leila haaparanta, jaakko hintikka (auth.), leila haaparanta, jaakko hintikka (eds.) frege...

FREGE SYNTHESIZED

SYNTHESE LIBRARY

STUDIES IN EPISTEMOLOGY,

LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE

Managing Editor:

JAAKKO HINTIKKA, Florida State University, Tallahassee

Editors:

DONALD DAVIDSON, University of California, Berkeley

GABRIEL NUCHELMANS, University of Leyden

WESLEY C. SALMON, University of Pittsburgh

VOLUME 181

FREGE SYNTHESIZED

Essays on the Philosophical and Foundational Work

of Gottlob Frege

Edited by

LEILA HAAP ARANT A

Academy of Finland

and

JAAKKO HINTIKKA

Department of Philosophy, Florida State University

D. REIDEL PUBLISHING COMPANY

A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP

DORDRECHT/BOSTON/LANCASTER/TOKYO

Library of Congress Cataloging·in-Publication Data

Frege synthesized.

(Synthese library; v. 181) Bibliography: p. Includes indexes. 1. Frege, Gottlob, 1848-1925. 1. Haaparanta,

II. Hintikka, Jaakko, 1929-B3245.F24F72 1986 193 8tHi523

Leila, 1954-

ISBN·13: 978·94·010·8523·6

e·ISBN·13: 978·94·009·4552·4

001: 10. 1007/978·94·009·4552·4

Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland.

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TABLE OF CONTENTS

PART I

INTRODUCTION

LEILA HAAP ARANT A and JAAKKO HINTIKKA / General Introduction 3

JOAN WEINER / Putting Frege in Perspective 9

PART II

SEMANTICS AND EPISTEMOLOGY

J. VAN HEIJENOORT / Frege and Vagueness 31 HANS SLUGA / Semantic Content and Cognitive Sense 47 THOMAS G. RICKETTS / Objectivity and Objecthood: Frege's

Metaphysics of Judgment 65 TYLER BURGE / Frege on Truth 97 LEILA HAAPARANTA / Frege on Existence 155

PART III

LOGICAL THEORY

MICHAEL D. RESNIK / Frege's Proof of Referentiality 177 NINO B. COCCHIARELLA / Frege, Russell and Logicism: A

Logical Reconstruction 197 ROBERT B. BRANDOM / Frege's Technical Concepts: Some

Recent Developments 253

PARl'IV

PHILOSOPHY OF MATHEMATICS

PHILIP KITCHER / Frege, Dedekind, and the Philosophy of Mathematics 299

VI TABLE OF CONTENTS

G REG 0 R Y CUR R IE / Continuity and Change in Frege's Philosophy of Mathematics 345

A. W. MOORE and ANDREW REIN / Grundgesetze, Section 10 375

INDEX OF NAMES 385

INDEX OF SUBJECTS 388

PART I

INTRODUCTION

LEILA HAAPARANTA AND JAAKKO HINTIKKA

GENERAL INTRODUCTION

Gottlob Frege's philosophical and foundational work was by any token a major factor in the development of contemporary analytic philosophy. Some say he was the grandfather of the whole tradition, some think of him merely as its godfather. In either case, one might expect that his work has been studied exhaustively. However, this turns out not to have been the case. Someone - it was probably Burton Dreben - once said that the worst-known period in the history of philosophy is always the time fifty to a hundred years ago. The intensive work on Frege which has been going on in the last decade and a half at first seems to belie this dictum, but in a looser sense it fits the facts well. For it is only the developments of the last couple of decades, largely of the last few years when Frege's work has reached the hundred-years mark, that have brought to light facts and issues which have shown that our understand-ing of Frege was seriously incomplete. In recent literature, one can also find a wealth of new and sometimes controversial viewpoints. For instance, Jean van Heijenoort has called our attention to an important but neglected aspect of Frege's attitude to logic and language that he calls "logic as language". Hans Sluga has challenged on a large scale the received view of Frege as a lonely figure in nineteenth-century phi-losophy whose ancestry goes to medieval objectivists rather than his German predecessors. Sluga wants to place Frege firmly in the middle of the German philosophical tradition of his day. It is indeed unmistak-able that there are, for instance, Kantian elements in his thinking that had earlier been overlooked. Indeed, the idea of logic as language is likely to be one of them. Another one is the sharp contrast between the realm of thinking and understanding and the realm of sense and intui-tion. Sluga's influence is illustrated amply in several papers in this volume. In an attempt to reverse the traditional priorities, Jaakko Hintikka has suggested, relying partly on van Heijenoort's interpreta-tion, that the crucial part of Frege's work in semantics lies in his ideas about the semantics of the familiar elementary logic (truth-functions and quantification) rather than in Frege's theory of sense and reference, which is merely intensional frosting on a more important extensional

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L. Haaparanta and 1. Hintikka (eds.), Frege Synthesized, 3-8. © 1986 by D. Reidel Publishing Company.

4 GENERAL INTRODUCTION

cake, even though it is typically given the pride of place in expositions in Frege's semantics. As a part of this attempted reversal of emphasis, Jaakko Hintikka has also called attention to the role Frege played in convincing almost everyone that verbs for being had to be treated as multiply ambiguous between the "is" of identity, the "is" of predication, the "is" of existence, and the "is" of class-inclusion - a view that had been embraced by few major figures (if any) before Frege, with the exception of John Stuart Mill and Augustus De Morgan. Hintikka has gone on to challenge this ambiguity thesis. At the same time, Frege's role in the genesis of another major twentieth-century philosophical movement, the phenomenological one, has become an important issue. Even the translation of Frege's key term "Bedeutung" as "reference" has become controversial.

The interpretation of Frege is thus thrown largely back in the melting pot. In editing this volume, we have not tried to publish the last word on Frege. Even though we may harbor such ambitions ourselves, they are not what has led to the present editorial enterprise. What we have tried to do is to bring together some of the best ongoing work on Frege. Even though the ultimate judgment on our success lies with out readers, we want to register our satisfaction with all the contributions.

The first paper included in this volume is an exploratory attempt to reach a general perspective on Frege's work. Joan Weiner stresses in her paper 'Putting Frege in Perspective' that once we start trying to under-stand Frege's work as a reaction to Kantian epistemology, it is urgent to pay attention to it also as an important part of the background of other philosophical work. According to Weiner, the problem will be "how one would connect Frege, the nineteenth-century neo-Kantian, with Frege, the first twe.ntieth-century philosopher of language and mathematics". Weiner suggests an answer to the question. She holds the opinion that Frege subscribed to most of Kantian epistemology. However, since he substituted his new logic, "the real formal rules of thought", for tradi-tional Aristotelian logic, he had to modify his Kantian This, according to Weiner, was the basic' content of virtually all of Frege's subsequent work. Weiner points out that what we regard as Frege's philosophical work was for him not a genuine philosophical theory but could only be a. set of SUbjective clues. On Weiner's view, it was designed to get us into the proper frame of mind "for understanding the purpose and significance of his real philosophical work - the work we think of today as his mathematical work." .

Weiner's paper thus essays into the same general direction Jaakko

GENERAL INTRODUCTION 5

Hintikka's reversal of the traditional view of Frege's semantical prior-ities mentioned in the beginning of this Preface.

This volume is divided in four parts. The second part consist of articles dealing with Frege's semantical and epistemological doctrines. Jean van Heijenoort discusses in his paper Frege's view of vagueness. He states that Frege's principle of completude (Grundsatz der Vollstiin�

digkeit), according to which any function must be defined for all objects, is closely connected with the requirement of sharpness of concepts, for Frege takes both lack of completude and lack of sharpness to be a failure of universal bivalence. Van Heijenoort concludes that Frege could have rejected the principle of completude, and still preserved bivalence, and that even if Frege had to ignore vagueness and other vagaries in his own project, it is perhaps time for us to look at them more carefully.

In his article 'Semantic Content and Cognitive Sense', Hans Sluga compares Russell-style theories of meaning with Frege-style ones. In the former theories, it is assumed that a satisfactory theory of meaning can be built by means of a binary relation, while the supporters of the latter theories regard a three-place relation (an expression e refers to an entity r through having a sense s) as necessary. Sluga discusses Frege's view, according to which a satisfactory theory of meaning must explain the difference between trivially true and informatively true identity state-ments, and attempts to show that Russell-style theories can, at least partially, satisfy Frege's requirement after all, provided that we are con-cerned with the difference between semantically trival and semantically

informative identity statements. Furthermore, Sluga argues in his paper that it was the problem of the status of arithmetical truths that made Frege take a critical stand against earlier semantic doctrines and intro-duce a cognitive notion of sense.

Thomas G. Ricketts argues in his article that, for Frege, ontological categories are secondary with respect to logical ones, and that under-standing the character of Frege's contrast between objective and sub-jective may help us see the primacy of judgment in Frege's thought. According to Ricketts, "our grasp of the notion of an object ... is exhausted by the apprehension of inference patterns and the recognition of the truth of the basic logical laws in which these [first-level] variables figure." Ricketts takes similar remarks to apply to the notion of concept. Thus he argues that, in Frege's philosophy, the objecthood of thoughts does not explain the objectivity of judgment but presupposes it.

lt is clear that Ricketts' starting-point is a number of observations


closely related to van Heijenoort's results mentioned in the beginning of this Introduction.

In his paper, entitled 'Frege on Truth', Tyler Burge suggests that Frege's odd-sounding conclusion about truth and falsity should be taken seriously. In the first section of his article he claims that too little attention has ben paid to the pragmatic basis of Frege's view that truth values are objects. According to Burge, Frege is committed to the doctrine that logic is primarily concerned with the normative notion of truth. The second section of Burge's paper consists mainly of the criticism of Dummett's interpretation of Frege's theses on truth values. In section III Burge purports to show how Frege's identification of the truth values with particular objects has its sources in "some of his deepest philosophical conceptions". He holds the view that "in particu-lar, it proceeds from a theory about the nature of logical objects, from a thesis about the aim and ordering of logic, and from his conceptions of assertion and truth."

In her article 'Frege on Existence' Leila Haaparanta emphasizes that Frege's greatest insight was the idea of first-order language, which, to a large extent, motivated the rest of his innovations. Haaparanta focuses her attention on Frege's concept of existence, which receives special attention in Frege's thought in connection with the thesis concerning the ambiguity of such words for being as the English 'is'. The ambiguity thesis was an important part of the Fregean paradigm of first-order logic. Haaparanta argues that Frege does not only assume the word 'is' to be ambiguous but that he considers 'exists', or the 'is' of existence, to be an equivocal word. She suggests that the equivocity view has a meta-physical and epistemological background in Frege's thought. Her paper thus pushes a great deal further the suggestions of laakko Hintikka mentioned earlier in this Introduction.

The third part of this volume is mainly focused on Frege's logical theory. Michael D. Resnik's paper 'Frege's Proof of Referentiality' deals with Frege's methodological principle according to which in a properly constructed scientific language every name must have a reference. Frege tries to prove in the Grundgesetze that his own system satisfies the principle. Resnik shows that Frege's proof contains a number of serious mistakes and that it would not prove what Frege wanted even if it were correct. Moreover, he argues that reasonable demands of rigor do not even require such a proof.

It is a widely held opinion that logicism is of no importance, as far as

GENERAL INTRODUCTION 7

current discussion of the foundations of mathematics is concerned. Nino B. Cocchiarella, for his part, believes that logicism can be defended in essentially the same philosophical context in which it was originally presented. He formulates separate reconstructions of Frege's form of logicism and of Russell's early form of logicism, which he takes to be closely similar. He reconstructs both of the logicisms as second-order predicate logics in which nominalized predicates are allowed to occur as abstract singular terms. The basic difference between Frege's form of logicism and Russell's early form of logicism is, according to Cocchiarella, that for Frege there is a sharp distinction between concepts denoted by usual predicates and concept-correlates denoted by nominalized predicates, whereas for Russell concepts are their own concept-correlates.

In the paper 'Frege's Technical Concepts: Some Recent Develop-ments' Robert Brandom praises Dummett's work on Frege, for Dummett realizes both the necessity of considering Frege's technical concepts in the framework of contemporary philosophy and the possibility of considering contemporary philosophical issues in relation to Frege's technical concepts. He reviews two recent books, David Bell's Frege's

Theory of Judgment and Hans Sluga's Gottlob Frege, against the back-ground of Dummett's work and of current interpretive controversies. Finally, he discusses the justification of the definition of a value-range (or a course-of-values, Werthverlauf) in the Grundgesetze, which yields a criticism of Frege's procedure in introducing such concepts as reference, sense, and function, as well as logical objects like numbers.

The articles included in the fourth part of this volume deal with Frege's philosophy of mathematics. In his paper, 'Frege, Dedekind, and the Philosophy of Mathematics,' Philip Kitchel' argues that Frege accepts most of Kant's philosophy of mathematics, but tries to improve it by showing how the whole of mathematics could be traced to the sources of a priori knowledge and how arithmetic can be traced to the sources of a priori analytic truths instead of pure intuition. Kitcher defends the claim that, due to his philosophical conviction, Frege bequeathed to his successors a misguided picture of the central prob-lems of the philosophy of mathematics. Instead of the Fregean view, Kitcher recommends us a Dedekindian view of mathematics, which is also Kantian but which differs from Frege's approach in certain im-portant respects. Frege tries to find a firm route from basic principles of logic to the theorems of arithmetic, while Dedekind tries to show why


our representations are inevitably arithmetically structured. It is true that Frege mentions in the Grundlagen that mathematics seems to be inescapable in our representation of experience, but unlike Dedekind, he does not develop the idea in more detail.

In his paper 'Continuity and Change in Frege's Philosophy of Mathe-matics', Gregory Currie presents a largely historical analysis of Frege's theory of real numbers in a framework which includes Frege's concept of a value-range and his view of the relation between arithmetic and geometry. More specifically, Currie discusses the role of the following three principles in the development of Frege's thought: (1) To every concept there corresponds an object, the extension of that concept. (2) The applicability of the real numbers to measurement marks a theore-tically important distinction between them and the natural numbers. (3) There is a sharp distinction between the sources of arithmetical and of geometrical knowledge. Currie concludes that the rejection of the first principle brought about the rejection of the third principle, which, for its part, brought about the rejection of the second principle.

The paper by A. W. Moore and Andrew Rein is an attempt to resolve a problem in Section 10 of Frege's Grundgesetze, which Michael Dummett calls Frege's permutation argument. The argument seems to show that, if there is one assignment of objects to value-range terms which satisfies Axiom V, then there are several. Frege suggests that we can overcome the resulting indefiniteness by demanding that it is specified for every function, when it is introduced, what values it takes on for value-ranges and other objects (if such there be) as arguments. However, this does not help us to ensure that a unique assignment of objects to value-range terms satisfies Axiom V. By strengthening the permutation argument, Moore and Rein establish a more radical form of indeterminacy, which says that "there is no distinguishing, from within a given theory, between isomorphic models of that theory". They argue that the real import of the argument is that it cannot be determined whether or not either truth-value is the value-range of any given function and that, elaborated this way, the problem suggests its own solution.

JOAN WEINER

PUTTING FREGE IN PERSPECTIVE

Until recently, there have been few attempts to read the work of Gottlob Frege in historical context. Many of those who read Frege's writings today believe that the usual reasons for reading a piece of philosophy in historical context do not apply to those pieces Frege produced. This may be a result of the popularity of two views concerning the nature of Frege's work. One is that Frege's work has no serious philosophical background. This view seems somewhat plausible given that Frege was trained as a mathematician, corresponded with mathematicians through�out his career, and seems to have done work only in the quite specialized areas of philosophy which are directly concerned with mathematics. According to the story which goes with this view, Frege began with philosophy of mathematics (investigating problems involved with his mathematical work which were not exactly mathematical problems) and saw that some work in the philosophy of logic (and later in the philosophy of language) was nef'essary. Thus, almost accidentally, Frege was pulled deeper and deeper into the problems of philosophy. If this story were true, it would not be unreasonable to assume that his worries and problems would not have been muddied by the sort of philosophi�cal assumptions and philosophically loaded terms whose prevalence in other philosophical writings makes it so important that one considers their historical context.

The tremendous influence Frege has had on contemporary analytic philosophy has also led people to ignore historical context. Frege is not only responsible for formulating modern logic, but also seems respon�sible for quite a number of the philosophical terms used and for the philosophical questions asked today. Many philosophers believe them�selves to be working on Fregean projects and at least one of Frege's philosophical papers, 'On Sense and Reference', is read in most intro�ductory courses in the philosophy of language. If we regard ourselves as Frege's direct philosophical heirs, it might seem unlikely that we should have to read his work in historical context in order to make sense of it. For, if we are involved in, or reacting to, Frege's projects and argu�ments, surely we understand their significance as well as we understand the significance and motivation of our own projects and arguments. 1

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L. Haaparanta andi. Hintikka (eds.), Frege Synthesized, 9-27. © 1986 by D. Reidel Publishing Company.

10 JOAN WEINER

However, in the years since Michael Dummett's book, Frege: Philo�

sophy of Language, first appeared, philosophers have begun increasingly to realize that Frege's work and its motivation are not as transparent as they traditionally have been taken to be. Of late, there have been a number of attempts to understand Frege's work by giving it a reading which is sensitive to historical context. In particular, there has been a great deal of interest in the extent to which Frege's work can, and should, be linked to Kant. Philip Kitcher has argued that there is an important sense in which Frege's work should be viewed as having been motivated by a concern with Kantian epistemology,2 and Hans Sluga has argued that Frege's work intended as a contribution to a 19th century German neo-Kantian movement. 3

I think this sort of reading of Frege's work must be correct. No sense can be made of Frege's motivation unless we view Frege as responding, in some serious way, to what he understood as Kantian epistemology. But, while some of these readings have provided a clearer pictUre of the philosophical background against which Frege's work should be read, there has been less attention paid to the sense in which Frege's work, given such a reading, might be taken as forming part of the background of other philosophical work. This becomes a pressing issue once we start trying to understand his work as a reaction to Kantian episte-mology. For it seems unlikely that the considerable impact Frege has had on twentieth century analytic philosophy can have been a complete accident; yet it is not obvious how one would connect Frege, the 19th century neo-Kantian, with Frege, the first 20th century philosopher of language and mathematics. To sketch the outlines of how such a connec-tion might be made, I want to examine certain tensions in Frege's views and the constraints which taking Frege's concerns as epistemo-logical (and, in some sense, Kantian) will impose on any Fregean responses to these tensions. There is little evidence that Frege was aware of these tensions; he certainly does not seem to have mentioned them explicitly. However, I believe many of his later writings can be read as responses I to these tensions which meet the epistemological constraints. When Frege's later writings are read in this way, an entirely different picture begins to emerge. This picture will end up coinciding in remarkable ways with a plausible interpretation of Wittgenstein's Tractatus. Indeed, this sort of reading of Frege's work might enable us to understand how the view of the Tractatus could have grown out of Wittgenstein's appreciations of tensions in Frege's work.

PUTTING FREGE IN PERSPECTIVE 11

I will begin with a brief sketch of the sense in which I think Frege's work should be read as having been motivated by both a dissatisfaction with some of the details of Kantian epistemology (as Frege understood it) and by a desire to make a version of Kantian epistemology work. Kant, as Frege understood him, believed that traditional Aristotelian logic set out the formal rules of all thought or, as Frege might have said, that all inferences which follow from the necessary laws of reason alone can be shown to be valid using Aristotelian logic. But Frege was convinced that Kant was mistaken here. Frege thought that the formal rules of all thought licensed more than those inferences licensed by Aristotelian logic and that his logical notation, the Begriffsschrift, was designed to set out the true formal rules of all thought. Frege's overall project was to modify Kantian epistemology in order to incorporate the real formal rules of all thought. This project might not seem to require much more than the formulation of the formal rules of all thought and the substitution of these rules for those of Aristotelian logic. However, the substitution of Frege's new logic for traditional Aristotelian logic created tensions in what Frege saw as the Kantian picture. I think that virtually all of Frege's subsequent work can be viewed as attempts to deal with the problems which he saw as having resulted from this substitution. But in order to make this plausible, it is necessary to say something about how Frege took himself to be a Kantian. I will try to do this in the next few pages. My aim will not be to detail and defend such a reading, but only to make clear, in broad outline, the perspective which makes the tensions in Frege's own views so revealing.

