives one explanation, although he discusses only laws, and not rules of inference.⁹ Logic, he writes, “can be expounded in a general way only by talking of forms of sentences.”¹⁰ It is not that logic is about language or about forms of sentences. Rather, it is that there is a problem with the sorts of generalizations required by logic. Consider the clause “time flies” in the sentence “If time flies then time flies.” Quine writes, We want to say that this compound sentence continues true when the clause is supplanted by any other; and we can do no better than to say just that in so many words, including the word ‘true’. We say ‘All sentences of the form “If p then p” are true’. We could not generalize as in ‘All men are mortal’, because ‘time flies’ is not, like ‘Socrates’, a name of one of a range of objects (man) over which to generalize. We cleared this obstacle by semantic ascent by ascending to a level where there were indeed objects over which to generalize, namely linguistic objects, sentences.¹¹

Several features of this passage are worth noting explicitly. First, if we want every sentence of the form “if p then p” to be a logical axiom, it will not be possible to list all axioms individually. The contemporary strategy is to use a schema. We might express Quine’s point in the paragraph just quoted by saying that “P!P” is a logical truth. But, in this case, the expression between the quotation marks is not to be understood as a sentence of the logical notation. Instead, it is to be understood as a schema in which “P” is a metalinguistic variable. The claim that “P!P” is a logical truth is simply shorthand for the claim that, whenever both occurrences of “P” are replaced by a (single) formula of ⁷ My characterization of contemporary justifications is not meant to apply to all contemporary approaches to logic. It does not apply at all, for example, to situation theory. The contemporary justifications that I discuss in this chapter are the model-theoretic justifications to which most contemporary introductory texts appeal. For a classic introduction, see Chang and Keisler (1990, Dover Publications, Third edn 2012). I concentrate on this conception of logic because it is the one that is taken as a development of Frege’s ideas. I do not think it is likely that anyone takes Frege, e.g., to be a proto-situation theorist. ⁸ Dummett (1991) p. 23. ⁹ Although many of Quine’s views about logic are idiosyncratic, the views under discussion here— that we need talk of forms of sentences and truth predicates in order to make general claims (e.g., of infinitely many axioms of the form “P!P,” that they are logical truths)—are not. See, for example, Jason Stanley (1996) p. 53, where Stanley claims that one reason that a truth predicate occurs ineliminably in discussions of the validity of rules of inference is that they are generalizations. And while it is true that many contemporary philosophers take truth to be a property of something other than sentences (propositions or statements, for example), the sort of model theoretic metatheory under discussion requires proof about a truth relation that holds between sentences and interpretations. Of course, Frege himself claims that truth holds primarily of thoughts, not sentences. However, as I also argue below, he does not exploit a truth predicate that holds of all and only true thoughts. ¹⁰ Quine (1960), p. 273. ¹¹ Quine (1992), p. 81.

, ,  ’ 

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the logical notation, the result is a logically true formula. A claim about the logical truth of a schema is really shorthand for a claim about the logical truth of infinitely many formulas of the formal language. One hallmark of this contemporary approach to logic, then, is the use of schemata. Another hallmark of this approach to logic is the use of the truth predicate, where truth is a property of sentences.¹² The importance of the truth predicate is a direct result of semantic ascent. For semantic ascent makes it look as if we are talking about language. The use of the truth predicate, Quine writes, “has precisely the purpose of reconciling the mention of linguistic forms with an interest in the objective world.”¹³ He writes, [T]he truth predicate serves, as it were, to point through the sentence to the reality; it serves as a reminder that though sentences are mentioned, reality is still the whole point.¹⁴

It makes sense, on this view, to talk of the laws of logic as the laws of truth and it makes sense to think that any general account of the logical laws must be metatheoretic. Finally, another noteworthy feature of Quine’s description is that semantic ascent in logic is motivated by the existence of an obstacle. This is particularly interesting because Quine’s explanation of why logic requires semantic theorizing is based on a philosophical position with which Frege disagrees. As Quine sees it, logic requires semantics because sentences (for example, “time flies”) are not names of objects. Yet on Frege’s view sentences are names of objects. As we shall see, this makes semantic ascent unnecessary for Frege, not only for logical laws, but also for rules of inference. In fact, as we shall see shortly, he does not use a truth predicate at all.

Ib. Schemata and Logical Laws The view that we cannot give the laws of logic without generalizing in this way is reflected in contemporary statements of logical laws. Let us consider the contemporary analogue to Frege’s Basic Law 1. An expression that gives the contemporary analogue, “(A!(B!A))”, uses two symbols that do not appear in the logical language. These symbols, “A” and “B,” are metalinguistic variables that range over sentential expressions. It is, strictly speaking, a mistake to say that the ¹² Or, to be more accurate, a relation between sentences and interpretations. I talk of truth being a property of sentences here because I am trying to state something that looks like a contemporary version of Frege’s view and Frege objects to viewing Begriffsschrift expressions as subject to multiple interpretations. ¹³ Quine (1986), p. 14. ¹⁴ Quine (1986), p. 11.

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contemporary expression is a statement of a law. Rather, it is a schema that has infinitely many instances, each of which is a statement of a law. From a contemporary point of view, we cannot write down a finite list of axioms (or basic logical laws) for classical logic—what we give is a finite list of schemata. All instances of the schemata are axioms. Thus to say what the axioms are, we must resort to metatheory. For example, we make a metatheoretic claim that all instances of the schema “(A!(B!A))” are axioms (or basic logical laws). Now suppose that we need—for whatever reason—some sort of justification of the axioms in question; some explanation of why they are good axioms. Since we cannot be expected to give infinitely many justifications, the only rational thing to do is to give a metatheoretic argument that all instances of our schema are true. Let us now compare the contemporary analogue of Frege’s Basic Law 1, with his actual law. (For convenience, in the discussion that follows I will continue to use the contemporary conditional symbol rather than Frege’s conditional-stroke.) Frege’s statement of Basic Law 1, in contrast to the contemporary statement, is not a schema but, rather, a statement of a single law directly expressible (and expressed) in Begriffsschrift. He asserts Basic Law 1 by writing something that looks like this: ⊢(—a → — (—b → —a)),

where the Roman letters ‘a’ and ‘b’ are not metalinguistic variables but actual symbols of his logical language. In order to understand this better, it will help to consider some of the features of Frege’s logic that have no contemporary analogues. As we saw in Chapter 2, the first version of Frege’s logical language has two symbols, the judgment stroke and the content-stroke, that have no contemporary analogues. Frege’s judgment stroke is a vertical line that, in the original version of his logic, can be attached to the left-most content-stroke in a Begriffsschrift expression. In the new version of his logic, the content-stroke is called the “horizontal.” The difference is more than terminological. The content-stroke of Begriffsschrift could only be prefixed to a sentential expression. In Basic Laws, however, there are no special formation rules for sentential expressions. They are simply proper names. Thus, when he introduces the new version of the content-stroke he introduces it as a first-level function sign. He writes, —Δ is the True when Δ is the True, and is the False when Δ is not the True.¹⁵

After pointing out that the horizontal function always has a truth-value as its value, he continues,

¹⁵ BLA vol. i, p. 9, emphasis added.

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Under this concept there falls the True and only the True.¹⁶

That is, the horizontal is a symbol for the natural language predicate “is the True” (or, “= the True”).¹⁷ Let us turn now to Frege’s judgment stroke. This symbol is used, he tells us, in order to assert something. What is asserted in the context of a proof will be a complete line of that proof. Thus, any complete line in a Begriffsschrift proof must begin with a judgment stroke. And the judgment stroke must be attached to the left-most horizontal in the Begriffsschrift expression. For current purposes, it will help to restrict our attention to the technical functions of this symbol in Frege’s logic. In addition to marking off the lines of a proof, there is another technical role played by the judgment stroke for the expression of generality. Frege begins Basic Laws by using Roman letters to express generality. However, in section 17, he notes that this will not do. He writes, Above we attempted to express generality in this way using a Roman letter [lateinischen Buchstaben] but abandoned it because we observed that the scope of generality would not be adequately demarcated.¹⁸

His solution is to introduce quantifiers, whose purpose is both to express generality and to mark off scope. These quantifiers are constructed by using a concavity (in what follows, I use parentheses for convenience) and gothic letters rather than Roman letters. But Frege does not give up the use of Roman letters for the expression of generality, he writes, We now address this concern by stipulating that the scope of a Roman letter is to include everything that occurs in the proposition apart from the judgementstroke.¹⁹

Thus, while there are no actual quantifier symbols in the above expression, this assertion is a universally generalized statement within Frege’s logical language. That is, the assertion of Basic Law I, which Frege expresses with Roman letters, is to be understood as the assertion of a universal generalization “for any a and b . . . ”. Frege’s statement of Basic Law I contains no metalinguistic variables.

¹⁶ BLA vol. i, p. 10. ¹⁷ See also FC, p. 21. ¹⁸ BLA vol. i, p. 31. ¹⁹ BLA vol. i, p. 31. The situation is actually more complicated than this indicates, since Frege also uses Roman letters to extend generality throughout several lines in a proof. Also, Frege adheres to this convention throughout his career from Begriffsschrift on. In 1896, in an article about Peano’s logic, he writes, Now when the scope of the generality is to extend over the whole of a sentence closed off by the judgement stroke, then as a rule I employ Roman letters. PCN, p. 377.

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Does Frege need a truth predicate to explain why Basic Law I is a good law? The actual explanation consists only of a few brief comments. He writes: According to section 12,

— (—Γ→ — (—Δ→—Γ)) would be the False only if Γ and Δ were the True while Γ was not the True. This is impossible; accordingly

⊢(—a → — (— b → —a)) (I)²⁰

It may seem that the absence of metalinguistic variables is beside the point, here. For Frege uses the predicate “is the True” in the above explanation of Basic Law I. And this, one might think, is clearly a truth-predicate. However, as we shall see in section IIb, “is the True” is not a truth predicate. Before we turn to the issue of how Frege understands the predicate “is the True,” let us consider a slightly different reason for thinking that Frege’s logic must have a metatheory: the necessity of rules.

II. Modus Ponens, Logical Laws, and Metatheory IIa. The Justification of Modus Ponens There is an important difference between laws of logic and rules of inference. As we have seen, Quine claims that there is no way of setting out the logical axioms without resorting to metatheory. But, even if we cannot offer a list of the axioms in the logical language, we can still set out any individual axiom in the logical language. In contrast, not even a single rule of inference can be stated in the logical language. Thus one might think that a discussion of logical laws is beside the point; that all we need do is think about rules of inference to see that Frege must have a metatheory. For instance, Jason Stanley writes, The justification of the inference rule of modus ponens lies in the fact that if the conditional is true, and its antecedent is true, then so is the consequent.²¹

Stanley also writes that the variables occurring in the statement of the validity of an inference rule “range over an infinite number of sentences.”²² That is, we might ²⁰ BLA vol. i, p. 34. ²¹ Stanley (1996) p. 47. In this claim, Stanley could be talking about thoughts rather than sentences. However, if what is it issue is metatheoretic justification of a particular syntactic rule in a logical system, the variables must be thought of as ranging over sentences. Moreover, as the next quotation shows, Stanley does describe statements of the validity of inference rules as containing variables that range over sentences. ²² Stanley (1996), p. 53.

, ,  ’ 

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also say: for any sentences A and B, if A ! B is true and A is true, then B is true. On this contemporary view, the justification of modus ponens requires both the use of a truth predicate and generalization over linguistic entities. Bearing this in mind, let us consider Frege’s explanation of the justification of modus ponens. The explanation, in its entirety, is: [F]or if Γ were not the True, then since Δ is the True (das Wahre ist)

—(—Δ →—Γ) would be the False.²³

Is this a metatheoretic justification of modus ponens involving both the use of metalinguistic variables to generalize over linguistic entities and a truth predicate? Let us consider, first, the issue of the truth predicate. Although Frege does not use a straightforward truth predicate (e.g., something that might be translated by “is true”) in the passage quoted above, he does use an apparently similar expression, “is the True” (das Wahre ist). Is Frege’s predicate simply a peculiarly worded truth predicate? To answer this question, we need to look again at Frege’s use of the expressions “the True” and “the False”.

IIb. Concepts as Functions; Why “is the True” Is not a Truth Predicate As we saw in Chapters 2 and 3, Frege’s function/argument analysis of sentences requires us to take predicates as function-names and sentences as object-names. Quine is among the philosophers who dismiss Frege’s assimilation of sentences to names of objects without argument; he simply states that sentences are not names.²⁴ This much is certainly true: sentences and proper names play different roles in natural language. But Frege’s logical language, Begriffsschrift, is not a natural language. And, as we saw in Chapter 2, Frege’s taking sentences to be proper names does not interfere with the purposes for which his logical language was explicitly designed. In particular, Begriffsschrift is intended to be a means for evaluating the legitimacy of any inference on any subject and for preventing any presupposition from sneaking into an inference unnoticed. Once our inferences are expressed in Begriffsschrift, it is supposed to be a mechanical task to determine whether an inference is correct and gapless, or whether it requires an unstated premise. We should be able to see by inspection whether or not a statement is a primitive logical law; whether or not the transition from one statement to another follows by Frege’s rule of inference.²⁵ Neither Quine nor Dummett (nor anyone ²³ BLA vol. i, p. 25. ²⁴ Quine (1986), p. 11. ²⁵ On Frege’s view, a rule of inference is a rule in which a new judgment is derived from more two or more judgments. However, there are also rules in Frege’s logical system that allow us to go from a

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else to my knowledge) offers any reason to suppose that the assimilation of sentences to proper names interferes with any of the aims of Begriffsschrift. The objection to taking sentences to be names is from the point of view of a theory of the workings of natural language. Let us continue, then, with our investigation of the consequences of the view that sentences are a kind of name. We saw earlier that Frege introduces the True and the False in order to make out his claim that a concept is to be understood as a sort of function. Since concept expressions are predicates, the expression for the value a concept has for a particular object will be a sentence. For example, “2 is a prime number” is an expression for the value the concept prime number has for 2. Since 2 is a prime number, “2 is a prime number” designates the True. Since it is true that 3+2=5, “3+2=5” also designates the True, as do all other true sentences. One consequence is that Frege would endorse the following identity: (2 is a prime number) = (3+2=5) Similarly, all false sentences designate the False. In particular, (1= 0) = (the Moon is made of green cheese). Of course, the two expressions set off above are not English sentences. Nor does Frege propose that we add such expressions to our list of natural language sentences. Rather, it is in Begriffsschrift that we find expressions that consist of two sentences flanking the identity sign. Were we to try to translate them into English, they would look like the expressions set off above. One upshot of this view is that there is no use of a truth predicate that holds of true sentences in Frege’s explanation of Basic Law I or in his explanation of modus ponens. For the predicate “is the True,” which appears in these explanations cannot be a predicate that holds of all true sentences. To see why, let us think again about Frege’s view that sentences are proper names. In “On Concept and Object” Frege writes, a name of an object, a proper name, is quite incapable of being used as a grammatical predicate.²⁶

statement to another that is equivalent to it. He does not identify such rules as rules of inference but, rather, as “transformation rules.” For more on this see section IIId. The only rule of inference in the first version of Frege’s logic is modus ponens (see BS, §6). In the later version, in order to provide shorter proofs, Frege includes more rules of inference. ²⁶ CO, p. 193.

, ,  ’ 

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It is not, Frege continues, that we cannot use predicates in which a proper name follows “is” (as we have already seen, Frege recognizes the legitimacy of writing, for example, “The Morning star is Venus”). It is that in these predicates “is” is not the copula but, rather, the “is” of identity. These comments apply to “the True” as well as to “Venus.” Since the True is an object, “the True” is an object-name. And the “is” in the predicate “is the True,” as in the predicate “is Venus,” is the “is” of identity. That is, what is expressed by the predicate “is the True” is just what is expressed by the predicates “is identical to the True” and “= the True.”²⁷ It follows, since there are distinct true sentences, that the predicate “is the True” cannot hold of all true sentences. To make this vivid, let us suppose for the moment that this predicate does hold of all true sentences.²⁸ It is important to note, first, that Frege himself introduces the use/mention conventions that we use today. He writes, in the introduction to Basic Laws, Someone may perhaps wonder about the frequent use of quotation marks. It is by this means that I distinguish cases in which I speak of the sign itself from cases in ²⁷ Jamie Tappenden has argued that this assertion about the meaning of “is the True” is unjustified. He writes, But even if we grant that Frege’s horizontal is not a truth-predicate (which seems to me a bit of a stretch) there’s nothing in these sections to indicate that Frege holds that the expression ‘is the True’ as it occurs in Grundgesetze is to be regimented as the horizontal, or as the expression ‘( ) = the True’, or as the predicate ‘is true’ introduced in section 8 of this paper, or anything else. Frege introduces ‘the True’ first at section 2, and then at section 4 defines the horizontal in terms of ‘denotes the True’. Tappenden (1997), p. 200. Several of Tappenden’s comments here are simply mistaken. When Frege introduces the horizontal (in section 5, not section 4 of BLA—there is a typographical error in the Tappenden paper), he does not define the horizontal in terms of “denotes the True.” There are two statements that look as if they could be definitions. The first is, —Δ is the True when Δ is the True (das Wahre ist), and is the False when Δ is not the True (nicht das Wahre ist). (BLA vol. i, p. 9) Frege then explains that it is a concept (a function whose value is always a truth value) and writes, “Under this concept falls the True and only the True.” (BLA vol. i, p. 10). These two statements are the only statements Frege makes that can be taken as a definition of the horizontal. Neither employs an expression that can be translated “denotes the True.” Moreover, I do not see how the latter passage can be read as saying anything other than that the horizontal is to be understood as expressing the predicate is (identical to) the True. Therefore, given that “is the True” is to be understood as “= the True”, it does not change the meaning of Frege’s initial introduction of the horizontal to re-write it as follows: —Δ is the True when Δ = the True (das Wahre ist); and is the false when Δ ≠ the True (nicht das Wahre ist). That is, the horizontal is a Begriffsschrift expression for “= the True.” ²⁸ Of course, since “is the True” does not hold of sentences on Frege’s view, the statements set off below are not correct. Rather, on his view, (1+1=2) = the True and (2 2’ refer to [bedeuten] the same truth-value, which I call for short the False, exactly as the name ‘2²’ refers to [bedeutet] the number Four. Accordingly, I call the number Four the reference [die Bedeutung] of ‘4’ and ‘2²’, and I call the True the reference of ‘3 > 2’.¹⁰³

One might think that this provides evidence that Frege takes the name/bearer relation as a model for how language is to be used to talk about the world; that the relation of language to a world that is independent of us is to be understood as

¹⁰² BLA vol. i, p. xiii.

¹⁰³ BLA vol. i, p. 7, emphasis is in the original.

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based on our interactions with objects. But, again, this reading does not survive scrutiny of the passage in question. What we have just seen in the discussion of the Yellow Sea is that, while our interactions with a particular object may form a part of our ability to refer to it or to have knowledge about it, they form no part of the general conception of what it is for a proper name to have Bedeutung. Similarly, there is no notion of interaction at play in his examples of proper names with Bedeutung in the passage from §1 (the numerals). He writes, without argument, that it is utterly incomprehensible why something that has being independently of the judging subject has to be actual, i.e., has to be capable of acting, directly or indirectly, upon the senses.¹⁰⁴

Frege clearly means to be talking about extra-linguistic entities in the following sense: he wants to make sure we distinguish numbers from numerals. Arithmetic does not belong to fiction and the topic of arithmetic is not linguistic expressions. But, as we have seen in this chapter, to determine that a particular linguistic expression has Bedeutung, what is required is not that we show that some sort of relation has been established between the expression and an extra-linguistic entity. Rather, what is required is that we show that the expression in question can be used to state truths, to figure in proofs. That is, Frege continues to hold the sentential priority view. It is not semantic theory that is needed to show us that we are not wandering aimlessly in the realm of fiction. No semantic task is required to show that (at least some of ) our sentences are true or false. What we need, rather, is science. In the pursuit of science, there may be times in which our arguments require data—data that must be obtained, in part, by interactions with objects. But there is no obvious reason to think that all arguments must be based on such data. And, as we will see in Chapter 8, the interactions can form only a part of our obtaining such data. What about Frege’s definitions of the numbers? To show that the truths of arithmetic are analytic is to show that they can be given gapless proofs from primitive logical laws. Since there are no number names in any primitive logical laws, these expressions must be defined. But what is required of definitions of the numbers? Let us turn to this topic in Chapter 6.

¹⁰⁴ BLA vol. i, xviii.

6 The Context Principle, Sentential Priority, and the Pursuit of Truth The official statement of what is often called the context principle—the statement that appears in the introduction to Foundations—is a statement of a methodological principle for defining expressions of natural language. But, as we saw in Chapter 5, this methodological principle is licensed by a view about language: [I]t is only in the context of a proposition that words have any meaning (Nur im Zusammenhange eines Satzes bedeuten die Wörter etwas)¹

And it is because of this, Frege continues, that our problem becomes this: To define the sense (den Sinn eines Satzes zu erklären) of a proposition in which a number word occurs.

It is important to see that the task Frege describes here is very different from that of satisfying what I have called the metaphysical requirement. There is no talk about identifying an extra-linguistic entity that a number word names. And, as we also saw in Chapter 5, Frege adheres to the same view in Basic Laws. For, according to Basic Laws, in order to fix the Bedeutung of a Begriffsschrift expression, what needs to be done is nothing more nor less than to fix the truth-values of all Begriffsschrift sentential expressions in which it appears. Moreover, Begriffsschrift is supposed to be a language to which new expressions can always be added. Thus, Frege’s view precludes the possibility of fixing the Bedeutung of an expression once and for all. For, with the addition of new expressions, there will be new sentences whose truth-values must be fixed.² As we saw in the previous chapter, this conception of having Bedeutung, which presupposes sentential priority, is very much at odds with the conception of having Bedeutung as referring to an (extra-linguistic) entity. It may be useful to ¹ FA, §62. ² One might suppose that, if a particular Begriffsschrift expression has a Bedeutung and if we add a new Begriffsschrift expression, stipulating, in natural language, what the expression is meant to stand for, then the truth-value of the relevant identity statement will have been determined. But Frege’s argument in §10 shows that this is not right. For he shows that, no matter what meaning is assigned to “η”, there is a problem.

Taking Frege at his Word. Joan Weiner, Oxford University Press (2020). © Joan Weiner. DOI: 10.1093/oso/9780198865476.003.0006

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say a bit more about this before addressing the issue of what Frege requires of definitions of the numerals and the predicate “number.” Let us begin by supposing that, on Frege’s view, an object-expression has Bedeutung just in case it names an (extra-linguistic) entity. And let us consider Frege’s view about two of his natural language terms of art, “the True” and “the extension of the horizontal.” Consider, first, “the True.” Frege offers no definition for this term and seems to think that none is needed. For, he writes, These two objects [the True and the False] are recognized, if only implicitly, by everybody who judges something to be true.³

If we combine Frege’s statements about the True and the False with the view that an object-expression’s having Bedeutung is its referring to an (extra-linguistic) entity, there is every reason to think that Frege regards “the True” (and “the False”) as having Bedeutung and his introduction of these expressions as an elucidation that makes it clear what he is talking about when he uses them. Similar considerations apply to the object-expression “the extension (valuerange) of the horizontal.” After all, presuming “the True” and “the False” refer to extra-linguistic entities, Frege offers a perfectly good definition of the horizontal function: its value is the True for the True as argument; the False for every other argument. Moreover, since this function always has truth-values as its values, it is a concept. And Frege writes, in Foundations, that he assumes it is known what the extension of a concept is.⁴ Thus, Frege would seem to regard the (natural language) object-name “the extension (value-range) of the horizontal” as having Bedeutung. Now consider the consequences of supposing that “the True” and “the extension of the horizontal” have Bedeutung and supposing that to have Bedeutung is to be correlated with an extra-linguistic entity in the universe. Then both expressions would be names of particular non-linguistic entities—either the same entity in both cases or different entities. Thus, the identity statement “The True = the extension of the horizontal” (and, hence, also the corresponding Begriffsschrift identity statement) should be either determinately true or determinately false. But, as we saw in the previous chapter, this is not consistent with what Frege says about the issue. For, on Frege’s view, we are entitled to stipulate that the Begriffsschrift identity statement is true. One might think, however, that the conception of Bedeutung that aligns with sentential priority conflicts with Frege’s avowed epistemological project.⁵ For, one ³ SB, p. 34. ⁴ FA, §§69, 107. Moreover, as his discussions of the notion of value-range in Basic Laws indicate, he assumes it is known what the value-range of a function is. ⁵ It may seem odd to introduce the issue of whether or not numerals are (already) names of extralinguistic entities given the argument, in Chapter 5, that Frege’s term “Bedeutung” should not be

  ,    

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might think, what distinguishes arithmetic from other sciences is its subject matter: arithmetic is about the objects named by our numerals, that is, numbers. Thus, it may seem that Frege’s project requires him to use his definitions to pick out particular objects in the universe that are (already) named by our numerals. For, if Frege’s definitions do not actually pick out the objects that form the subject matter of arithmetic, how can he claim to be showing the truths of arithmetic are analytic? How can he not be simply changing the subject? There are several distinct issues here. One is an interpretive issue. I shall argue, first, that Frege does not view his project as requiring the formulation of definitions that pick out objects in the universe that are (already) named by our numerals and that, nonetheless, he did not regard his procedure as changing the subject. I shall also argue that his view is entirely reasonable. If we are committed to saying that Frege’s definitions of the numbers change the subject, we should likewise be committed to saying that many unexceptionable scientific investigations of empirical phenomena are subject-changing as well. And I shall argue that there is abundant evidence that we do not regard these investigations as subject-changing. Insofar as it seems to many contemporary philosophers that Frege’s project sodescribed changes the subject, it is because of mistaken views about truth. In Chapter 7, I shall illustrate how this works in one kind of contemporary philosophical literature: the literature on the semantics of languages with vague predicates.

I. Changing the Subject and the Logicist Project: What makes Logicism about our Arithmetic? What is required of a definition of the number one that is to provide part of a foundation for arithmetic? It would seem that when we use, say, the numeral ‘1’ it must be to talk about an extra-linguistic entity. Thus, when Frege asks what “the number one is, or what the symbol 1 means,”⁶ it is natural to suppose that the

conflated with the term “reference.” But the point of the Chapter 5 argument is that it is a mistake to take Frege to be using “Bedeutung” to introduce a kind of semantic theory, that is, to introduce a theory on which the truth-value of a sentence is determined by the reference relations that hold between its subsentential linguistic expressions and extra-linguistic entities. I argued, in particular, that there is no role to be played in Basic Laws by a demonstration that all Begriffsschrift expressions refer to extralinguistic entities. But it remains reasonable to talk about whether or not a linguistic expression names an extra-linguistic entity—we might still want to say that e.g., “Julius Caesar” names someone, but “Odysseus” does not. ⁶ FA, p. i. Although, in the English translation, the fact that these are two characterizations of the same question is only implicit, it is explicit in Frege’s actual words. Frege’s sentence begins, “Auf die Frage, was die Zahl eins sei, oder was die Zeichen 1 bedeute . . . ” Both characterizations are characterizations of die Frage.

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answer to this question is a matter of fact, not stipulation. And the second of these two formulations of the question (what the symbol 1 means) suggests that he is asking for a definition that picks out what it is that we have been—already—using the symbol ‘1’ to talk about. When we say, for instance, that 0 ≠ 1 (that 1 is not an even number; that 1 is the successor of 0, that the Earth has 1 moon, etc.) we are, presumably, talking about something extra-linguistic. It seems reasonable to suppose, then, that an acceptable definition of the number 1—a definition that avoids changing the subject—must contain a description that picks out precisely the object that is already mentioned in all these statements. Similarly, it seems reasonable to suppose that a definition of the concept of number must contain a description that picks out precisely those objects that are numbers. That is, it may seem only reasonable to attribute to Frege the following apparently obvious faithfulness requirement: The apparently obvious faithfulness requirement: A definition of an object expression (concept expression) must pick out the object to which the expression already refers (objects of which the expression already is true).

This faithfulness requirement requires Frege to come up with definitions that pick out the extra-linguistic entities that we are (already) talking about when we use the expressions of arithmetic. For convenience, I will characterize this requirement as the requirement that the definition preserve reference. For all the apparent obviousness of this requirement, Frege never explicitly says that his definitions must satisfy it; nor does he ever argue that his definitions do satisfy it. There is, however, an interpretation that is consistent with Frege’s holding the apparently obvious faithfulness requirement and that also explains why he never explicitly stated it. On this interpretation, when he wrote Foundations, he was groping for a somewhat different constraint on definitions, but one that he could not yet express: that the definitions must express the sense these expressions already have. This idea fits with his later statements that a definition determines, not just the Bedeutung of an expression, but also its sense. He writes, for instance, that the sense of the definiendum “is built up out of the senses of the parts of the definiens.”⁷ Moreover, since the sense of an expression picks out the object (if any) to which an expression refers, definitions of the numerals that preserve sense and pick out objects will thereby pick out objects that we have been talking about all along. That is, on this interpretation, the apparently obvious faithfulness requirement must be satisfied by any acceptable definition. And, on this interpretation, we can see why Frege does not bother to articulate the apparently obvious faithfulness requirement: the satisfaction of this ⁷ Frege (1914), PW, p. 208/NS, p. 224. See also, Frege OFG II, p. 323; (1895), p. 75; (1918/19b), p. 150.

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requirement will be a consequence of the real faithfulness requirement, the preservation of sense. There are, however, a number of problems with this interpretation.⁸ First, as with the apparently obvious faithfulness requirement, the requirement in question is one that Frege never articulates. Second, given his very limited explanations of the notion of sense, it is far from clear what sense is and, consequently, what a sense-preservation requirement really amounts to. It is important to recall, for example, that the most common contemporary view⁹— that sense is “mode of presentation” (Art des Gegebenseins)—is both wrong and unhelpful. In the passage typically quoted in support of the claim that sense is mode of presentation, Frege says, not that sense is mode of presentation, but that sense contains mode of presentation. Moreover, the expression “mode of presentation” appears only rarely in Frege’s writings. In particular, it does not appear when Frege first introduces the notion of sense (which is not in “On Sinn and Bedeutung,” but in Function and Concept). In “On Sinn and Bedeutung,” Frege mentions mode of presentation only four times, after which he seems to have dropped the locution.¹⁰ But, even were there evidence that, by “sense,” Frege means mode of presentation, it would not be much help. Were sameness of mode of presentation Frege’s criterion for the acceptability of his definitions, we would need a way to recognize sameness of mode of presentation. As we shall see shortly, the real problem with this interpretation of Frege’s faithfulness requirement—as with the apparently obvious faithfulness requirement—is that it conflicts with many of Frege’s explicit statements. In particular, both interpretations are based on an assumption that, I shall argue, is inconsistent with Frege’s writings. The assumption is that there is a definite particular object that we are already talking about when we use the numeral “1.” Supposing both preservation of sense and preservation of reference are ruled out as faithfulness requirements. What faithfulness criteria must Frege’s definitions satisfy? And how can these definitions teach us anything about our science of arithmetic? To answer these questions, we need to begin with what it is that Frege thinks we need to learn about the science of arithmetic.

⁸ I have discussed some of the problems with this interpretation at more length in Weiner (2007). ⁹ For example, Tyler Burge writes, “A sense is a mode of presentation that is ‘grasped’ by those ‘sufficiently familiar’ with the language to which an expression belongs” Burge (1990), p. 243. For some encyclopedia articles in which this is stated, see e.g., Kevin Klement’s Frege entry in The Internet Encyclopedia of Philosophy [https://www.iep.utm.edu/frege/]; Jill Buroker’s 2014 entry on the Port Royal Logic to the Stanford Encyclopedia of Philosophy [https://plato.stanford.edu/entries/port-royallogic/]. ¹⁰ There is, however, a related locution that appears in his later work. He mentions, in “Thoughts,” how an object gegeben ist, (1918/19a), pp. 65–6. and, he continues, such a Weise corresponds to a particular Sinn.

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Ia. Why Define the Number One and Concept Number? As we saw earlier, Frege wants definitions of the number one and the concept number in order to determine the epistemological classification of the truths of arithmetic. This project requires proofs of the truths of arithmetic from primitive truths. But how are we to recognize which truths are primitive? And how are we to identify which place a primitive truth occupies in Frege’s classificatory scheme? One of the striking features of Frege’s discussion of his system for classifying truths is that he never answers these questions. As we saw in Chapter 4, however, we can make some inferences from what he does write about how these questions should be answered. In particular, because the point of proving truths of arithmetic from primitive truths is to enable us to determine the correct classification of the arithmetical truths, there will be eligibility conditions that determine what truths can be taken as primitive. One eligibility condition is that it must not require proof—its truth must be evident without proof.¹¹ Another is that there must be some means, other than examining a proof, of determining whether the truth in question is a fact about particular objects (synthetic a posteriori), a primitive general truth of some special science (synthetic a priori), or a general logical law (analytic). For, even if we take each primitive truth to constitute its own trivial one-line proof, the fact that it constitutes its own trivial proof will not, on its own, tell us how to classify this truth. Thus, if a truth is to be primitive, it must be evident, without proof, whether it is analytic, synthetic a priori or a posteriori. Given that the project is to classify the truths of arithmetic, we must either recognize the simplest truths of arithmetic as primitive truths or prove these truths from primitive truths. Where do definitions of the number one and the concept number come in? Frege writes that, in the process of proving a truth from primitive truths, [W]e very soon come to propositions which cannot be proved so long as we do not succeed in analysing concepts which occur in them into simpler concepts or in reducing them to something of greater generality. Now here it is above all Number which has to be either defined or recognized as indefinable.¹²

He suggests that all positive integers can be defined from one and increase by one.¹³ Hence, one might think, some of the axioms that underlie all truths of

¹¹ For more on the issue of self-evidence and its role in Frege’s project, see Chapter 4, n74. ¹² FA, §4. ¹³ This is somewhat misleading since, when Frege actually gets to the point of defining the numbers, he begins with 0 rather than 1. I suspect that Frege’s reason for directing our attention to the number 1 rather than the number 0 is that he thinks that some of his audience, particularly those with an empiricist bent, will be suspicious of the number 0. See, for instance, FTA, p. 97; FA, §8.

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arithmetic will be about the number one. Other likely candidates are general laws about numbers (e.g., the commutative law for addition).¹⁴ Why should we not recognize such truths as primitive-eligible? That is, why should we need definitions of the number one and the concept number? In order to address this question, it may help to compare simple truths about numbers to a truth that, Frege indicates, clearly is eligible to be taken as primitive: that every object is identical to itself.¹⁵ Since the truth of this law, which Frege calls the “law of identity,” is self-evident, it satisfies the first of the eligibility requirements for primitive laws.¹⁶ Supposing it to be a primitive law, is it analytic—that is, is it a general logical law? As we have seen, the hallmark of a logical law is its generality. Logic is independent of the special sciences. The laws of logic, Frege writes, are those “upon which all knowledge rests.”¹⁷ He also distinguishes analytic truths from other truths on the grounds that they cannot be denied in conceptual thought—that is, cannot be denied “without involving ourselves in any contradictions when we proceed to our deductions.”¹⁸ This is in contrast to the axioms of geometry, which are synthetic on Frege’s view because we can assume the contrary of an axiom of geometry without involving ourselves in contradictions.¹⁹ Given these criteria, it should be clear that the law of identity is analytic; it belongs to logic. First, this law surely tells us not just about every actual (spatio-temporal) object or every intuitable object, but about every object. Second, it seems that

¹⁴ I mention these candidates for primitive truths because they appear in Foundations discussions. But one might also think that there are more obvious candidates: the Dedekind–Peano axioms. Using these axioms, we seem to be able to get arithmetic from two primitive undefined terms (“0” and a sign for the successor function). Although Frege does not actually discuss these axioms, it is pretty clear that the considerations I raise below apply in exactly the same way. In fact, of course, Frege begins his definitions with “0”, not “1,” and the Dedekind–Peano axioms are among the truths he wants to prove from logical laws and definitions. ¹⁵ Note that although this law is easily derivable from two of Frege’s basic laws, it is not itself one of his basic laws. Since one of Frege’s aims is to prove everything from the smallest possible number of primitive laws (see, for example, BLA vol. i, p. vi), many laws that are eligible to be taken as primitive laws of logic are derived, rather than primitive, laws of Frege’s system. For additional discussion, see Chapter 4, section III. ¹⁶ BLA vol. i, pp. xvi–xvii. ¹⁷ BS, preface. ¹⁸ FA, §14. There is also an apparently stronger version of this statement in Foundations, Frege suggests that the truths of arithmetic must be analytic because “Here, we have only to try denying any one of them, and complete confusion ensues” (FA, §14). But this sounds a bit too psychological and, perhaps for that reason, Frege seems to have retreated from it as a criterion. He writes, in Basic Laws that we may say we must acknowledge the law of identity “if we do not want to lead our thinking into confusion and in the end abandon judgement altogether” (BLA vol. i, p. xvii), but then adds “I neither want to dispute nor to endorse this opinion”. On the other hand, he writes, in one of his last papers, The assertion of a thought which contradicts a logical law can indeed appear, if not nonsensical, then at least absurd; for the truth of a logical law is immediately evident of itself, from the sense of its expression. (CT, p. 50) It is not obvious, however, that these statements conflict with one another. For the passage I have quoted above in the main text can be viewed as providing a non-psychologistic interpretation of the remark about confusion—that the confusion involved in denying such a truth is simply that it forces us to involve ourselves in contradictions when we proceed to our deductions. ¹⁹ FA, §14.

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we cannot deny it, without involving ourselves in contradictions, in conceptual thought. Can we say something comparable about the simple truths about numbers and the number one? As with the law of identity, Frege seems to think that we can see, without proof, that the fundamental truths of arithmetic are true. He suggests without argument that the fundamental propositions of the science of number have the same status as logical laws—that denying them will involve us in contradictions. But do they have the requisite maximal generality? Frege writes that the truths of arithmetic govern the widest domain of all.²⁰ For we can, and do, use numbers in discussions of virtually every subject matter. Everything that can be an object of thought, including mental and physical phenomena, spatial and non-spatial phenomena, can be counted.²¹ It seems to follow that these truths belong to logic. But, for all that, Frege never makes the decision to add any basic truth of arithmetic—for example, the Dedekind–Peano axioms—to his list of primitive logical laws. Why not? It is easy to see an important difference between number statements and the law of identity. The law of identity is obviously general. Were someone to ask why this statement applies to everything, it would be difficult to see how to give a substantive answer. But, for all Frege’s remarks about the universal applicability of arithmetic, the truths of pure arithmetic, at first glance, seem clearly not to have the requisite maximal generality. Frege writes, For a truth to be a posteriori, it must be impossible to construct a proof of it without including an appeal to facts, that is, to truths which cannot be proved and are not general, since they contain assertions about particular objects.²²

Frege argues at length that numbers are objects.²³ Thus, the claim that 0 is not equal to 1, for instance, would seem to be—not a general truth, not a truth that tells us something that holds for every object—but a particular truth that tells us something about two particular objects. If we take this to be a primitive truth it would, by the above criterion, be an a posteriori truth. Nor do the general laws of arithmetic seem maximally general. These laws seem to govern, not the widest domain of all, but a specific domain, the domain of numbers. Indeed, Frege writes

²⁰ [das umfassendste], Frege FA, §14. This is not, however, to contrast the variables of arithmetic with variables of other sciences. For Frege, all variables range over an unrestricted domain (see Chapter 4, section IIIc). Rather, to say that the truths of arithmetic govern the widest domain of all is to say that they do not express, for example, “the peculiarities of what is spatial” as Frege says in FTA, pp. 94–5. In his later writings, Frege describes this maximal generality somewhat differently. Laws of logic, he says, can be called “laws of thought,” “only if thereby it is supposed to be said that they are the most general laws, prescribing how to think wherever there is thinking at all” (BLA vol. i, p. xv). This is opposed to laws of geometry or physics, which provide a guide to thought only in restricted fields. See also, Frege (1897), PW, pp. 145–6/NS, pp. 157–8. ²¹ See, for example, FA, §§24, 48. FTA, p. 94. ²² FA, §3. ²³ FA, §§55–61.

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that inferences by mathematical induction appear to be “peculiar to mathematics” and hence that one of his tasks is to show that such an inference is “based on the general laws of logic.²⁴ Thus, if there is anything to Frege’s claim that the truths of arithmetic do have the generality of analytic truths, no recognizable truth of arithmetic should be a primitive truth. But why think all truths of arithmetic could be provable from truths that do not mention particular numbers? Frege’s answer requires us to see, first, that statements of number can be understood as assertions about concepts. To say that Venus has 0 moons, for instance, is to say nothing more nor less than what is captured by the (partially regimented) statement: ~(Ǝx)(x is a moon of Venus). As is clear from the regimentation this tells us something about the concept moon of Venus (i.e., that nothing falls under it).²⁵ There is no mention of a particular number. Moreover, concepts will be involved in any statement about any subject matter—it is not insignificant that Frege calls his logical notation “Begriffsschrift,” or “concept-script.” Thus once we recognize that statements of number are assertions about concepts, it makes sense that arithmetic should be applicable in every domain. One difference, then, between primitive logical laws and truths of arithmetic, is that the generality of primitive logical laws—but not of simple truths of arithmetic—is evident. However, there is a sense in which the discovery that a statement of number gives us an assertion about a concept introduces a new mystery. It looks as if Frege is offering us a recipe for eliminating numerals. For, when he redescribes the statement that Venus has 0 moons so that it will be apparent what the assertion about a concept is, the numeral “0” disappears. Similarly, the numeral “1” would disappear from the statement that Venus has 1 moon, etc. But the recipe works only for a certain sort of statement in which numerals appear. It tells us nothing about the elimination of numerals from statements of pure arithmetic (e.g., “1+1=2”). What, then, is the relation between the “1” that appears in “Venus has 1 moon” and the same symbol in “1+1=2”? Given Frege’s interest in the applications of arithmetic, this cannot be ignored. For we must be entitled to make inferences that involve both statements of number and statements of pure arithmetic. Given that Venus has 0 moons, that the Earth has 1 moon, and that 0 < 1, for example, we must be entitled to infer that the Earth has more moons than Venus. But unless there is some strategy for redescribing the content of “0 < 1”, there is no obvious explanation of why this is a good inference. And it is far from obvious that Frege’s discovery that statements of number can be understood as statements about concepts will help us out here.

²⁴ FA, p. iv. ²⁵ As is well known today, a similar sort of regimentation can be used to express what is stated by assertions of other numbers. For example, to say that Venus has 1 moon is to say: (Ǝx)[(x is a moon of Venus) & (y)(y is a moon of Venus ! y = x)].

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This discovery shows us why we “have no choice but to acknowledge the purely logical nature of arithmetical modes of inference.”²⁶ But, Frege continues, “Together with this admission, there arises the task of bringing this nature to light wherever it cannot be recognized immediately, which is quite frequently the case in the writings of mathematicians.” The discovery that statements of number are assertions about concepts does not show us how it can be that the claim that 0 is not equal to 1 is a maximally general statement. Nor does it show us that the laws of arithmetic are maximally general—they still look like statements that apply to a specific domain, the domain constituted by the natural numbers. One of the important achievements of Begriffsschrift was to have shown that mathematical induction—a law that appears to be, and is ordinarily taken to be, peculiar to arithmetic²⁷—can be derived from laws that are recognizably laws of logic. The same must be done for all statements of arithmetic that appear to be about particular objects or to be laws governing a limited domain. In order to do this, he needs to show how to define numerals and the concept of number from recognizably logical notions and to prove the truths of arithmetic using only these definitions and logical laws. Given Frege’s project, the definitions must make it possible to provide gapless proofs of the truths of arithmetic from primitive truths—from primitive logical laws, if he is to substantiate his conviction that they are analytic. But, as we noted earlier, if the proofs are to show us the epistemological nature of the truths of our arithmetic, rather than the truths of a different and foreign science, then the definitions must satisfy some sort of faithfulness requirements.

Ib. Why Frege Would Reject the Apparently Obvious Faithfulness Requirement On the apparently obvious faithfulness requirement, Frege’s definition of, say, the number one, must give us a description that picks out the object we have been talking about all along when we used the numeral “1.” This, obviously, presupposes that there is a definite particular object that we are already talking about when we use the numeral “1.” Yet, as I indicated earlier, there are statements in Frege’s writings that conflict with this assumption. One such statement appears in his critique, in Foundations, of a proposed definition of the concept of number. He writes “it must be noted that for us the concept of number has not yet been fixed, but is only due to be determined in the light of our definition of numerical identity.”²⁸ This is an odd choice of words if we suppose that each numeral already has a determinate sense—hence already refers to a particular

²⁶ FTA, p. 96.

²⁷ FA, p. iv, §§80, 108.

²⁸ FA, §63.