It is not easy to pin down the sense in which Frege took himself to be a Kantian. Many of Frege's explicit claims seem to indicate that his pretensions were quite modest - that his aim was merely to patch up trivial problems in the comers of Kantian epistemology. For instance, on those questions which Frege does not discuss (e.g. the source and justification of our knowledge of geometry) he refers us directly to Kant's answers. He also says about Kant 4

I feel bound, therefore, to call attention also to the extent of my agreement with him, which far exceeds any disagreement.

and, If Kant was wrong about arithmetic, this does not seriously detract, in my opinion, from the value of his work. His point was, that there are such things as synthetic judgments a priori; whether they are to be found in geometry only, or in arithmetic as well is of less importance.

12 JOAN WEINER

Since Frege claims that the project of the Grundlagen is part of an attempt to show that arithmetic is not synthetic a priori, these comments might be taken to indicate that Frege views his work as an attempt to correct an almost incidental mistake of Kant's. But it would be wrong to take this too seriously. While Frege uses many Kantian terms and distinctions in his writings, he reinterprets them. One such distinction is the analytic/synthetic distinction. On Frege's definition a proposition is analytic if and only if it can be proved from definitions of the concepts involved using only general logical laws, and this is certainly not Kant's definition. Thus Frege's claim that Kant was wrong in saying that arith�metic was synthetic a priori requires some explanation. We must understand the claim in light of Frege's reinterpretation of the Kantian analytic/synthetic distinction, and this reinterpretation must be regarded as more than a minor emendation of a detail of Kantian epistemology. It is important to keep these considerations in mind when we attempt to give an account of the meaning and significance of Frege's claims of allegiance to Kant.

I think that Frege's explicit claims that he is following Kant can be explained without much difficulty. Frege believed that our knowledge could be divided into three categories. The first of these categories is knowledge which is justified, if we understand the concepts involved, merely by virtue of the rules without which thinking is impossible. The second category is knowledge which is possible only with the aid of pure intuition, but for which sense experience is unnecessary. Finally, the third category is knowledge which cannot be justified without appealing to sense experience. The purpose of Kant's a priori/a posteriori and analytic/synthetic distinctions was, Frege thought, to mark off these three categories. An analytic proposition was one which could be known through the mere form of reason. But knowledge of a synthetic proposition must have either pure intuitiort or experience as its source. Kant's point, Frege says 5 is to show that there are synthetic judgments a priori; i.e. that we can make judgments for whose support experience is not necessary (or, which are a priori) but, since they are not justifiable immediately by our understanding the concepts involved, for whose support something additional is necessary (or, which are synthetic). Thus when Frege says in the conclusion to the Grundlagen that Kant did a great service in drawing the analytic/synthetic distinction 6 but that he drew it too narrowly 7 and also when Frege earlier claims only to be making clear what Kant meant by analytic and synthetic 8 he is talking


about the failure or success of Kant's distinction in marking off sources of knowledge. Frege's point is that, assuming traditional Aristotelian logic sets out the formal rules of all thought, Kant's distinctions do

characterize propositions according to the sources of our knowledge of them. However, on the introduction of Frege's new logic, Kant's distinc-tions will not be able to play this role. For, given that Frege's logic sets out the true formal rules of all thought, Kant's distinctions have been drawn too narrowly. Frege's redefinition of the analytic/synthetic distinction is designed to characterize propositions appropriately according to sources of knowledge. Frege can thereby claim to be draw-ing the distinction so that it plays the role which Kant intended it to play or, saying what Kant really meant by analytic and synthetic.

The most obvious objection to this sort of reading has to do with the purported relation between Frege's and Kant's analytic/synthetic distinctions. For Frege and, on this reading, Frege's Kant, both the analytic/synthetic and a priori/a posteriori distinctions have to do with justification. However, Kant's actual formulation of the analytic/synthetic distinction (unlike that of the a priori/a posteriori distinction) concerns content, not justification, of propositions.

There is a straightforward answer to this objection, for it is easy to find evidence that Frege did in fact take Kant's analytic/synthetic distinction to have something to do with justification. For instance, it is clear from Frege's discussion of Kant is section 12 of the Grundlagen that he takes Kant's claims that arithmetic is synthetic a priori to be a claim that pure intuition is ''the ultimate ground of our knowledge of such judgments". And Frege not only quotes from Kant's Critique of Pure Reason - many of his remarks echo passages from it. Thus when Frege reworks Kant's analytic/synthetic distinction in an attempt to draw the line between those judgments which can be made merely on the basis of understand-ing the concepts involved and those for which some additional support (i.e., "truths of some special science" 9) is needed, it does not seem unreasonable to take this as Frege's version of such passages as 10

It would be absurd to found an analytic judgment on experience. Since, in framing the judgment, I must not go outside my concept, there is no need to appeal to the testimony of experience in its support.

If we take this as Frege's understanding of the Kantian analytic/ synthetic distinction, his redefinition of the distinction can be motivated, in part, by a careful consideration of some of Kant's passages which

14 JOAN WEINER

seem to conflict when the Begriffsschrift is substituted for the traditional logic. In a passage which Frege cites, II Kant seems to hint that analytic truths can be seen to be true from the principle of contradiction alone while for synthetic truths some additional support (i.e. another synthetic proposition) is necessaryP But Kant also says that our knowledge can be extended only by synthetic truth.13 Since traditional Aristotelian logic yielded no surprises, it may well have seemed that a truth of logic (or a truth which follows from the principle of contradiction) cannot extend our knowledge, and thus there would be no conflict between these two characterizations. And, since the application of the traditional logic could result in no new knowledge, its results could not be counted as additional support (or synthetic propositions). Thus we could under�stand analytic truths as those which can be seen to be true from logic alone. But while these remarks do not conflict at all if we understand by "logic" traditional logic, Frege's new logic makes the conjunction of these passage look odd. As Frege notes,14 some of the truths of the new logic (which should, thereby, be analytic by the first characterization) seem to extend our knowledge (and ought, therefore, to be synthetic by the second characterization). As Frege formulated it, the central role of the analytic/synthetic distinction was to separate those judgments for which support in addition to the concepts involved was necessary from those which can be made without additional support (or truths of a special science). However, it is not immediately clear whether or not Frege's new logic should be counted as additional support.

Frege would have been able to preserve many of Kant's actual words by counting his new logic as a special science (i.e. as synthetic). How�ever, this position is not open to Frege since it conflicts with his view of what his logic does. Frege took his logic to provide a unified account of how the parts of a proposition contribute to determining its truth or falsity. Such an account cannot be broken up into substantive and nonsubstantive parts; it is meant to be nothing more nor less than a means of setting out the formal rules of all thought, the rules which make the use of the understanding possible. But such rules can hardly be counted as synthetic or as truths of a special science, for they must underlie all knowledge. Given the aims of his Begriffsschrift, Frege has no choice but to take its results to be analytic. Thus Frege characterizes analytic judgments as those which follow from logic and an understand�ing (or analysis by means of definitions) of the concepts involved.


Frege's characterization of analyticity introduces new complexities into the project of categorizing a proposition as analytic or synthetic. Thus Kant's arguments for taking arithmetic to be synthetic a priori seem too simple, and it is not at all surprising that Frege found them inadequate. One of the explicit motivations of the Grundlagen was to determine the source of arithmetic knowledge, to show whether arith�metic was a priori or a posteriori, synthetic or analytic. But the explicit project of the Grundlagen is to define the number one and the concept of number. It is not immediately obvious that such a definition will help in determining the source of our knowledge of arithmetic. I believe that the first chapter of the Grundlagen should be read as an argument for Frege's claim that definitions of the number one and the concept of number will be necessary for determining the source of our knowledge of arithmetic.

In his first chapter, Frege indicates that there are primitive notions (e.g. the notion of point) which cannot be defined. And a claim that something is not definable has epistemological consequences. In particu�lar, consider the consequences of assuming that the numbers cannot be defined. Then arithmetic is the science of the peculiar irreducible objects we call numbers. Now consider the claim that zero is less than one. Since "zero" and "one" are not definable, this claim cannot be justified by logic from the definitions of the concepts involved. Further�more, since no number words appear in Begriffsschrift, it cannot be a straightforward logical truth. Thus, if the numbers are not definable, arithmetical truth cannot be analytic. And Frege is quite convinced that arithmetical truth is analytic. The reason for this seems to be that Frege believes that everything thinkable can be numbered and thus that our knowledge of arithmetic should not be dependent on pure intuition or sense experience (since not all thought is). This, then is Frege's reason for being dissatisfied with Kant's account of arithmetical truth as synthetic a priori. If Frege could give definitions of the numbers which showed arithmetical truth to be analytic, he would have succeeded in correcting another of what he saw as the errors in Kantian epistemology. It should be noted that we can understand Frege's dissatisfaction with Kant's account without understanding Frege's Begriffsschrift. I believe that this dissatisfaction with Kant's account of the source of our knowl�edge of arithmetical truth was one of Frege's original divergences from Kant. We can view Frege's Begriffsschrift as an attempt to show that

16 JOAN WEINER

analytic truths can be substantive and also as having been motivated, at least in part, by Frege's original conviction that arithmetical truth must be analytic.

If the above account of Frege's motivations is accurate, we can give a very easy explanation of the purpose of Frege's arguments against Kant and Mill. The importance of the Grundlagen project of defining the numbers results from Frege's conviction that arithmetical truth is analytic. Although Frege gives his reasons for thinking this in the Grundlagen, the only conclusive argument would consist of defining the numbers and showing that arithmetic is in fact analytic. But Frege needs to motivate the project of defining the numbers for those who are not antecedently convinced that arithmetic is analytic. Frege's strategy is to use Kant's and Mill's accounts of the source of our knowledge of arith-metic to shake his readers' confidence in the claim that the numbers are well-understood and indefinable.

If numbers were indefinable and well-understood, one would expect it to be easy to give an adequate account of the source of our knowledge of arithmetic. In the first chapter of the Grundlagen Frege argues that the available accounts of our knowledge of arithmetic are either clearly wrong (Mill's) or inadequate (Kant's). Thus it makes sense to take seriously the possibility that numbers are not primitive and irreducible and to try to come up with acceptable definitions. Frege's first chapter can be viewed as an argument that the source of our knowledge of arithmetic has not yet been determined and that the most plausible way of undertaking the investigation of the source of this knowledge is to attempt to define the numbers.

Clearly all numbers must be defined for Frege's project to succeed. Thus Frege notes in the Grundlagen that all numbers can be defined from the number one, increase by one, and the concept of Number. And the projects of giving definitions of these three notions are intertwined. 1 5

Hence Frege describes the project of the Grundlagen as that of defining the number one and the concept Number. But while this is clearly a part of the overall project of determining the source of our justification of arithmetical propositions, the definitions alone will not suffice. Having given these definitions in the Grundlagen, Frege's next step is to show that all arithmetical truths can (or cannot) be derived from these definitions by logic alone. This step was to be accomplished in the Grundgesetze. Had no error been discovered, this work would have completed Frege's overall project.


In the last few pages I have given an outline of the views of Frege, the historical person. In order to understand what these views come to, however, it is necessary to focus on some of the tensions which Frege, the historical person, may not have recognized. I will outline some of these below. After the Begriffsschrift, Frege's central task is to give rigorous logical proofs of the truths of arithmetic from the primitive truths on which these arithmetical truths depend. Once such proofs are given, the status of the arithmetical propositions will be determined by whether the primitive truths on which they depend are analytic or synthetic. In order to carry out this project, we must answer two questions. First, what are the primitive truths? Second, how do we determine whether the primitive truths are analytic or synthetic? The answers to these questions are deceptively simple. A primitive truth is either a proposition all of whose terms are primitive (i.e. indefinable), or a definition in which the only definable term is the definiendum. And this notion of primitive truth allows an easy way of answering the second question, for it follows that determining whether a primitive truth is analytic or synthetic is trivial. If all its primitive terms are logical terms (terms which are part of Begrijfsschrift), the proposition is analytic. If some of its primitive terms are nonlogical terms, the proposi�tion is synthetic.

There are important gaps in the above description which cannot be filled in by simple references to passages in the Grundlagen. For instance, if we are to have general criteria for determining whether Frege has given proofs adequate for establishing arithmetical truth as analytic, an account of definability seems necessary. But there is no explicit account of definability in the Grundlagen. And, given some of the remarks in the Grundlagen, it is not entirely clear what definition is for Frege. In particular, Frege indicates that definitions often require justification.16 Is the justification required for a definition which appears in a proof part of that proof? In the Grundlagen Frege also gives no explicit accounts of the notion I)f definition, the sort of justification required for a definition, or the role the justification of a definition might play in a proof which uses the definition.

Frege does have accounts of definition and of definability which will serve the requisite epistemological roles, but these are not explicit in his writings. Such an account of definition, for instance, is a consequence of the views expressed in Frege's correspondence with Hilbert and some of his papers, in particular, two papers entitled "On the Foundations of

18 JOAN WEINER

Geometry" which grew out of that correspondence. I think that this account of definition is implicit in the Grundlagen. However, it would have been awkward to make this explicit in the Grundlagen because, without the sense/reference distinction, it would follow that all ordinary language is meaningless.

While it is beyond the scope of this short paper to indicate the arguments for taking Frege to have the views on definition I attribute to him, they can be easily summarized here. One of Frege's most explicit statements appears in a letter to Hilbert. l ?

Every definition contains a sign (expression, word) which previously has had no reference and which is given a reference only through this definition. Once this has happened, one can make out of this definition a self-evident proposition which is then to be used like an axiom. But we must adhere to the tenet that in a definition nothing is asserted: rather, something is stipulated. Therefore what requires a proof or some other reasoning to establish its truth ought never to be presented as a definition. 18

There are three ways in which this seems to conflict with Frege's Grundlagen, although there is little evidence that Frege was aware of any of these conflicts. The first is that in this passage Frege seems to say that definitions require no justification, while in the Grundlagen Frege said that definitions do require justification. The second conflict is that from the above passage it follows that, if Frege defined "one" in the Grundgesetze, then it must have previously had no reference. But Frege did take himself to have defined "one" in the Grundgesetze, and yet it seems rather implausible that "one" should have had no reference before Frege did this. Finally, the third conflict is that Frege does not give an arbitrary definition of "one" in the Grundlagen, yet the views on definition expressed in the above passage do not seem to leave room for any constraints on definitions. In the next few pages I will briefly indicate how these problems are to be answered and how the answers to these problems lead to a completely new way of viewing Frege's work.

The first question concerns the justification necessary for a definition. As I read the above passage, Frege is claiming that it is never necessary to justify assigning a particular referent to a particular word. However, this does not rule out requiring some justification to show that a purported definition actually meets the criteria which Frege requires of definitions. Frege says 19

I demand from a definition of a point that by means of it we be able to judge of any object whatever - e.g. my pocket watch - whether it is a point.


Frege says this because he requires that any definition of a concept-word must give us a description, in primitive terms, which either holds or does not hold of each object. Similarly a definition of an object-word must consist of a description, in primitive terms, which picks out one and only one object. These are Frege's - admittedly very strict -requirements on definitions. Thus some justification may be required for a definition; it may be necessary, for instance, to prove that a certain description holds of one and only one object. And Frege explicitly applies this criterion of adequacy. In section 64 of the Grundlagen, he rejects a definition of direction because

it will not, for in,tancc, decide for m whether England is the the direction of the Earth's axis.

Thus it is not that the truth of a definition may require justification, but rather that it may be necessary to show that the description used in a definition is actually a defining description in Frege's strict sense.

Let us now turn to the second problem. Frege seems committed to saying that "one" and "I" had no reference prior to his investigation in the Grundlagen. How serious a problem this is clearly depends on the nature of Frege's notion of reference. I think it is possible to argue that Frege's notion of reference is a very peculiar one and that, given this notion of reference, it would not be untoward to take "one" to have had no reference before the work of the Grundlagen.

The passage I quoted earlier was written after "On Sense and Refer-ence", and thus we cannot take the notion of reference which is used in that passage to be the notion of Bedeutung in the Grundlagen. However, these notions are related. In fact, I believe that the development of the later notion of reference can be viewed as a response to tensions between Grundlagen-Bedeutung and Grundlagen-definition. To see this tension, we should first note that, in the Grundlagen, a definition fixes the meaning (Bedeutung) of a word.20 Furthermore, we can ask for the meaning (Bedeutung) of a word only in the context of a proposition; to fix the Bedeutung of a word, one must fix the meaning of the proposi-tions in which it appears.21 Defining (i.e. fixing the Bedeutung of) the number one entails fixing the sense of "zero is less than one". Thus it seems that Frege may be committed to saying that "zero is less than one" is meaningless before his work in the Grundlagen. Furthermore, our use of most ordinary words does not determine a definition in Frege's strict sense. For instance, there are people who are neither

20 JOAN WEINER

clearly bald nor clearly not-bald. Thus no description of our ordinary concept of baldness will meet Frege's criteria. It is not obvious that this is a serious problem, for the sense in which Frege is committed to holding that such propositions are meaningless is not clear, and he certainly does not seem to believe that he was committed to this. But I think this is symptomatic of an important tension in the Grundlagen.

We can regard Frege's sense/reference distinction as providing a partial response to his apparent commitment to the view that ordinary language is meaningless. If the notion of sense plays the intuitive role of meaning, we can take the notion of reference to play the highly technical role which is needed for working out Frege's project. To have reference, then, is to be either primitive or used in a way which determines a definition, in Frege's strict sense, from primitive terms. Thus, given the vagueness of our ordinary language, most non-primitive terms will have reference only if an explicit Fregean-style definition has been given. Having recognized this distinction, we can take "zero is less than one" to have (hence to be meaningful) but no reference before Frege's work. It does, of course, follow that the sentence had no truth-value before Frege's work. But this, I will argue, is a view to which Frege is firmly committed.

The final problem has to do with the constraints to which Frege subjects his definition of the number one. Given Frege's discussion of definition, it would seem that any way of fixing the reference of "one" is as good as any other. But Frege is only willing to consider a certain sort of definition. However, this is not because there are any constraints on what referent can be stipulated for "one", rather it is because Frege wants to use his definition for a particular purpose.

In order to see that it is legitimate to impose constraints on defi-nitions, it is useful to consider Frege's paradigmatic example of a definition the mathematical definition of the continuity of a function. Clearly we could assign any number of descriptions to "fis continuous" which would not enable us to express the mathematical theorems which the standard definition helps us express. But such definitions would not have served the purpose for which the standard definition was drawn. Thus it is perfectly reasonable for Frege to impose constraints on his definition of "one" if he wants the definition to play a certain role. And he does. Frege wants to give a definition of "one", from primitive terms, which will make 'true all those propositions containing "one" which we have already judged to be true. In this way, the definition will provide

the sort of foundation for arithmetic which Frege requires any science to have. This is the sense in which the definition will constitute an analysis of our concept of what it is to be the number one. Frege says 22

The real importance of a definition lies in its logical construction out of primItive clements. And for that reason we should not do wIthout it, not even in a case like this. The insight it pcrmlts into thc logical structure IS a condition for insight into the logIcal linkage of truths. A definition is a constituent of the system of a sCIence.

The problem with this is that the propositions we take to be the truths of arithmetic do not determine a definition, in Frege's sense, of the number one. Frege deals with this problem in a 1914 paper, 'Logic and Mathematics'.23

Now we have to consider the difficulty we come up in giving a logical analy,is when it is problematic whether this analysis is correct. Let us a,sume that A IS

the long established sign whose sense we have attempted to analyse logically by constructing a complex expression which the analysis. Since we are not certain whether the analysis is we are not prepared to pre,ent the complex expression as one whieh can be replaced by the simple sign A. If it is our intention to put forward a definition proper. we are not entitled to choose the sign A, which already has a but we must choose a fresh sign B, say, which has the sense of the complex expression only in virtue of the definition. The question now is whether A and B have the same sense. But we can bypass thi, question altogether if we are constructing a new system from thc bottom up; in that case we shall make no further use of the A - wc shall only use B ... If we have managed in this way to con<;truct a system for mathemalles 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.

This tells us how we are to understand the project of the Grundlagen. Before the Grundlagen, "one" did not have a referent, in Frege's strict sense, Frege wants to build arithmetic up from primitive terms. Thus he is required to construct a description of an object, in primitive terms, which can replace "one" or "I" in the propositions of arithmetic without appreciably altering mathematical practice. If this can be done, there is no harm in using that description as a definition of "one". However, assigning this referent to "one" rather than to some newly formulated word should be viewed as a heuristic, not as an important elucidation of our ordinary use of "one".

The view outlined above contains a significant gap - the lack of an account of primitiveness. While we can recognize and give definitions,

22 JOAN WEINER

Frege has given no explicit criteria for distinguishing between definable and indefinable terms. And here, it may seem, we hit a real dead end. Frege does not seem to have said what it is for a term to be primitive, although he sometimes gives examples of primitive terms or hints about what primitiveness comes to. On my reading of Frege's work, there is an important reason for this. It is that the notion of primitiveness is itself primitive, that is, indefinable. In order to understand fully the signifi�cance of this gap, it is important to consider the relation between primitiveness and Frege's notion of objectivity.

Frege says very little about objectivity. His most sustained discussion of the subject seems to be the discussion in section 26 of the Grundlagen. In that section, he seems to offer three explicit criteria for objectivity. He says that what is objective is what is expressible in words, what is subject to laws, and what is independent of sensation, intuition, and imagination, but not what is independent of the reason. While the connection between these criteria may not be immediately apparent, I have argued elsewhere that they are tightly bound together and that they all delimit the same realm of objectivity.24 I have argued that, for Frege, to be objective is to be subject to the laws of Begriffsschrift and that it is this view which binds the above criteria.

If what is objective is what is subject to the laws of Begriffsschrift, then a good deal of work is required to show that a proposition is objective. In particular, if a proposition is subject to the laws of Begriffsschrift, all its terms must have fixed reference. But, as I mentioned earlier, few of our ordinary terms have fixed reference in Frege's sense. It seems, then, that definitions of each definable term are required. But how are we to know when this has been done? We can only recognize the completion of such a task if we can recognize primitive terms. Frege assumes that we can recognize primitive terms, and it remains to be seen whether this assumption will get him into serious trouble. However, there is a more immediate problem. It is not clear that the task has been completed once the original proposition has been translated into a proposition whose terms are all primitive. We must still show that all its terms have fixed reference. Thus we must say something about what it is for a primitive term to have reference and how we can show that a primitive term has reference. Until now, the only means available for showing that a term has reference have been the use of definitions. But primitive terms cannot be defined.


While Frege has given us no explicit account of primitiveness, this does not mean that we know nothing about what it is to be a primitive term. The primitive terms are the indefinable terms which underly all terms usable in communication. While we often communicate by using vague terms, the success of such communication is possible only because what is communicated can be made unambiguous. Thus one of the conditions necessary for communication is that what is communi-cable must be expressible in terms which either have fixed reference or can be defined from terms which have fixed reference. Primitive terms cannot be defined thus, if they can be used in communicating, they must have fixed reference. For instance, the terms "function" and "object" are primitive and are necessary for the explication of the logical regimenta-tion of propositions in Begriffsschrift. Furthermore, there is an answer to whether or not a proposition has been correctly regimented. The regimentation must be objective and hence the terms "function" and "object" must have reference. But what does this sort of reference come to?

It is important to note that the above does not amount to saying that the words "function" and "object" refer to whatever they must refer to if Begriffsschrift regimentation is to be objective. One of Frege's most fundamental assumptions is that Begriffsschrift regimentation is objective - this is the foundation of Frege's very understanding of objectivity. It follows that "function" and "object" must have reference. But there is no way of specifying, describing, or giving structure to what these words refer to. The closest we can come is to give hints, i.e., to use these words in propositions. The claim that these terms have reference is nothing

more than the claim that regimentation is objective. Thus, in the end, the notion of reference drops out altogether.

One of the most striking features of this notion of objectivity is that it seems that everything is objective. Certainly, for instance, we communi-cate about the character of our mental images and if what is objective is what is communicable, it would seem that these claims concerning our mental states must be objective. But this is especially surprising since Frege says that the character of our mental images is subjective. This is best explained by carefully considering the one discussion in the Grundlagen which deals with the subjective realm.

Frege gives an example in the Giundlagen of two beings who intuit only projective properties and relations. One of them intuits as a plane

24 JOAN WEINER

what the other intuits as a point, etc .... Thus what one will intuit as a line joining two points, would be intuited by the other as a line of intersection of two planes. Now it is important that the two beings intuit projective, not Euclidean, properties and relations. In projective geome�try, what is true of planes is precisely what is true of points. Thus these two beings would agree on all axioms and theorems of geometry.25 Since they obey the same geometrical axioms and could not discover any difference in their intuitions through language, the objective meaning for both beings of "point" and "plane" is the same. Thus, although there seems, from Frege's description of these beings, to be a difference in the characters of their intuitions, they are both intuiting the same things. The point of this example was to convince us that pure intuitions have both objective and subjective sides. The objective side is what is express�ible and the subjective side is the character of the intuitions which is not expressible. How is this supposed to work? Because the axioms of geometry are known by pure intuition, they are justified by the construc�tion of mental pictures. By setting the situation out as he has, Frege has asked us to construct the mental pictures we would imagine each of the two beings constructing and to examine the difference between the two pictures. I will discuss the images the two beings would construct in order to assure themselves of the justification of the axiom which tells us that any two points determine exactly one line. Frege's assumption, I believe, is that we will construct two pictures which look something like my Figure 1:

A.

, , ,

POINTS -;_' B. , i,' LINE ( , ,

Fig. 1

POINT

These pictures are very different and, in some way, this difference is supposed to exhibit the subjective difference. At first glance, however, this might seem somewhat suspect. For the difference between these two pictures is an objective, physical difference which can be described in

words. And one might suspect that the two beings <lescribed by Frege could discover their difference by examining these pictures. But this creates no problem for Frege. It is possible to imagine that one of the beings, when looking at picture A, has the same experience the other has when looking at picture B.26 The point is that the objective difference between A and B is a difference between the geometrical configurations of pen marks on paper, not a difference between mental images. Frege is not asking us to examine these objective differences, but rather to construct these two pictures in our minds and examine the difference in appearance (that is, the subjective difference) between the two pictures.

Now it is important to realize that it follows that Frege is not making an objective argument. Frege's example works only if his readers construct two mental images which are different in appearance, that is, subjectively different. But, although the pictures he constructs might be subjectively different, there is no way for him to guarantee that the pictures constructed by his readers are subjectively different. This is a case in which we must grant Frege the grain of salt he asks for in "On Concept and Object". For Frege has nothing to say to someone who considers his example and claims to experience no subjective difference in the images she constructs. Subjective difference, after all, cannot be described. Frege's example cannot be an argument, it can only be considered a hint.

I have grawn numerous conclusions about Frege's motivations and about underlying views which would allow Frege to respond to tensions in his explicit picture; however, it is not entirely clear that Frege was aware of these tensions. I will end by discussing a consequence of my reading of the example discussed above. This consequence is something of which, I believe, Frege was almost certainly unaware.

I have argued that, for Frege, it is not possible to do more than hint at the difference between the subjective and objective realms. It is certainly not possible to describe this difference objectively. But it follows from this that the difference is not objective. This may not seem too prob-lematic. The subjective realm is, in an important sense, empty and it does not seem all that strange to say that everything is objective for, since we cannot talk about the subjective, everything we can discuss must be objective. Furthermore, Frege has said very little about objectivity, and very little seems to be at stake for him. We can regard what Frege does say as a few innocent hints. But this problem does not apply only to Frege's comments about objectivity. Frege's discussions of the differ-

26 JOAN WEINER

ence between objects and concepts, and of why number statements are about concepts - in fact almost all of what we might be tempted to call Frege's theoretical framework - also must have the status of hints.

If we read this consequence back into Frege's project, the perspective shifts dramatically. I began by arguing that Frege's mathematical work was philosophically motivated and that Frege intended us to draw philosophical morals from it. What we think of as his philosophical work was intended to show how we are to draw philosophical morals from the mathematical work. None of this has really changed. However, the implication of reading Frege's work in that way seemed to be that the most important work was what we call his philosophical work, for this is where Frege set out his philosophical theories. But it now appears that Frege cannot have any philosophical theories. Far from being the presentation of the philosophical theory within which Frege's mathe-matical work was to play a role, what we think of as his philosophical work can only have had the status of subjective hinting designed to get his readers into the proper frame of mind for understanding the purpose and significance of his real philosophical work - the work we think of today as his mathematical work.

University of Wisconsin - Milwaukee

ACKNOWLEDGMENT

I would like to thank Burton Dreben, Warren Goldfarb, Mark Kaplan, Hilary Putnam, and Thomas Ricketts for criticism and advice.

NOTES

1 It is not easy to find support for this sort of view in writing, since people who believe this tend to be pursuing Frege's questions,.not worrying about what he meant. However, I have found that quite a number of people will defend such a view in conversation. 2 Philip Kitcher, 'Frege's Epistemology', The Philosophical Review 88 (1979), 235-262. 3 Hans Sluga, Gottlob Frege, (London: Routledge & Kegan Paul, 1980). 4 Gottlob Frege, The Foundations of Arithmetic, translated by J. L. Austin (Evanston: Northwestern University Press, 1978), pp. 10 1-102. 5 Frege, Foundations, p. 102. 6 Frege, Foundations, p. 101. 7 Frege, Foundations, p. 99. 8 Frege, Foundations, p. 3.


9 Frege, Foundations, p. 4. 10 Immanuel Kant, Critique of Pure Reason, translated by Norman Kemp Smith (New York: St. Martin's Press, 1929), A 7 /B12. II Frege, Foundations, p. 100. 12 Kant, Critique of Pure Reason, B14. 13 Kant, Critique of Pure Reason, A8, B12. 14 Frege, Foundations, p. 101. 15 Frege, Foundations, p. 25. 16 See, for instance, Frege, Foundations, pp. 4, 77. 17 See the Frege-Hilbert correspondence in On the Foundations (!If Geometry and Formal Theories of Arithmetic, translated by Eike-Henner W. Kluge (New Haven and London: Yale University Press, 1971), p. 7. 18 While Frege is talking about mathematical definitions here, it is important to note that his definition of the number one is intended to be used in proofs and hence is mathematical. 19 Frege, 'On the Foundations of Geometry' in On the Foundations of Geometry, p. 63. 20 See, for instance, Frege, Foundations, p. 9. 21 Frege, Foundations, p. 73. 22 Frege, 'On the Foundations of Geometry' in On the Foundations of Geometry, p. 60. 23 Frege, 'Logic and Mathematics' in Posthumous Writings, edited by Hans Hermes et al., translated by Peter Long et al. (Chicago: The University of Chicago Press, 1979) p. 210. 24 Joan Weiner, Putting Frege in Perspective (Doctoral Dissertation, Harvard Univer-sity,1982). 25 A projective plane can be constructed from a Euclidean plane by adding, for each line in the original plane, exactly one point at infinity. Lines which were parallel in the original plane share a point at infinity in the projective plane. Furthermore, the points at infinity in this plane determine a single line at infinity. Two planes which are parallel in Euclidean space share a line at infinity when the points at infinity are added to each. Frege's doctoral dissertation, 'Uber eine geometrische Darstellung der imaginaren Gebilde in der Ebene', is concerned with providing the kind of foundation for these infinitely distant (or imaginary) points which he later attempts to provide for the integers. 26 This is undoubtedly what Frege would have us imagine if his two beings could see. Of course, since Frege follows Kant in including sensation under intuition and since, by hypothesis, these beings intuit only projective properties and relations, it follows that they cannot see.

PART II

SEMANTICS AND EPISTEMOLOGY

J. VAN HEIJENOORT

FREGE AND VAGUENESS

If it is Yes, say Yes; if it is No, say No. Anything else comes from the EVil One.

Matthew 5, 37

Whoever undertakes to scrutinize logical arguments conducted outside mathematics cannot but wince at the flightly conduct of words. Their meanings have no exact lines of demarcation. What an adjective conveys may depend on the noun that follows ('white wine'). We often have difficulty in grasping the exact meaning of a verb if we do not know its subject and complement(s). When we look up a word in a dictionary, we see how the word's uses have come to diverge ever more from its primary meaning and have branched out in many directions. Metaphors lose their sparkle and, like dead stars, come to lead an obscure existence ('source of grief). Words in ordinary language are far from having neat definitions, like that of 'even' in arithmetic. Every predicate is vague, in the sense that there are individuals for which it is intrinsically indeter�minate whether the predicate holds or not.

In his first logical work, Frege had to point out how the vagueness of a predicate can wreck a logical argument. Theorem 81 in §27 of his 1879 can, in modern language, be stated as follows: Let R * xy be the closed iterate (to use Quine's expression) of a binary predicate Rxy; that is, R*xy is to Rxy what 'x is an ancestor of y' is to 'x is a parent of y'.

Then, for any unary predicate Px which is inductive over Rxy, that is, such that

VxVy«Px& Rxy)::::l Py),

we have

VxVy«Px& R*xy) ::::l Py).

Let Px be interpreted as: x is a heap of beans; and Rxy as: y is obtained from x by the removal of one bean. Then Theorem 81 would imply that a single bean, or even none at all, would be a heap of beans.

31

L. Haaparanta and J. Hintikka (eds.), Frege Synthesized, 31-45. © 1986 by D. Reidel Publishing Company.

32 J. VAN HEIJENOORT

To overcome the difficulty, Frege claims that the predicate Px is not inductive over Rxy because, u being one of certain objects, the sentence Pu 'cannot be taken as being a judgment' (,ist unbeurtheilbar') on account of the 'indeterminateness' (' Unbestimmtheit') of the predicate Px. The definition of the closed iterate of Rxy contains a universal quantifier ranging over unary predicates. If a certain unary predicate is not included in that range, then Theorem 81 does not apply to that predicate. The criterion that Frege uses for excluding a predicate from the range is its' Unbestimmtheit', which leads to 'unbeurtheilbare'

sentences. With these few remarks Frege puts vague predicates outside the pale of logic.

Let us specify our terms. For us, a predicate is a linguistic expression; thus, if unary, it corresponds to Frege's Begriffswort and, if binary, to his Beziehungswort. Since Frege maintains that a concept whose delimita-tion is 'blurred' ('verschwommen') is in fact no concept at all (1969, p. 133), we shall speak of vague predicates, and not of vague concepts.

Let us briefly recall what Frege's ontology is. He has a fixed universe, which comprehends all objects whatsoever. An object may be abstract, like a natural number, or concrete, like the Moon. Over these objects there are first-level functions of one or two arguments and, since he has to quantify over these functions, he has first-level variables. The impor-tant point for us here is that Frege does not divide his universe into various sorts and he requires that any function (in particular, any concept) be defined for all objects. This is what he calls his principle of completude (,Grundsatz der Vol!standigkeit') , presented and defended in §56-§65 of 1903 and subjacent in all his work.

In his presentation of the principle Frege takes exception to what he calls piecemeal definitions (,stuckweise Definieren'). An example of such a definition would be addition, originally defined for natural numbers, then successively extended to signed integers, rational numbers, and so on. Weare dealing here with different operations: we should in each case use a different sign and speak of the isomorphism between a system of numbers and a certain proper part of the new system. There never was any misunderstanding on that score in the mind of qualified mathematicians. But some expositors chose to stress how the definitions of addition for various kinds of numbers successively grew out of one another, sometimes speaking of a genetic definition. Frege severely criticizes such procedure: 'Concept-like constructions that are still in a state of flux, that have not yet received final and sharp limits cannot be

FREGE AND VAGUENESS 33

recognized by logic as concepts; and that is why it must spurn all piecemeal defining' (1903, p. 71). And: 'Thus nowhere do we have firm ground under our feet. Without final definitions, no final theorems. One would not come out of imperfection and vacillation' (1903, p. 74). If these strong words were directed only against some occasional defective presentations, there would be little to say, except that they perhaps overshoot the mark. It turns out, however, that what Frege condemns in the piecemeal definition is its being restricted to a certain domain of mathematical objects. Every predicate, every operation has to be defined for all objects in the Universe. If addition were not defined for the Moon and the Moon, 'the question whether the sum of the Moon and the Moon is 1 or whether the Moon falls under the concept something that added to itself gives 1 could not receive an answer; in other words, what we called a concept would in fact not be a genuine concept, because the sharp delimitation is lacking' (1903, p. 76).

A second consequence of Frege's principle of completude is his rejection of conditional definitions. Consider, for example, for a binary predicate Rxy of natural numbers, a definition that would read as follows:

(1) VxVy«Nx& Ny) (Rxy= . .. )),

Nx being interpreted as: x is a natural number. Peano had called such definitions 'deJinizioni condizionate' or 'deJinizioni con ipotesi'. Frege calls them 'bedingte Erkliirungen' and rejects them because, in the case of definition (1), if a and b are not natural numbers, the sentence

(2) (Na& Nb) (Rab == ..• )

has the truth value t for whatever syntactic object we put at the place of the three dots; hence, for objects a and b that are not natural numbers, (1) does not specify anything for Rab.

In a letter to Peano dated September 29, 1896, Frege writes (1976, pp. 182-183, or 1980, p. 114; I modified the translation): 'A condi-tional definition of the sign for a concept decides only for some cases, not for all, whether an object falls under the concept or not; it does not therefore delimit the concept completely and sharply. But logic can recognize only sharply delimited concepts. Only under this presupposi-tion can it set up precise laws. The logical law that there is no third case beside "a is b" and "a is not b" is really just another way of expressing our requirements that a concept (b) must be sharply delimited. 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 really the case'. Frege's ban of conditional definitions directly springs' from his principle of completude.

In his defence of that principle, Frege asks whether 'x + y' may have a Bedeutung only when the two objects considered as arguments are natural numbers, and he answers negatively. If the answer were posi�tive, the sentence 'The sum of the Moon and the Moon is l' would be neither true nor false and, he claims, 'no scientific investigation can culminate' in such a sentence (1903, p. 76). In that whole discussion Frege's arguments are somewhat repetitious and circular. Why would there be something unscientific in the sentence? Because some truth value would be left indefinite. We are back to completude. Frege also argues that, if addition were not defined for all objects in the Universe, we would not know the number of solutions of the equation x + x = 1; some a such that a + a = 1 might pop up in a comer where we do not expect it. A strange and circular argument! Once we have defined the function x + yover a definite domain, all solutions of x + x = 1, if any, are in that domain.

Frege finally presents a more elaborate argument. From (2), by sentential logic, we obtain

(3) «Rab =f: ••• ) & Na):::> - Nb,

hence we are led to consider an object, namely b, which is not a natural :pumber, or, as Frege writes, 'here it is impossible to maintain the restric�tion to the domain of numbers. The state of affairs has a force that irresistibly acts so as to break down such barriers' (1903, p. 78). But the argument remains inconclusive; it certainly does not establish the necessary existence of a universal domain comprehending all objects. In (1) the quantifiers may very well range over a sufficiently large domain of mathematical objects, and in that domain a subdomain is cut out by the predicate Nx. In fact, when natural numbers are defined as certain sets, the quantifiers in (1) range over sets. A mathematician would not use definition (1) unless a definite mathematical domain has been introduced for the quantifiers; if no such domain is under consideration, he would replace (1) by

(4) VxVy(Rxy == ••• ),

with the understanding that the quantifiers now range over the natural numbers. With (4), Frege's argument cannot get started.