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object—and that to be a natural number is simply to be one of these objects. For Frege writes that all that can be demanded of a concept is that it should be determined, for each object, whether or not it falls under the concept.²⁹ If each numeral already refers to a particular object, and if the numbers are simply the objects to which the numerals refer, then the concept of number is already fixed. We may be lacking a definition that identifies this fixed concept of number.³⁰ We may imperfectly grasp the conceptual content (or, the sense) that is already associated with the term “number” (and already determines the objects to which it applies). But it certainly does not follow that the concept of number is due to be determined in the light of our definition. Indeed, on the interpretation under discussion, it is not the concept of number that is to be determined by the definition; it is our understanding of the senses that our sentences already express. On this interpretation, the role of the definition is to get us to understand a sense that is already associated with the word “number.” Were this Frege’s only remark of the sort, one might dismiss it as merely an odd choice of words. But it is not. The numbers on which Frege concentrates in most of the discussions of Foundations are the natural numbers.³¹ However, definitions of the natural numbers will not suffice to accomplish his ultimate goal—to show that analysis is analytic. In the last part of Foundations, Frege turns to the issue of defining the complex numbers. He considers the possibility of stipulating that the time-interval of one second is the square root of –1, and he adds, in a footnote, that we are entitled to choose any one of a number of objects to be the square root of –1. The reason is that the meaning [Bedeutung] of the square root of –1 is not something which was already unalterably fixed before we made these choices, but is decided for the first time by and along with them.³²

Here there is no ambiguity. If the choice decides for the first time what the square root of –1 is, then it cannot simply be that we have imperfectly grasped the (already determinate) sense of our statements about the complex numbers. It is not consistent with the above passage to take “the square root of –1” as already ²⁹ FA, §74. ³⁰ It is also odd that Frege says that “for us” the concept is not fixed rather than, for example, “by this procedure.” ³¹ He actually says that he is writing about “positive whole numbers” (see, for example, FA §109) but 0 is among the numbers defined. ³² FA, §100. It is not entirely clear how “Bedeutung” should be translated in this passage. But, however, it is translated, the passage implies that one of the things that was not fixed before the choices in question, is what it is to which “the square root of –1” refers. It is also clear that, insofar as the expression in question does have a sense or conceptual content antecedent to the choice being made, once the choice has been made, the sense or conceptual content that is chosen will not be the same as whatever sense or conceptual content was originally associated with the expression.

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having a determinate (referent-fixing) sense. Rather, it has to be that our symbols for complex numbers—antecedent to Fregean definitions—do not have determinate referents. Indeed, he goes on to suggest as much in the next section. Moreover, since the choice in question is of a referent-fixing sense, insofar as the symbols for complex numbers have fixed and determinate senses or conceptual contents, these senses or conceptual contents will not be entirely preserved by the definitions. One might suspect that this marks a difference between the complex and natural numbers. But Frege gives us no indication that there is such a difference. The language he uses is virtually the same in both discussions. This is not to say that there are no constraints on what will do as a definition of the square root of –1. For he goes on to suggest that there is a problem with defining the square root of –1 as a time-interval. To do so is to import into arithmetic “something quite foreign to it, namely time” and to make arithmetic synthetic.³³ In order to show that arithmetic is analytic, Frege suggests using the same solution for complex numbers that he used for natural numbers: to define them as extensions of concepts. The notion of the extension of a concept is a logical notion, on Frege’s view,³⁴ and definitions of the numbers as extensions of concepts should make it possible to prove truths about the numbers from logical laws. He ends Foundations with the following remark about offering such definitions: Once suppose this everywhere accomplished, and numbers of every kind, whether negative, fractional, irrational or complex, are revealed as no more mysterious than the positive whole numbers, which in turn are no more real or more actual or more palpable than they.³⁵

This would be an odd remark if, for example, “1” had all along referred to a particular extension of a concept while the symbol “i” refers to an extension of a concept only because of an arbitrary stipulation. But, again, this may be simply an odd choice of words. What other evidence is there? As we have seen, Frege views the definiens of a definition as expressing a sense. When he defines the numbers as extensions of concepts, it becomes (if it is not already) a part of the sense of each numeral that the number in question is an extension of a concept. If we assume, as we must on the interpretation under discussion, that Frege’s definitions are meant to capture a determinate sense that our numerals already express (whether or not we grasp this sense), then it is already part of the sense of our numerals and of the term “number” that numbers are extensions. His definition is, in part, a discovery about the nature of numbers. But a closer look at Frege’s discussions of definitions in later writings reveals a number of conflicts with this interpretation.

³³ FA, §103.

³⁴ See Chapter 4, section IIIc.

³⁵ FA, §109.

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First, Frege does not think that, in general, the fact that a term is used in everyday science (science that precedes the sort of systematic treatment that he advocates) is any indication that the term has determinate sense. For in a discussion of the enterprise of defining a scientific term in “On the Foundations of Geometry II”, he writes, Now it may happen that this sign (word) is not altogether new, but has already been used in ordinary discourse or in a scientific treatment that precedes the truly systematic one. As a rule, this usage is too vacillating for pure science.³⁶

Since our scientific use of arithmetic does precede Frege’s systematic treatment, it is plausible that, on his view, the presystematic use of numerals is too vacillating for systematic science. If this is so, a proper definition must introduce a sense distinct from the sense it has due to our pre-systematic scientific use. It may be that, nonetheless, there is a unique correct sense for each of the symbols of arithmetic—a unique sense that is appropriate for pure science and that fits with what we are doing in our pre-systematic science. But Frege’s comments here make it clear that he does not regard this as in any way an automatic consequence of the fact that our pre-systematic science works. If he thinks that our use of the symbols of arithmetic does have this character, it would not be something that goes without saying. And supposing the symbols of arithmetic do have this character (whether or not it goes without saying), then there is another problem with attributing to Frege the view that his definitions are meant to express the unique correct sense of each symbol of arithmetic that is consistent with our use of it. Frege is adamant that definitions (as opposed to axioms) are stipulative.³⁷ But he does acknowledge that sometimes we may want to define a simple term that is already in use, in which case we cannot give an arbitrary stipulation. In the 1914 course notes, he identifies two different situations in which we might be said to be giving a definition. In the first, the definition is stipulative. We construct a complex sense and use a new sign to express the sense. Such a definition, he says, may be called constructive [aufbauende] but, he adds, “we prefer to call it a ‘definition tout court’.”³⁸ He also describes a second sort of situation—one in which we want to give a logical analysis of the sense of a simple sign with a long established use. In such a situation, he says, we might speak of an “analytic [zerlegende] definition.”³⁹ But, he continues,

³⁶ OFG II, pp. 302–3. ³⁷ Frege is most explicit and adamant in his series of papers on the foundations of geometry. See, OFG I, pp. 320–1; OFG II, pp. 294–9, 383. ³⁸ LM, PW, p. 210/NS, p. 227. ³⁹ LM, PW, p. 210/NS, p. 227. It is important to note that the word “analytic” here is a translation of “zerlegende.” The issue here is not about the epistemological status of definitions, since there could be no such thing, on Frege’s view, as a synthetic or a posteriori definition.

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[I]t is better to eschew the word ‘definition’ altogether in this case, because what we should here like to call a definition is really to be regarded as an axiom. . . . So we shall stick to our original way of speaking and call only a constructive definition a definition.⁴⁰

Thus if our pre-systematic science does determine a unique correct sense for our numerals, we should have a systematic science that begins, not with definitions, but with axioms expressing the senses of these terms. Moreover, the truth of such axioms, Frege claims, is recognizable “only by an ‘immediate insight’ ” [unmittelbares Einleuchten].⁴¹ These remarks are difficult to square with the idea that it might be a substantive discovery that numbers actually are extensions of concepts—a discovery that we can be made to understand but will not see immediately. But these remarks are completely in accord with Frege’s treatment of his definitions in Foundations. In the discussions that follow his introduction of definitions of the natural numbers, Frege acknowledges that the correctness of his definitions is not evident for, he says, we “think of the extensions of concepts as something quite different from numbers.”⁴² Were it a substantive discovery that numbers are extensions of concepts, we would expect his defense of his definitions to include an argument that shows that the statements we make about the numbers commit us to the view that numbers really are extensions of concepts. But no such argument appears among Frege’s defenses of his definitions. Frege’s observation that we think of extensions of concepts as different from numbers appears in §69 of Foundations and the rest of the section contains a brief response—a response that seems to exploit the context principle and to rely on the sentential priority view. For, instead of addressing the question “Are numbers extensions?” he considers the question, “Are the assertions we make about extensions assertions we can make about numbers?” There are, he says, two sorts of assertions we can make about extensions. The first sort is an identity statement. Just as we can assert the identity of extensions, he says, we can assert the identity of numbers. The second sort of assertion is that one extension is wider than another. This, he admits, is not something we say of numbers. But it is also, he notes, not a relation that can hold between the extensions that, according to

⁴⁰ It is interesting to note that Frege says something very similar in his review of Husserl’s Philosophy of Arithmetic. He writes, A definition is also incapable of analysing the sense, for the analysed sense just is not the original one. In using the word, to be explained, I either think clearly, everything I think when I use the defining expression: we then have the ‘obvious circle’; or the defining expression has a more richly articulated sense, in which case I do not think the same thing in using it as I do in using the word to be explained: the definition is then wrong. (1894, p. 319) The words used here are almost exactly the same as the words used in his later course notes. In this case, however, it is not entirely clear that Frege is speaking in his own voice, rather than Husserl’s. ⁴¹ LM, PW, p. 210/NS, p. 227. ⁴² FA, §69.

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his definitions, are numbers. It follows that the assertions we make about the extensions that are numbers (given Frege’s definitions) are also assertions that we can make about numbers. Frege turns next, in §§70–83 to the “testing and completion” of his definitions, where he tries to show that the definitions of numbers as extensions allows us to derive the appropriate number statements. There is, then, no argument that it is part of the nature of numbers to be extensions. And there is no indication, here or elsewhere, that the definition of each numeral picks out a unique object to which the numeral already refers. Indeed, there is a remark that looks to be a denial of this. Frege claims later in Foundations that he attaches no decisive importance to bringing in the extensions of concepts.⁴³ This claim is completely mysterious if we assume that, when we use the numerals in our current pre-systematic language, we are talking about particular objects, and if we assume that Frege’s task is to provide definitions that pick these objects out. For if so, either we are already talking about (our numerals already refer to) extensions of concepts (in which case it would be essential to bring in extensions) or we are already talking about (our numerals already refer to) objects other than extensions of concepts (in which case it would be wrong to bring in extensions). Frege’s comments are simply not consistent with the assumption that his definitions are meant to pick out objects that we have been talking about all along. Unless we are prepared to engage in interpretive contortions, the appropriate conclusion is that, whatever Frege is doing when he asks for a definition of the concept number, it cannot be that he is asking for an explicit description of objects to which our numerals already refer (or an explicit description of the sense that the numerals already have). And, given this, it is implausible to attribute to Frege the view that there is a unique concept to which “number” refers, and unique objects to which the numerals refer, antecedent to his introduction of his definitions. That is, antecedent to Frege’s introduction of his definitions, the concept number is not fixed. Of course, if the view we get from Frege’s explicit remarks is absurd, there may be a compelling reason to engage in interpretive contortions. But is this view absurd? A number of contemporary philosophers subscribe to least one part of this view—that our numerals do not refer to particular objects.⁴⁴ One reason is that there are many distinct set theoretic definitions of the numbers that fit our

⁴³ FA, §107. Frege does come, later in his career, to attach more importance to bringing in extensions of concepts. But his reason is not that numbers really are extensions. It is, rather, that without them “one would never be able to get by” (BLA vol. i, p. x). ⁴⁴ Many of these views have their roots in Paul Benacerraf ’s (1965), where Benacerraf concludes, from the fact that there are distinct, apparently acceptable, set theoretic definitions of the natural numbers, that numbers are not objects. Some examples of philosophical views on which arithmetic is not about objects to which numerals refer include Field (1980); Kitcher (1985), chapter 6; Resnik (1997); Shapiro (1997).

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understanding—both everyday and scientific—of the numbers. Nothing in our understanding of the truths of arithmetic seems to offer grounds for deciding between alternative systems of set theoretic definitions or, for that matter, grounds for saying that numbers are (or are not) sets. Given this, Frege’s explicit remarks do not seem absurd at all. There is every reason to believe that the numerals, prior to his definitions, do not refer to particular objects and, consequently, that the content associated with the numerals can be captured by offering definitions that are at least partly stipulative.

Ic. What Are Frege’s Actual Faithfulness Requirements? Were Frege to offer truly arbitrary stipulations as definitions, his proofs might very well tell us nothing about arithmetic. Were he, for example, to define the number 2 as Julius Caesar, he would thereby have changed the subject. There must be some kind of faithfulness requirement. But I have argued that Frege did not regard either preservation of sense or preservation of reference as requirements that constrain his definition. What is left? The answer is to be found in §§70–83 of Foundations, which he labels “the completion and testing [Ergänzung und Bewährung] of our definition.” For Frege is explicit that his definitions must pass the following test: The definitions must allow us to derive ‘the well-known properties of numbers.’⁴⁵

More specifically, he wants to prove that the numbers, as defined, have the properties that seem to underlie the uses we make of arithmetic, both in science and in everyday life. For example, it must be possible to prove, from the definition of 0, that 0 is the number that belongs to a concept if and only if no object falls under it (the number that belongs to a particular concept is the number of objects that fall under the concept).⁴⁶ We must be able to prove such claims as: if 1 is the number that belongs to a concept, then there exists an object which falls under that concept.⁴⁷ Or, as he also says, The definitions must provide a basis for an arithmetic that meets the demand ‘that its numbers should be adapted for use in every application made of number’.⁴⁸

Another way of describing these requirements is to say that what we take to be simple truths and applications of our arithmetic must be reproducible in an arithmetic based on Frege’s definitions. No acceptable definitions of “0” and “1”

⁴⁵ FA, §70.

⁴⁶ FA, §75.

⁴⁷ FA, §78.

⁴⁸ FA, §19.

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will make it true that 0=1, or false (failing new astronomical events or discoveries) that the Earth has 1 moon. How, then, should we characterize Frege’s faithfulness requirements? It would be misleading to say that they amount to the requirement that the definitions preserve truth (or, at least, what we antecedently think of as the simple truths of arithmetic). For that requirement is too modest. Frege does not simply try to convince us that it is true, given his definition of “1,” that if 1 is the number which belongs to a concept, then some object falls under the concept. He tries to show us that it is provable.⁴⁹ The definitions must provide support for the inferences of pre-systematic arithmetic. If, before Frege’s definitions are offered, we regard ourselves as entitled to make inferences about the numbers of moons of Venus and the Earth, the introduction of these definitions should enable us to replace our original, enthymematic arguments with gapless arguments.⁵⁰ What all this suggests is that we have the answer to our question about Frege’s faithfulness requirements. Faithful definitions must be definitions on which those sentences that we take to express simple truths of arithmetic turn out to express provable truths and on which those series of sentences that we take to express correct inferences turn out to be enthymematic versions of gapless proofs in the logical system. Supposing, then, that Frege can give definitions that satisfy these faithfulness requirements, is this sufficient for us to agree that he has not changed the subject? Surely his definitions would amount to a change of subject if, for example, in our pre-systematic use of “1” we have been talking about a particular object all along, an object that is not an extension of a concept. But this is not the situation. For there is nothing in our pre-systematic science of arithmetic that determines whether or not the number one is an extension. Frege’s explicit faithfulness requirements are meant to preserve everything that we can identify as being intrinsic to arithmetic and the numbers pre-systematically. A science based on definitions that satisfy these requirements deserves to be called “arithmetic” because it is able to do all the work that arithmetic has always done. To use such a science is not to change the subject.

⁴⁹ In FA, §78, Frege lists a number of simple facts about numbers that “are to be proved” by means of his definitions. His words are “Ich lasse hier einige Sätze folgen, die mittels unserer Definitionen zu beweisen sind.” ⁵⁰ For these reasons, one might be inclined to think that the criterion of correctness for Frege’s definitions is that certain claims of ordinary pre-systematic arithmetic be logically equivalent to statements in Frege’s systematic science of arithmetic. That such truths should be logically equivalent, in the sense that each is derivable from the other in Frege’s logic, is defended by Patricia Blanchette (2012). Although this is not the place for an extended discussion of Blanchette’s claim, it is useful to note that since, from Frege’s point of view, all truths of arithmetic are truths of logic, all truths of arithmetic are logically equivalent (and are logically equivalent to simple logical laws). Thus, this is not the requirement I have described above.

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II. How Does this Square with the View that the Sentences Expressing the “Well Known Properties of the Numbers” are True? IIa. The Sentential Priority View Revisited I argued in Chapter 5 that Frege’s explicitly stated views commit him to rejecting the subsentential priority view (the view that a sentence can have a truth-value only if all its subsentential constituents already refer to entities in an extra-linguistic world) in favor of sentential priority (the view that to fix the Bedeutung of a subsentential expression, it is both necessary and sufficient to fix the truth-values of all sentential expressions in which it appears). As we saw there, this fits Frege’s procedures in his Basic Laws argument that all Begriffsschrift expressions have Bedeutung. For his actual argument is that all Begriffsschrift sentential expressions have truth-values. As we also saw, the sentential priority view fits with one of Frege’s explicit statements about his project of defining the numbers, Since it is only in the context of a proposition that words have any meaning (Nur im Zusammenhange eines Satzes bedeuten die Wörter etwas), our problem becomes this: To define the sense (den Sinn eines Satzes zu erklären) of a proposition in which a number word occurs.⁵¹

However, the textual evidence for sentential priority is not entirely consistent. For example, in the second volume of Basic Laws, Frege explicitly claims that a sentence containing a proper name that does not designate anything has no truthvalue.⁵² This appears in an argument that it is not legitimate to define the addition function only on numbers. He writes, [T]he proposition, “The sum of the Moon and the Moon is One”, is now neither true nor false; for in either case the words, “The sum of the Moon and the Moon”, would have to refer to something, which is exactly what the suggested stipulation denies. Our proposition is rather to be compared with the proposition, “The Scylla had six dragon gullets”. This proposition too is neither true nor false but fiction, since the proper name “Scylla” designates nothing. Such propositions can be the object of a scientific treatment, e.g., one concerned with mythology, but no scientific investigation can be carried out using them.⁵³

He makes a similar remark in the first volume, where he writes,

⁵¹ FA, §62.

⁵² BLA vol ii, p. 76.

⁵³ BLA vol ii, p. 76.

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For the proposition, “Nessus did not carry Deïanira across the river Euenus”, could likewise only be true if the name “Nessus” had a bearer.⁵⁴

And both of these passages echo a remark from “On Sinn and Bedeutung,” where he says, of the sentence “Odysseus was set ashore at Ithaca while sound asleep,” [A]nyone who seriously took the sentence to be true or false would ascribe to the name ‘Odysseus’ a meaning [eine Bedeutung zuerkennt], not merely a sense; for it is of what the name means [der Bedeutung dieses Namens] that the predicate is affirmed or denied.⁵⁵

If, as we have seen, Frege thinks that the numerals do not designate particular objects antecedent to his providing his definitions, the above passages suggest that it is his view that, antecedent to his definitions, the sentences in which numerals appear had no truth-values but, rather, belong to fiction. One might think, then, that he owes us the kind of fictionalist account that appears in some contemporary writings about mathematics. Yet he gives us nothing of the sort. This cannot be a mere oversight, for the problem is not limited to numerals. On Frege’s view, both early and late,⁵⁶ a concept determinately holds or not of each object or, to use his locution, concepts have sharp boundaries.⁵⁷ Moreover, ⁵⁴ BLA vol i, p. xxi. ⁵⁵ SB, pp. 32–3. ⁵⁶ Although most of the passages discussed below come from the second volume of Basic Laws the sharp boundary requirement for concepts appears in many of Frege’s earlier writings. The sharp boundary criterion is stated explicitly in Foundations §74, and is part of his reason for raising the question of whether or not Julius Caesar is a number. It is also emphasized in “The Law of Inertia” (1891), pp. 158–60. In FC, p. 20, he mentions the sharp boundary requirement for concepts and the consequence that functions must have values for all objects. In BLA vol i, p. 11, he writes that, in order for “x.(x–1)=x²–x” to name the True, the notations for multiplication, subtraction, and squaring must be defined “to apply also to objects that are not numbers.” The topic is also discussed at length in §§56–67 of Basic Laws vol ii. He may also be alluding to this feature of concepts when he mentions the notion of the heap in Begriffsschrift (1879, §27). ⁵⁷ Patricia Blanchette (2012) claims that Frege does not really mean to say what he actually says: that a (first-level) function must be defined for all objects. Instead, she offers two accounts of what he really means to be saying. One of these is that every well-formed object-name must have reference. Thus all function-expressions must be defined over every object that has a name in the language. But, in a language with, say, no name for Julius Caesar, there is no need for a function-expression to be defined in a way that gives a value for the function when Julius Caesar is taken as argument. That is, a functionsign in such a language can refer to a kind of entity, a partial function that does not have values for all objects as arguments. This might seem to fit with Frege’s claim, in §29, that a first-level function-sign has Bedeutung provided every result of filling its argument place with an object-name has Bedeutung. But, as we have seen in Chapter 5, it is a mistake to understand having Bedeutung (in the §29 sense) as being associated, via a reference relation, with an extra-linguistic entity. That is, the discussion in question does not provide support for the claim that Begriffsschrift first-level function-signs can refer to partial functions. Moreover, while Blanchette claims repeatedly that Frege’s Begriffsschrift function-signs are not everywhere defined, this is, at least, misleading. To see why suppose, as Blanchette does, that there are Begriffsschrift expressions that refer to the True. Now consider the horizontal function. On Frege’s view, the horizontal designates the function of being identical to the True. If the True is a particular object in the universe, then any object (including Julius Caesar and the Moon) is either identical to the

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just as a legitimate proper name must designate an object, a legitimate predicate must designate a concept. But do the predicates of arithmetic designate concepts? Frege writes, in the second volume of Basic Laws, “Would, for example, the proposition ‘Every square root of 9 is odd’ have any graspable sense at all if square root of 9 were a concept without sharp boundaries?”⁵⁸ A sentence that does not have graspable sense, of course, cannot have a truth-value. One might suspect that Frege takes it to be obvious that “any square root of 9 is odd” does have a comprehensible sense and hence, also, that the concept square root of 9 does have a sharp boundary. However, a look at the context in which the question appears shows that this interpretation is incorrect. For example, in order for greater than zero (or positive) to be a proper concept, Frege says, “That would require, e.g., that it also be determined whether the Moon is greater than zero.”⁵⁹ He continues,

True or not. That is, the horizontal is everywhere defined. Similar arguments can be given for the other primitive first-level function-names. For all that, as we saw in Chapter 5, Frege recognizes a problem here: it seems that there is no determinate answer to the question about whether a particular valuerange is identical to the True. See Chapter 5 for an account of how Frege’s discussions of the problem are to be understood. Blanchette also gives another account: when Frege says that a function must be defined for all objects, he does not mean all objects but, rather, all objects that belong to a particular science. However, the textual support for this view is slim. She offers two citations: (1) to a comment that appears in Carnap’s notes for Frege’s 1914 course and (2) to a passage from Function and Concept. But (1), on its own, is not very convincing, given that Frege’s own notes for his 1914 course do not include the comment from Carnap’s notes. What about the passage from Function and Concept? Frege writes, So long as the only objects dealt with in arithmetic are the integers, the letters a and b in ‘a+b’ indicate only integers; the plus-sign need be defined only between integers. (FC, p. 19) In isolation, this passage appears to provide strong evidence for Blanchette’s interpretation (although one might wonder why Frege would have said this only once in his writings). However, the evidence looks substantially weaker when we try to fit this into Frege’s general views. There are a number of respects in which this conflicts with many of Frege’s oft stated views. One example is the conflict with Frege’s understanding of logic. Among the objects that belong to logic, and are named in logical laws, are the True and the False (neither of which is a number) as well as numerous other (non-number) value-ranges. Thus, since arithmetic belongs to logic, it would not do for the plus-sign to be defined only on integers, it must at least be defined also on all value-ranges. Since Blanchette accuses Frege of being “incautious” in stating the sharp boundary requirement, she might be inclined to say something similar here. Perhaps, he meant to say the plus-sign need only be defined on integers as well as other objects for which logical names can be constructed. But, again, this is an idea that only works if we read the relevant passages in isolation from Frege’s understanding of logic. For he views logic as underlying all science and his logical language is designed to be used in all rigorous sciences. Logical objects are named in, and belong to, all sciences. Suppose, for the moment, that Blanchette is right that Frege thought that a function need be defined only for objects belonging to the science in which that function is used. Even on this view, all functions of any science must be defined on all value-ranges. If there is to be, say, a rigorous science of obesity then we need to define obesity in a way that determines whether or not the True has obesity. Blanchette’s reinterpretation of Frege’s claim that functions must be everywhere defined is meant to avoid such “absurdities” as the requirement that it be determined whether or not the Moon is greater than 0 (BLA vol. ii, §62). But it is difficult to see why this is any more absurd than requiring that it be determined whether or not the True has obesity. ⁵⁸ BLA vol. ii, p. 69.

⁵⁹ BLA vol. ii, p. 74.

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One could now stipulate that only numbers can stand in this relation, and then conclude from it that the Moon, since it is not a number, is not greater than Zero. To accomplish this, however, would require a complete explanation of the word “number”, which is usually lacking.

Indeed he argues, not just that such concepts must be defined, but that the sort of definitions typically offered by mathematicians are unacceptable. In the discussions of Basic Laws that immediately follow, he suggests that such expressions as “greater than” and “+” are used by mathematicians in such a way that they have neither determinate sense nor determinate Bedeutung.⁶⁰ Thus, insofar as there is a problem here for Frege, it cannot be understood as a special problem for the sentences of arithmetic. For, it was surely evident to Frege that everyday predicates not belonging to arithmetic are no more likely to have sharp boundaries than those belonging to arithmetic. He mentions the problem with the predicate “heap.”⁶¹ He also writes, “The only barrier to enumerability is to be found in the imperfection of concepts. Bald people for example cannot be enumerated as long as the concept of baldness is not defined so precisely that for any individual there can be no doubt whether he falls under it.”⁶² What we have just seen is that it follows from at least some of Frege’s statements that the sentences of everyday language—whether belonging to arithmetic or not—do not have truth-values. But is this really Frege’s view? There are at least some passages in which he explicitly states that at least some everyday sentences do have truth-values. For example, in Function and Concept, Frege claims, without adverting to any definitions of “0” and “1”, that “(1)²=1” is true and “0²=1” is false.⁶³ Indeed while, as we have seen, the discussions of §§55–67 of Basic Laws vol. ii imply that the everyday sentences of arithmetic do not have truth-values, only a few pages later, he writes, The obvious question is of course: how is it [arithmetic] distinguished from a mere game? Thomae answers by pointing to the service it can provide in the

⁶⁰ Most of the discussions in these sections are about Bedeutung. For example, Frege claims that the expression “something the half of which is less than one” has no Bedeutung because the function 1/2x is not defined for the Moon as argument. But there are some mentions of sense. In §56, he says that the sentence “all square roots of 9 are odd” does not have sense if “square root of 9” does determinately hold or not of each object. The implication of later remarks is that it does not—if we do not know, for example, whether the Moon is less than 1⁄2 of 1, then we presumably also do not know whether it is a square root of 9. In §65 he also mentions sense, claiming that, because the plus sign is not defined for all objects, “if a + b is not equal to b + a” is senseless. ⁶¹ See, BS, §27, The Argument for my stricter Canons of Definition, PW, p. 155/NS, p. 168 and letter to Peano September 29, 1896 WB, pp. 181–8 /PMC, pp. 112–18. ⁶² Letter to Marty, August 29, 1882. WB, p. 163/PMC, p. 100. ⁶³ FC, p. 13. Also, in “Peano’s Conceptual Notation” he says, of “1+3=2.2” and several other everyday sentences of arithmetic, that they name the True—from which it follows immediately (given his explanation of the expression “the True”) that these sentences are true. PCN, p. 370.

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explanation of nature. This can rest only on the number-signs referring to something; . . . If one accords arithmetic a higher honour than chess, it can only be founded in this.⁶⁴

And, while statements in §§56–67 imply that the sentences of arithmetic do not express thoughts, in §91 he writes, Why can no application be made of a configuration of chess pieces? Because, obviously, it does not express a thought. . . . Why can one make applications of arithmetical equations? Solely because they express thoughts.⁶⁵

That is, in the space of a few sections of Basic Laws, we can find almost immediate contradictions on the issue of whether or not our everyday sentences have truthvalues and about whether or not these sentences express thoughts. What should we do in light of these contradictions? Every philosopher makes mistakes and one of our goals in interpreting a text is to make the best case we can for the views stated in it. It may seem that, if there is a contradiction in a writer’s corpus, our job is to identify a view that needs to be rejected and then fit together the remaining views. But how are we to decide which view should be rejected? And, once the decision has been made, are there any further obligations to the orphaned view? It may seem that the choice should be determined by plausibility: if one of the views in question is absurd, this is surely the view that should be rejected. And it may seem that if, once we have rejected this view, we can fit the rest of the author’s views together into a satisfying picture, nothing more is required—there is no need to consider the role the now-orphaned view played for the author. But one of the problems with this strategy is that it leaves out any consideration of the relative importance that the philosopher in question places on these conflicting views. This is especially evident when we consider the conflict, in Frege’s writings between the sharp boundary requirement and the view that the sentences of everyday life have truth-values. As I noted earlier, the view that our numerals do not actually name particular objects—and hence, similarly, that there is no determinate answer to the question of what are the objects to which the predicate “number” applies—is commonplace today among philosophers. And it may seem obvious that a sentence can have a truth-value only if its constituents (including its proper names and predicates) bear a referential relation to the world. Thus there is, it seems, a problem with taking such sentences as “0≠1” to have truth-values. But this is widely assumed to be a special problem that is limited to mathematical language. Everyday proper names and predicates, most philosophers think, are not problematic in the way

⁶⁴ BLA vol. ii, p. 99.

⁶⁵ BLA vol. ii, p. 100.

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expressions of mathematics are. It seems obvious that, for the most part, our everyday sentences do have truth-values—it is a virtual necessity to presuppose this both in our everyday lives and in the context of inquiry. And there is, of course, a large literature devoted to the contributions everyday proper names and predicates make to determining the truth-values of everyday sentences. But what we have just seen is that, for Frege, the problem is not limited to mathematical expressions. Given his standards—his sharp boundary requirement—everyday predicates do not have fixed extensions. And yet he presupposes, as we all do, that (most of ) our everyday natural language sentences have truth-values. I will call this combination of views—that everyday predicates do not have fixed extension and that everyday proper names do not name particular objects, along with the assumption that (most of) our natural language sentences have truth values— Frege’s apparently absurd view. Why might it seem apparent that this view is absurd? It does seem absurd if we assume that Frege means to be contributing to the kind of compositional theory of truth for natural language described in the passage from Devitt and Sterelny that we looked at in the previous chapter.⁶⁶ For it is part of such a theory that the sentence “0 ≠ 1” has a truth value only if “0” and “1” are names of particular objects. Since, as we saw in section Ib, it is Frege’s view that “0” and “1” were not— antecedent to his definitions—names of particular objects, it is mysterious that he should have regarded this sentence as true and, indeed, wanted to prove it from logical laws. But, as I have argued at length in earlier chapters, there is abundant evidence that Frege does not mean to be contributing to a theory of truth. One might be inclined to take this as evidence that Frege was not as prescient as he has long been taken to be. But, as I shall argue in Chapters 7 and 8, this is the wrong moral to draw. For it is perfectly normal in many scientific investigations to take a sentence to have a truth-value even if it has subsentential constituents that do not bear the requisite referential relations to the extra-linguistic universe. Still, even if we abandon the view that Frege means to be giving (a part of) a compositional theory of truth, we need to figure out the purpose of his claims that appear to be meant as part of an account of truth, including his sharp boundary requirement. I shall argue that these claims are articulations of methodological standards for inquiry. To say that each predicate or proper name must have Bedeutung is to say that securing Bedeutung for each expression is part of the burden of engaging in inquiry. And, on Frege’s sentential priority view, to say that each subsentential term must have Bedeutung is not to say that there must be a way, short of identifying the truth-values of sentences in which the term appears, of identifying an extra-linguistic entity to which it refers.

⁶⁶ Devitt and Sterelny (1999), p. 22.

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IIb. Methodology: Natural Language and Inquiry How are these methodological principles supposed to work in practice? On the subsentential priority view, in order to say anything about the world, we need, first, to come up with proper names and predicates that are correlated in the appropriate way with extra-linguistic entities. Only then can we use these expressions to construct sentences that have truth-values. In Chapter 5, we saw reasons for believing that some of Frege’s statements commit him to rejecting the subsentential priority view. Frege also makes statements that seem to be more explicit criticisms of the methodological consequences of subsentential priority. For Frege criticizes views in which “the logically primitive activity is the formation of concepts.”⁶⁷ He writes, As opposed to this, I start out from judgements and their contents, and not from concepts. . . . I only allow the formation of concepts to proceed from judgements.⁶⁸

This is in line with the sentential priority view and his claim that words have meaning [bedeuten etwas] only in the context of a proposition. It is also in line with his brief discussion of vernacular language in a letter to Peano, where he writes, The task of our vernacular languages is essentially fulfilled if people engaged in communication with one another connect the same thought, or approximately the same thought, with the same proposition. For this it is not at all necessary that the individual words should have a sense and meaning of their own, provided only that the whole proposition has a sense.⁶⁹

Moreover, to say that one of the burdens of inquiry is to ensure that all predicates and proper names have Bedeutung is not to say that this burden must be discharged first. Indeed, on Frege’s sentential priority thesis, to discharge this burden requires us to identify, first, the truth-values of sentences in which these predicates and proper names appear. Of course, if we do start with statements and their contents, we will be starting with logically defective language.⁷⁰ But there is ⁶⁷ BLC, PW, p. 15/NS, p. 16. Unlike much of what can be found in NS/Posthumous Writings, this passage comes not from a fragment or from early drafts but, rather, from a full article about Boole’s logic that Frege tried (and failed) to publish. See, also, the letter to Marty, in which he writes about getting concepts “by decomposition from a judgeable content.” PMC, p. 101/WB, p. 164. ⁶⁸ PW, p. 16/NS, p. 17. ⁶⁹ Frege to Peano, September 29, 1896, WB, p. 183/PMC, p. 115. ⁷⁰ We saw in Chapters 3 and 4, that Frege identifies certain natural language expressions (e.g., “function” and “concept”) as defective. But the kind of defect exemplified in “function” and “concept” is very different from the kind of defect that is the subject of this and Chapters 7 and 8. For example,

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no reason to think that we have the option of avoiding this problem and starting, instead, with predicates and proper names that have Bedeutung. For there is an essentially linguistic component of the enterprise of securing Bedeutung for a term. And the language we would need to use for this, antecedent to our having terms with Bedeutung, will be a logically defective language. Frege is well aware of this, for he writes, [O]ne might think that language would first have to be freed from all logical imperfections before it was employed in such investigations. But of course the work necessary to do this can itself only be done by using this tool, for all its imperfections. Fortunately as a result of our logical work we have acquired a yardstick by which we are apprised of these defects. Such a yardstick is at work even in language, obstructed though it may be by the many illogical features that are also at work in language.⁷¹

One of the interesting features of this passage is that Frege not only acknowledges the importance, for scientific purposes, of natural language, he also acknowledges that, even in imperfect natural language, we can (and already do) apply a logical yardstick. Logically defective natural language sentences can (and already do) function in our investigations. These include sentences that contain expressions without Bedeutung and that according to Frege’s strict criteria, as a consequence, have no truth-value. On Frege’s view we can (and should) start with the assumption that certain sentences are true (or false) even though we know that, strictly speaking, many (if not all) of their terms fail to have Bedeutung. It is worth noting that it should not be any surprise that we can apply a logical yardstick—that is, evaluate arguments—in which defective sentences appear. For, after all, it is only by making use of our ability to distinguish good from bad arguments that it is possible to construct a logically perfect language. Indeed, the evaluation of arguments including fictional sentences is a staple of introductory logic classes. It is only Frege’s insistence on his very strict requirements that give us any reason to think otherwise. And the apparent conflict with Frege’s strict requirements is solved once we realize the methodological role of the requirements. They are to be regarded as goals, rather than prerequisites. One might suspect that, once it is acknowledged that, say, the numeral “1” does not name a unique object, there are no constraints—that any sentence in which it “bald” is defective because it does not satisfy Frege’s requirement that a predicate hold or not of each object. And this kind of defect is rectifiable by making the appropriate sort of stipulation: the kind of stipulation that will allow us to do the work that “bald” is meant to do for us. In contrast, Frege’s view is that any attempt to explain the notions of function and concept will fail, for reasons that we saw in Chapters 3 and 4. ⁷¹ 1924, PW, p. 266/NS, p. 285.

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appears can be taken as either true or false. But a look at our practice indicates that we think this is wrong. For we already know that there is no unique object picked out by “1” and yet it is unlikely that anybody would be inclined to posit that 0=1. For, it is not just in introductory logic classes that we apply logical yardsticks to sentences of our logically defective natural language. We do this in all our presystematic sciences, including pre-systematic arithmetic. And we cannot do this without presupposing that its sentences have truth-values. At the same time, we need to be able to recognize, and correct, logical defects. And, in fact, Frege provides us with a model of how this should work. The model is his treatment of the numerals and the expression “number.” Each of these terms must be defined—it must be secured a Bedeutung. But these terms are not defined in isolation. Frege tests his proposed definition of “0,” for example, by making sure that it allows us to prove the right things. It must be possible to prove from the definition that if 0 belongs to a concept nothing falls under it. To follow this procedure, we begin with sentences, not proper names or predicates, and we begin with the presupposition that sentences of our logically defective language have truth-values. But to say that Frege’s apparently semantic statements are meant to be methodological is not, on its own, to remove all mysteries about his views. After all, the demand for sharp boundaries might be methodologically mad. I shall argue that it is not. For in stating the sharp boundary requirement, Frege is making explicit something that does in fact play a role in inquiry and in the evaluations of theories. Indeed, I shall argue that some apparently intractable contemporary philosophical problems are largely the result of ignoring this role.

PART IV

P U T T I N G F R E G E’ S L E S S O N S TO WORK

7 Why Frege’s Apparently Absurd View Is not Absurd at All On the interpretation I have been defending in this book, Frege’s views are very much at odds with those attributed to him by Standard Interpretation. They are also very much at odds with many contemporary views about language. One might think, then, that, on such an interpretation, Frege’s views can only be of historical interest. In this and the next chapter, I shall argue that this is a mistake. Frege’s views, correctly understood, have a great deal to teach us about philosophical issues that are of concern to us today. In the remainder of this book, I will use Frege’s views to draw morals about how to understand two issues of contemporary interest: vagueness in natural language and mathematical truth.

I. Vagueness It should be immediately evident that Frege’s sharp boundary requirement conflicts with most of today’s writings about vagueness. This is manifest in Frege’s comments about Sorites. His attitude is unequivocal: vagueness, when it appears in natural language, is a logical defect.¹ He writes, for instance, The fallacy known as the ‘Sorites’ depends on something (e.g. a heap) being treated as a concept which cannot be acknowledged as such by logic because it is not properly circumscribed.²

And The fallacy known by the name of “Acervus” rests on this, that words like “heap” are treated as if they designated a sharply delimited concept whereas this is not the case.³

In contrast, many contemporary philosophers who write about vagueness think it is logic that should answer to the demands of natural language. On these views, we ¹ See, for instance, the discussion of “heap” in BS, §27. ² The Argument for my stricter Canons of Definition, PW, p. 155/NS, p. 168—the term that is translated “Sorites” in this passage is “Acervus.” ³ Letter to Peano September 29, 1896, WB, p. 183/ PMC, p. 114.

Taking Frege at his Word. Joan Weiner, Oxford University Press (2020). © Joan Weiner. DOI: 10.1093/oso/9780198865476.003.0007

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should not use classical logic either to evaluate natural language or to evaluate inferences involving vague predicates. Instead, we should search for a logic that is more appropriate for evaluating inferences in a language with vague predicates. And even among those who continue to advocate for classical logic in the face of Sorites, there is little sense that the problem lies with natural language or that our scientific investigations require a version of the sharp boundary requirement.⁴ That is, the contemporary consensus seems to be that Frege’s sharp boundary requirement is absurd. I shall argue, however, that the mistake lies in the contemporary views, rather than in Frege’s way of thinking about vagueness.

Ia. Vagueness and Deference to Experts I begin, not with a look at either contemporary or Fregean views about vagueness, but with some of Hilary Putnam’s views about language; views that, although they are very influential, are almost invariably ignored—even by Putnam himself— when the topic is vagueness. One of these views is about the role played by the division of linguistic labor and deference to experts in our use of language.⁵ As Putnam has informed us, he cannot tell an elm from a beech. But he does not think that this should prevent him, or the many who share his ignorance, from making confident assertions about elms and beeches. What licenses this behavior is that there are people who can distinguish elms from beeches. When we say that elms are good shade trees, we mean to be talking about the trees that, according to these people, count as elms. We do not take polls to determine what counts as an elm, a tiger, or a lemon. We defer to experts, to people who are carrying out relevant research in the special sciences. And one reason we defer to experts is that we think both that what it is to be a tiger or lemon and what things are true of tigers or lemons depend, not just on what people think, but (in part) on nonpsychological, nonlinguistic facts about, for example, biology. Indeed, the view that nonpsychological, nonlinguistic facts have this significance, at least for proper names and natural kind terms, is the basis for many of Kripke’s and Putnam’s renowned arguments against description theories. Why have these influential views been ignored in the philosophical literature on vagueness?⁶ Perhaps it is because it has seemed obvious to so many writers that ⁴ See, for example, Williamson (1996), Graff (2000). ⁵ See, for example, Putnam (1973) and (1975). ⁶ Two notable exceptions to this are Kamp (1981) and Sorenson (2001). Kamp’s discussion may explain the widespread neglect of Putnam in the vagueness literature, since he assumes that the Putnam considerations apply only to natural kind terms and that it is reasonable to take natural kind terms not to be vague. I think both of these assumptions are dubious. I argue against the first in this chapter. Sorenson, in contrast, assumes a more thoroughgoing externalism in his epistemicist view of vagueness. See, for instance, Sorenson (2001), pp. 31–2. I argue, below, that a thoroughgoing externalism does not commit one to the epistemicist position.

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science has no purchase on the linguistic phenomena of import to the philosophical problems surrounding the issue of vagueness. After all, how could we find empirical evidence that a particular object is or is not a heap, or that the movement of a single grain of sand can make (or destroy) a heap?⁷ There is no science of heaps; there are no experts to whom deference is owed. But while “heap” may be an example of a vague predicate that does not currently pick out a topic of empirical investigation, other vague predicates do. Baldness, for example, is a topic of empirical investigation. Indeed, it is an established result of epidemiological research that baldness is associated with increased risk of heart disease.⁸ Yet these medical investigations of baldness do not figure in the numerous discussions of baldness that appear in the vagueness literature. Nor do people who use the term “thin” as an example of a vague term discuss the epidemiological literature on body weight, fat, and size. Rather, there seems to be a widespread assumption that, in our discussion of the semantics for vague predicates, no empirical research need be taken into account. The views that notoriously lead to paradox—that a single grain of sand cannot make the difference between a heap and a non-heap, that a single hair cannot make the difference between a bald and non-bald person, etc.—are simply widely shared intuitions. And, while these intuitions can be (and often are) challenged, the challenges are typically based, not on empirical evidence but, rather, on fuller discussions of the intuitions involved. The evidence adduced in such discussions is limited almost exclusively to the judgments that nonspecialist competent speakers—typically the authors of articles about vagueness—would make (for example) about the meaning of “bald” and about who is or is not bald. When empirical evidence is mentioned, it is typically evidence about competent speakers.⁹ What significance does this have for our understanding of vague predicates? It is important to realize that Putnam’s point is not just that we should defer to experts about such terms as “elm” and “tiger”, but that we do. This is, in fact, part of the way in which natural language works. And this deference to experts about the meanings of terms is in no way limited to natural kind terms. There is ample evidence that we defer to experts about the meanings of vague terms that form the subject of empirical research.¹⁰ And there is, similarly, ample evidence that we ⁷ This view about heaps is widely, but not universally, held. Although he does not suggest that “heap” is an actual topic of any science, W. D. Hart (1992) offers an elaborate argument (which includes scientific evidence) that four is the minimum number of grains of sand in a heap. ⁸ See, e.g., Herrera et al. (1995), Lotufo et al. (2000). ⁹ See, for example, Kenton F. Machina’s suggestion in “Truth, Belief and Vagueness” for determining the people to whom “bald” applies by determining “at approximately what point people begin to feel unsure whether sample baldish men are really bald”. In Keefe and Smith (1999), pp. 187–8. See also, Diana Raffman (1994), Scott Soames Truth, (1999), ch. 7, and Max Black, in Keefe and Smith (1999), pp. 69–82. Raffman cites some empirical evidence from research in psychology. ¹⁰ For example, in epidemiological research and the guidelines currently accepted by, among others the CDC and the WHO, obesity is defined as body mass index (BMI) >30. This definition is also widely used, without comment or objection in newspapers and television reports as well as countless

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defer to experts about established results of this research. There does not seem to be anything wrong with this practice. And, given this practice, it is hard to see how it can be reasonable to defer to the relevant experts about what counts as an elm or tiger but not to defer to the relevant experts about who counts as bald or thin. Supposing, then, that we do defer to experts about who counts as bald or thin, this deference carries with it substantial, and not very surprising, commitments. In particular, there would be something wrong with an account of vagueness or of the semantics of vague predicates that tells us that we cannot address questions about baldness via empirical research. For, on such an account, the text of Rogaine commercials would make no sense. Moreover, our views about vague predicates should be consistent not only with the correct verdicts about individuals (e.g., about whether or not a particular person is bald), but also with the correct verdicts—when we have them—about such things as whether people who are bald are at increased risk of heart disease; whether thin people have a lower risk of heart disease; whether rich people are healthier than poor people. In the remainder of this chapter, I shall argue that a variety of proposed accounts of the semantics of vague predicates conflict with both the results and methodology of empirical research. It is important to be clear about the significance of such conflicts. I do not mean to say that any acceptable account of the semantics of vague predicates must agree with all results of empirical work or even with all widely accepted practices of the special sciences. For philosophical arguments can elucidate problems with a scientific practice. A conflict between empirical research and a view about the semantics of vague predicates could show that there are hitherto unrecognized difficulties with the methodology of some empirical research. But empirical non-medical websites. For some recent examples, see the CBC news website, http://www.cbc.ca/news/ bigpicture/obesity/bmi.html: USA Today (January 22, 2004), Newsweek (December 8, 2003). This is, at least, evidence that reporters defer to experts. What of the rest of us? Since I have been emphasizing the importance of empirical studies, I am on somewhat shaky ground asserting, without the benefit of such studies, that the rest of us agree with these reporters. It does, however, seem somewhat unlikely that reporters would continue to write such articles if the view of the general public is simply that the results of using such measures as BMI >30 are statements that are not really about obesity. For all that, it is not true that everyone defers to experts, if “expert” is understood as “professional”. As I note, below, there is controversy about whether or not this definition of obesity is a good one. Many of the contributers to the controversy are not medical researchers. But this, I maintain, does not show that we don’t defer to experts. Rather, it is a fact that people who are not professionals can enter into debate with professionals and can thus be considered experts. For example, both William Herschel (1757–1822), who was an amateur astronomer when he discovered Uranus, and Alfred Percival Maudsley (1850–1931), an amateur who argued that Mayan writing had a syllabic character, are considered experts. There continue to be amateurs who make contributions to both astronomy and our understanding of Mayan writing. It is also important to note that the objections to the definition of obesity are directed, in part, at changing the definition used by the experts and at challenging scientific conclusions that are based on the definition to which they object. Indeed, the WHO has changed some of its definitions in response to these objections. These cases are very different from the cases of philosophers who make claims about the meanings of vague predicates while not engaging with the science and, indeed, accepting the conclusions of epidemiologists that are based on definitions that conflict with the philosophers’ claims.