Frege's requirement of completude is intimately connected with that

of sharpness. For him, in fact, the two requirements seem to fuse into

one. Countless times in his writings, we find the words 'complete' and

'sharp' conjoined. Discussing completude, all of a sudden he brings in

examples of vagueness (lack of sharpness), like the word 'heap' in his

letter to Peano quoted above or the word 'Christian' at the end of § 56

of 1903. For Frege, lack of completude and lack of sharpness are both a

failure of universal bivalence and thus seem to merge one into another.

Let us try to distinguish them. For an atomic sentence Pa, universal

bivalence may fail in three cases:

(1) Although 'a', by its form and its grammatical role, appears to

name an object, it actually does not; this is the well-known problem of

empty descriptions raised by Bertrand Russell. We shall say that here

the individuation has failed; as no individual is apprehended by 'a', the

question whether the predicate 'P' holds or not does not arise, and Pa has no truth value. (If 'a' is a definite description, we may, as Russell

does, look at Pa as a relation between predicates and not analyze it as a

subject-predicate sentence.)

(2) An object is denoted by 'a', but this object is not one of which the

question whether the predicate' P' holds can be raised. Is the number 7

blue? A good dictionary says of an adjective 'is said of ... " thus

indicating its domain of applicability. This pigeon-holing is constantly

questioned by metaphoric ('sharp mind') or poetic (Rimbaud: A black,

E white, ... ) uses of words, but it certainly is an integral part of the

working of any natural language.

(3) A definite individual is apprehended by 'a' and for that individual

the question whether the predicate 'P' holds is meaningful. Nevertheless,

the criteria connected with the application of 'P' are not such that a

truth value is assigned to Pa. For example, I see this vase perfectly well,

the light is good, but is the vase blue or green? I don't know how to

answer; I grope for a new word: 'bluish', 'blue-green'.

Failure of individuation may occur in mathematical language ('the

largest natural number') or in ordinary language ('the present king of

France'). In mathematics the proper use of the singular definite article

requires an existence and uniqueness proof. In ordinary language failure

of individuation is likely to stop communication. If I start a sentence

'The present king of France is ... ',my interlocutor is liable to stop me:

'What are you talking about? France has no king now'. As for Frege, he

proscribes nondenoting individual constants from any properly con-

structed language. In 1893, p. 19, he introduces the slanted boldface


stroke for the description operator; applied to (the Werthverlauf of) a concept, it yields the object falling under the concept in case there is one and only one such object; otherwise, it yields the Werthverlauf itself.

Mathematics knows no completude in Frege's sense. A universe embracing 'everything' is repugnant to mathematicians. Hilbert (1904, p. 175, or van Heijenoort 1967, p. 130) already reproached Frege for imposing no restriction on the range of his universal quantification, and he introduced a system in which the quantifiers range over objects recursively constructed from two basic objects; another mathematician, Poincare (1906, p. 18, or 1908; pp. 181-182), was prompt to stress the difference between the well-delimited domain thus introduced by Hilbert and the all-embracing universe of Frege and Russell. The closest approximation that mathematics would have to a universal domain would be the (a?) universe of sets. But one cannot speak today of a fixed, well-delimited universe of sets in which every mathematician would embed the entities he is working with. Moreover, such a universe would still be far from Frege's universe since it would not comprehend nonmathematical objects. As to natural language, it respects sorts and natural kinds. We do not feel obliged to ascribe a truth value to the sentence 'The number 7 is blue'. Even in Frege's system, it is not clear why completude, in his sense, is required. He wants bivalence, but bivalence, in the sense that each predicate has a definite domain of applicability and either holds or does not hold of each object in that domain, can obtain in a many-sorted logic. No, Frege is adamant, a predicate or a function not defined for all objects whatsoever in the Universe would somehow be deficient. It would not be impossible, it seems, to reconcile Frege's ontological assumptions with a many-sorted bivalent logic. In imposing his brand of complete bivalence, Frege seems to have fallen prey to some kind of extremism.

We can, of course, assign a truth value to the sentence '7 is blue'. Using

'v' x( x is blue :::J x is a physical object),

where the quantifier ranges over all objects, and

7 is not a physical object,

we obtain

7 is not blue.


This argument may be in need of some improvement. Although the sky is hardly what we would call a physical object, it is blue. The problem is to circumscribe a fixed and well-delimited universe of discourse con-sisting of all objects, that is, of all ground-level entities. Frege assumed the existence of such a universe, without paying much attention to the problem hidden in this notion. Once a universe is postulated, predicates can, of course, be defined, or considered to be defined, for each object in it. We make the blanket decision that, if a is not in the domain of applicability of'P', Pais false.

The situation, though, is somewhat different with functions, or opera-tions. Consider the predicate Sex, y, z), taken to mean: x, y, z are natural numbers and z = x + y. Take the predicate to be false whenever an argument is not a natural number. Then, for every z, S(the Moon, the Moon, z) is false, and the equation z = the Moon + the Moon has no solution. Hence, though the predicate S is defined over the whole universe, the function + is not. Frege, who wants functions, just as well as predicates, to be defined over the whole universe, must, to the predicate associated with a function, assign the value t for some objects that are outside what we would consider to be its domain of applicabil-ity, and the blanket solution of equating meaninglessness to falsity is not open to him.

What are the criteria that govern the assignment of truth values outside the domain of applicability? For Frege, it seems, the assignment is arbitrary. The point I want to make here is not that such an enterprise cannot be carried out, but rather that neither mathematics nor ordinary language proceeds thus.

Since vagueness pervades natural language, Frege is led to postulate, behind each vague predicate of ordinary language, an exact 'objective' predicate, so that logic can operate without a hitch. In 1884, §26, or 1953, pp. 36 and 36e, he writes: 'The word "white" ordinarily makes us think of a certain sensation, which is, of course, entirely subjective; but even in everyday speech it often bears, I think, an objective sense. When we call snow white, we intend to express an objective quality, which we recognize, in ordinary daylight, by a certain sensation. [ ... 1 Even a color-blind man can speak of red and green, in spite of the fact that he does not distinguish between these colors in his sensations. [ ... 1 By being objective I understand being independent of our sensa-tion, intuition [Anschauen] and imagination [Vorstellen], and 9f all construction of mental pictures out of memories of earlier sensations,


but not independent of reason'. Frege is saying that what an empirical word of our natural language means for us is not constituted by the training acquired through ostension learning ('memories of earlier sensations'), but that, behind the word, we comprehend some objective property. A bit earlier in the same section (1953, pp. 35 and 35e; I modified the translation), Frege had written: 'Objective here is what is subject to law, conceptual, adjudicable, what can be expressed in words' ('Objektiv ist darin das Gesetzmiipige, BegrijJliche, Beurtheilbare, was sich in Warten ausdriicken liipt'). Frege's words ('subject to law', 'con-ceptual', 'can be expressed in words', 'not independent of reason') suggest that the objective property he assumes behind every empirical term has a definition not unlike that of 'even' in arithmetic. If there were such a definition of 'red' or 'green', then, of course, a color-blind man would comprehend these words. Frege has to adopt this panrationalism because he wants to fit the natural language to his bivalent logic. His use of the word 'beurtheilbar' already suggests that the objective property behind the empirical term is bivalent. This is reinforced by the use of 'objective' at other places in Frege's writings (for instance, in 1892, p. 34, or 1952, pp. 63-64, where objectivity is connected with bivalence). What Frege is in fact suggesting is that one can disregard the vagueness prevalent in a natural language because many words are learned by ostension. But this conception of the objective property is so contrary to the actual working of a natural language that he has to hedge ('often', 'I think').

In this letter to Peano quoted above, Frege writes (1976, p. 183, or 1980, p. 115; I modified the translation): 'The task of our vernacular languages is essentially fulfilled if the persons having verbal intercourse with one another connect, with one sentence, the same thought, or approximately the same thought. For this it is not at all necessary that the individual. words should have a Sinn and a Bedeutung of their own, provided only that the whole sentence has. a Sinn. Whenever inferences have to be drawn, the case is different; it is then essential that the same expression occurs in two sentences and that it has exactly the same Bedeutungin both.'

Here Frege deals with a pressing problem, namely the statUs of logical arguments conducted in ordinary language; but we may wonder, since the lines are taken from a letter written with a running pen, how much exegesis each word can bear. When some sentences are used in a logical argument, each part of a sentence is to have the same Bedeutung


at each of its occurrences, and each sentence has a Bedeutung, that is, a truth value. Speaking a bit more precisely, we could say that, if we consider the premiss( es) and the conclusion of a rule of inference, a predicate symbol is to occur in more than one of these formulas; and the correctness of the rule would vanish if, to two different occurrences of the predicate symbol, we were to assign different subsets of (a Cartesian product of) the universal domain when we interpret the formulas. On the other hand, when ordinary language is used only for communication, parts of a sentence may have no Sinn and no Bedeutung, although the sentence has a Sinn. Since no Bedeutung is mentioned for the sentence, does it mean that the sentence is truthvalueless?

How can a word have no Sinn and/or no Bedeutung? This could be because the word is used syncategorematically. But, first, not all words of a sentence can be syncategorematic. And, second, this would not fit well with Frege's systems, in 1879 and in 1893-1903, since in these systems no sign is syncategorematic; each has a Bedeutung (and a Sinn); the sign for negation, for example, denotes a certain function. We would seem to be more justified, in the light of Frege's writings, in assuming that a word has no Bedeutung because it is vague. Remember, a pre�dicate whose delimitation is 'blurred' denotes no concept. So Frege seems to think that words of ordinary language have neither Sinn or Bedeutung because they are vague, but that, somehow, the whole sentence has a Sinn (though perhaps no Bedeutung, that is, no truth value). All that remains quite schematic, and the manner in which a sentence composed of words having no Sinn has itself a Sinn is certainly obscure.

In a manuscript written two or three years after his letter to Peano and left unpublished, Frege wrote (1969, p. 168, or 1979, p. 155; I modified the translation):

Inference from two premisses often, if not always, rests on there being a concept common to both. If a fallacy is not to occur, not only must the sign for the concept be the same, but this sign must also have the same Bedeutung [dasselbe bedeuten]. It must have a Bedeutung independent of the context, and not first obtain one in the context, something that is, however, very often the case with words of [ordinary] language.

Here Frege repeats what he has said in our quotation from his letter to Peano, but only in part. He no longer speaks of words without Sinn constituting a sentence that has Sinn. The initial remark, that in an inference from two premisses a predicate symbol is to occur in each,


reminds us of the role of the middle term in an Aristotelian syllogism and is not correct for modern systems; if, however, we consider all the premisses and the conclusion of any rule of inference in any modern system, we can safely say, I think, that every predicate symbol occurs at least twice. And, in any interpretation, the same set has to be assigned to each occurrence of the symbol.

Frege's view is that vagueness does not prevent communication, but wrecks logical inferences. A word, at its two occurrences in two different sentences, has different Sinne, that is, strictly speaking, has no Sinn. Within a sentence, each word has a local Sinn, and these Sinne succeed in combining in such a way that the sentence as a whole has a Sinn; but, as we go from sentence to sentence, the Sinn of a word varies, and these changes play havoc with logical inferences, because these necessarily involve more than one sentence.

Frege's conception would involve, it seems, a considerable amount of elaboration and justification. But it may be right. Vagueness is omni-present in ordinary language, in the sense that every word has a fringe of indeterminacy. The word 'man' is vague (and that in at least three directions: 'adult' versus 'boy', 'Homo sapiens' versus 'ape', 'male' versus 'female'), but the word is nevertheless a serviceable component of our language. Ordinary language is quite nimble at taming vagueness. It makes use of all the information that the context can provide in order to curb the vagaries of words. If that does not succeed and the indefinite-ness of a word threatens the mutual understanding between speaker and hearer, we drop the word and shift to another way of speaking.

Logical inference puts more severe constraints on language than does the mere impartment of information; it tolerates less wobbling in the senses of words, perhaps no wobbling at all. The Greeks began to worry about the Heap and the Bald when their language became an instrument of argumentation, whether mathematical or philosophical. Since the best we can do, in ordinary language, is to locally reduce vagueness, but not eliminate it, and since logic has been built on the same lines as mathe-matics, that is, with initial bivalent predicates plus rules that maintain bivalence (set-theoretic semantics), what is the status of logical argu-ments conducted in ordinary language, with vague words? Shady, some would say. Behind a vague word we perceive (imagine? assume?) a bivalent predicate, and the argument is actually conducted with that predicate in mind; when the words are again taken as belonging to ordinary language, the conclusion, it is to be hoped, will remain valid.

FREGE AND VAGUENESS

The enterprise has an 'as if' character. In the syllogism

All Athenians are men, All men are mortal,

Therefore, all Athenians are mortal,

41

the predicates ' ... is an Athenian', ' ... is a man', ' ... is mortal' are vague; but nobody would question the validity of the argument. On the other hand, when we try to apply mathematical induction to the predicate ' ... is a heap', we reach an absurd conclusion, and precisely because of the vagueness of the predicate. Here, if we would simply argue as if the predicate ' ... is a heap' were exact, our convention would seem glaringly artifical and leave us with a feeling of dissatisfaction. The first Greek who drew our attention to the Heap wanted us to reflect on how logical arguments can at all be conducted with vague words.

When we introduce logic to students, we are prone to gloss over vagueness and we will not demur, perhaps, at giving the sentence 'It is raining or it is not raining' as an example of the law of the excluded middle. We carry out, without always saying so, Quine's regimentation, that is, we assume a universe of discourse with bivalent predicates. Quine, however, is more sophisticated than Frege about this regimenta-tion. For him (1977, p. 195), 'the regimentation is not a matter of eliciting some latent and determinate content of ordinary language'. Eliciting a latent and determinate content is precisely what Frege is doing when, behind the word 'white', he discovers the bivalent 'objec-tive' whiteness. Quine continues: 'It is a matter rather of freely creating an ontology-oriented language that can supplant ordinary language in serving some particular purpose one has in mind'.

Although Frege and Quine give a different ontological status to the universe of discourse, both agree that ordinary language has to be supplanted if logic is to be able to function. Ordinary language is somehow too weak to stand the stress of bivalence and should not be asked to bear up against the requirements of logical rigidity. A logical argument conducted in ordinary language is to be successful only whenever we can discern, behind ordinary words, bivalent predicates, as we perceive the watermark behind the surface of a sheet of paper.

Russell comes to a similar conclusion. According to him, we are to imagine a precise meaning for the words of ordinary language, and logic is 'not applicable to this terrestrial life, but only to an imagined celestial


existence' (1923, pp. 88-89). And some years later (1937, p. xi) he writes: 'none of the raw material of the world has smooth logical properties, but [ ... ] whatever appears to have such properties is constructed artificially in order to have them'.

Russell's artificial construction of a bivalent world by imagination is closer to Quine's free creation of an ontology-oriented language than to Frege's objective realism. But the three of them, Frege, Russell, Quine, agree on one point, namely that ordinary language has to be supplanted by a bivalent regimented discourse if logic is to function properly.

Regimentation comes in various forms. Frege's universal domain is a fixed Universe that comprehends all objects, the number 7 as well as Julius Caesar, with functions on top. One could claim, as Bachmann did in his 1975, that the individuals whose existence is imposed by the axioms of Frege 1893 and 1903 constitute a denumerable universal domain of purely logico-mathematical objects. But, first, even in 1893-1903 Frege constantly adduces persons or heavenly bodies as examples of objects and uses them as counterexamples to theses that he wants to refute; and, second, his discussion of a number of key notions, Wirklichkeit for example, clearly indicates that he puts all objects, whether abstract or concrete, in the same ground domain of individuals. Hence the denumerable domain isolated by Bachman constitutes a minimal core, but it is difficult to maintain that Frege restricts himself to that core. (The point deserves further discussion and I hope to come back to it on another occasion.) Russell has an infinitely stratified hierarchy, which is not, it seems, ontologically anchored. to a fixed ground domain. As he writes (Whitehead and Russell 1910, p. 169, or 1925, p. 161), 'lilt is unnecessary, in practice, to know that objects belong to the lowest type, or even whether the lowest type of the variable occurring in a given context is that of individuals or some other. For in practice only the relative types of variables are relevant; then the lowest type occurring in a given context may be called that of individ-uals, so far as that context is concerned'. As for Quine, he has theories, changed according to our needs, each of which having, it seems, a universal domain comprehending everything that there is (in the theory).

For Frege, regimentation is global, in the sense that the supplanting of ordinary language is done en bloc. Perhaps also for Russell, with the difference that Frege's universe is unique and fixed, while Russells' hierarchy can, so to speak, move up and down. But Russell's quotation, with its reference to a given context, opens a path that we may enter and


along which we may go farther than Russell. Logic is to be used locally to test the correctness of arguments in ordinary language. Only the few sentences that constitute the argument are considered. We pick the universal domain that suits us locally, introduce predicates and connec-tives, and check the argument. We now have a logic utens rather than a logic magna. Logic has no longer any ontological import. The question whether the number 7 and Julius Caesar are to be put in a common uni-versal domain of objects belongs perhaps to ontology or metaphysics, but not to logic. Frege, no doubt, would have considered such a use of logic to be mere bricolage. But his global supplanting of ordinary language by a system of bivalent predicates everywhere defined over a fixed universal domain takes us far away from the problems of ordinary words. These words are inherently vague because the process through which we learn them involves a finite number of instances. We have learned how to live with that vagueness, and we may want to understand how ordinary language works, in particular how it works in spite (or, perhaps, because) of vagueness. But this is not a task that Frege welcomes. 'A large part of the philosopher's task consists - or at least should consist - in a struggle with language', he writes (1969, p. 289, or 1979, p. 270). Frege carries on that struggle, and brushes aside vagueness.

What he does is to introduce an ontology that would allow his new logical laws to function in a way that should be unobstructed and, moreover, as simple as possible; it should be so simple that he is even reluctant to divide his universe into sorts. He could not have tackled the problems of vagueness at the moment he was introducing these logical laws without extremely complicating his ontological view of the world. We can even say more: he would not have been able to formulate these laws without leaving vagueness out of the picture. The only way to proceed at an early stage in the development of a science is by bold simplification and abstraction. As Kreisel reminds us, this was Galileo's successful strategy, he 'found immensely manageable laws by simply excluding the vagaries of friction and air resistance in suitable experi-mental setups' (1984). So Frege's disregard of vagueness and other vagaries was, in a way, inevitable. But his logical laws have been formulated more than hundred years ago, and it is now perhaps time to look at the vagaries.

Stanford University


REFERENCES

Bachmann, Friedrich 1975 'Frege als konstruktiver Logizist', in Thiel 1975, 160-168.

Frege, Gottlob 1879 Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen

Denkens; English translation by Stefan Bauer-Mengelberg in van Heijenoort 1967, 1-82.

1884 Die Grundlagen der Arithmetik, eine logisch-mathematische Untersuchung uber den Begriff der Zahl; see Frege 1953.

1892 '(fuer Sinn und Bedeutung,' Zeitschrift fUr Philosophie und philosophische Kritik, new series, 100,25-50; English translation in Frege 1952,56-78.

1893 Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, vol. 1. 1903 Grundgesetze der Arithmetik; begriffsschriftlich abgeleitet, vol. 2. 1952 Translations from the Philosophical Writings of Gottlob Frege, edited by Peter

Geach and Max Black; 2nd ed., 1960. 1953 The Foundations of Arithmetic, German text and English translation of Frege

1884 by J. L. Austin, 2nd revised edition. 1969 Nachgelassene Schriften und wissenschaftlicher Briefwechsel, vol. 1, Nachge-

lassene Schriften. 1976 Nachgelassene Schriften und wissenschaftlicher Briefwechsel, vol. 2, Wissen-

schaftlicher Briefwechsel. . 1979 Posthumous Writings. 1980 Philosophical and Mathematical Correspondence.