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research does impose constraints on our philosophical views. It will not do to hold on to a philosophical view from which it follows that there is a problem with some of our scientific practices unless we are also prepared to reject those practices. Suppose we are prepared to accept an account of vagueness that has as a consequence, for example, that baldness cannot be studied by empirical means. Then we must be prepared to say that the research in question is about something else— that baldness and its effects really cannot be studied by empirical means. Claims about meaning must not issue in practical advice that we are unwilling to follow. One might suspect that these points are correct but of no particular interest for those who want to figure out the semantic role of vague predicates. For one might suspect that the consequences of a general account of everyday vague predicates will surely be neutral towards empirical research. This, however, is a view that can be tested. And, as I shall argue, it fails the test. There are many contemporary views about vagueness that are not neutral towards that research at all. In the discussion that follows, I shall argue that several popular views of vagueness commit us to rejecting not just accepted, but unexceptionable, scientific methodologies. In contrast, on Frege’s view—properly understood—there is no such conflict. Indeed, we can understand Frege’s insistence on the sharp boundary requirement as stemming from unexceptionable scientific practice.

Ib. Vague Predicates, Semantics, and Empirical Investigation The view that most clearly conflicts with Frege’s is the widely held view that the recognition of vagueness in natural language gives us reason to reject classical logic. On this line of thought, it is a necessary part of the semantics of vague predicates that some sentences in which they appear fail to have truth-values.¹¹ This view is one of the cornerstones, for example, of the supervaluationist semantics for vague language. But it is also the foundation of a critique of supervaluationism by Jerry Fodor and Ernest Lepore.¹² Fodor and Lepore claim that there will be individuals who cannot be correctly classified as bald or non-bald. And this is more than a mere matter of fact. They write, We claim that, if there is no matter of fact about whether someone one ninth of whose head is covered with (his) hair is bald, then it is necessary (indeed

¹¹ This is not, of course, meant to be a claim about the context-dependence of vague predicates. The view is that, even when the context is fixed, there will be such cases. And, as is well known, this view is not universally accepted. For example, Sorenson and Williamson are well known dissenters. For expositions of their views, see, e.g., Williamson (1996) and Sorenson (2001). I discuss Williamson’s so-called epistemicist views in sections IIb.ii and IIb.iii below. ¹² Fodor and Lepore (1996).

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conceptually necessary) that there is no matter of fact about whether someone one ninth of whose head is covered with hair is bald.¹³

The point is not specific to the figure of one ninth—they invite their readers to choose a different fraction if this seems wrong. Their claim is that there is some amount of hair coverage for which there is, necessarily, no matter of fact about whether someone with that amount of hair coverage is bald.¹⁴ Fodor and Lepore take this to be characteristic of all vague terms and to yield a constraint on correct semantics for a language with vague predicates. Any attempt to tidy up the extension of “bald” and use the result in an account of its semantics will likewise violate their constraint. Indeed, their target is not so much supervaluationism as it is the conviction that we can use the model-theoretic interpretations of classical logic in explanations of the contribution vague predicates make to the truth-values of sentences in which they appear. It is, they claim, extremely dubious that a language that violates conceptual truths of natural language can do this sort of work. But do the statements in question really express conceptual truths? In defense of the putative conceptual nature of the claim that there is no matter of fact about whether or not a borderline-bald person is bald, Fodor and Lepore offer only the following challenge: If you doubt that this is necessary, ask yourself what fact about the world (or about English, for that matter) would convince you that, by gum, people one ninth of whose heads are covered with hair are definitely bald after all. If you doubt the necessity is conceptual, remember that baldness does not have a “hidden essence.”¹⁵

Fodor and Lepore seem to think that this challenge is unanswerable. Yet, insofar as the challenge appears unanswerable, it is largely a result of the restricted data that have appeared in the vagueness literature. Most philosophers who write about vagueness have restricted their attention to two sorts of examples. One sort of example is of a sentence that expresses a particular classification of a particular object or individual, for example “Al is bald”. The other consists of sentences—for

¹³ Fodor and Lepore (1996), p. 523. Fodor and Lepore do not devote a great deal of argument to this claim, which they take to be evident. And they are not alone. In a response to Fodor and Lepore, Michael Morreau writes that his aim is to show that supervaluation can respect “the conceptual truths that they [Fodor and Lepore] have in mind.” (1999), p. 148. ¹⁴ It is worth noting that there is something very odd about Fodor and Lepore’s example. They concede that they are uncertain about the fraction in question. What is odd is that they acknowledge that they (and, perhaps, the rest of us as well) don’t really know whether this is true. This is rather different from the sorts of statements usually taken as conceptual truths—statements such as that what is pink is not red—which virtually everyone agrees are true. ¹⁵ Fodor and Lepore (1996), p. 523.

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example “there are people who are borderline bald,” “the loss of a single hair cannot make a non-bald person bald,” “a person with no hair growing out of her/ his head is bald”—that seem to be conceptual truths. The data typically used to address questions about the semantics of vague predicates concern our views about the correct evaluation of such sentences. And the considerations that are adduced in discussions of these sorts of sentences are almost invariably limited to mining intuitions about the meaning of the predicate in question. Now, it is certainly important to give some consideration to these sorts of examples. But given our everyday concerns, such examples are not the only (or even the most interesting) sorts of statements involving vague terms. In everyday life, we do not generally ponder or attempt to determine the correctness or incorrectness of classifying a particular person as bald. As for the other examples, most of them have their life primarily in discussions of semantics. Even those who are worried about their own impending baldness do not tend to concern themselves with the possibility that the loss of a single hair might make a difference between baldness and non-baldness. Nonetheless, someone who is concerned about hair loss may very well be concerned with the issue of whether such drugs as minoxidil can prevent or reverse baldness. If our interest is in the contributions that such predicates as “bald” make to the truth-values of sentences in which they appear, then the available data are not limited to the correct evaluation of the standard sorts of examples. The available data also include our views about the correct pursuit of empirical research in which these predicates play a role. Of course, the restriction to philosophers’ examples may seem reasonable if we suppose, as many contemporary philosophers do, that no examination of empirical truths or of work in the special sciences (with the exception of psychology and linguistics) is required for our investigations of the significance of vague predicates.¹⁶ But it is easy to see why this cannot be entirely correct. Consider the difference between Fodor and Lepore’s conception of baldness as supervening on hair coverage with the more common view that it is the number of hairs growing out of an individual’s head that determines whether that individual is bald. Kit Fine writes that one example of a truth that is to be “retained throughout all growth” of a language is: if Herbert is to be bald, then so is the man with fewer hairs on his head.¹⁷ Fodor and Lepore’s view might be entirely consistent with Fine’s view. But it probably is not. And it is empirical facts that can show us that it is not. Were all human hairs exactly (or almost exactly) the same width, it is unlikely that there would be any conflict between the Fine view and Fodor and Lepore’s

¹⁶ Hart (1992) is a notable exception. ¹⁷ Fine (1975), pp. 275–6. Fine seems to think that this view is sufficiently obvious as to require little defense. Nor do Fodor and Lepore devote a great deal of argument to their claim about hair coverage, which they take to be evident.

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view. In fact, however, the width of a human hair can vary widely—from less than 40 microns, to over 100 microns. And, if asked to classify someone as bald or not, most of us would rely on appearance of hair coverage, rather than a hair count. But, given the difference in hair width, it is plausible that one might count Herbert as bald yet also count someone with fewer (but coarser) hairs as non-bald. I say that this is plausible, not that it is correct. Perhaps it is not. But it is at least conceivable that two features on which baldness is thought to supervene—hair coverage and number of hairs—sometimes give conflicting verdicts. If so, the correctness or incorrectness of Fine’s statement about Herbert (or of baldness’ supervening on hair coverage) is dependent in part on empirical matters. Another problem with this view is that, as I have indicated above, baldness is a topic of empirical research. Such research could be based on a mistake—there might be no laws or regularities involving baldness. But, if this is so, it would not be a conceptual truth but a matter determined by empirical investigation. We might, for example, infer that there are no laws or regularities involving baldness if no attempts to do research on baldness yielded any substantive results. In fact, however, research on baldness has yielded results: for example, that baldness is associated with increased risk of heart disease.¹⁸ Baldness may—as Fodor and Lepore claim—have no hidden essence, but there is certainly evidence of underlying laws and regularities. Once we take these data into account, the challenge issued by Fodor and Lepore becomes much less compelling. Perhaps there is no fact about the world that would convince Fodor and Lepore that people one ninth of whose heads are covered with hair are definitely bald after all. But it is not difficult to imagine situations in which many of us would be convinced. Suppose, for example, that research is underway to test a drug, call it Vancobald, for its efficacy in reversing hair loss. And suppose a threshold is found: if a person has hair coverage of more than one ninth, treatment with Vancobald dramatically reverses hair loss. However, if the coverage is one ninth or less, the hair loss has advanced too far—Vancobald will be entirely ineffective. In these circumstances, it does not seem unreasonable to consider changing our evaluation of someone with one ninth hair coverage from borderline-bald to bald. A sufficient number of such results may well convince us that our initial judgment that someone with one ninth hair coverage is not determinately bald was incorrect. These considerations suggest that it is not unreasonable to suppose that Fodor and Lepore’s purported conceptual truth might actually not be true at all. Fodor and Lepore might respond that, if there is such a threshold, this just shows that “bald” was not vague after all. But this hardly bolsters their case. For consider, again, their claim that “if there is no matter of fact about whether

¹⁸ See n8 above.

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someone one ninth of whose head is covered with (his) hair is bald, then it is necessary (indeed conceptually necessary) that there is no matter of fact about whether someone one ninth of whose head is covered with hair is bald.” On the view we are now considering, we need to engage in empirical research to determine whether or not there is a matter of fact about whether someone with oneninth hair coverage is bald. How, then—particularly given Fodor and Lepore’s insistence that baldness is not a natural kind—can it be a conceptual truth that there is no matter of fact about this? This brief examination of Fodor and Lepore’s rhetorical challenge, far from giving us a compelling reason to agree with their remarks about conceptual truths, suggests that many of their purported conceptual truths may not be conceptual truths and, indeed, may not be true at all. Moreover, as I shall argue shortly, if Fodor and Lepore’s claims about baldness are correct, such everyday notions as baldness and obesity simply cannot be the subject of scientific research. Indeed, if they are correct, there are no good techniques for determining such things as whether obesity is associated with increased risk of heart disease or whether minoxidil can prevent baldness. In fact, one of the interesting features of Fodor and Lepore’s criticism of supervaluationism is that it focuses on something important that the supervaluationist proposals get right. The meaning of vague predicates (at least those that might be used in empirical investigations) is tied to the enterprise of precisification—ways of sharpening the bounds of a predicate. But the supervaluationist strategy, I shall argue, mistakes the significance of precisification.

II. Precisification IIa. Supervaluationism, Precisification, and the “Homophonic” Objection Given a particular precisification, on the supervaluationist view, each person must be either bald or not bald.¹⁹ And the supervaluationist counts a sentence containing a vague predicate as true just in case it is true given any admissible precisification. One condition on the admissibility of a precisification of an everyday term is that it agree with our everyday verdicts in all determinate cases. Consider, for example, the sentence “Al is bald.” If this sentence is determinately true given our everyday understanding of “bald,” then Al will be bald on each admissible precisification. Thus, on every admissible precisification of baldness, someone who

¹⁹ Here I am concentrating on Fine’s version of supervaluationism, the version on which Fodor and Lepore focus. See Fine (1975). Morreau (1999) offers an account on which precisifications need not classify every individual.

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has not a single hair growing out of her/his head is bald. Similarly, if this sentence is determinately false, given our everyday understanding of “bald,” then Al will be non-bald on each admissible precisification. If Al is a borderline case of baldness, a precisification must still classify Al as bald or non-bald, but there will be admissible precisifications on which he is bald and admissible precisifications on which he is not bald. The classification of borderline cases, however, cannot be entirely arbitrary. One of the purposes of this approach to the semantics of vague predicates is to preserve what Fine calls penumbral truths. Fodor and Lepore give the following example: If, in a precisification, Al goes into the extension of ‘bald’ and Bill has the same number or fewer hairs that Al, then Bill must go into that extension of ‘bald’, too;²⁰

That is, supposing Al and Bill are both in penumbra of “bald” (i.e., both count as borderline-bald), the admissibility of precisifications of baldness is constrained by Al’s and Bill’s relative lack of hair. The result of precisifying all vague predicates is a classical valuation—a valuation on which, for any predicate, and any member of the domain, that member is either inside or outside its extension. The supervaluationist notion of truth for vague languages is super-truth, truth in every admissible classical valuation.²¹ For classifications of individuals, super-truth and truth are indistinguishable. If “Al is bald” is determinately true (false), it is true in every admissible precisification, hence super-true (-false). If “Al is bald” is borderline (neither true nor false), there will be admissible precisifications in which it is true and admissible precisifications in which it is false. Hence it is neither super-true nor super-false. That is, supervaluationism gives us a three-valued semantics. But supervaluationism is distinguished by its treatment of the quantifiers and sentential connectives. For example, in the penumbral case described above, “if Al is bald, then Bill is bald” will be true (super-true)—despite the fact that both “Al is bald” and “Bill is bald” count as neither true nor false. Moreover, since on every classical valuation each person either is or is not bald, etc., the law of the excluded middle is super-true—despite the fact that many of its instances are neither true nor false. Fodor and Lepore’s objection is not to the supervaluationist verdicts about the truth of particular sentences. They object, rather, to the fact that the verdicts are reached by examining precisifications. Since each precisification of baldness

²⁰ Fodor and Lepore (1996), p. 519. ²¹ Although Fodor and Lepore focus on this particular version proposed by Fine, their criticism is meant to apply to virtually all attempts to use the notion of precisification in an account of truth conditions for sentences in vague languages.

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eliminates vagueness, the precisifications are valuations that, on their view, violate conceptual truths about baldness. On their view, once we advert to a classical valuation in our understanding of the term “bald,” we have left the English word behind in favor of a homophonic non-English word. There is a response to this objection in Fine’s article, which Fodor and Lepore characterize as follows, So the language for whose semantics precisification preserves the truths of classical logic is not, strictly speaking, English. But, gee, it is a lot like English. Like enough so that we can learn interesting things about what ‘bald’ means in English by attending to the behavior of its not-quite-English counterpart.²²

They reply that it is unclear how the meaning of English expressions “is illuminated by investigating the homophonic expressions in a language that is not English and none of whose terms is vague”.²³ Fodor and Lepore are committed to dismissing the resulting precisified sentences as non-English sentences whose investigation can tell us nothing about the homophonic English sentences with which we began. This is a view that can seem reasonable in the abstract. But, as I suggested earlier, it seems a good deal less reasonable when we look at other contexts. For many people (epidemiologists, for example) precisification is part of the everyday use of English. Indeed, most of us typically take the results of epidemiological research—research that involves precisification of an everyday term—as teaching us about what we are talking about when we use that term. To see this, it will help to say a bit about how such research proceeds. Because I will be discussing epidemiological results at some length in the remainder of this chapter, it is important to emphasize that the statements I make about researchers are not the results of a philosopher’s speculations about what researchers “must think” or “would say.” My characterizations are of what many actual epidemiologists do think and do say.²⁴ I will focus on the significance epidemiological research has for our understanding of a particular vague predicate, “obese.”

²² Fodor and Lepore (1996), p. 528. The response Fodor and Lepore mean to characterize can be found in Fine (1975), p. 275. ²³ Fodor and Lepore (1996), p. 528. Actually, they say that it is unclear how the “vagueness of English expressions” is to be illuminated by such an investigation. However, as the previous quote indicates, the view they are criticizing is not a view about illuminating the vagueness of “bald” but rather about illuminating its meaning. Thus, it is not unreasonable to assume that it is appropriate to speak of “meaning” here. ²⁴ It may be useful to note that I write, not as a philosopher who has read a few epidemiology articles on obesity, but rather as someone with professional experience in the field. I interrupted my work in philosophy for several years to study and work in epidemiology. After finishing a master’s degree in biostatistics and clinical epidemiology in 1993, I worked for half a year as a biostatistician for the Medical College of Wisconsin. My research during that period (as well as for the Master’s thesis) concerned obesity.

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IIb. “Obese”: A Case Study IIb.i. Is “Obese” a Technical Scientific Term? Uses of the term “obese” go back to the sixteenth century, a time in which there was no epidemiological research of the sort described here.²⁵ The term appears in myriad novels. Some examples that pre-date current epidemiological research include, The Good Soldier, Ford Madox Ford; Ulysses, James Joyce; Jane Eyre, Charlotte Bronte; Kim, Rudyard Kipling; The Woman in White, Wilkie Collins; Vile Bodies, Evelyn Waugh; The Forsyte Saga, John Galsworthy; and Joan of Arc, Mark Twain.²⁶ There are also numerous contemporary writings in which the term “obese” is used in descriptions that have nothing to do with scientific research— for example, in a BBC review of “Casablanca,” Sidney Greenstreet is characterized as “obese.”²⁷ There is, in fact, considerable evidence that, for many people, “obese” is merely a substitute for “fat” (which clearly is an everyday term from natural language). Moreover, the adjective “obese” is often used interchangeably with “fat” in newspaper accounts of medical research. Some headlines of stories reporting on such research include, “We’re fat and that’s that,”²⁸ and “Land of the Fat.”²⁹ In a Cox News Service story, the following sentence appears: “There is a statistical correlation between being fat and living in the land of fried chicken, cornbread, grits with red-eye gravy, sweet iced tea, pecan pie, porch swings and Sunday afternoon naps.”³⁰ “How obese are we?,”³¹ an op-ed piece that appeared in the LA Times written by two scientists, is described as an “attempt to define the parameters and significance of our modern fatness.” As all this indicates, among obesity researchers the term “obese” is regarded as an expression for someone who weighs a lot relative to her/his height and, for the most part, this is also true of people whom many would call “fat”.³² It should be evident from all this, that “obese” is not a term of art, invented for the purpose of carrying out research and later imported into natural language.

²⁵ See, e.g., the entry in the Oxford English Dictionary. Epidemiology is thought to have its starting point in the nineteenth century discoveries about the relation between cholera epidemics and water supply. ²⁶ Detailed references can be found by going to amazon.com searching for “obese” in the searchable full texts for these novels. ²⁷ See the BBC website. This website also has a review of a movie titled “Shallow Hal” in which the word “obese” is used to describe a character. ²⁸ Indiana Herald Tribune, August 24, 2005. ²⁹ Indiana Star, August 24, 2005. ³⁰ August 24, 2005. This conflation of “fat” and “obese” also appears regularly in BBC stories. See, for instance, a story titled “Fatbusters” from January 24, 2002, in which the following sentences appear, “In 1994, research into a fat mouse was the starting point for a revolution in the science of obesity. The obese mouse was missing a hormone called leptin, which turns off the feelings of hunger.” ³¹ September 17, 2007, LA Times. ³² Why, then, one might ask, should researchers not simply use the word “fat”? I don’t think that it is particularly difficult to see why it might be useful to substitute “obese” for “fat.” One reason is that “fat” is a pejorative. Another is that “fat” has other uses in what is more or less the same field of research (e.g., how is obesity related to the percentage of fat in one’s diet?).

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Rather, “obese” is an everyday natural language term and the research in question is an attempt to find out about the effects of obesity. For all that, it does not follow that obesity, in the everyday sense, is something that can be investigated by such means; it may be that the nature of this research has resulted in a change of subject. In order to discuss this issue, however, it is necessary to look at the kind of techniques involved in obesity research.

IIb.ii. Precisification and Empirical Studies Let us turn, now, to the methods involved in the investigation of the claim that, for example, obesity is associated with increased risk of heart disease. In the initial formulation of such a hypothesis, the use of the term “obesity” is clearly the everyday use. But how do we go about determining whether this hypothesis is true? In the early stages of investigation, the most efficient strategy is to conduct a case-control study. The first step is to identify a number of individuals who suffer from heart disease (cases) and a number of individuals who do not (controls). The next step is to determine the proportion of each group that has obesity. The results of the study will come from a comparison of the proportions in the two groups. In order to determine the proportion of obese subjects in each group, each person in the study must be classified as having obesity or not. But our everyday term ‘obesity’ is vague. There is always the possibility that there will be subjects in the study who are not determinately classifiable given the everyday understanding of obesity. What is the researcher to do? On Fodor and Lepore’s view, it is a conceptual truth that such subjects cannot be classified. Nor are they the only ones whose views imply that there is nothing to do in such a case. For example, Hartry Field asks us to consider someone, Joe, who is borderline rich and writes, Suppose that we have enough information about Joe’s income, his assets, his liabilities, the economy of his society, and so forth, to be confident that no further such information could help us decide whether he is rich.³³

In such a case, Field writes, it is “pointless” or “misguided” to investigate the issue. There simply is no correct answer. Timothy Williamson, who—unlike Field— believes that there is a correct answer in these cases, also believes that it is pointless to investigate the issue. For, according to Williamson, we can never find the answer. He writes, No one knows whether I am thin. I am not clearly thin; I am not clearly not thin. The word ‘thin’ is too vague to enable an utterance of ‘TW is thin’ to be

³³ Field (2001), p. 283.