Hilbert, David 1904 'Uber die Grundlagen der Logik und der Arithmetik', Verhandlungen des

Dritten 1nternationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904; English translation by Beverly Woodward in van Heijenoort 1967, 129-138.

Kreisel, Georg 1984 'Frege's Foundations and Intuitionistic Logic', The Monist 67, 72-91.

Merrill, Kenneth R. See Shahan, Robert W., and Kenneth R. Merrill.

Poincare, Henri 1906 'Les mathÄmatiques et la logique', Revue de metaphysique et de morale 14,

17-34. 1908 Science et methode.

Quine, Willard Van 1977 'Facts of the Matter', in Shahan and Merrill 1977, 176-196.

Russell, Bertrand 1923 Vagueness, The Australasian Journal of Psychology and Philosophy 1, 84-92. 1937 The Principles of Mathematics, 2nd ed. See 'Whitehead, Alfred North, and Bertrand Russell.

Shahan, Robert W., and Kenneth R. Merrill 1977 American Philosophy from Edwards to Quine.

FREGE AND VAGUENESS

Thiel, Christian (ed.) 1975 Frege und die moderne Grundlagenforschung.

Van Heijenoort, Jean (ed.) 1967 From Frege to Godel, A Source Book in Mathematical Logic, 1879-193l.

Whitehead, Alfred North, and Bertrand Russell 1910 Principia Mathematica, vol. 1. 1925 Principia Mathematica, 2nd ed.

45

HANS SLUGA

SEMANTIC CONTENT AND COGNITIVE SENSE

1.

We can characterize the disagreement between adherents of Russell-style theories of meaning and those of Frege-style theories as follows:

RS theorists assume that a satisfactory theory of meaning can be built with the binary relation - e refers to r - whereas FS theorists maintain that a three-place relation - e through having sense s refers to r - is required. l

It should be clear from this characterization that anything an RS theory can do can also be done by an FS theory, since the crucial notion of RS theories, the binary reference relation, is available in any FS theory; for we can define

e refers* to r= df' 3 s (e through s refers to r).2

The question in the dispute between the two theories is then whether there are conditions of adequacy only FS theories can satisfy.

On my account the disagreement is not over the issue of how definite descriptions are to be analysed. According to Russell a semantic theory should not be expected to assign meaning directly to expressions of the from "the f," but only to sentences in which such phrases occur. A sentence of the form "the f is g" is then to be taken to have the same meaning as a sentence of the form "there is one and only one thing which is both f and g." On Frege's analysis, on the other hand, expres-sions of the form "the f" are to be taken as proper logical constituents of propositions which say, in effect, that the semantic theory should assign meaning directly to such expressions.

It might seem at first sight that there is a logical link between FS theories of meaning and the Fregean analysis of definite deSCriptions and likewise between RS theories and the Russellian analysis of definite descriptions. That is, however, not the case. For fairly obvious reasons an FS theorist could consistently adopt the Russellian analysis since it is structurally equivalent to Frege's analysis of expressions of the form "all

47


48 HANS SLUGA

f" and "some f" which, in tum, is presumably compatible with an FS theory.

But the claim that a Fregean analysis of definite descriptions is incompatible with RS theories of meaning is initially more plausible. If we treat expressions of the form "the f" as referring expressions and not merely as having contextual meaning, we must, it seems, distinguish between their reference and how they characterize that reference. The expression "the teacher of Alexander the Great" refers to Aristotle and characterizes him as a teacher of Alexander whereas the expression ''the author of the Prior Analytic" refers to that same person, but charac�terizes him as an author rather than a teacher. Such a distinction between the reference of an expression and how that reference is characterized seems to demand an FS account in terms of the notions of the reference and the sense of expressions. I will try to show that this is not so.

A satisfactory theory of meaning in the sense in which I have spoken of it in characterizing the disagreement between the two parties is a theory which satisfies certain conditions of adequacy. Any such condi�tion met by an RS theory will obviously be satisfiable by an FS theory. But Frege argues in the essay "On Sense and Reference" that there are conditions of adequacy which only his style of theory can satisfy. One of them, he claims, is the following:

Condition A: A satisfactory theory of meaning must explain the (intuitively obvious) difference between trivially true and informa�tively true identity statements.

For the reasons given by Frege himself I assume that theories of this style satisfy the condition. My question is whether RS theories also can satisfy it. If they can, condition A cannot serve as a criterion for determining which of the two styles of theory is preferable.

I will argue that RS theories can, in fact, satisfy condition A, but only partially so. We can then ask whether the condition is valid only to the extent to which both styles of theory can satisfy it. Reasons for that claim can be produced. Even if those reasons are ultimately rejected, we can learn something about the basic difference between RS and FS theories by considering them.

There are other conditions of adequacy which Frege claims only his style of theory can satisfy. Thus, he argues, that only an FS theory can account for the logic of belief contexts and that only an FS theory can

SEMANTIC CONTENT AND COGNITIVE SENSE 49

explain the use of names which lack reference. For the purpose of the present discussion we can set their consideration aside.

2.

In order to show that RS theories can, at least, partially satisfy condition A, I introduce a notion of semantic content. The claim is that this notion can be explained by means of syntactic terms and the binary relation "e

refers to r" but can do much of what Frege's notion of sense is meant to do. I will first illustrate the notion informally.

Given two mathematical functions f and g which we explain as

f(x) = x + 7 and g(y) = 12 - y

we can form the true equation

f(2) = g(3).

It is here obvious that the two functional expressions "f(2)" and "g(3)",

while both referring to the number 9, do- so in different ways. But in order to describe the difference in their way of specifying the number 9, we have no need to talk about the supposed sense of the two functional expressions. All we need to consider is the reference of the two func-tional terms as laid down in the initial specification, the reference of the expressions "2" and "3" and the way the functional terms and the number terms are combined into the complete functional expressions "f(2)" and "g(3)". There is then a notion of semantic content, specifiable in referential and syntactic terms alone, which allows us to say that in the equation "f(2) = g(3)" the functional expressions have the same reference, but differ in semantic content.

In order to make the notion of semantic content precise we need to determine both when two expressions have the same semantic content and when they have different ones. We must do so first for simple expressions (proper names and functional terms) and then for complex ones formed from them.3

Since intuitively we mean by semantic content the information con-veyed by the composition of an expression given the reference of its constituents, it seems plausible to say that for simple expressions, i.e., those without composition, semantic content varies with reference. In other words:

Two simple expressions have the same semantic content if and only if they have the same reference.

50 HANS SLUGA

And for complex expressions we can lay down:

Two complex expressions have the same semantic content if and only if they are constructed in the same way out of constituents with the same semantic content.

Thus, if "f" and "g" have the same semantic content and so do "a" and "b", then "f(a)" and "g(b)" have the same semantic content. But if "a"

and "b" have different semantic content, then "f(a)" and "g(b)" also have different semantic content, though they may have the same refer-ence. Given a two-place relation "h", the two expressions "h(a, b)" and "h(b, a)" have different semantic content since they are built up in different ways, though they are built from the same constituents and may have the same reference.

It should be obvious how we can give a Fregean analysis of definite descriptions in terms of this notion of semantic content. Treating definite descriptions as expressions of the form ''f(g)'' we can distinguish between the reference and the semantic content of the definite descrip-tion. Thus, ''The teacher of Alexander the Great" and "the author of the Prior Analytic" are expressions with the same reference, but with different semantic content.

It should also be immediately clear how we can use the notion of semantic content to distinguish between trivial and informative identity statements. An identity statement is informative if the two expressions connected by the identity sign have the same reference, but different semantic content. Thus, the sentence "The teacher of Alexander the Great is identical with the author of the Prior Analytic" is informative because the first expression refers to Aristotle as a teacher and the second refers to him as an author.

3.

It remains to be shown that the distinction between trivial and informa-tive identity statements is only partially characterizable by means of the notion of semantic content. That will become clearer, if we consider what different kinds of identity statement there are.

Assuming "a" and "b" to be simple names and "m" and "n" complex referring expressions, we can distinguish three kinds of identity state-ment:


(1) thoseoftheform "a= b"

(2) those ofthe form "a = m" or "n = b" and (3) those of the form "m = n".

When we test for each of those three kinds of statement how well the notion of semantic content accounts for a distinction between trivial and informative ones, we discover that it seems to fit the third kind of statement, but that there are problems with the first and the second.

According to our characterization of the notion of semantic content, all true identity statements of the first kind must be trivial. And similarly no true identity statements of the second kind can be trivial. That seems not to conform to our intuitions and it is here where an FS theory appears to have an advantage. According to Frege every simple name has a sense as well as a reference. An identity statement of the first kind is informative for him, if the two names "a" and "b" have the same reference but different sense. Similarly an identity statement of the second kind is trivial for him, when the name "a" in it has the same sense as the complex expression "m" occurring in the statement. Such distinctions are not available in the account built on the notion of semantic content as we have defined it, since we have laid down that two simple names have the same content if and only if they have the same reference.

We can remove part of the problem by changing our original defini-tion to say:

When an expression has been explicitly introduced by means of a definitional specification, the semantic content of the simple expression will be the same as that of the original specification. Two simple expressions not introduced in that way have the same content if and only if they have the same reference.

For all identity statements of kinds (1) and (2) which contain names to which the first part of this characterization applies the distinction can now be made in terms of the notion of semantic content. Consider the sentence "Cicero is identical with Tully." According to the original characterization of the notion of semantic content the statement is trivial; but according to the new one it is informative, if one of the proper names has been introduced by a definite description. If both have been introduced by definite descriptions, the statement may again tum out to be trivial, namely when the two descriptions have the same semantic content.

52 HANS SLUGA

There are other possible and plausible modifications of the notion of semantic content that make it possible to draw the trivial/informative line of distinction closer to the intuitive one. We might, for instance, specify that two complex expressions have the same semantic content if one is a transformation of the second according to specifiable rules and if the corresponding constituents in the two expressions have the same semantic content. Such rules might state that expressions of the form "m

and n" and "n and m" have the same semantic content and likewise for "m or n" and "n or m", as well as for "m" and "not-not-m".

Such modifications are, however, unlikely to change the fact that we cannot completely capture the intuitive distinction between trivial and informative identity statements in terms of such a (modified) notion of semantic content. The case seems to be different for an FS theory, and it is easy to see why. Frege holds that every expression with a reference must also have a sense, for the sense is meant to determine the way the object is given. He argues with some plausibility that in order for us to identify or re-identify an object, we must recognize it as a something or other. It is this mode of recognition that constitutes the sense of the sign referring to the object. In terms of this notion of sense we can explain why an identity statement of the first kind can be informative, and one of the second trivial. If the two kinds of identity statement contain proper names not introduced through definitional specifications, the account of,the trivial/informative distinction in terms of the notion of semantic content will not coincide with the Fregean account. And since it is the Fregean account that is closer to the intuitive distinction we seem to have a criterion for choosing between FS and RS theories of meaning. The former can satisfy condition A more fully than the latter and on that account seems to be preferable.

4.

But this conclusion follows only if we uphold condition A without restriction. One might argue instead that the condition is, in fact, not a completely suitable criterion for choosing between semantic theories. One might argue that the notions of triviality and informativeness are, in fact, not at all unified concepts, that there are hidden semantic and cognitive elements which must be separated when we consider require-ments of semantic theories.


Consider the propositions "Aristotle is Aristotle" and "Aristotle was the teacher of Alexander the Great." The former is trivial because it is formally an instance of the principle of identity; we need not consider how Aristotle is to be identified in order to recognize the truth of the proposition; all we need to know is that the two occurrences of the name refer to the same man. The second sentence, on the other hand, is not at all trivial in this sense. It can convey information; but to an Aristotle scholar or a Greek historian the sentence may still be a triviality.

I call the first sentence semantically trivial and the second semanti�cally informative. But what is semantically informative may still turn out to be cognitively trivial for a particular speaker, as illustrated above. The sentence "Aristotle is Aristotle" is, for most of us, both semantically and cognitively trivial; but there are also sentences which are semantically trivial though cognitively informative; "war is war" comes to mind. It is less clear whether there can also be sentences which are semantically informative and cognitively trivial. I will argue below that Frege, at least, thought that there could and that this conviction motivated his introduc�tion of the theory of sense and reference.

Having distinguished the two sets of notions we are now in a position to say why condition A may not be a suitable criterion for judging semantic theories; for we may reasonably hold that a semantic theory should have to explain only semantic distinctions and not cognitive ones. Condition A should then give way to:

Condition A': a satisfactory theory of meaning must explain the difference between semantically trivial and semantically informa�tive identity statements.

This condition, we can argue, is in fact satisfied by both RS and FS theories and can therefore not serve as a criterion for choosing between them.

What reason have we for considering the notion of sense a cognitive rather than a semantic notion? Consider again the sentence "Aristotle was the teacher of Alexander the Great." According to Frege the sentence is trivial, if we connect with the name "Aristotle" the sense "the teacher of Alexander the Great." There is nothing in the expression itself nor in the fact that this expression is a name of a particular person which determines the sense in which the name is to be taken. If semantic

54 HANS SLUGA

notions are those which can be explicated in terms of the nature and structure of linguistic expressions and what they stand for, the notion of sense is obviously not a semantic notion at all. Frege's argument for holding that names must have sense is this: that for speakers to be able to refer to objects they must be able to identify and re-identify objects, objects must be given to them in particular ways. Such considerations seem epistemic in character, rather than semantic.4

It might be useful to distinguish more sharply than the philosoplftcal literature has done so far between the semantics and the epistemology of language. Recent work in the theory of meaning has shown how complex issues can become in that field and for that reason alone it might be helpful to keep its boundaries as narrowly defined as possible.

I formulate this suggestion in hypothetical terms, because I have certainly not shown beyond doubt that a sharp distinction between semantic and cognitive issues can be drawn in the way I have suggested. My intention here was simply to argue that condition A is not an uncontroversial criterion for choosing between semantic theories. If we consider it unrestrictedly valid we commit ourselves to a view that does not distinguish sharply between semantic and cognitive aspects of language. My conclusion is simply that FS theories cater to such an amalgamation, whereas RS theories demand a separation of the two aspects.

We can see how FS theories lead to an amalgamation of semantic and cognitive issues by considering how Husserl used the Fregean notion of sense and by generalizing it to the notion of noema generated a whole cognitive theory from it.5 In recent work John Searle and Michael Dummett illustrate two alternative directions in which such an amalga-mation can lead. Searle, travelling a path parallel to Husserl's, has now generalized the theory of sense into a theory of intentionality; the theory of meaning has become embedded for him in a cognitive theory.6 Dummett, on the other hand, has made the theory of meaning absorb the theory of knowledge.7 In either case the boundary between the theory of meaning and the theory of knowledge has been traversed.

5.

The notion of semantic content appears to me of obvious interest for systematic work in the theory of meaning. But I have introduced it primarily for another purpose - to clarify the origin of Frege's doctrine


of sense and reference. That doctrine is closely tied to Frege's under�standing of the logical character of identity statements which, in turn, is related to his concern with the nature of arithmetical truths.

Frege's conviction that arithmetical truths are reducible to logic, a conviction that shaped so much of his work, is almost certainly derived from Hermann Lotze's Logik of 1874.8 There Lotze develops the reducibility thesis in conjunction with a theory of identity that is closely akin to Frege's own early theory in the Begriffsschrift. In order to understand the latter it is useful to go back to Lotze's account.

Lotze, in contrast to Kant and the Kantian tradition, holds that "all calculation is a kind of thought and that the fundamental concepts and principles of mathematics have their systematic place in logic." (p. 26) At the same time he agrees with Kant that arithmetical equations do not "rest simply upon the principle of identity." (p. 504) He agrees with Kant against Hume that such truths are necessary, but not tautologous. Lotze, like Kant, takes the principle of identity to say no more than "a = a", and it is obvious that the usual arithmetical truths cannot be derived from this principle alone. Nevertheless, Lotze asserts:

We must not forget that calculation in any case belongs to the logical activities, and it is only their practical separation in education which has concealed the full claim of mathematics to a home in the universal realm of logic (p. 110).

For Kant the principle of identity is, of course, the one on which all logic rests. If arithmetical truths are not derivable from that principle alone they cannot be logical truths for him. Lotze, on the other hand, takes a somewhat different line by introducing a distinction between the form and the content of expressions in identity statements. And it is this distinction which he believes will reconcile the claims that arithmetical truths are logical, but no mere instances of the principle of identity.

If two different expressions refer to the same thing, they have for Lotze the same content, but different form. As a matter of fact Lotze's distinction of form and content corresponds precisely to the previously given distinction between semantic content and reference.9 The distinc�tion allows Lotze to say that an arithmetical equation like "7 + 5 = 12" has the same content as the principle of identity, "a = a", but a different form. He says of this equation that we have in it "a perfectly identical judgment as regards its matter, and it is only synthetical formally because it exhibits the number 12 first as the sum of two quantities and then as determined by its order in the simple series of number" (p. 64).

56 HANS SLUGA

As far as its content is concerned every identity statement (and every arithmetical equation) is, thus, a necessary truth which says the same as "a = a." But this necessary truth can be represented under different forms. According to its form an identity statement can be empirical or a priori. There can thus be empirically necessary and a priori necessary identity statements. lO Lotze says that as far as the form is concerned all nontrivial identity statements are synthetic, some synthetic a posteriori and some synthetic a priori. Arithmetical equations are characteristically both identical (and hence necessary) propositions and synthetic a priori.

That seems to bring Lotze close to the Kantian view of arithmetic, but the impression is misleading, since he does not hold that arith-metical truths are based on an intuition of space and time. He says rather of the exemplary equation "7 + 5 = 12":

For that which all turns upon is in fact nothing more than the assertion which is contained in the sign of addition - viz. that quantities can be summed so as to compose another and a homogeneous quantity; a proposition the importance of which we may once more be tempted to ignore, because it seems to us self-evident and a mere identical proposition defining the nature of numerical quantity as such. And so it undoubtedly is, but how do we arrive this piece of self-evident knowledge? (p. 508).

And the answer to that question is that,

This very fact, that there is such a thing as quantity to be found in the world of ideas ... is a fact of immediate perception . .. The proposition therefore that quantities can be summed is undoubtedly an identical proposition; but that the subject and predicate of that proposition appear as valid in the world of ideas . . . does not follow from the principle of identity (p. 509).

The realm of objective ideas is for Lotze the realm with which logic is concerned. The truths that obtain in it are the truths of logic. But our knowledge of those truths, the forms under which they appear, is synthetic a priori knowledge. It depends on our direct apprehension of what is to be found in the world of ideas. In order to understand the necessity of the arithmetical truths we must apprehend that summable quantities belong to that world. Such apprehension is synthetic a priori in so far as it is intuitive and immediate rather than conceptual and discursive.

With this argument Lotze has, in fact, concluded that the truths of arithmetic cannot be derived from the principle of identity alone, that additional logical principles, a richer logic that Kant's are required in order to carry out the derivation. He writes:


Turning to the discovery of mathematical truth, we shall not dispute the validity nor yet the importance of the principle of identity, but we must dispute its fruitfulness; we must insist that if it were the only principle we had to start from, mathematical truth could never be discovered at all (p. 505).

Lotze is convinced that mathematics is an "independently progressive branch of universal logic" (p. 26), but he never tries to spell out the additional logical principles that he considers necessary for the logical derivation of the mathematical and, in particular, the arithmetical truths. That is a task Frege takes upon himself.

6.

In so far as we can speak of a conception of meaning in Frege's early work, it was an RS conception built on the binary relation of a name to its content. Following Lotze's usage Frege employed the term "content" in this period to mean as much as the reference of a name, that which the name stands for. The account of identity statements which he offers in his earliest publication, the Begriffsschrift, presupposes such an RS conception and is essentially equivalent to the account given here in terms of the notion of semantic content.