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recognized as true or as false, however accurately my waist is measured and the result compared with vital statistics for the rest of the population. I am a borderline case for ‘thin’. If you bet someone that the next person to enter the room will be thin, and I walk through the door, you will not know whether you are entitled to the winnings.³⁴

If TW is the next person to walk through the door, he writes, “we do not even have an idea of how to find out”³⁵ who has won the bet. But the plausibility of Williamson’s and Field’s examples depends, at least in part, on their abstractness. Williamson, for instance, assumes that the only issues that will arise are those of his measurements and of how they compare with the measurements of other individuals in the relevant population. But, as with baldness and obesity, thinness is a property that we might well want to investigate. Thinness is a highly coveted property in our society and we might wonder about what benefits it really confers. We might, for example, want to know if thin people are generally perceived as more attractive or more virtuous than others. We might wonder whether or not thinness is associated with higher salaries. And, in addition to the social values conferred by thinness, we might want to know if there are other values: for example, are thin people healthier than those who are not thin?³⁶ Now let us consider the consequences Williamson’s view has for our attitudes toward attempts to answer such questions. In particular, let us elaborate Williamson’s example by supposing TW to have walked in the door of a researcher who is recruiting subjects for an investigation of the effects of thinness. Let us suppose TW to have volunteered to be part of her sample population and let us suppose she is scheduled to meet and interview, individually, each volunteer. According to Williamson, our researcher will not even have an idea of how to find out whether she is entitled to classify TW as thin or not. Suppose he is right. What is the researcher to do when TW walks through the door? One might infer that the only option is to throw TW out of our subject pool. But it should be obvious that this is no solution to the problem. If the researcher needs to eliminate everyone who is borderline-thin from the subject pool, then the original problem is not solved, but multiplied. Each time a prospective subject walks in the door, the prospective subject must be classified as thin, not thin, or borderline-thin. And the researcher is in no better shape to distinguish between the thin and the borderlinethin than she is to distinguish between the thin and the non-thin. How, then, should this sort of research be carried out? Rather than speculating about what the researcher might do, let us consider what researchers actually have

³⁴ Williamson (1996), p. 185. ³⁵ Williamson (1996), p. 185. ³⁶ One might claim that what counts as thinness, obesity, low birth weight, etc. varies with context. But, in each case, supposing the context is fixed, the vagueness remains. In the discussions that follow, I will be supposing that the context is fixed.

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done. In the 1980s and 1990s, epidemiologists understood obesity to be some characteristic associated with high relative weight and with increased risk of morbidity and mortality. BMI, a measure of relative weight,³⁷ was used to classify people as obese or not. For the most part, the associations found between BMI and mortality (or various types of morbidity) turn out to be describable as U-shaped curves. The nadir of these curves—the point associated with the lowest rates of morbidity and mortality—tends to appear somewhere between 22 and 23. On either side of this interval, the rates begin to rise—slowly at first and gradually increasing as BMI gets farther from the ideal level.³⁸ People with BMI >25 are at increased risk of morbidity and mortality,³⁹ although the risk is nowhere near as high as the risk for people with BMI >30. But there are no sharp discontinuities. How, then, was a line drawn between the obese and the non-obese? How was it decided how much increased risk is sufficiently significant?⁴⁰ In 1985, as a result of an influential National Institute of Health (NIH) conference, an arbitrary line was adopted and, subsequently, widely used. Obesity was defined as 20 percent above ideal BMI. This was calculated separately for men and women, resulting in definitions of BMI >27.3 (for women) and BMI >27.8 (for men). For people in this category, according to the NIH documents, treatment was “strongly advised.” In 1995, the World Health Organization (WHO) recommended a classification in which the cutoff point for obesity was 30,⁴¹ which was subsequently adopted by most researchers and public health officials. One reason for this change is that it is more useful to have a single cutoff for men and women. Other reasons were more significant. Some treatments for obesity, drug therapy, and gastrointestinal surgery, have serious risks.⁴² Thus, it is important to limit the people receiving these treatments to those who can be expected to benefit substantially from them.⁴³

³⁷ BMI is defined as weight in kilograms divided by height in meters squared. ³⁸ See, e.g., Calle et al. (1999). ³⁹ Although this is the standard view, a recent study found lower all-cause mortality rates among people with BMI >25. See, Flegel et al. (2013). ⁴⁰ Note that “significantly higher” does not mean statistically significant. After all, there can be a statistically significant, but very small, increased risk of mortality for, say, someone with BMI =24. If it is risk of mortality that determines who is to be classified as having obesity and if it turns out that someone with BMI =24 is .001 greater than that of someone with ideal BMI, then it would be absurd to categorize someone with BMI =24 as having obesity. ⁴¹ “Physical status: The use and interpretation of anthropometry.” Report of a WHO Expert Committee. World Health Organization: Geneva, 1995 (WHO Technical Report Series; 854). The WHO has also introduced a category of “pre-obese” that describes those people with BMI >25 and 30 is a good one. Were the researchers in question truly only concerned with relative weight, in the form of BMI, it is difficult to understand these controversies. For another, if we think that a good account of the meaning of “obese” must answer to the way speakers of English use and understand this term, we should be taking the researchers’ use of “obese” at face value—not redescribing it. The data to which an account of the meaning of “obese” should answer surely include the widespread belief—shared by epidemiologists, the NIH, the WHO, the journalists who write about science, and those of us who take the reports at face value—that epidemiological studies have shown that obesity is associated with increased risk of heart disease. Moreover, in many cases the view that the research in question is really about measurable properties conflicts, not just with researchers’ view of what their topic is, but with their decisions about how to carry out that research. To see this, let us consider a (literal) textbook example of what outcome variables should be used to test the relation between passive smoking by pregnant women and low birth weight.⁴⁵ One option, the authors of the text Designing Clinical Research say, is simply to use weight as the outcome variable. Another option is to choose the dichotomous (2-valued) variable: low birth weight vs. non-low birth weight. There is, the authors write, an advantage to using birth weight as the outcome variable. But the advantage is not that the real topic of interest is the correlation of birth weight with passive smoking. Rather, because weight is a continuous variable, it is possible to get statistically significant results with a smaller sample size. However, the authors describe the dichotomous variable as “more meaningful.” They write, The study of passive smoking and low birth weight discussed in Appendix 2 might be more concerned with babies whose weight is so low that their health is compromised than with differences observed over the full spectrum of birth weights. In this case the investigator is better off with a large enough sample to be able to analyze the results with a dichotomous outcome like the proportion of babies whose weight is below 2500 G.⁴⁶

That is, sometimes, as in this case, the elimination of the dichotomous variable will obscure the issue that is of concern. In this case, decisions about how to carry

⁴⁵ As with the predicate “thin,” standards used to classify individuals as having low birth weight might vary with context. But to restrict our attention to the use of the expression “low birth weight” in the context in question is not to eliminate vagueness. Hulley and Cummings (1988), pp. 32–3. ⁴⁶ Hulley and Cummings (1988), pp. 32–3.

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out the research are determined, in part, by the fact that the concern is with the correlation of passive smoking and low birth weight—rather than of passive smoking and birth weight. This concern, the authors tell us, may make it worth spending a great deal of extra money to get a large enough sample. One aim of this research is to find a good cutoff point. The authors go on to write, it is still best to collect the data as a continuous variable. This leaves the analytic options open: to change the cutoff point for low birth weight (we may later decide that 2350 G is a better value for discriminating babies at increased risk of developmental abnormalities).

It is, in part, a consequence of this sort of research that low birth weight is now defined as a birth weight of 30 is a good definition of obesity. Yet this question has been a topic of epidemiological research.

IIb.iv. Changing the Subject—the Homophonic Objection Revisited As we have seen, if we are to conduct empirical investigations on the effects of something like obesity, thinness, low birth weight (that is, if we are to investigate a characteristic that we mean to be talking about when we use a vague predicate), it will be necessary to draw a sharp boundary, a boundary that will, of necessity, be partly stipulative. It seems that Fodor and Lepore’s view about this kind of research is that it has nothing to teach us about, for example, obesity. For its topic (let’s call it “newobesity”), is not simply different from the old topic— conceptual truths about obesity are violated. What this really highlights, however, is not a problem with precisification but, rather, the oddness of Fodor and Lepore’s understanding of conceptual truths. Fodor and Lepore’s rhetoric suggests that there is no interesting relation between the meaning of a vague term and that of a homophonic precisified term—that the distinction is much like the distinction between the everyday word “charm” and the physicists’ word. But this is obviously false. It is not as if most of the objects that possess charm in the everyday sense also possess charm in the physicists’ sense. There is no overlap at all. The physicist does not mean to be saying anything at all about charm in the everyday sense. The epidemiologist, however, does mean to be saying something about obesity in the everyday sense. And, while the epidemiologist does not attempt to determine exactly who is obese in the everyday sense, the choice of how to draw the line cannot be entirely arbitrary. There should be considerable overlap between those who have obesity in the everyday sense and those who have obesity given the epidemiologist’s definition. If we choose, for example, to categorize as having obesity just those people whose initials are from the beginning of the alphabet then it is highly unlikely that the result will—on anybody’s view—tell us anything about obesity.⁴⁸ For all that, the epidemiologist may have gone astray. There are actual examples in which researchers have—at least according to some critics—so distorted the criteria for satisfying a predicate that the definition does amount to changing the subject.

⁴⁸ It is, of course, conceivable (although unlikely) that this definition might turn out to pick out exactly the same individuals as some acceptable definition, e.g., the definition of obesity as BMI >30. And in that case we might want to say that the results of a study using this criterion does tell us something about obesity. However, as unlikely as it is that this could work on one occasion, it is difficult to imagine that it could work on repeated tests. Unless, that is, there is some lawlike relationship between body fat and names.

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Many believe that this is exactly what happened with psychological research on intelligence. As with obesity, if we are to get empirical results about intelligence, intelligence must be measured. But the IQ tests that are used for this purpose, many believe, tell us little if anything about intelligence. Indeed, some have argued that there is reason to believe that the original subject—intelligence—is really not the sort of subject that can be studied in this way.⁴⁹ If they are right, the subject has been changed. Is there, similarly, reason to think that the subject has been changed by obesity research? The definition of obesity as BMI >30 has been widely criticized. Most criticisms of the definition are driven by the imperfect correlation between body mass and body fat. Since muscle weighs more than fat, some highly muscled athletes—including some with low percentages of body fat—are classified as having obesity by the current system of classification.⁵⁰ For this reason, and because of the stigma associated with obesity, the use of BMI to classify people as obese has come under fire, with a number of people objecting to the classification on the grounds that it classifies such athletes as obese. But the grounds of these criticisms are not that the definition relies on precise lines or on stipulation. Moreover, one of the interesting features of these kinds of criticisms is that they show that the subject has not changed. Were the researchers simply to be regarded as investigating a different subject, there would be no reason for them to pay any attention to such objections. They could reasonably regard the objections as having been based on a misunderstanding. In fact, however, the controversy has led to some rethinking on the part of researchers. On the current web site of the WHO, we find “Overweight and obesity are defined as abnormal or excessive fat accumulation that presents a risk to health.”⁵¹ High BMI is now taken to be a somewhat flawed marker of high levels of body fat. In line with this, they write, “A person with a BMI of 30 or more is generally considered obese.” That is, the investigation of obesity is not, in the view of obesity researchers, in the view of the WHO, or in the view of the critics, to be understood as an activity in which we change the subject of our study. This is not to say that researchers have moved from the initial project of investigating the effects of obesity to a new project of investigating the effects of a particular precisification of obesity, that is, having BMI >30. Rather, the investigation both early and late is to be understood as the investigation of the effects of

⁴⁹ Many people believe that there is a single, well-defined and measurable entity that can reasonably be viewed as general intelligence and that is measured by IQ tests. This view about intelligence and IQ tests played, notoriously, a central role in Herrnstein and Murray (1996), which gave rise to an enormous critical literature. However, this view has been controversial for some time. One of the most well-known critiques, first published in 1983 and now in its 3rd edition, is Gardner (2011). Two other books, which take different sides on the issue are, Goleman (1995), and Carroll (1993). ⁵⁰ See, for example, the historical remarks in Kuczmarski and Flegal (2000). Myriad internet web pages give examples of athletes who count as obese on this definition (favorite examples are Sylvester Stallone and Arnold Schwarzenegger). See, for example, http://www.obesityscam.com/myth1.1.htm. ⁵¹ http://www.who.int/topics/obesity/en/.

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obesity. It is because we take the research to address these questions, that we do not want to introduce criteria for identifying people as having obesity that amount to a change in subject. Insofar as there are conceptual truths about obesity, it is among the aims of such research to give a criterion that fails to violate those truths. Moreover, as we saw earlier, it is easy to come up with a large number of magazine and newspaper articles that purport to describe obesity research to the general public, that make repeated use of the terms “fat” and “obese,” and that include the definition of obesity as having BMI >30.⁵² But what about Williamson’s claim that such stipulations change the meaning of a term “on just about any view”? Can we not save the idea that there are truths about obesity without insisting on such stipulations? The supervaluationist view is designed to address just these concerns. For the supervaluationist urges us to replace our original notion of truth with that of super-truth. On this view, the original meaning of, for instance, “obese” determines, not a unique extension, but a group of best candidate extensions—the admissible precisifications. And if it turns out that every one of these candidate extensions is a group with higher rates of heart disease, then there is no need to consider a revised meaning for “obese.” The only extensions that fit with the original meaning of the term, then, are groups at increased risk of heart disease. That is, we are entitled to say that it is (super-) true that obesity is associated with increased risk of heart disease without introducing a meaning-changing stipulation. But this supervaluationist solution conflicts in all sorts of ways with unexceptionable epidemiological practice. To see some of these conflicts, suppose we want to engage in a first-ever epidemiological investigation of the effects of obesity. Since the identification of an effect of obesity must identify an effect on any admissible precisification, what counts as an admissible precisification cannot be determined partly by which precisifications have (which) effects. Admissibility must be entirely dependent on the views of competent (but untutored) speakers. Thus, one crucial requirement, if we are to get at the truth about the effects of obesity, is to engage in a linguistic investigation; we would need to determine what are the admissible precisifications of “obese.” It is difficult to imagine how this investigation could be carried out—competent speakers may, after all, disagree about cases. But it is even more difficult to imagine the point of engaging in such an investigation. And, leaving these problems aside, this conception of the relation between admissibility and truth seems to preclude the possibility that our research into the effects of obesity can tell us something about the extension of the term. Suppose, for instance, that a linguistic investigation reveals that one of the admissible precisifications of “obese” is having BMI >28 and that another is BMI > 30. On the supervaluationist view, then, it is not possible for epidemiological investigation

⁵² See section IIb.ii above.

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to tell us that people with BMI =29 do not have obesity. Suppose, now, that our research reveals a sharp cut-off: people are at increased risk of heart disease, diabetes, gall bladder disease, and a host of other problems only when their BMI exceeds 30. If we accept the WHO characterization of obesity as sufficiently high body fat to have deleterious health implications, we ought to be entitled to say both that obesity is associated with increased risk of heart disease and that someone with a BMI of 29 is not obese. But on the supervaluationist view, we are entitled to neither of these conclusions. A person whose BMI is 29 is neither obese nor non-obese. It is not true that every precisification of “obese” marks a group that has increased risk of heart disease, hence not true that obesity is associated with this increased risk. It may help, at this point, to turn back to the issue that launched this discussion: the issue of whether or not a certain combination of views that Frege holds is absurd. As we saw earlier, one of these views—the view that the numerals do not already, antecedent to his investigation, name particular objects—is actually held by many philosophers today. And it follows, of course, that the predicate “number” does not, already, have a determinate extension. But the conclusion Frege draws from this is very different from those drawn by contemporary philosophers. On Frege’s view, the fact that the sharp boundary requirement is not, already, satisfied by the predicate “number” is no reason to conclude that arithmetic is not about numbers. Nor is it a reason to abandon, as useful fictions, the statements we regard as stating truths of arithmetic. Rather, Frege’s conclusion is that, since the demands of science require us to satisfy the sharp boundary requirement, it is entirely appropriate to make stipulations; or, at least, stipulations that satisfy certain constraints. The view that we identified earlier as Frege’s apparently absurd view was the combination of this conclusion with the view that statements in which “number” and the numerals appear already have truth values. Let us now consider the significance of the parallel between Frege’s treatment of “number” and the epidemiologist’s treatment of “obese.” I have argued above that, on the views of most of us (with the obvious exception of some of the philosophers whose views are under discussion) the stipulations made by the epidemiologist do not change the meaning of a term, or the subject of investigation. Moreover, it is worth noting that, if we do take these stipulations to change the meanings of such terms as “obese,” then the constraint that meaning be preserved is a special, fine-grained kind of fidelity to the original term; a kind of fidelity that is more fine-grained than is demanded by our investigations into truths about the topics these terms are used to label. Why is it not of particular interest to preserve meaning, in this fine-grained sense, in our investigations? If we assume that the meaning of a predicate (along with relevant facts about the objects or individuals of which the predicate might be true or false) fixes its extension, then a change in what is taken to be the extension of a predicate will result in a change in its meaning. However, as Putnam has argued, meaning cannot both be what we understand when we understand an expression and fix the extension of

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the predicate. It should be clear that if we take the meaning of “obese” to be, as the WHO currently does, “abnormal or excessive fat accumulation that presents a risk to health,” its meaning, even given the relevant facts about individuals, is not sufficient to fix the extension of “obese. “A look at the history of obesity research makes it pretty clear that the aim of our definitions of obesity is to preserve meaning in this sense rather than to preserve meaning in the extension-fixing sense. Moreover, preserving meaning in this sense amounts to taking the truthvalues of certain sentences to be fixed, already, antecedent to defining the terms in question. That is, the extension of “obese” need not be fixed antecedent to our accepting a particular statement in which “obese” occurs to be true: that someone who has obesity has an abnormal or excessive fat accumulation that presents a risk to health. One might be inclined to think, however, that this is a result of the peculiar nature of obesity research. In particular, one might be inclined to think that the leeway in defining obesity might be a result of its not being a natural kind or having, as Fodor and Lepore would put it, “an essence.” Thus, it may be helpful to consider another kind of research that does seem to involve an essence: AIDS research. We now know that AIDS is caused by HIV infection. But that is not to say that AIDS is HIV infection. There is a difference between someone who has been infected by HIV but has a normally functioning immune system and someone who has AIDS, that is, someone whose immune system has been seriously compromised by the HIV infection. This is, obviously, an important difference and a great deal of research has been done to find treatments that will prevent immune damage in HIV infected people. Just as the various case definitions of obesity are designed to pick out individuals who satisfy the WHO description of having abnormal or excessive fat accumulation that presents a risk to health, the various case definitions of AIDS that have been adopted from the earliest definitions in the 1980s, to the current 2014 definition, are designed to pick out individuals who have severely compromised immune function due to HIV infection. Although the history of the AIDS case definitions is a more complicated story than the history of obesity case definitions, the different AIDS case definitions result in the same kinds of changes in extension that are exhibited in the different case definitions of obesity. Someone is categorized as having AIDS on the basis, not only of being infected with HIV, but also as exhibiting certain signs of immune deficiency, including low levels of CD4 cells and/or having suffered from one of a list of opportunistic infections.⁵³ Among the changes of this case definition over time are a number of ⁵³ An opportunistic infection is an infection that is not contracted by people with healthy immune systems. The discussion that follows is very oversimplified. For example, it ignores (as current definitions do not) the difference in tests for identifying HIV infection, the difference between a surveillance definition and a definition that is diagnostic, the difference in definitions for adults vs. children. However, the point in question is not affected by the oversimplification.

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changes in the list of opportunistic infections. For example, invasive cervical cancer is on the list of all the case definitions from 1993 onward but not on any of the case definitions from the 1980s. That is, some people who did not fit the 1980s case definitions (i.e., did not have AIDS in the sense of the 1980s definitions) would now be categorized as having AIDS.⁵⁴ According to the standards of sameness of fine-grained meaning discussed above, then, the meaning of “AIDS” changed between the 1980s and 1993. But for most of us there is no interesting sense in which this should be viewed as a change in meaning. It is, rather, evidence that we have learned more about AIDS. In particular, when AIDS was originally discovered in the 1980s it was thought to be a disease that affected primarily gay men. Now we know that women are actually more susceptible than men and that invasive cervical cancer is one of the first opportunistic infections to appear in a woman whose immune system has been compromised by HIV infection. What we have here is a parallel with the history of the case definitions of obesity. Just as the case definition of obesity is somewhat arbitrary, the case definition of AIDS is somewhat arbitrary. In the interests of keeping the list short enough to be easily remembered and consulted, some opportunistic infections will need to be left off the list. Moreover, the significance of certain opportunistic infections has been re-evaluated: on the 2014 version of the case definition for children, one of the opportunistic infections, Lymphoid interstitial pneumonia, was left off the list on the grounds that it is indicative only of moderate rather than severe immunodeficiency.⁵⁵ These changes in case definitions over time are perfectly normal and there is no obvious reason to claim that they introduce changes in meaning. As I have argued earlier, the existence of such changes should not be regarded as a change in topic. Moreover, if one is committed to regarding them as changes in meaning then it is important to note that the existence of such changes in meaning over time is explicitly acknowledged by researchers and public health officials and is in no way to be regarded as a defect of the research on the topic.

III. Explication and Vagueness as a Logical Defect Let us return now to the bet that Williamson asks us to imagine, the bet about whether or not TW is thin. Earlier I suggested that, in order to pursue the investigation, the epidemiologist who is investigating the effects of thinness must classify TW either as thin or not-thin. But, as we have seen, this is not

⁵⁴ For the 1985 case definition, see https://www.cdc.gov/mmwr/preview/mmwrhtml/00000567.htm For the 2014 case definition, see https://www.cdc.gov/mmwr/preview/mmwrhtml/rr6303a1.htm. ⁵⁵ Revised Surveillance Case Definition for HIV Infection—United States, 2014, MMWR Recommendations and Reports April 11, 2014 / 63(RR03); 1–10.

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quite right. The better strategy is to record each subject’s BMI and salary. The epidemiologist can then consider, simultaneously, whether there is a “thinness” effect on salary and which people should be classified as thin. Indeed, as I suggested earlier, it is not unreasonable to believe that a variety of general truths distinguishes the population of thin people: they may be at less risk of heart disease and have higher salaries than non-thin people. Let us suppose that there are recognizable and discoverable trends. Let us suppose that all the benefits that, we antecedently suspected, accrue to thin people really can be found in our statistical studies: members of the population consisting of people whose BMI is under 21 really are at lower risk of heart disease as well as myriad other diseases and really do have higher salaries than people with higher BMIs. What this suggests is that we might well find that there is an answer to the question of whether or not TW is thin. If, say, TW has a BMI of 20, it seems perfectly reasonable to classify him as thin, regardless of what his body measurements or those of the general population are. It is useful to think about why studies like this can convince us that someone whose BMI is less than 21 is thin. We want our words—including our vague predicates—to be suitable for describing the world. Some of the work that the predicate “thin” does for us is to be found in the connections we can state and discover between thinness and other features of the world. Our propensity for being reluctant to classify a particular individual or for claiming that she or he is a borderline case may well be—contrary to Field’s and Williamson’s suggestions—a reason to try to discover the correct classification. This kind of discovery is not the discovery of the kind of facts that, Williamson suggests, are the only relevant ones (e.g., BMI or girth measurements of the individual), but of general facts about, say, thinness and other properties. The discoveries in question are the discoveries of truths that belong to a good theory of the topic in question, whether it be of the effects of low birth weight, of being thin, or of being rich. Williamson advocates for a view, epistemicism, on which vagueness is to be understood in terms of ignorance. On this view, our vague predicates, already, have determinate extension. A borderline case of, say, thinness, is a case about which we are ineradicably ignorant. It may seem that a version of Williamson’s epistemicism is entirely compatible with the story I have just told about epidemiological research. Williamson might concede that there are some mistakes in his example. He might concede that his assumption about the facts relevant to determining whether TW is thin—that they are limited to facts about TW’s girth measurements and about those of the general population—is mistaken. And, having conceded this, he might well also concede that we could be in position to determine whether or not TW is thin. But these concessions need not undermine the epistemicist view that, for any vague predicates, there will be some cases of whose status we are ineradicably ignorant. Indeed, he might argue that the very fact that our lines between, for example, the thin and non-thin, must

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be partly stipulative rather than entirely determined by our investigations, shows our ineradicable ignorance. On this view, there is some fact of which we are ignorant. It is a cornerstone of Williamson’s epistemicism that there is—already, pre-stipulation—a matter of fact for each individual, about whether or not that individual is thin. But why think this? Williamson’s answer is that the price for giving up the view that vague predicates are bivalent is too high. He writes, of vague utterances, One can no longer apply classical truth-conditional semantics to them, and probably not even classical logic. Yet classical semantics and logic are vastly superior to the alternatives in simplicity, power, past success, and integration with theories in other domains.⁵⁶

This claim is remarkably abstract, however. It is true that, for instance, a classical semantics for a language with vague predicates is simpler than a supervaluationist semantics. But it is not that much simpler. It is only if we want to do something further with the semantics—say, if we think it should constrain our practice in inquiry—that there might be a serious advantage to using the classical semantics. And Williamson does not give an example of the advantages classical logic has in actual practice. When he asks us to imagine a bet about whether TW is thin, all that is at stake is winning or losing the bet. There is no real use made of semantics or logic, classical or otherwise. And it is not obvious that classical logic has advantages if our only interest is in evaluating the truth or falsity of isolated claims of this sort. Indeed, it is precisely because there are no theories at stake in Williamson’s example that there is no obvious difficulty with his claim that we must throw up our hands and admit ignorance in this case. For, as we have seen, this strategy cannot be used in epidemiological investigations. For all that, this much is right about Williamson’s claim: if our philosophical investigations were to require us to give up classical logic in our empirical studies and statistical arguments, this would be a very high price to pay. It is easy to see, for instance, that in obesity research we are better off with classical logic than we would be with some more complicated logic, designed to take into account the vagueness of our predicates. But the view that classical logic is the appropriate logic to be used in our investigation does not in any way support Williamson’s claim that our vague predicates, already, have determinate extension preinvestigation. It is, after all, routine for epidemiologists to assume that there is no unique correct line dividing, say, the obese from the non-obese, the thin from the non-thin. Yet that assumption does not prevent them from using classical logic. The actual price we pay in our practices for our use of classical logic is that

⁵⁶ Williamson (1996), p. 186.

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we are required to draw lines that are partly stipulative and that we must be prepared to change our case definitions when it seems reasonable to do so. Williamson does not consider this—probably because it is so obvious to him that such stipulations change the meaning of the term. But what we have seen in the above discussions of epidemiological practice is, first, that it is not clear that there is a change of meaning and, second, that even if there is, it is a kind of meaning-change that is perfectly acceptable given what we want to accomplish— that it does not amount to a change of subject. There is no reason to think that the use of classical logic in our actual practice requires us to assume that the vague predicates of natural language have meaning that, already, in advance of any sort of stipulation, determine fixed extensions. Williamson takes vagueness to be a logical defect. But his conclusion—that natural language, as it is, must already be free of logical defects—is unwarranted. I suggested earlier that what has made it seem plausible that it is pointless to investigate borderline cases is the lack of attention to the use of vague predicates to give informative theories about the world. Once we consider our attempts to give theories about the world, however, it is immediately apparent that, even if we can identify a unique measurement on which a vague classification supervenes, facts about this measurement cannot be the only facts relevant to the classification of an individual or object. Low birth weight, for example, clearly supervenes on a single measurement, weight. Yet information about the birth weight of an infant, even in combination with information about birth weights of other infants, may not suffice to determine whether or not the infant’s birth weight is low. We also need facts about the relations between birth weight and various sorts of morbidity and mortality. For there is abundant evidence that we do in fact believe that information about morbidity and mortality can be relevant to this kind of verdict. This evidence appears not only in journals reporting research in various fields, but also in newspaper reports of this research. Is it, to use Field’s words, beyond belief that a rich person could, by losing a cent, become no longer rich? If we think of the meaning of words like “rich” or “obese” as answering only to our untutored dispositions (or untethered beliefs) but not to our serious investigations of the world, then it may seem so. It may also seem so if we think that the primary function of words like “rich” or “obese” is that of describing an individual. But there is another use that these words have. Once we start thinking about what determines the extension of a predicate in terms of overall theories rather than our dispositions for applying the predicate in particular cases, the apparently obvious sentences that create the Sorites problems (e.g., a thin person could not, by gaining a single ounce, become non-thin; that the loss of a cent could not result in a rich person’s becoming no longer rich) are no longer obvious. If we think of these words as belonging to a vocabulary that enables us to give good theories of some part of the world then, I maintain, such claims are not beyond belief. They are exactly what we do believe.

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Let us conclude by comparing Frege’s stipulations with the stipulations of the obesity researcher. In both cases, there are troublesome expressions—“obese,” in one case, and “number” (and the numerals) in the other. The trouble is that these words are defective in ways that interfere with the researchers’ attempts to answer certain questions. But these terms also serve important purposes. In particular, there are statements that we already recognize as statements of established truths about obesity and numbers. And these statements place constraints on our definitions. But in both cases the constraints will not suffice to give us a definition. Any definition that will eliminate the defects must be partly stipulative. The researchers use the terms—defined in this partly stipulative way—to answer the questions with which they began. It is tempting to think that these definitions must work as logical analyses of the meaning the terms in question already have. But, as we saw earlier, the WHO acknowledges that the case definition of obesity does not do this. Frege makes a similar acknowledgement.⁵⁷ After asking how one can judge whether a logical analysis is correct, he writes, But does it coincide with the sense of the word with the long established use? I believe that we shall only be able to assert that it does when this is self-evident.

In the next paragraph he writes That it agrees with the sense of the long-established simple sign is not a matter for arbitrary stipulation, but can only be recognized by an immediate insight.

But this is not a rejection of stipulative definitions. For Frege goes on to say that a so-called definition that is not stipulative—that is, that agrees with the sense of the long-established simple sign—is not really a definition at all, it can only be an axiom. In contrast, Frege’s definitions are meant to be real definitions. They are not self-evident—as he indicates in Foundations, we do not think of numbers as extensions—and they are not axioms of his systematic science. He elaborates as follows,

⁵⁷ I say that this is a similar acknowledgment, not that it is the same. There is an obvious difference between the WHO’s attitude towards the case definition of obesity and Frege’s attitude towards his definitions of “number” and the numbers. Frege, unlike the WHO, means to be fixing the sense of the terms he defines. There is a good reason for this difference. Frege means to be giving once-and-for-all definitions—his assumption is that we already know what the constraints are (i.e., which sentences must be made true by his definitions). In contrast, we do not know what all the constraints on a definition of obesity are—the case definition may need to be changed due to future discoveries. Thus, for the epidemiologist it is better to limit the sense of the expression to something vague like “abnormal or excessive fat accumulation that may impair health.”

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Let us assume that A is the long-established sign (expression) whose sense we have attempted to analyse logically by constructing a complex expression that gives the analysis. Since we are not certain whether the analysis is successful, we are not prepared to present the complex expression as one which can be replaced by the simple sign A.

He then proposes that we choose a fresh sign, B, “which has the sense of the complex expression only in virtue of the definition.” He continues, If we have managed in this way to construct a system for mathematics, without any need for the sign A, we can leave the matter there; there is no need at all to answer the question concerning the sense in which—whatever it may be—this sign had been used earlier. . . . However, it may be felt expedient to use sign A instead of sign B. But if we do this, we must treat it as an entirely new sign which had no sense prior to the definition.

He concludes by insisting “We stick then to our original conception: a definition is an arbitrary stipulation by which a new sign is introduced to take the place of a complex expression whose sense we know from the way it is put together. A sign which hitherto had no sense acquires the sense of a complex expression by definition.”⁵⁸ This kind of definition is taken up by Carnap and Quine, who label it “explication.” Just as Frege suggests that we eliminate the sign A, Quine describes the enterprise of explication as elimination and writes: We have, to begin with, an expression or form of expression that is somehow troublesome. It behaves partly like a term but not enough so, or it is vague in ways that bother us, or it puts kinks in a theory or encourages one or another confusion. But it also serves certain purposes that are not to be abandoned. Then we find a way of accomplishing those same purposes through other channels, using other and less troublesome forms of expression. The old perplexities are resolved.⁵⁹

⁵⁸ Taken in isolation, these passages may seem to show that Frege’s view of definitions conflicts with the attitude that epidemiologists take towards their definitions; these passages may seem to show that on Frege’s view the definitions of the numerals are entirely unconstrained by earlier uses of the numerals. But this is not right. Frege’s point here is that the sense of the defined expression is determined from the way it is put together. That is, once we choose to use, e.g., Frege’s definitions of the numbers in a systematic science of arithmetic, we cannot appeal to anything that we “knew” about the numbers pre-definition. As we saw in Chapter 6, Frege’s actual choices of definitions are constrained by extra-systematic faithfulness requirements. And, as I argued in Chapter 6, these faithfulness requirements are what is required, on Frege’s view, for him not to be changing the subject. As I have argued in this chapter, the epidemiologist’s claim not to have changed the subject is dependent on her definitions’ satisfying similar faithfulness requirements. ⁵⁹ Quine (1960), p. 260. For a more recent account of explication and a defense of its use, see Ebbs (2011), p. 623. See also, Ebbs (2009).

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This description of explication is Quine’s generalization of the circumstances surrounding the definition of the expression “ordered pair.” The notion of ordered pair, which was originally used without a definition, can be helpful for some of the things we want to say and prove about functions and relations. For example, we can take a two-place relation to be a set of ordered pairs. The notion of ordered pair is, in Quine’s words, “a device for treating objects two at a time as if we were treating objects of some sort one at a time.”⁶⁰ But Quine also claims that the notion of ordered pair, understood in this way, is defective. The defect stems from the fact that the normal occurrences, both of the expression “ordered pair” and of expressions meant to name particular ordered pairs, are “limited to special sorts of context.”⁶¹ The special sorts of context require only one constraint on the notion of ordered pair. It is that, (1) for any ordered pairs, and , = if and only if x=z and y=w. What about other contexts? In particular, if “” is to be a name of a particular object, we need more. There should be an answer to the question of whether or not is identical to, say, {0,{1}}. Yet (1) provides us with no determinately correct answer to this question. This, from Quine’s point of view, is a defect. And the defect can be addressed by a stipulative definition, for example, that for any x and y, the ordered pair = {{x}, {x, y}}. This stipulative definition solves the problem. If we define ordered pairs in this way, (1) turns out to be true. Moreover, there is a determinate answer to the question of, for example, whether the ordered pair is a set (on this definition it is) and which set it is (on this definition it is {0,{1}}). Quine’s story about the explication of “ordered pair” is a simplified version of the stories of both Frege’s definitions and the epidemiologist’s definitions of obesity. Both the numerals and the predicate “obese” serve purposes that are not to be abandoned. But in each case, there is something that is an obstacle to further research. For Frege, we need a definition of the predicate “number” in order to identify the source of our knowledge of the truths of arithmetic. For the epidemiologist, we need a way to classify each individual as having obesity or not in order to perform our statistical tests. It is also useful to think of the notion of explication in connection with Frege’s context principle. It is not that there was, from the moment anyone used the notion of ordered pair, a determinate extension of this notion. Although currently the standard definition is Kuratowski’s ( = {{x}, {x,y}}), other definitions have been proposed by Wiener and Haussdorff.⁶² Kuratowski’s definition has clear advantages, but it is highly improbable that anyone would seriously claim ⁶⁰ (1960), p. 257. ⁶¹ (1960), p. 258. ⁶² For some of the history of the definition of ordered pair, see Scott and Mccarty (2008).

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that the Kuratowski definition is what everyone, all along, has meant by “ordered pair.” Rather, what comes first is not the establishment of the extension of the term but a truth: that, for any ordered pairs, and , = if and only if x=z and y=w. It is only the assumption that this statement is true that makes the notion of ordered pair of any use to us. And there is nothing absurd about both accepting this as a truth prior to giving a definition that fixes the extension of “ordered pair” and yet demanding that the extension be fixed. Similarly, what makes the numerals and the term “number” of use to us is our assumption of such truths as that 1 is the successor of 0, every number has a successor, etc. And there is nothing absurd about accepting these as truths and yet demanding (as Frege does) that the extension of “number” be fixed. Indeed, when we look for how Frege employs his methodological principle “never to ask for the meaning of a word in isolation, but only in the context of a proposition,” what we see is that what constrains any identification of, say, the number one or the extension of “number” is the sentences that we take to express simple truths of arithmetic. I have argued that we should attribute to Frege a combination of views that may seem absurd to many contemporary readers. These views are that the numerals did not name particular objects and “number“ did not have fixed extension antecedent to his offering his definitions and yet sentences in which such terms appear have truth-values. I have also argued, in this chapter, that there is nothing absurd about holding this combination of views. From a methodological point of view, this is exactly how we do think about words and sentences in many empirical investigations. This distinction of methodological and semantic concerns may make it seem that some of the plausibility of attributing these views to Frege also rests on the argument that he is not giving a semantic theory. But this is not right. For if a semantic theory tells us that there is something wrong with a perfectly good empirical investigation, this tells against the semantic theory. And I have argued, in the discussions of epidemiological practice, that several widely accepted views, (e.g., about the semantics for languages with vague predicates) should be rejected on just such grounds. The interest of Frege’s unfamiliar views is not simply historical. Frege’s unfamiliar views tell us something about language; something that should constrain any attempts to develop a semantic theory. What about arithmetic? Here, there is no question that Frege has much teach us—even though, on his own view, his project of showing that the truths of arithmetic are analytic resulted in failure. But just as we can learn from some of Frege’s unfamiliar views about language, we can also learn about arithmetic, I shall argue, from Frege’s unfamiliar views. I turn, in the next chapter, to the issue Benacerraf raises in his 1973 article, “Mathematical Truth.” There is a Fregean response to this issue that is, I shall argue, far more compelling than the more widely accepted contemporary responses. Indeed, on a Fregean view, there is no problem at all.

8 Mathematical Knowledge and Sentential vs. Subsentential Priority In his 1973 paper “Mathematical Truth” Paul Benacerraf introduces a puzzle about truths of arithmetic. There is, he argues, at least an apparent conflict between what is required for a good semantic theory and what is required for a good epistemological theory. The puzzle is based on several explicit assumptions, semantic and epistemological. The first semantic assumption, the semantic homogeneity assumption, is that there should be a unified semantic theory for mathematical and non-mathematical language. A second, less highlighted, semantic assumption, the grammar of arithmetic assumption, is that the grammar of everyday sentences of arithmetic is not misleading. Supposing these assumptions are satisfied, Benacerraf writes, numerals must be recognized as object names and numbers as objects. So far, so good. But this, he argues, conflicts with an epistemological assumption. He writes, about knowledge of everyday medium-sized objects, “such knowledge (of houses, trees, truffles, dogs, and bread boxes) presents the clearest case and the easiest deal with.”¹ He thinks that knowledge about such objects is easy to explain in terms of an easily seen link between objects in the real world and our cognitive faculties. And he thinks that an account of this kind of knowledge is to be regarded as a model for any account of any kind of knowledge. His epistemological assumption is that an account of our knowledge

¹ Benacerraf (1973), p. 672. Benacerraf also gives a stronger characterization of the epistemological assumption: that some sort of causal theory of knowledge is correct. However, it is the weaker version that figures in Benacerraf ’s argument. The irrelevance of causal theories is manifest, for example, in Benacerraf ’s discussion of Gödel’s view that we can have knowledge about mathematical objects. Gödel’s explanation of our knowledge of these objects is that we have a non-sensory faculty of perception, mathematical intuition, that allows us to perceive these objects. Benacerraf does not object to this explanation on the grounds of Gödel’s appealing to a non-sensory faculty of mathematical perception. Nor is Benacerraf ’s objection that, on Gödel’s view, it appears that we cannot have causal interactions with mathematical objects. Rather, Benacerraf simply thinks that Gödel has not given a convincing account of the link between our cognitive faculties and mathematical objects. Among those who have argued that Benacerraf’s problem does not depend on the assumption that a causal theory of knowledge is correct are Penelope Maddy (1990) and Hartry Field (1989), both of whom address versions of Benacerraf’s problem in recent writings on mathematics. Other attempts to address versions of Benacerraf’s problem appear in writings by Hellman (1989), Hodes (1984), Kitcher (1985), Resnik (1981), and Wright (1983). In this chapter, I argue that we should respond by dropping an underlying assumption, an assumption that I have labeled “subsentential priority.” This is very similar to Philip Ebert’s suggestion in (2007).

Taking Frege at his Word. Joan Weiner, Oxford University Press (2020). © Joan Weiner. DOI: 10.1093/oso/9780198865476.003.0008

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of any sort of objects (including mathematical objects) requires “an account of the link between our cognitive faculties and the objects known.”² The conflict can now be described as follows. Supposing numbers are objects—as the semantic assumptions tell us they are—they are not physical objects; they have no location and no physical properties. Numbers, if they are objects, must be abstract; they must be causally inert. The problem is that it is difficult to see what link there can be between our cognitive faculties and objects that are causally inert.

I. The Logical Notion of Objecthood There is something misleading about Benacerraf ’s characterization of his problem as the result of a conflict between semantic and epistemological assumptions. What makes this characterization misleading is that the epistemological problem, if there is one, is independent of whether or not numbers are objects in the sense of ‘object’ that figures in Benacerraf ’s semantic assumption. To see this, let us take a closer look at the notion of objecthood that figures in the semantic assumptions. The argument that numbers are objects in this sense stems from the idea, which has its roots in Frege’s writings, that number words function as proper names rather than, for example, predicates, quantifiers, or function-names. Frege writes, Precisely because it forms only an element in what is asserted, the individual number shows itself for what it is, a self-subsistent object. I have already drawn attention above to the fact that we speak of “the number 1”, where the definite article serves to class it as an object.³

But this is not to give a characterization of those things that are objects. Frege denies, for example, that it follows from taking numbers to be self-subsistent objects that we can have ideas or images of numbers or that numbers are spatiotemporal objects. He also denies that it follows from taking numbers to be selfsubsistent objects that numerals designate numbers outside the context of a proposition, writing, The self-subsistence which I am claiming for number is not to be taken to mean that a number word signifies something when removed from the context of a proposition, but only to preclude the use of such words as predicates or attributes, which appreciably alters their meaning.⁴

² Benacerraf (1973), p. 674.

³ FA, §57.

⁴ FA, §60.

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What, then, is the content of the statement that numbers are objects? As we saw in Chapter 3, Frege’s view about number words is based, not on the grammar of our natural language statements about numbers, but on the correct logical regimentation of statements about numbers. A disagreement about how a claim is correctly regimented is a disagreement about which are the correct inferences in which it can figure. Suppose, for instance, we are interested in what counts as a correct regimentation of a claim that 4 is an even integer. To say that 4 should be represented by an object expression in this regimentation is to provide information about the sort of inferences we can make about 4. For example, this claim licenses the inference that there is at least one even integer. From the additional claim that 4 is the positive square root of 16 we are entitled to infer that the positive square root of 16 is an even integer, and so on. Or, to put it another way, to be an object is to be the value of a certain sort of variable.⁵ On this view, our categorization of numbers as objects depends on our understanding of complete statements about numbers and what we can infer from them. Moreover, Frege tells us that numbers are not the only objects that, “in principle . . . cannot be imagined.” For, he writes, “Even concrete things are not always imaginable.” One example is the Earth. The Earth is a concrete object. But when Frege discusses the claim that the Earth is an object, he does not bolster the claim with talk of ideas or images of the Earth or its status as something spatiotemporal. Nor is there mention of any causal relation or cognitive link we can have to it.⁶ Rather, we know that the Earth is an object because of what we can say and infer about it. He writes, Time and time again we are led by our thought beyond the scope of our imagination, without thereby forfeiting the support we need for our inferences.⁷

From the claim that the Earth is a planet we are entitled to infer that there is at least one planet. And these claims about what inferences we can make about the Earth are indicated, in part, by grammar. In natural language, “the Earth” is a singular term, it plays the same grammatical role as proper names. And this should be substantiated in any correct regimentation of statements about the Earth. This Fregean notion of objecthood appears to be exactly the notion of objecthood that is operative in Benacerraf ’s discussions of the semantic homogeneity assumption. As in Frege’s discussions, in Benacerraf ’s discussions of the semantic homogeneity assumption, there is no mention of any feature that is common to

⁵ In a second-order logic, not all variables are first-order, i.e., not all variables range over objects. ⁶ Or, to use more Fregean language, what is important is not the representations or ideas we have of the Earth (or that we have them). ⁷ FA, §60.