There are, to be precise, two aspects to the Begriffsschrift account of identity which it is important to separate. The first is that identity "applies to names and not to contents ... for it expresses the circum�stance that two names have the same content."l1 The second is that

statements can be informative because "the same content can be completely determined in different ways." The informative identity statement says that two ways of determining the content can yield the same result. Frege writes:

Before this judgment can be made, two distinct names corresponding to the two ways of determining the content must be assigned to what these ways determine. The judgment, however, requires for its expression a sign of identity of content, a sign that connects these two names. From this it follows that the existence of different names for the same content is not always merely an irrelevant question of form; rather, that there are such names is the very heart of the matter if each is associated with a different way of determining the content (p. 21).

There is no question that Frege's distinction between the content of a name and its way of determining a content is derived from Lotze's account of identity and like it corresponds to our distinction between reference and semantic content.

58 HANS SLUGA

At the same time there is one important difference between Frege's and Lotze's account. For Frege an identity statement is a statement about signs, whereas Lotze makes no such assumption. One reason for Frege's departure from Lotze's account suggests itself immediately. The latter has the peculiar consequence that identity statements can turn out to be both identical (and thus analytic in their content) and synthetic in form. In the Begriffsschrift Frege may have considered it possible to simplify the story by considering identity statements as statements about signs. In that case, he can say that statements of the form "a = a" are analytic and that identity statements in which two expressions of differ�ent form stand for the same content are synthetic. This is, in fact, what he indicates when he says about the latter:

In that case the judgment that has the identity of content as its object is synthetic, in the Kantian sense (ibid.).

But this simplification has the unfortunate consequence that Frege is no longer able to explain how any but the most trivial arithmetical equa�tions can turn out to be logical truths. This, however, was a point he came to appreciate only later.

While Frege allowed in the Begriffsschrift that signs can sometimes stand for what they signify and sometimes for themselves, he later came to reject such a "bifurcation" violently. It was, presumably, the confron�tation with mathematical formalists who assumed that number state�ments are statements about numerical signs which made Frege aware of the need to distinguish sharply between the sign and the signified. Given the fact that he believed that his early account of identity statements depended on taking them as statements about the signs, we can see, at least, one reason why he may have wanted to modify his early theory. He writes in the essay 'On Sense and Reference':

What is intended to be said by a = b seems to be that the names "a" and "b" designate the same thing, so that those signs themselves would be under discussion; a relation between them would be asserted.i2

It is clear that he finds this no longer plausible; but the reason he offers for why the earlier account is implausible is peculiar. He continues:

But this relation would hold between names and signs only in so far as they named or designated something. It would be mediated by the connexion of each of the two signs with the same designated thing. But this is arbitrary. Nobody can be forbidden to use any arbitrarily producible event or object as a sign for something. In that case the


sentence a = a would no longer refer to the subject matter, but only to its mode of designation; we would express no proper knowledge by its means (ibid.).

This explanation of what is wrong with the earlier account is peculiar because if consists in a mere repudiation of what he had previously affirmed without really adding new arguments. In the Begriffsschrift he had said that "the existence of different names for the same content is not always merely an irrelevant question of form," but he now asserts blandly that "nobody can be forbidden to use any arbitrarily producible event or object as a sign for something."

It seems at this point that the earlier account is, in fact, more plausible. Every simple word in the language could, of course, have a reference different from the one it has, but once the reference of simple words is fixed, the reference of complex terms formed from them seems no longer arbitrary. Why then did Frege feel himself forced to abandon his earlier account? Surely not just because he had come to reject the idea the identity statements are statements about the signs.

7.

There is, in fact, no way in which we can understand the motivations for Frege's shift by looking exclusively 'On Sense and Reference.' Instead we must turn to the essay 'Function and Concept,' published shortly before 'On Sense and Reference,' where the new semantic theory is discussed for the first time.

'Function and Concept' is clearly part of Frege's work on the thesis that arithmetic is reducible to logic. In it he introduces the notion of the value-range of a function, meant as a generalization of the notion of class. And he argues for a principle which he considers essential for his reductionist program which says that statements of the form "the value-range of the function f is identical with that of the function g" are equivalent to "the functions f and g have the same value for identical arguments." He writes:

The possibility of regarding a general statement to the effect that the values of functions are equal as an equality, namely as an equality of value-ranges is, I think, not capable of proof, but must be considered a basic law of logic (p. 26).

In the Basic Laws of Arithmetic, Frege's main work, the principle recurs as one of the logical axioms (Axiom V) from which arithmetic is to be

60 HANS SLUGA

derived. In accordance with his practice of regarding equivalences as identity statements, it is stated there as:

(x/(x) = yg(y» = 'VZ(j(Z) = g(Z»14

Since Frege was aware of the fact that this axiom was essential for his derivation of arithmetic, and since his project was to show that arith-metic is nothing but an extended logic, he must have asked himself immediately upon formulating the axiom what reasons there were for considering it a logical principle at all.

The Begriffsschrift account of identity statements can, in fact, provide no such reasons. The expressions on the two sides of the identity sign obviously have different semantic content and, according to the Begriffsschrift, statements of that sort are synthetic rather than analytic. A new account of identity statements and a new account of meaning is required, if Axiom V is to come out as a logical (and analytic) truth.

Frege begins the essay 'Function and Concept' by recalling the Lotzean distinction between form and content (p. 22). But this termi-nology is now set aside in favour of another one, the sense-reference terminology, which occurs for the first time in the context of the discussion of the supposed basic law of logic concerning value-ranges. Considering the step from a statement of the form x/ex) = yg(y) to one of the form 'Vx(j(x) = g(x», Frege writes that the latter "expresses the same sense, but in a different way" (p. 27). It is at this point that the new, technical notion of sense is made to do work for the first time.

The point seems clear. If we are to regard a statement of the form "a = b" as a logical truth, we must assume that the two expressions "a" and "b" do not only stand for the same thing, they must also have the same meaning. But the notion of sameness of meaning that is here required is not one that can be spelled out in terms of the notion of semantic content. The whole point of Axiom V is to say that statements of different form (and hence of different semantic content) can be logically equivalent. Axiom V is meant to be semantically informative, but cognitively obvious.

A different notion of meaning, one not explicable in purely semantic terms, is required in order to assure us that Axiom V is indeed a logical principle. The notion of sense is that new concept. According to Frege, Axiom V shows us how the same content can be apprehended in two different ways. This apprehension of sameness, which guarantees the status of the axiom as a logical principle, does not express itself in the formal structure of the axiom. It is an immediate, intuitive apprehension.


It is clear then that the notion of sense serves here as a cognitive notion; it concerns our conception or apprehension of a particular truth.

Michael Dummett has recently said that Frege never repeated the claim made in 'Function and Concept' that the two sides of Axiom V have the same sense. He writes that

in the Grundgesetze, where much space is devoted to justifying the introduction of value-ranges, the assertion is not repeated .... The remark in Function und Begriffis surely a residue from his earlier style of thought, before sense and reference had been distin-guished; when he had reflected further on the new distinction, he realized that he could not sustain the claim.15

Dummett's assertion is, in fact, open to doubt. In the Basic Laws of

Arithmetic Frege writes:

I use the words "the function f(x) has the same value-range as the function g(x)" as gleichbedeutend with the words "the functions f(x) and g(x) have the same value for identical argument (p. 36).

The word I have left untranslated seems, at first sight, appropriately rendered by "denoting the same as" or by "standing for the same thing as."16 But there is a question whether such renderings are, in fact, right. In § 27 of the Basic Laws Frege explains:

We introduce a name by means of a definition by stipulating that it is to have the same sense and the same reference as some name composed of signs that are familiar. Thereby the new sign becomes gleichbedeutend with that being used to explain it (pp. 82f.),17

Here the word "gleichbedeutend" obviously means as much as "synony-mous" and not just "having the same reference." And if the former is the correct translation, then Frege is clearly still convinced that the two sides of Axiom V have the same sense.

Even if ultimately he changed his mind on this point, the fact remains that he initially introduced the notion of sense in order to explain why Axiom V is not a synthetic truth and not because of the unsatisfactory argument at the beginning of the essay 'On Sense and Reference.'

8.

Dummett is, of course right that in the Basic Laws Frege did not justify the introduction of the sense-reference distinction in terms of the needs of his philosophy of mathematics. In the introduction to that book he

62 HANS SLUGA

gives a different account of what he considers to be the most important aspect of the new semantic theory. He writes:

Formerly I distinguished two components in that whose external form is a declarative sentence: (1) the acknowledgment of truth, (2) the content that is acknowledged to be true .... This last has now split for me into what I call "thought" and "truth-value," as a consequence of distinguishing between sense and reference of a sign .... How much simpler and sharper everything becomes by the introduction of truth-values, only detailed acquaintance with this book can show (pp. 6f).

It may at first seem strange that Frege puts no emphasis at all in this text on the new semantics of names and definite descriptions which he had described in the essay 'On Sense and Reference'; but then it becomes clear from looking more carefully at the Basic Laws that this part of the sense-reference theory is of no significance for the logic he lays out in that book or for the attempted derivation of arithmetic from the logic. Going back to the essay our attention is drawn to the fact that the largest portion of it is devoted to the defence that declarative sentences under normal circumstances have truth-values as references, and not to the semantics of names and definite descriptions.

The question remains why Frege should have considered the introduc-tion of truth-values such a significant achievement. Michael Dummett once said that Frege's doctrine that sentences refer to truth-values "proves to have very implausible consequences."18 He was thinking then of the assimilation of sentences to names and of predicates to functions that goes with that doctrine. Frege obviously considered the possibility of such an assimilation one of the great advantages of the new theory. While he had made a distinction between semantic content and refer-ence in his early work (or mode of designation and content, as he said in the Begriffsschrift), he had not applied that distinction to the semantics of complete declarative sentences. In the case of such sentences he had simply spoken of their content. In the 1890s when he introduced his new semantic theory he came to think that various technical simplifica-tions would be possible, if the distinction of sense and reference was systematically applied to all meaningful expressions, names, definite descriptions, functional expressions, and sentences alike. In retrospect he treats that discovery as the important breakthrough of the doctrine of sense and reference.

But it should be clear that it was only a historical accident that had stopped Frege from adopting the same kind of assimilation in the

Begriffsschrift. He could have distinguished there between the semantic content and the reference of declarative sentences and he could have taken all true sentences to have the same reference and all false ones, too. The new semantic theory was, thus, not at all necessitated by the introduction of truth-values. It was rather, as I have shown, the question of the status of arithmetical truths that made the break with the earlier semantic doctrines inevitable for him and forced him to introduce a cognitive notion of sense.

University of California Berkeley

NOTES

I The fact that I am describing here a conflict between RS theories and FS theories does not mean that I assume all theories of meaning to be of one or the other of the two kinds. The verificationist theory of the Vienna Circle is an example of a totally different sort of theory. 2 I introduce the term "refers*" here in order to obviate possible objections from FS theories that the newly defined term is not the same as the binary term used in RS theories. It is sufficient for my purpose to show that RS theories can be modelled in FS theories. 3 I make the Fregean assumptions here that all expressions are either names or functional expressions and that either kind can have reference. If we allow functional terms to refer to partially defined functions (Le., functions that lack values for certain possible arguments) we can account for the fact that expressions can have semantic content but no reference. 4 Michael Dummett writes that "to give an account of the sense of an expression is ... to give a partial account of what a speaker knows when he understands that expression." ('Frege's Distinction of Sense and Reference' in Truth and Other Enigmas, Cambridge, Mass.: Harvard University Press, 1978, p. 122.) I know, in fact, of no better arguments against taking the notion of sense as a semantic notion than the ones Dummett produces in this essay in support of that idea.

Arguments for taking the notion of sense as a cognitive notion are also provided by Tyler Burge, 'Sinning Against Frege,' Philosophical Review, vol. 88, 1979, pp. 398-432. While I agree with much of what Burge says, I do not quite agree with him that Frege never intended to give a semantic theory in terms of the notion of sense. It was just that his reflections drove him to amalgamate semantic and epistemi<; questions. 5 The best account of this is given in Dagfinn F0llesdal, 'Husser!'s Notion of Noema,' in B. Dreyfus, ed., Husserl, Intentionality, and Cognitive Science, Cambridge, Mass: MIT Press, 1982, pp. 73-80. 6 John Searle, Intentionality, Cambridge University Press, forthcoming. 7 Michael Dummett, 'Can Analytical Philosophy Be Systematic and Ought it to Be?' in Truth and Other Enigmas, pp. 437-458; as well as in many other places, Vol. 65, 1964-65.

64 HANS SLUGA

8 Hermann Lotze, Logic, trans!. B. Bosanquet, Oxford: Clarendon, 1884. All refer-ences will be to this edition. I try to justify the claim that Frege was influenced by Lotze in Hans Sluga, Gottlob Frege, London: Routledge & Kegan Paul, 1980. 9 It is unfortunate that my term "semantic content" is so close to Lotze's (and Frege's) term "content," when the terms mean quite different things. I retain my use here for the sake of terminological continuity with an earlier paper discussing these issues. Cf. Hans Sluga, 'On Sense,' Proc. of the Arist. Soc., vo!., 651964-65, pp. 25-44. 10 Lotze's account has here obvious similarities to that given recently in Saul Kripke's Naming and Necessity. 11 Gottlob Frege, Begriffsschrift, in J. v. Heijenoort, ed., From Frege to Godel, Cam-bridge, Mass.: Harvard University Press, 1967, p. 20. All further references will be to this edition. 12 Peter Geach and Max Black, Translations from the Philosophical Writings of Gottlob Frege, Oxford: Blackwell, 1977, p. 56. All references to the essay 'On Sense and Reference' will be to this edition. 13 References will be to the Geach-Black volume. 14 G. Frege, The Basic Laws of Arithmetic, ed., M. Furth, Berkeley/Los Angeles: University of California Press, 1967, p. 105. Notation slightly altered. 15 M. Dummett, The Interpretation of Frege's Philosophy, London: Duckworth, 1981, p. 532. 16 Furth gives the former translation (Basic Laws, p. 36); Geach and Black give the latter (Translations, p. 154). 17 Furth is aware of the problem of how to translate "gleichbedeutend" and renders it here as "having the same meaning." He adds in a footnote: "In view of the previous sentence, it seems best to translate this in the manner of the ordinary German gleich bedeuten and not to restrict it to Frege's technical use." He does not, however, see the relevance of this point to the earlier passage. 18 Michael Dummett, 'Frege's Philosophy,' in Truth and Other Enigmas, p. 107.

THOMAS G. RICKETTS

OBJECTIVITY AND OBJECTHOOD:

FREGE'S METAPHYSICS OF JUDGMENT*

The question famed of old, by which logicians were supposed to be driven into a comer ... is the ques-tion: What is truth?

To know what questions may reasonably be asked is already a great and necessary proof of sagacity and insight. For if a question is absurd in itself and calls for an answer where none is required, it not only brings shame on the propounder of the question, but may betray an incautious listener into absurd answers, thus presenting, as the ancients said, the ludicrous spectacle of one man milking a he-goat and the other holding a sieve underneath.

Critique of Pure Reason A 58 = B 82

The first of three fundamental principles Frege enunciates at the beginning of The Foundations of Arithmetic bids us "always to separate sharply the psychological from the logical, the subjective from the objective."! As commonly understood, this principle represents little more than Frege's insistence on the distinction between mind-indepen-dent objects and mind-dependent states, and so expresses his rejection of subjective idealism. Such an ontological construal of the objective-subjective distinction in its turn supports a very common reading of Frege according to which he is the archetypical metaphysical platonist. The mind-independent existence of things is for Frege a presupposition of the representational operation of language: it explains how our state-ments are determinately true or false apart from our ability to make or understand them. On this reading, Frege's novelty lies in his theory of language, a theory that offers an impressively general and precise account as to how the truth-value of a sentence is determined by the reference of its well formed parts. Frege is thus canonized the father of formal semantics. Furthermore, application of this theory to particular stretches of discourse enables us to uncover the ontological presupposi-tions of the discourse. Frege, in The Foundations of Arithmetic, provides the paradigm for philosophical application of formal semantics in

65

L. Haaparanta and 1. Hintikka (eds.), Frege Synthesized, 65-95. © 1986 by D. Reidel Publishing Company.

66 THOMAS G. RICKETTS

arguing that the objective truth of the statements of pure and applied arithmetic requires the mind-independent existence of numbers.

The crucial feature of this line of interpretation is its taking ontologi-cal notions, especially that of an independently existing thing, as prior to and available apart from logcal ones, from notions of judgment, assertion, inference, and truth. The explanatory priority of ontological notions renders intelligible and inevitable the questions, "How does language hook on to reality?" and "How do we know that the ontologi-cal presuppositions of our discourse are satisfied?" The contemporary consensus is that such answers to these questions as can be extracted from Frege are patently inadequate. Frege's most promising response might appear to lie with his doctrine that the sense expressed by a word determine to what, if anything, the word refers. But a moment's thought shows that Frege's doctrine, whatever other uses and motivations it has, only splits our original question into two equally intractable ones: how do words become associated with senses, and how do the senses, our words express determine their referents? Reading Frege like this, we read much subsequent philosophy of language as attempts to bridge the gulf between language and the world that now looms so starkly. Russell's logical atomism resting on direct acquaintance with sensory items was an early attempt to give semantics an epistemological foundation. Causal theories of reference are more recent attempts to do the same.

There is another philosophically more interesting and historically more apt construal of Frege's work, one which denies to ontological notions the independence and primacy they have on the platonist interpretation. As I read Frege, ontological categories are wholly super-venient on logical ones. This supervenience is the product of the fundamental status Frege assigns to judgment. That judgment should be the starting point for Frege's philosophy is unsurprising, given his animus toward the naturalism and empiricism prevalent in mid-nine-teenth century German philosophy and his corresponding sympathy with Kant and Leibniz.2 The priority of judgment is to guarantee its objectivity, as exhibited in the linguistic practice of assertion, against any general challenge. Thus, it is meant to render unintelligible the chasm between thought and reality that is the consequence of the platonist reading.

This is not to say that Frege's metaphysics of judgment is entirely coherent. The priority of judgment shapes Frege's conception of logic and motivates the identification of his Begriffsschrift as logic. There are,

OBJECTIVITY AND OBJECTHOOD 67

however, deep tensions in Frege's thought that arise from the failure of Frege's logical doctrines, harnessed as they are to the execution of his logicist program, to be true to the conception of judgment that moti-vates them. These tensions can even be said to open up a gap between language and reality, though one whose character is rather different from that just described. Moreover, I hold that Wittgenstein in the Tractatus exploits these tensions to motivate the distinctive logical doctrines of that work. By systematically criticizing Frege's philosophy, Wittgenstein, unfettered by Frege's constructive interests, attempts to rework a Fregean conceptiori of judgment into a complete and coherent account of logic, language, and the world. We shall not, however, be in a position to identify the difficulties in Frege's thought or to assess their influence on subsequent philosophy without understanding the sources of Frege's notions and doctrines in judgment. It will emerge that anything like formal semantics, as it has come to be understood in the light of Tarski's work on truth, is utterly foreign to Frege.

Frege's conception of judgment is best approached through an exami-nation of his rejection of psychologism. Frege's extensive polemic is more than the deserved dismissal of a nest of scientifically crude and philosophically confused views. By scrutinizing the terms of criticism he brings to bear against psychologism, we shall see that the contrast between objective and subjective is not an ontological one. Understand-ing the character of this contrast will prove to be the first and decisive step toward appreciating the primacy of judgment in Frege's philosophy. Indeed, it is in this primacy that Frege himself locates his break with traditional logic:

For in Aristotle, as in Boole, the logically primitive activity is the formation of concepts by abstraction .... As opposed to this, I start from judgments and their contents, and not from concepts.3

Psychologism maintains that logic is a branch of psychology: accord-ingly, logical laws are generalizatiqns about the inferential operations of the human mind. Frege, of course, has no quarrel with the empirical study of human cognitive faculties, but he takes the psychologistic logician's denial that there are non-psychological laws of logic to mani-fest a confusion of the objective and the subjective. For this denial, we


shall see, forces an assimilation of assertions to the ventings of inner states, to cries of pain and shouts of joy. This contrast is, I hold, the source of Frege's conception of objectivity and logic.

To see how psychologism involves confusing the subjective and the objective, let us consider a line of argument extractable from the intro-duction of The Basic Laws of Arithmetic. The psychological logician who maintains that the laws of logic are empirically established general-izations is committed to the intelligibility of encountering and recogniz-ing beings whose own thought is governed by laws different from those that describe our own. So, to use Frege's example, we might encounter beings who do not accept the principle of identity, who sometimes or always deny the statements of the form "t = t" that we unhesitatingly affirm. The psychological theory describing our inferential habits says, as Frege puts it, "It is impossible for people living in the year 1893 to acknowledge an object to be different from itself."4 The psychological theory describing the logical aliens denies this of them. There is, of course, no inconsistency between the two theories, as the claims of each are suitably circumscribed so as to apply only to the appropriate population. The possibility of logical aliens raises the question as to which inferences are correct, ours or theirs. This seems to be an extra-psychological question; for neither the psychologist's description of our inferences nor his description of the aliens' addresses the question of the self-identity of every object. Were the psychological logician to admit the legitimacy of this question, he would concede the existence of a nonpsychological study of inference. This concession would be lethal to his position.

The psychologistic logician can turn aside this question only by identifying the content of the principle of identity with that of the psychological law that asserts the universal acceptance, by us, of the principle. This identification is ill founded because, as Frege says:

There is no contradiction in something's being true which everyone takes to be false .... If it is true that I am writing this in my chamber on the 13th of July, 1893, while the wind howls out-of-doors, then it remains true even if all men should subsequently take it to be false.5

We should not, however, take this observation - one the psychologist would find question-begging - as the resting point for Frege's criticism. The imagined encounter with the logical aliens serves rather to highlight the special role logic plays in discourse, a role belied by the psycholo-gistic account.


What makes radical disagreement over logic so peculiar is that the principles of logic provide us with a shared background, context, or framework that enables us, first, to recognize disagreement, and second, to arbitrate the consequent debate. Once two of us recognize that we have made contradictory judgments, we each put forth reasons for our opposed claims. Application of the principles of logic enables us to measure the relevance of these further claims to the original dispute. It also enables us to locate the source of our controversy within the bodies of belief we each bring to bear on the dispute. Once we have done this, characteristically, we will be in a position to use the basic laws and methods of the particular subject matter under consideration to resolve the difference. So, when we imagine logical aliens, we are imagining beings with whom we cannot reason. Small wonder that Frege remarks that to encounter logical aliens would be to discover a new type of insanity and warns that the psychologistic logician, in both adhering to the standards of consistency logic provides but refusing to reject the aliens' thought as contradictory, is attempting to jump out of his own skin.6

The psycho logistic logician, if he is to maintain his monopoly on the study of logic, must adopt a more tolerant attitude toward the aliens. But how does the psychologist resist Frege's suggestion that he ought to reject the aliens' thOUght as contradictory while investigating the causal antecedents of their malady? The tolerance of the psycholo-gist stems from his identification of logical principles with psychological laws. From the vantage point of his theory, our appeal to the principle of identity in rejecting the aliens' claims as contradictory amounts to no more than an insistence that we think like this - a claim whose content is dispassionately spelled out in the causal laws describing our thought. This identification reveals that the psycho logistic logician is treating disagreement in judgment as merely a species of psychological difference. As such, the recognition of disagreement no more raises an issue of correctness than does the acknowledgment of any other personal idiosyncracy. Disagreement receives special treatment only in that if I have judged p to be true and you have judged p to be false, then, according to the laws describing our thinking, neither of us can adopt the judgment of the other without relinquishing his previous one.

It is this treatment of disagreement that reveals the psychologistic logician's confusion of the objective and the subjective. Consider three diners who have just shared a sumptuous repast. Two sigh with satiated


contentment and one groans with distended discomfort. The groan in no way contradicts the sighs. It merely gives public expression to a state that is causally incompatible with the the sighs express. The incompatibility by itself raises no issue among our diners. One may accuse another of shamming a feeling, but such an accusation does not contradict the supposedly dishonest expression. Nor do the two diners who sigh agree with each other. Each diner expresses his own feeling; and even if the feelings are exactly similar, nonetheless, each in sighing gives vent to his own. This last observation is just the reverse side of the preceding one: the possibilities of agreement and disagreement go hand in hand. If I cannot contradict your cry of pain, neither can I affirm it.

The psycho logistic treatment of judgment and assertion is perfectly parallel to the preceding description of inner states and their expression. Judgments are assimilated to inner states and assertions to the ventings that manifest them. The price of this identification is the conflation of contradiction with causal incompatibility. This conflation underlies the psychologist's refusal to see any issue between the logical aliens and us. But it also forces on us a falsified conception of judgment, one whose misconstrual of the possibilities of agreement and contradiction im-manent in discourse leads to a sort of solipsism of judgment. Frege rejects this solipsism in a memorable passage:

If every man designated something different by the name "moon", namely one of his own ideas, much as he expresses his own pain by the cry "Ouch", then of course the psychological point of view would be justified; but an argument about the properties of the moon would be pointless .... If we could not grasp anything but what was within our own selves, then a conflict of opinions [based on] mutual understanding would be lacking .... There would be no logic to be appointed arbiter in the conflict of opinions?

On the interpretation I have been presenting, the absurdity Frege is here evoking is not the picture of the self trapped within the veil of its own ideas unable to claw its way through to epistemic contact with the material world. The solipsism Frege has in mind is one of communica-tion, in which it is impossible for people to contradict each other and hence impossible for them to agree either - the solipsism of refined society in which every assertion is met with the demurral, "That's your opinion." Frege thus rejects the psychologistic account of judgment for its conflation of assertions with the ventings of inner states. These two acts are entirely different, as the impossibility of contradicting another's groan or affirming another's sigh shows.


It should be noted in passing that Frege thinks that the psychologist's confusion is natural enough; for the vast portion of our uses of language serve both to manifest judgments in assertion and also to give vent to feelings, emotions, and ideas. Frege's tirelessly repeated admonition to distinguish the ideas or images associated with a word or sentence from its content, its sense and meaning, is to re-enforce our appreciation of the radical difference between these two functions of language.8 He goes so far as to urge that logicians master several foreign languages to assist them in readily applying this distinction:

It is true .hat we can express the same thought in different languages, but the psycho-logical trappings, the clothing of the thought, will often be different. That is why the learning of foreign languages is useful for one's logical education. Seeing that the same thought can be worded in different ways, we learn better to distinguish the verbal husk from the kernel with which in any given language, it appears to be organically bound Up.9

I have maintained that Frege's distinction between the subjective and the objective lodges in the contrast between asserting something and giving vent to a feeling. We now need to consider how this contrast gives rise to Frege's conception of objectivity. Let us begin with an examination of the interconnections Frege draws among a raft of notions - assertion, judgment, content of judgment or thought, understanding, and inference. None of these notions can be understood apart from the others, and it is by attention to language and our linguistic practices that these notions are to be collectively elucidated. For example, Frege's characterization of assertion as the manifestation of a judgment does lend a certain priority to the latter notion. The sentences we utter when we make assertions, considered as series of sounds, have no communicative powers in their own right. Nevertheless, it would be a mistake to think that we have any understanding of what an act of judgment is apart from the given by the formula that judgments are what assertions manifest.

Frege's characterization of judgment as the recognition of the truth of a thought gives a structure of judgment. Let us begin with the distinction between the act, jUdging or recognizing to be true, and the object or content of the act. Frege introduces this distinction by observing that we need to distinguish between understanding or grasping a thought and recognizing it either to be true or to be false. The precedent for this distinction lies in the intimate relation between yes-no questions and the assertions that answer them. Frege describes this relation by saying that


an assertion that affirmatively answers a yes-no question manifests the recognition of the truth of the thought that the question puts forward for consideration. To distinguish between understanding or grasping a thought and recognizing its truth is to acknowledge that we can genuinely ask a question without being in a position then and there to answer it, even if we know what sorts of reasoning and investigations would answer it.

This way of thinking about judgment is bound up with Frege's insist-ence that several people can all grasp the same thought and judge it true or false. In this respect judging differs from feeling: no one can have my pain. So Frege says things like the following:

Thus I can also acknowledge thoughts as independent of me. Other men can grasp them just as much as I: I can acknowledge a science in which many can be engaged in research. We are not owners of thoughts as we are owners of our ideas.! 0

Passages like this should not be taken to fund the platonist reading of Frege. The platonist interpretation supposes the identification of thoughts as abstract mind-independent objects to give an ontological foundation to the distinction between subjective and objective. Frege's language for talking about judgment is rather a means for systematically redescribing selected features of our linguistic practices, those which elucidate the various aspects of Frege's conception of objectivity. I talk here of redescription to preclude the idea that Frege is presenting the ontological underpinning that secures the objectivity of our judgments. From the perspective Frege acquires in starting from judgments and their contents, the distinction between objective and SUbjective exhibited in our linguistic practice needs no securing and admits of no deeper explanation. We have seen how the distinction between the act and content of judgment is enforced by our appreciation of the relation between questions and their answers. Talk of several people grasping the same thought just restates the possibility of agreement that Frege takes to be intrinsic to assertion.

We have considered the genesis of Frege's distinction between the act and content of judgment. Let us now focus on the act, in particular on die significance of the phrase "recognize the truth of". Frege remarks, "A propositional [yes-no] question contains a demand that we should either acknowledge the truth of a thought or reject it as false."!! To grasp a thought is then to be faced with the question of whether it is to be affirmed or denied; or, as Frege will say in order to avoid the


unnecessary introduction of two parallel species of judging, it is to be faced with the question whether the thought or its opposite, its negation, is to be recognized as a truth. To talk of opposition here is to observe that the same thought may not both be affirmed and denied; one and only one of a thought and its opposite may be recognized to be a truth. This observation should be distinguished from the claim that a person can not affirm and deny the same thought, that it is psychologically impossible to do so. This second claim is an empirical claim about our mental constitution concerning which Frege is agnostic. Frege distin-guishes between these two claims by locating our appreciation of the impermissibility, the incorrectness, of affirming outright contraditions in our understanding, and hence the incorrectness itself in the opposition between the thoughts grasped. On this picture, the contents of judgment impose standards on our acts of judging. These "standards", unlike the sort of standards that apply, say, to gymnastic performances, are in-escapably applicable to any judgment; we cannot opt out of them and still take ourselves to be making judgments. This point will receive closer examination in Section II.

Frege takes his stand against psychologism by insisting on the trichotomy of agreement, disagreement, and mere difference in judg-ment. We noted how the possibilites of agreement, disagreement, and mere difference in judgment come together. We now see the distinction between the act and the content of judgment together with Frege's talk of thoughts and their opposites as providing a structured way of marking out this distinguishing feature of judgment.

There is more to our understanding of a thought than just the appreciation that it and its opposite may not both be affirmed. A person who understands,. for instance, both the thought that every philosopher is wise and the thought that Socrates is a philosopher but not wise, realizes that these two thoughts are contradictory. Here though, the second thought is not the opposite of the first; nothing in our grasp of them precludes their joint denial. I have been speaking of our appre-ciation of patent inconsistencies. The other side of this appreciation is our awareness of elementary implications, of the basis that the recognition of the truth of one thought provides for further judments. Frege's primary given is this awareness of obvious implications and contradictions - I speak loosely and nontechnically here. It is this awareness that enables us to discern agreement and disagreement, 10 reason together, and so, in the fullest sense, to communicate.


It is in this context that Frege distinguishes causal explanation of person's beliefs from justification of those beliefs, and so separates logic and psychology. According to Frege, there are two ways in which a judgment may be justified: "We justify a judgment either by going back to truths that have been recognized already or without recourse to other judgments. Only the first case, inference, is the concern of Logic."12 Characteristically, to justify an assertion, one presents reasons for the assertion: that is, one puts forward further assertions from which the given assertion may be inferred. Such a procedure, of course, may only push the question of justification back a step, to the justification of the premises. Eventually, we arrive at primitive truths whose assertion is not inferentially justified. These will presumably include the basic laws of the discipline within whose purview the original assertion lies and, in the case of empirical claims, observationally established assertions concern-ing the properties and relations of material objects. Frege has almost nothing to say about ·noninferential justification, His interest lies exclu-sively with the first species of justification, with inference.

II

While we have the capacity to recognize elementary implicational relations and with it the capacity to construct chains of reasoning that justify various judgments, our exercise of this last ability is often haphazard. The problem Frege sees with our argumentative practice lies not so much in our unwitting fallacies as with enthymatic reasoning. Often the truth of a conclusion drawn from an explicity stated premise depends on a body of unstated assumptions. Because even our explicit reasoning leaves so much unspoken, what passes for a demonstration of a claim may leave us in doubt as to the judgments that justify the claim and hence in doubt as to the ultimate epistemic status of the claim. Frege is particularly exercised by this carelessness on account of his desire to maintain against Kant that pure intuition plays no role in mathematical reasoning. Frege speaks of " ... chains of deductions with no link missing such that no step in it is taken which does not conform to some one of a small number of principles of inference recognized as purely logical," and goes on to complain:

To this day, scarcely one single proof has ever been conducted on these lines; the mathematician rests content if every transition to a fresh judgment is self-evidently correct without inquiring into the nature of this self-evidence, whether it is logical or intuitive.13


Logic must aim to state general principles whose application enables us to determine if one statement, by itself, implies another, and so forces us to identify all the premises of any proof. Application of the principles of logic will then enforce on us the standards of explicitness, clarity, and rigor that are the prerequisites for rational communication.

In order to play this role, the principles of logic must be applicable to any topic whatsoever. This universal applicability of the principles of logic constrains the form the statement of these principles may take. The principles must not draw on the terms of any special science in such a way as to preclude the application of logic to other subjects. Logic cannot lie in the purview of any other subject. Nor does logic have a restricted subject matter all its own.

Frege says, "Just as 'beautiful' points the way for aesthetics and 'good' for ethics, so do words like 'true' for logiC."14 It might seem from this remark that just as ethics provides us with a compilation of principles by which to assess the moral goodness of actions, so logic enables us to assess the truth or falsity of thoughts. Frege quickly rejects this sugges�tion, noting in the next sentence:

All sciences have truth as their goal; but logic is also concerned with truth in a quite different way. Logic has much the same relation to truth as physics has to weight or heat. To discover truths is the task of all sciences; it falls to logic to discoverer the laws of truth.

Frege's negative point is clear enough. Every special science is con�cerned with truth insofar as it aims to state those general principles which enable us to infer the truth or falsehood of further claims that lie within the province of the discipline. How then are we to understand logic's special and defining interest in truth?

It is tempting to read the previously quoted passage from "Thoughts" to be suggesting that the laws of logic are about truth in much the same way that the laws of physics are about weight and heat. The logician as well as the physicist aims the state truths; however, the statement of the logician's truths, unlike the physicist's, requires the use of a truth predicate. This interpretation seems confirmed by the examples of provisional statements of principles of inference that Frege provides in 'Compound Thoughts' for instance:

(If B, then A) is true; Bistrue;tl1erefore Ais true.15


It looks then as if the principles of logic take the form of the identifica-tion of elementary valid inference patterns. These principles may be conceived as asserting that any statement or thought of some given form is true. From such a principle and the identification of some particular thought as having the requisite form, the attribution of truth to that thought follows. Then by use of an instance of a quasi-disquotational paradigm like that enshrined in Tarski's Convention T, the f>tatement itself follows. Thus, from the principles of logic together with the auxiliary premises just mentioned, follow claims couched in the vocabu-lary of the special sciences. The universal applicability of logic to the provision and evaluation of argumentation in any discipline is thus secured. The principles of logic, so conceived, generalize over the forms of statements as regards their truth; this character is rendered patent by the use of schematic letters and a truth predicate in their statement. Frege's conception of the form of logical laws then appears close kin to Quine's. 16

This impression, however, is mistaken. For Frege as well as early Russell, the generality of logic is substantive, not schematic. l ? On Frege and Russell's view, the basic laws of logic generalize over every thing and every property. These laws do not mention this or that thing; nor do they generalize over things with respect to various properties or rela-tions, at least no ordinary property - for example, being mortal - that belongs to the subject matter of a special science. Any letters that appear in the statement of logical laws must be understood as variables, not as schematic letters. Moreover, the quantifiers that appear in these laws, and in any conclusion drawn with their aid, are understood to be unrestricted over entities of the appropriate logical type. This view of logic contrasts sharply with the more contemporary one just canvassed. That conception depends on a notion of a logical schema subject to varying interpretations; logical laws are thus applicable to a range of languages without regard to their (nonlogical) vocabulary or the range of their quantifiers. But the notion of a logical schema that admits of multiple interpretations is foreign to Frege's thought.

Nor is it possible, through reasonable emendations, to read the contemporary view back into Frege.18 For the contemporary view requires the ineliminable use of a truth predicate. Such a use is anti-thetical to Frege's conception of judgment. This conception of judgment precludes any serious metalogical perspective and hence anything properly labeled a semantic theory.


Frege, we have seen, draws our attention to the distinguishing feat]Jres of judgment by use of the slogan that judgment is the recogni-tion of the truth of a thought. He denies, however, that such remarks constitute an informative definition of judgment.!9 If we now take truth to be a property of thoughts, then we must take the slogan to be offering just such a definition. So understood, it tells us that judgment is a special kind of recognitional capacity. We have, for example, the ability, under suitable circumstances, to identify the colors of things. Judgment now seems to be the capacity to recognize that some thing (a thought) has some property (truth). But this is absurd. Judgment is not itself a special recognitional capacity, a species of some genus. Judgment is itself the genus - to recognize anything to have a property is ipso facto to make a judgment.2o Nor can judgment be characterized by comparing it with our particular recognitional capacities. As these capacities are capacities to recognize the truth or falsehood of some range of thoughts, they do not provide an independently understandable model through which judgment is elucidated. How, though, does taking truth to be a property of thoughts force on us this confusion of genus and species?

The answer to this question emerges from a consideration of Frege's curious objection to the correspondence theory of truth?! The corre-spondence theorist treats truth as a property of thoughts and goes on to offer a definition of this property in terms of some favored relation between the contents of judgment and the world. Let us use the phrase 'corresponds with Reality' as a placeholder for whatever definition the correspondence theorist offers. If truth is a matter of correspondence with Reality, then in order, for example, to determine whether (it is true that) Socrates is mortal, it is necessary to inquire whether the thought that Socrates is mortal corresponds with Reality. But once again, to determine whether (it is true that) the thought that Socrates is mortal corresponds with Reality, one should inquire whether the thought that the thought that Socrates is mortal corresponds with Reality itself corresponds with Reality. And so on. A parallel regress arises for judging as well as inquiry.

This brief argument is unsatisfying. It is obvious neither that the regress it turns on is unavoidable, nor that the regress, even if inevitable, is vicious. Frege's objection to the correspondence theory is an all too casual articulation of the fundamental status he assigns to judgment. It is cogent only in the context of the conception of judgment outlined in the previous section. Moreover, in that context, the availability of the


objection illuminates the content of this status. Let us then consider the genesis of the regress more closely.