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numbers and such everyday medium-sized objects as houses, trees, truffles, dogs, and bread boxes. The issues that arise in Benacerraf ’s discussions of whether numbers are objects have to do with our use of object-names in our expressions of statements and inferences. What is at issue for Benacerraf, as well as for Frege, is correct logical regimentation rather than everyday grammar. In particular, Benacerraf raises a question that is logical, in Frege’s sense, when he asks whether the words “there are at least three” in the sentence “There are at least three perfect numbers greater than 17” can be eliminated “in the usual way in favor of quantifiers, variables and identity.”⁸ Benacerraf ’s assumption that the grammar of our everyday sentences of arithmetic is not misleading is the assumption that a number name that appears in one of the everyday sentences will be represented by a singular term in a correct logical regimentation of the statement. As we have seen, this logical conception of objecthood, which has been called a “thin conception of objecthood,”⁹ is thin in the following respect: to say that numbers are objects in the logical sense is not to identify any feature that numbers share with more familiar objects. There is also another sense in which Frege’s conception of objecthood is thin. Frege identifies the term “object” as belonging to logic and he claims that it is not definable. Thus, it may seem only natural to suppose that his logical notation should include a primitive term for predicating objecthood. But, as we have seen in earlier chapters, it is no accident that Frege’s logical notation has no such term. There would be no point to having a term for objecthood, unless predicating objecthood of something were to distinguish it from other things that are not objects. But, on Frege’s view of objecthood, there can be no substance to the claim that some particular thing is an object. For in order to say, of some particular object, that it is an object, we need to use either an object name for it or an object-description that picks it out uniquely. And any sentence that consists of an object name (or description) followed by the predicate “is an object” will be trivially true. Indeed, as we saw in Chapter 3, on Frege’s view of objecthood, the sentence “everything is an object” turns out to be true.¹⁰ There would be no point to adding an objecthood predicate to Begriffsschrift. ⁸ Benacerraf (1973), p. 663. ⁹ See Charles Parsons (2004), p. 75. However, it is worth noting that Parsons is actually mistaken. Frege’s conception of objecthood, on Parsons’ interpretation, is not thin in this sense. For Parsons claims that Frege has a theory of functions as unsaturated. On Parson’s interpretation, Frege’s distinction between objects and functions must be an important and substantive part of any theory of functions as unsaturated. Thus, on the theory Parsons attributes to Frege, to say that something is an object is to make a claim of substance: it is to say that it is not a function. Indeed, were Parsons correct about this, Frege’s conception of objecthood would be “thicker” than the metaphysical conception of objecthood. Moreover, this aspect of Parsons’ characterization of Frege’s conception of objecthood cannot be reconciled with Parsons’ characterization of Frege’s conception of objecthood as a formal concept. I think, as I have argued elsewhere (see, e.g., Weiner (2001)) that there is something to the latter characterization: Frege’s attitude towards his notion of objecthood shares much of Wittgenstein’s characterization of formal concepts. There is, moreover, overwhelming evidence that Frege does not have a theory of functions as unsaturated [For arguments see, e.g., Weiner (1995a) and (1995b)]. ¹⁰ See Chapter 3, section IIb.ii.

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But, for all that, there is something odd about describing Frege’s conception of objecthood as thin. For, as we saw in earlier chapters, he argues, at length, that numbers are objects. What kind of content, then, does the claim that numbers are objects have? One of Frege’s aims in Foundations is to give guidelines for constructing number definitions that can be used, in turn, in the construction of a systematic arithmetic. Acceptable definitions must be usable in expressing and evaluating inferences about numbers. And not all words (and, more significantly, not all Begriffsschrift expressions) are object-names. Begriffsschrift and natural language also contain function-names, both of first and second level. Object- and function-expressions play different roles in the expression of inference. For example, suppose one has proved a universal generalization from Frege’s logical laws, say (x)Φx, where Φ is a function-expression. Given that an object expression, a, has been introduced into his notation by an appropriate definition, one can then prove Φa. But an expression that results from similarly juxtaposing two occurrences of function-names (e.g., something like: ΦΦ) does not just fail to be provable, it fails even to be a Begriffsschrift expression. Although Frege does not say it quite this way, it is not inappropriate to read the universal generalization as saying “Φ holds of every object.” What makes the numbers objects is that the numerals must be given object-expression definitions if Begriffsschrift is to do its job in the expression and evaluation of the inferences of arithmetic. Frege’s use of the term “object” has an unusual purpose that is, as Benacerraf suggests at one point, connected to the use of variables of quantification. To say that numbers are objects in Frege’s sense is to make a claim about the proper regimentation of statements about numbers. And insofar as there is content to Benacerraf ’s semantic homogeneity requirement, it relies on a version of Frege’s logical conception of objecthood.¹¹ This explains the fact that Frege devotes a considerable amount of space in Foundations to an argument that numbers are objects. It also explains the fact that, while Frege argues that numbers are objects, it is no part of his theory of arithmetic that numbers are objects. The argument that numbers are objects belongs to the pre-theoretical elucidation of his logic and to his attempt to show that arithmetic belongs to logic. Thus, the Fregean notion of objecthood, while not a part of any metaphysical theory, is not so thin as to be devoid of content. Is there an epistemological difficulty with taking numbers to be objects in this logical sense? One of the examples that Benacerraf contrasts with our knowledge about numbers is the knowledge that someone, Hermione, who is holding a truffle, has of the truffle. The truffle, of course, is an everyday medium-sized object. And Hermione knows things about the truffle via perception. The epistemological difficulty with our knowledge of truths about numbers is supposed to ¹¹ This is not to say that Benacerraf shares Frege’s view of truth-values as objects. Rather, Benacerraf shares Frege’s view that objecthood is determined by inferential role.

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be that, unlike Hermione’s knowledge about the truffle, there appears to be no perception-like link between our cognitive faculties and numbers. Supposing this is an epistemological difficulty, why is it a difficulty? One might simply think that all knowledge requires support from perception or some sort of perception-like link between our cognitive faculties and whatever it is that we are talking about. But if this explains why there is a problem with our knowledge about numbers, then whether or not numbers are objects is beside the point. Can we identify a special problem for our knowledge of arithmetic, a problem that stems from our understanding of numbers as objects (in the logical sense)? If we suppose— as, of course, Frege does—that not all knowledge requires anything like perceptual support, why should knowledge about numbers require this kind of support? It may help to consider our knowledge about color. Color, like the truffle, is something observable. It does not seem unreasonable to suppose, then, that just as perception seems to underlie Hermione’s knowledge of the truffle, perception underlies Hermione’s various judgments of color. Is there, similarly, a feature shared by numbers and the truffle? We have already noted that the fact that both the truffle and the number 4 are objects in the logical sense gives us no reason to suppose that they share any common feature. Thus, on its own, the fact that both the truffle and the number 4 are objects in the logical sense gives us no reason to think that accounts of knowledge of the number 4 must resemble accounts of knowledge of the truffle. As long as, by “objecthood,” we mean “objecthood in the logical sense,” it is not obvious that there is a conflict between the demands of epistemological and semantical theories. Benacerraf ’s discussion of semantics is a red herring. But is there another conception of objecthood that gives rise to the problem? The conception of objecthood that figures in Benacerraf ’s semantic assumptions is not the only conception of objecthood that runs through his paper. There is another conception that I will call “the metaphysical conception of objecthood.” On this conception, to be an object is simply to be something we can talk about— using any kind of expression. Benacerraf writes, I favor a causal account of knowledge in which for X to know that S is true requires some causal relation to obtain between X and the referents of the names, predicates and quantifiers of S.¹²

If we think of predicates and quantifiers (as well as names) as having referents, then these referents—these entities that we can use names, predicates, or quantifiers

¹² Benacerraf (1973), p. 671. I quote this passage by way of drawing attention to the way Benacerraf advocates a unified treatment of names, predicates, and quantifiers. The talk about causal theory is not important to the point at issue here. As I have noted in n1 above, it is widely believed, today, that Benacerraf neither needs nor exploits this assumption in his argument that there is a problem.

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to talk about—are objects in the metaphysical sense. It is important to note that this metaphysical conception of objecthood is, arguably, thinner than Frege’s logical conception of objecthood. For, as we saw earlier, there is content to Frege’s claim that numbers are objects, although not content that belongs in a logical or metaphysical theory. In contrast, it is not obvious that there is any content to the claim that numbers are objects in the metaphysical sense. In order to argue that numbers are objects in the metaphysical sense, what seems to be required is an argument that number-talk makes sense. It is important, of course, that statements in any field of inquiry should make sense. But to show that such talk makes sense need not be tied to any notion of objecthood. And this is in line with some of Benacerraf ’s other discussions of objecthood, for example, “Entity”, “thing”, “object” are words having a role in the language; they are placeholders whose function is analogous to that of pronouns (and, in more formalized context, to variables of quantification).¹³

Insofar as “object” is being used as a place-holder, there is no substance at all to this notion of objecthood.

II. Metaphysical Objecthood and Subsentential vs. Sentential Priority Let us consider the view that every substantial linguistic expression (including every predicate and quantifier) that appears in a sentence with a truth-value must refer to some entity. Suppose we combine this view with the view that our knowledge of everyday medium-sized objects should provide a model for any account of any sort of knowledge about any sort of entity. Benacerraf ’s problem then arises without any mention of the logical notion of objecthood. It suffices for number words to play an important role in stating truths of arithmetic. To many contemporary readers, however, it may seem appropriate to disregard Benacerraf ’s statement about referents of predicates and quantifiers—for it is a ¹³ Benacerraf (1965), p. 66. Thus he also suggests that expressions of the form “x=y” have sense “only in contexts where it is clear that both x and y are of some kind or category C” (1965, p. 64), that is, only in contexts where it is clear that x and y share some characteristic. And he concludes that there is no general notion of objecthood: that “the notion of an object varies from theory to theory, category to category” (1965, p. 66) and that Frege’s mistake “lay in his failure to realize this fact” (1965, p. 66). But if this is right, it makes no sense to ask, in general, whether or not the numbers are objects. And, since objecthood varies from theory to theory, it seems that the only question one should ask is whether or not there is a theory whose objects are numbers. Or, to put this question more correctly: is there a theory of numbers? And the answer is obvious: there is a theory of numbers. Thus, on this conception of objecthood, there should be no problem of the sort introduced in “Mathematical Truth.” Moreover, Benacerraf ’s claim, in the earlier paper, that numbers are not objects at all makes sense only if we do have a general notion of objecthood.

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commonly held view that predicates, quantifiers, etc. do not refer. But to look at Benacerraf ’s statement in this way is to miss an important point. Even if we deny that the predicate “is a truffle” has a referent, we cannot explain how Hermione knows that the object she is holding is a truffle without ascribing to her some sort of knowledge of what trufflehood is. On Benacerraf ’s view, this requires a link between Hermione and some part of the external world. It is not obvious what kind of thing this part of the external world should be. However, it needs to be connected in an important way with the contribution that “is a truffle” makes to determining the truth-value of sentences in which it appears. Thus it might do, for instance, for Hermione to be connected with the extension of the predicate. It should be evident that Benacerraf ’s problem presupposes subsentential priority: in order for a sentence to have a truth-value, its subsentential constituents must be connected in appropriate ways to pieces of the world. Or, as we have already seen, in Dummett’s words, The referent of an expression is its extra-linguistic correlate in the real world: it is precisely because the expressions we use have such extra-linguistic correlates that we succeed in talking about the real world, and in saying things about it which are true or false in virtue of how things are in that world.¹⁴

Benacerraf ’s problem also relies on an epistemological version of subsentential priority: if we can have knowledge that is expressed by a sentence, there must be some link—something like perception—between our cognitive faculties and something connected with the semantic value (perhaps something like an extension) of each of its substantive constituents. The constituents in question include singular terms, predicates, and quantifiers. The assumption is that, in order to give an account of knowledge about some object (in the metaphysical sense of objecthood), we must be able to answer the question: what is the link between this object and our cognitive faculties? For simplicity, in what follows I am going to abbreviate this question. I will say that the assumption is that, if we can have knowledge about some object (in the metaphysical sense of objecthood), we must be able to answer the question “how is it given to us?”

IIa. Predicates and Carving Nature at the Joints It may be tempting to think that the question “how is it given to us?” need be asked only about objects (where “object” is understood in the logical sense—as something that is named by a proper name). It is not difficult, however, to see why this ¹⁴ Dummett (1981a), p. 198. The expressions Dummett has in mind are subsentential expressions, since he thinks it is a mistake to take sentences to have extra-linguistic correlates.

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is false. There are two reasons. First, every statement requires a predicate. Consider, for instance, Hermione’s knowledge that the object she is holding is a truffle—or, in other words, her knowledge of the truth of “This object is a truffle.” Hermione’s knowledge depends, in part, on her knowledge of what it is to be a truffle. That is, there must be a link between her cognitive faculties and the semantic value of “is a truffle.” Second, even if we ignore propositional knowledge and limit our attention to object names (in the logical sense) the explanation of how the object in question is given to us will require links to something like the semantic value of a predicate. To see this, let us suppose, say, that I try to introduce a new name by gesturing in front of me while delivering a lecture and saying “that’s Amy.” Have I, thereby, succeeded in securing a unique referent for “Amy” and, if so, what accounts for that success? It may be tempting to think that my gesture and utterance, along with our abilities to perceive physical objects of roughly the right size guarantees this success. But this will not suffice. For, while my gesture may well be in the direction of a person, it will be every bit as much a gesture in the direction of the chair she is sitting on, her computer and her left index finger. If the dubbing is successful, one part of the explanation is to be found in our presuppositions—that, for example, I mean to name a person, not a chair, computer, or index finger. In another context—say a context in which it is understood that we mean to be anthropomorphizing electronic devices—the same gesture and utterance could, presumably, be used to secure the computer as the referent of “Amy.” Thus, the actual success of such a dubbing is dependent on my and my audience’s already dividing the world up into persons and nonpersons. Or, as Frege says about Boole’s logic, it will be successful only on the presupposition that we have ready to hand a logically perfect concept of personhood—that is, a concept of personhood that determinately holds or not of each object.¹⁵ Thus, even if we try to limit our account to links between cognitive faculties and everyday medium-sized objects, the requisite links require that we also have links to something like a Fregean concept. Consequently, we also need an account of links between our cognitive faculties and concepts. Do we have a logically perfect concept of personhood? And, if we do, how did we get it? One might think that, here too, that the answer is to be found in some kind of ostensive definition. There is a traditional metaphor that might be taken to suggest that perception, on its own, can give us concepts: the metaphor of carving the beast of reality at the joints. The metaphor suggests that these joints (or properties) are much like joints of a turkey. The turkey’s joints are among its parts. We can perceive the turkey and we can likewise perceive its parts, including

¹⁵ He writes, I believe almost all errors made in inference to have their roots in the imperfection of the concepts. Boole presupposes logically perfect concepts as ready to hand, and hence the most difficult part of the task as having been already discharged. (BLC, PW, pp. 34–5/NS, p. 39)

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not only its legs but, equally, its joints. Now suppose the joints of reality are parts of reality in just the way the joints of the turkey are parts of the turkey. And suppose the difference between, say, persons and non-persons, is among these joints. It may seem, then, that personhood is a joint of reality that is available to be perceived. But this falls short of what is needed to explain the links between our cognitive faculties and the concepts that we need in order to talk about particular objects. As we saw in the previous paragraph, we need the concept of personhood to show that “Amy” refers to a person rather than, say, a computer. Now consider both personhood and computerhood. If they are both joints that are available to be perceived, then mere ostension provides no guarantee that Amy is the person, rather than the computer. On the other hand, if personhood is a joint of reality and computerhood is not, then it is not obvious that there is any way for me to have, playfully, given the name “Amy” to the computer. What seems clear here is that nature, along with my act of ostension, is not sufficient to determine who or what Amy is. Something more is needed. To see what is involved, it will help to consider a few additional examples. Hilary Putnam writes, Suppose I point to a glass of water and say “this liquid is called water.”¹⁶

Putnam assumes that, by this act, he has established the necessary and sufficient conditions for being in the extension of “water”: that to bear a relation (the ‘sameL’—or same liquid—relation) to the substance in the glass at which he has pointed is both necessary and sufficient. This is the point at which one might think that perception gives us an advantage. For there is something, the liquid, that one perceives. And the assumption is that, given that one has latched on to that particular piece of the world, nature steps in and does the requisite disambiguation. However, while Putnam’s example is often taken as a demonstration that nature determines which liquid is in question, there are problems with this view. Indeed, Putnam himself explicitly states that the sameL relation is a theoretical relation. Why is this so important? Supposing the liquid is, indeed, H₂O, nature still presents us with distinct alternatives. There are three isotopes of oxygen, oxygen-16, oxygen-17, and oxygen-18 and hence three isotopic compounds of H₂O: H₂O¹⁶, H₂O¹⁷, H₂O¹⁸.¹⁷ Putnam presumably means his term “water” to apply to all isotopic compounds of H₂O, since he does not give us any indication that his sameL relation distinguishes different isotopic compounds.

¹⁶ Putnam (1973), p. 702. ¹⁷ For purposes of clarity, I am limiting the discussion to compounds formed from different isotopes of Oxygen. However, it is worth noting that Hydrogen, too, has isotopes that are found in nature. Strictly speaking, H₂O has at least eighteen different forms.

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And here we see the significance of Putnam’s claim that the sameL relation is a theoretical relation. For, just as nature draws a line between H₂O and liquids that are not H₂O, nature draws lines between H₂O¹⁶, H₂O¹⁷, and H₂O¹⁸. Nature also draws lines between liquids composed of different ratios of these isotopic compounds. Indeed, a study of the change over time of ratios of H₂O¹⁸ to H₂O¹⁶ in a certain body of water can give us information about climate change.¹⁸ With this in mind, let us return to Putnam’s scenario: Suppose I point to a glass of water and say “this liquid is called water.”

It should be clear now that nature alone does not determine what the sameL relation is. Without factoring in our interests and concerns, there can be no answer to the question of whether or not the liquid in another glass counts as water.

IIb. Another Example: The Case Definition of AIDS Suppose, then, we add to nature’s joints and our ties to the everyday world another feature, our interests. Will this suffice to pick out a unique extension for the predicates of importance for scientific purposes? There is reason to think otherwise.¹⁹ To see why, it may be helpful to consider a recent episode in the history of medical research: the discovery of AIDS. In the period between October 1980 and May 1981, five young men in Los Angeles were diagnosed with pneumocystis pneumonia. Pneumocystis pneumonia is considered an “opportunistic infection”—that is, an infection that is not contracted by people with healthy immune systems. There was no recognized reason for the men in question to have compromised immune systems. What was the cause of their immune deficiency? In 1982 the CDC gave the disease a name, “Acquired Immune Deficiency Syndrome” (AIDS) and published the first case definition of AIDS.²⁰ AIDS was described as “a disease, at least moderately predictive of a defect in cell-mediated immunity, occurring in a person with no known cause for diminished resistance to that disease.”²¹ Even without reading it against a Fregean background, the ¹⁸ See, e.g., Riebeek (2005). ¹⁹ In the above discussion of H₂O, I haven’t mentioned the significance of the fact that our interests change in different contexts. There may, for example, be situations in which we care about distinguishing isotopes or ratios of isotopes of H₂O and others in which we do not. But this kind of variation is a less serious kind of variation than the problem I wish to highlight. Were the only problem a problem with this kind of change in context or interests, however, one could suppose that, given a fixed context, our predicates will have determinate extension. ²⁰ A case definition is a set of standard criteria for classifying a person as having (or not having) a particular disease, syndrome, or other health condition. This comes from the CDC’s Introduction to Epidemiology. See, https://www.cdc.gov/ophss/csels/dsepd/ss1978/lesson1/section5.html. ²¹ CDC (1982).

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problems with this case definition should be apparent. First, it is much too vague to distinguish between those who have the disease and those who do not. What is it to be moderately predictive? What is the defect in cell mediated immunity? How much diminishment is required for diminished resistance and how is the diminishment to be measured? Second, when this definition was first published, in 1982, the cause of the disease was unknown. The discovery that HIV causes AIDS only came in 1984. The status of the predicate “has AIDS” in 1982 was not so different from that of the predicate “is a number” when Frege wrote Foundations (or now). Just as we appear to know things about numbers (e.g., every number has a unique successor) it seems that in 1982 we knew things about AIDS (e.g., someone who has AIDS has an increased risk of histoplasmosis). Just as it seems that we can not determine which sets really are numbers (or whether numbers are sets), it seems that in 1982 we could not determine which people really had AIDS. One might think, however, that there is an important difference. Benacerraf argues that, no matter how much number theory we learn, there can be no answer to the questions about which objects are numbers. That is, the problem is not one of ignorance but of indeterminacy. In contrast, one might think, the AIDS classification problem in 1982 was not a problem of indeterminancy but a problem of ignorance. Supposing that in 1982 we were already talking about something determinate, then the inadequacy of the 1982 case definition is simply the inadequacy of the definition to pick out unambiguously, in words, what it is that we were—already—talking about. While the disease was discovered in 1981, one might think, only facts that were discovered later would tell us that someone who has AIDS must have been infected by HIV. After 1984, on this view, the problem of ignorance of the cause of the disease was solved. But it is not obvious that the indeterminacy was eliminated in 1984. Nor is it obvious that addressing some additional ignorance would eliminate the indeterminacy. To see why, let us look a bit more at some of the issues involved in giving a case definition of “AIDS.” In particular, how does our knowledge that AIDS is caused by HIV infection affect the earlier definition of AIDS? One might suspect that the correct definition of AIDS is that someone has AIDS if and only if they have been infected with HIV. This is wrong, however, and for a simple reason: some people are highly resistant or virtually immune to HIV. What, then, distinguishes someone who has been infected with HIV and has AIDS from someone who has been infected with HIV but does not have AIDS? Again, it might seem that the answer is simple: a person who has AIDS is someone who has been infected with HIV and has an immune deficiency. But what is it to have an immune deficiency? One obvious way to address this—obvious because it led to the discovery of AIDS in the first place—is to advert to opportunistic diseases. This is the strategy that was followed in the formulation of the 1986 case definition. The 1986 case definition includes a list of opportunistic infections and classifies any person who tests positive for HIV

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infection and suffers from one of the diseases on this list, as having AIDS.²² One might think, however, that if our interest is in constructing a definition that identifies those people who have AIDS, this definition is not quite right. For, the opportunistic infection is a result of immune deficiency.²³ Thus, it would seem that there might be someone who is immune-compromised due to HIV infection but has not (yet) suffered any opportunistic infections. What puts someone in the category of having AIDS should be the existence of immune deficiency rather than suffering from an opportunistic infection. As it turns out, we now have a way of measuring immune deficiency due to HIV infection. HIV destroys CD4 cells, which play an important role in cell-mediated immunity. Thus, one marker for an immune deficiency due to HIV infection is a low CD4 count. The normal range is from 500 to 1200 CD4 cells per microliter. And the 1993 revised case definition includes a clause for measuring CD4 concentration. As of the most recent definition in 2014, HIV infection is now classified in one of 5 stages. To have AIDS, on this definition, is to satisfy the criteria for stage 3 in this classification system. There is a group of opportunistic infections that are stage-3 defining. Otherwise stages for adults with HIV infection is determined by CD4 count as follows: CD4 500, stage 1; 200–499, stage 2;