Why, on the correspondence theory, must one inquire whether the thought that Socrates is mortal corresponds with Reality in order to determine whether Socrates is mortal? The connection between judg-ment and truth is not casual. We saw in the previous section how our appreciation of the trichotomy of agreement, contradiction, and mere difference in assertion gives us a conception of standards of correctness imposed by the contents of judgment on our acts of judging: a thought is either to be affirmed or to be denied; it is either true or false. We cannot take someone to be making assertions in complete disregard of the correctness of what he asserts; such a person would be understood to be play-acting or perhaps merely mouthing words. To take truth to be definable forces a particular construal on this talk of standards of correctness. For to have a definition of truth is to have a general description of the conditions that have to be satisfied for a judgment to be correct. So, if truth is definable, then any person who makes a judgment must have ascertained, or taken himself to have ascertained, whether these conditions, applied to the thought under consideration, are satisfied. If I judge that Socrates is mortal, then I must have deter-mined that the thought that Socrates is mortal corresponds with Reality. But to ascertain that some condition holds is to make a judgment. So the regress begins. Given the definition of truth, a person cannot have judged that the thought that Socrates is mortal corresponds with Reality unless he has judged that the content of this second judgment corre-sponds with Reality. On the correspondence theory then, a person is never in a position to make a judgment; for no one is ever in a position to have satisfied, or even think of himself as having satisfied, the standards for judgment that would ipso facto be provided by its definition of truth.

Frege concludes his argument against the correspondence theory with the somewhat hesitant statement, "So it seems likely that the content [Inhalt] of the word 'true' is sui generis and indefinable."22 Frege's talk of indefinability in this context should be understood differently from his other uses of this notion. Elsewhere Frege talks of one concept's being definable in terms of other prior concepts. Through such definitions the grounds for one branch of science may be seen to lie in some other body of knowledge. Frege's definition of the concept of number is an example of this sort of definition. Correlative to this talk


of definition, Frege speaks iIi a number of places of primitive concepts, concepts that admit of no definition in any more basic terms.23 Truth, I suggest, cannot be understood to be a primitive unanalyzable property of thoughts; for this position conflicts with the argument just examined. We observed that the argument does not turn on the details of the correspondence theorist's definition; indeed, Frege himself generalizes the argument to any proposed definition. In fact, nothing in the argu�ment turns on taking truth to be definable in terms of any other concepts. The vitiating regress is generated by the assumption that truth is a property of thoughts. The proper conclusion to the argument is not that truth is a primitive property, but that truth is not a property at all. Frege cautiously voices this conclusion two pages later in 'Thoughts'. Having observed that the statement 'I smell the scent of violets' has the same content as the statement 'It is true that I smell the scent of violets.' Frege goes on to say:

So it seems then that nothing is added to the thought by my ascribing to it the property of truth. . . . May we not be dealing here with something which cannot be called a property in the ordinary sense at all? In spite of this doubt I will begin by expressing myself in accordance with ordinary usage, as if it were a property, until some more appropriate way of speaking is found.24

Returning now to the topic that sparked this discussion of truth, let us examine how the strong conclusion I draw from Frege's objection to the correspondence theory motivates his identification of logical principles with maximally general truths.

It is an upshot of the regress argument that the standards of correct�ness we think of our judgments as satisfying cannot be thought of as a set of general conditions on the contents of judgment. The conception of judgment in which this argument is embedded does not permit any real metaperspective. All of this is not, however, to say that the standards of correctness are ineffable. In denying that logical principles have a normative status that distinguishes them from the principles of physics or geometry, Frege observes, "Any law asserting what is can be con�ceived as prescribing that one ought to think in accordance with it."25 To think in accordance with a law is not just to affirm the law; it is also to refrain from affirming any thought that contradicts the law. As we observed in Section I, the other side of this prohibition is the license that these general laws give us for making further judgments on the basis of them. Frege thinks that the content of any developed discipline is


constituted by a body of basic laws, laws which are basic in that they cannot be justified by appeal to other laws couched in the vocabulary of the discipline that admit of independent justification. These laws thus present standards for making judgments in that discipline. They consti�tute a framework to which other judgments must conform on pain of contradiction. Frege's conception of judgment does then admit of general standards of correctness, but the generality of these standards does not involve any metaperspective. The general standards for the judgments of a discipline are not provided by statements about the discipline. They are provided by judgments within the discipline.

We are now in a position to appreciate the identification of the laws of logic with maximally general truths. Maximally general truths are truths that do not mention any particular thing or any particular property; they are truths whose statement does not require the use of vocabulary belonging to any special science. Thus, in the same way that the basic laws of chemistry give standards for judgments in chemistry, so maximally general truths give us standards for judgment in any subject matter. No judgment about any individual thing and no judg�ment that generalizes over individuals with respect to a particular property may contradict a maximally general truth. Frege clearly articu�lates this conception of logic in an unpublished writing:

How must I think in order to reach the goal, truth? We expect logic to give us the answer to this question, but we do not demand of it that it should go into what is peculiar to each branch of knowledge and its subject matter. On the contrary, the task we assign logic is only that of saying what holds with utmost generality for all thinking whatever its subject .... Consequently we can also say: logic is the science of the most general laws of truth.26

To say that the laws of logic are the most general laws of truth is to say that they are the most general truths. The laws of physics are the laws of truth about mass, motion, heat, and light. In mastering these laws we come to understand physical truth. The laws of logic generalize over every thing and every property. It is then in studying logic that we learn whatever there is to be learned about truth simpliciter. As Frege puts the point, "The meaning [Bedeutung] of the word 'true' is spelled out in the laws of truth."Z7

If we thus identify the principles of logic as substantive laws in their own right and so fundamentally similar to the laws of the special


sciences, how are we to understand the applicability of these principle:: in providing gap-free proofs to supplant commonplace enthymatic reasoning? Frege's basic picture is clear enough. Suppose on the basis of having judged p, I wish to prove q. In the simplest case, I might begin by finding a logical truth whose variables may be instantiated by constants in such a way as to yield the conditional whose antecedent is p and whose consequent is q. The proof then begins with an assertion of this law. On the basis of this assertion, the desired conditional is obtained. Then p is asserted. On the basis of the assertion of the conditional and the assertion of p, q is asserted. And characteristically, the logical law with which the proof begins will be inferred from other simpler, more perspicuous general truths. There is, as far as Frege is concerned, nothing to be said about the justification for our recognition of those basic laws of logic to be truths:

The question why and with what right we acknowledge a law of logic to be true, logic can answer only by reducing it to another law of logic. Where that is not possible, logic can give no answer.2M

Moreover, the maximal generality of these laws precludes their infer-ence on the basis of the truths of any other discipline.

The universal applicability of the laws of logic is then secured by their generality. This conception of generality is original to Frege and is the product of his construal of the quantifier-variable notation of his begriffsschrift. As traditionally conceived, logic is concerned with the form rather than the content of judgment. A sharp distinction between the form· and content of judgment was to explicate the sense in which logic abstracted from the content of the claims of the special sciences. So, for example, Kant remarks:

That logic should have been thus successful is an advantage which it owes entirely to its limitations, whereby it is justified in abstracting ... from all objects of knowledge and their differences, leaving the understanding nothing to deal with save itself and its form.29

Such talk of form insinuates that logic is somehow concerned with judging itself, and this suggestion in turn can serve as the thin edge of the wedge of psychologism. The conception of generality that Frege's quantifier-variable notation makes available transforms the old distinc-tion between form and content. The claims logic makes still "abstract"


from the differences that distinguish the content of the claims of the various sciences, but this abstraction does not give logic a special subject matter of its own - the forms of judgment. Nor does this

rob the claims of logic of content, as though they concerned merely the "empty forms" of judgment. Frege's variables enable him to take his analogues of the laws of traditional logic to be making substan-tive claims in their own right, claims that are, nevertheless, applicable to reasoning in the special sciences on account of the relation between generalizations and their instances.

So far, the principles of logic have been identified with the basic laws of logic. These laws get applied in the provision of proofs only through our discerning implicational relations between logical laws and their instances, and among the claims of the special sciences. Or to speak in a more Fregean vein, the application of the laws of logic in proofs depends on our making one assertion on the basis of another. If these proofs are to be taken to be gap-free with all the premises necessary for the proof of the conclusion explicity stated, then we will need to have isolated a small number of principles of inference, asserting one thing on the basis of another only when the conditions laid down in these prin-ciples are satisfied. These principles of inference are then as necessary as the basic laws of logic for the replacement of everyday argumentation with proofs. Indeed, seeing how the basic laws of logic are statements like any other that get used in proofs by our asserting further claims on the basis of them, it appears that the inference rules rather than the logical laws more properly deserve the title "principles of logic". For it is the inference rules that deal with what is the defining concern of logic -the assertion of one thought on the basis of another.

Frege's statement of rules of inference follows the pattern exhibited by the following formulation of modus ponens:

From a conditional and the antecedent of that conditional, the consequent of th§ conditional may be inferred.30

What licences this permission ? We recognize the inference rule to be valid; and to talk of recognizing the validity of an inference appears to making a judgment. Our question then is: what thought do we affirm when we recognize this inference rule to be valid and what basis, if any, is there for this judgment?

It is tempting, at first blush, to identify the apprehension of the validity of the inference rule with the recognition of the truth of the


corresponding logical law. In the case of MP this law reads something like:

If P and also if p then q, then q.3!

However, this identification obviates the distinction between logical laws and rules of inference. Frege is clear on the difference here between asserting a conditional and asserting one thing on the basis of a previous assertion.32 Indeed, this difference is another of the central linguistic precedents lying behind Frege's distinguishing the content of judgment from the act of judging. If I assert q on the basis of having asserted p and you deny q, you contradict me. You do not contradict me, if you deny q in the face of my assertion of the conditional 'If p, then q'. Moreover, were use of an inference rule to be justified by the judgment of a general law, we would encounter the vicious regress in the provision of proofs that Lewis Carroll pointed out. For then, in order to make a proof complete, any use of an inference rule would have to be accompanied by an assertion of a corresponding logical law. Only in this way would all the premises on whose correctness the conclusion depends be explicitly stated. But this added statement creates the need for further inferences, each of which would need to be similarly accompanied by assertion of justifying laws. This regress would make completed proofs impossible.

At this point, recourse to a metaperspective seems inescapable. On the contemporary conception of logic, the acceptance of MP as a correct rule of inference is vouchsafed by our metalogical judgment that if a conditional is true and its antecedent is true, then so is the consequent. This judgment is not supposed to appear as a premise in every proof employing MP; it rather is a part of our reason for accepting derivations in some given formal system as proofs. But a statement of the validity of MP unavoidably involves taking truth to be flatly a property of thoughts. This treatment of truth is precluded by Frege's conception of judgment. The proper conclusion for him is that our apprehension of the validity of MP is not a judgment. It is not mani�fested in any single assertion and so is, in this important sense, ineffable. TID" apprehension is, however, manifested linguistically in the inference we make and accept.

Once truth is excluded as a property, Frege has no nonsyntactical metalogical vocabulary. It is then misleading to have talked earlier of recognizing a thought to be contradictory, or of recognizing one thought


to imply another. The recognition of a thought to be contradictory is not a judgment to the effect that a thought has some property. It is rather an apprehension manifested by the refusal to affirm the thought while affirming the opposite. Similarly to recognize that one thought implies another is to be prepared to accept the second on the basis of the first. Again, there is no judgment that gives expression to this notion of on the

basis of, implication is no more a relation between thoughts than truth is a property of them.

We have seen how the attempt to make our apprehension of the validity of an inference rule over into a judgment leads to the use of the truth predicate that the regress argument excludes. This flawed attempt at a statement is the best that can be done; and its failure shows why the essence of inference, of logic, is ineffable. In an unpublished jotting entitled "My basic logical Insights" Frege notes the redundancy of attributions of truth and then says:

But it is precisely for this reason that this word ['true') seems fitted to indicate the essence of logic ... it allows what corresponds to the assertoric force to assume the form of a contribution to the thought. And although this attempt miscarries, or rather through the fact that it miscarries, it indicates what is characteristic of logic. . . what logic is really concerned with is not carried in the word 'true' at all but in the assertoric force with which a sentence is uttered.33

III

It remains to examine how ontological categories are supervenient on logical ones for Frege. The treatment of this topic provides the oppor-tunity to apply, albeit in an incomplete and only partially defended manner, the interpretive perspective of the previous two sections to Frege's central and celebrated distinctions and doctrines.

We noted at the beginning of Section II Frege's demand that proofs be gap-free. Having made this demand, Frege goes on to observe two obstacles to satisfying it. First, there is the comparatively minor barrier posed by the sheer tedium of advancing in argument step by step in conformity with a small number of antecedently marked out inference patterns. Second and much more important, there is, as regards every-day language, the difficulty of isolating out some small number of logical laws and inferences sufficient for Frege's logicist goals. Frege complains, "the excessive variety of logical forms that has gone into the shaping of our language makes it difficult to isolate a set of modes of inference


which is both sufficient to cope with all cases and easy to take in at a glance."34 He continues:

To minimize these drawbacks, I invented [erdenken] my Begriffsschrift. 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.

Frege's function-argument analysis of language with its associated talk of the truth-value of a statement being determined by the meanings of its parts must be understood against the backdrop of this project.35

How then can Frege proceed - or think of himself after the fact as having proceeded - with the construction of a logically perspicuous language? We observed that the starting point for Frege's conception of objectivity is our appreciation, at least in straightforward cases, that one statement may be asserted on the basis of the assertion of another statement. This datum was presented at the level of the apprehension of individual inferences. Reflection on these individually correct inferences leads the logician to the isolation of inference patterns, patterns like those that are eventually set forth in the inference rules. Frege is describing the logician's reflection when he says:

Kerry holds that no logical rules can be based on linguistic distinctions; but my own way of doing this is something that nobody can avoid who lays down such rules at all. For we cannot come to an understanding with one another apart from language, and so in the end we must always rely on other people's understanding words, inflexions, and sentence-constructions in essentially the same way as ourselves ... to this end I appealed to the general feeling for the German language.36

The apprehension of inference patterns involves thinking of statements as logically segmented into significant parts. Consider, for example, the inference pattern set forth in Leibniz's law. Inferences that instance this pattern lead from the assertion of a statement containing occurrences of a term t and the assertion of an ·equation 'i = S', to the assertion of the result of replacing occurences of t in the original statement by occur-rences of s. The appreciation of the validity of this inference pattern goes hand in hand with seeing statements to be segmented into proper names and predicates, a predicate being the result of removing one or more occurrences of a proper name from a statement. Of course, our grasp of these two categories is not exhausted by Leibniz's law; reflec-tion on inferences concerning generality plays a crucial role in imposing this segmentation into proper names and predicates. In the end, it is not


the apprehension of a single inference pattern, but of an interlocking series of basic inference patterns, that constitutes the recognition of parts and features of statements as logically functioning units that belong to various logico-syntactic categories.

This observation is indeed the point of Frege's enigmatic Context Principle that bids us "never to ask for the meaning of a word in isola-tion but only in the context of a statement [Satzzusammenhang)."37 It is ultimately our appreciation that one statement may be asserted on the basis of another and the ensuing apprehension of inference patterns that gives us the idea of words having a meaning.38 To ask after the meaning of a word is then, first of all, to ask after its logical category; such inquiry is answered by reflection on the implicational relations of state-ments in which the word occurs. Thus it is that, in the body of The

Foundations of Arithmetic, Frege recurs to the Context Principle at the conclusion of a discussion of statements of the form 'There are n P, a discussion that argues in effect that the numeral must be reckoned as a proper name.39

By attention to selected simple inference patterns within everyday language, Frege discerns amidst the ill-defined welter of inferences found there, the logical segmentation that provides the scaffolding for pruning and reforming everyday language into a Begriffsschrift. The segmentation of statements into proper names and predicates is a crucial aspect of this enterprise. This segmentation enables Frege to explain the (first-level) quantifier as a second-level predicate - an expression which, when completed by a first-level predicate, yields a statement. Exploiting this insight, Frege eliminates the grammatically chaotic and ambiguous ways everyday language has of expressing generality in favor of the perspicuity of the quantifier-variable notation. The way is then prepared for Frege's greatest achievement - the complete and accurate depiction of the patterns of polyadic quantifica-tional inference.

To think of statements as logically segmented is to think of the expressions thus segregated as logical units, as playing a uniform role in the determination of the truth or falsehood of the statements in which logical analysis discovers them. Of course, it is not the expressions themselves - the bare marks and noises - that so contribute. Frege is always thinking of statements and their component expressions as significant. Talk about the meaning of expressions of various categories in terms of the contribution they make to the truth-value of statements


points toward the source of logical segmentation in inference patterns. Such talk is of a piece with the attempt mentioned at the end of Section II to state the warrant for making inferences in accordance with syn-tactically stated rules. It tries to say what cannot be said but only shown in the inferences we make and accept. However, such gesturing is Frege's indispensible means for instructing us in the use of the Begriffsschrift. By talking about truth-value determination, he obtains our affirmation of the basic laws of his formal system, our apprehension of the validity of his inference rules, and, generally, an understanding of the notation that extends to those Begriffsschrift formulas for which there are no readily intelligible analogues in everday language.

In setting forth his logical notation, Frege talks of proper names as meaning objects and predicates as meaning concepts. Against the backdrop of the connection just surveyed between logical segmentation and significance, the phrase 'means an object' serves to label that feature proper names share by which they all make the same sort of contribu-tion of the truth-value of statements. So, as regards proper names by themselves, ,we could use the less opaque phrase 'significant proper name' instead. Similar remarks hold for predicates. The dichotomy between objects and concepts comes into its own not as regards proper names and first-level predicates but rather in introducing and distin-guishing first-level generality (over objects) from second-level generality (over concepts). This logical difference is masked in everyday language, including the language of mathematics.

We observed that Frege's first-level quantifier is a second-level predicate. The meaning of this quantifier may be specified as follows. Completion of the empty position in the quantifier by a predicate yields a truth just in case every object falls under the concept meant by the predicate. In this elucidatory context, unlike those that concern just proper names ,and predicates, 'object' is used outside the phrase 'means an object'. Moreover, the phrase 'every object' is being used in contrast with 'every concept' to call attention to a second sort of generality, generality over concepts. Frege's quantifier over concepts is a third-level predicate that stands to second-level predicates as the qUantifier over objects stands to first-level predicates. However, the use of 'object' and 'concept' to contrast these two levels of generality can seriously mislead. The phrases suggest the availability of a quantifier over entities of which objects and concep\s are genera; for, in Frege's elucidating remarks, the words 'object' and 'concept' alike appear to be first-level predicates.4o


Frege's conception of generality precludes any such quantifier, any such notion of entity.

This last point is worth some elaboration, as it is essential for under-standing how the basic laws of logic abstract from the particular content of the statements of the special sciences. This abstraction is glossed in terms of the relation between generalizations and their instances. A variable may be substituted for any significant expression in a statement, a transformation that yields a statement that speaks generally where the given one spoke specifically. The mark of this generality lies in the replaceability of the variable in its turn by any expression of the same category as the original expression to obtain an instance of the general-ization. So to each logical category of expression there corresponds a type of variable.

Suppose ,Jlow that we attempt to get by with one level of generality. Even restricting attention to simple statements from which generality is absent, we observe that the replacement of the grammatical subject by a grammatical predicate produces nonsense. This fact presumably indi-cates that the grammatical predicate is combining two distinct roles - . the presentation of a concept and the expression of the copula. The understanding of the relationship between generalizations and their instances, variables and constants, just surveyed requires that these two roles be separately discharged in a logically perspicuous notation. To this end, an expression, 'falls under', might be introduced to express the copula. So, we obtain as renderings of everyday statements, statements like

Socrates falls under the concept of being wise,

and

If x falls under the concept of being a philosopher, then x falls under the concept of being wise.

Generalizing on the direct object position of our new expression as well as its subject position, we can use it in the obvious way to state logical laws.

However, our new relational predicate is an expression of a distinct logical category; after all, no statement results from replacing it with a proper name, with an expression that may occur in the positions on which we are generalizing. In thus admitting an expression of a distinct logical category, we ipso facto allow for second-level generality and with


it, further logical laws. But once second-level generality is acknowl-edged, there is no call to treat predicates as playing the same logical role as proper names and so forming with them a single logical category. The acknowledgment of the distinct logical category of 'falls under' might be resisted by maintaining that this relational predicate suffers from the same duality of logical role it was introduced to correct - it both presents a relation and is a copula. It should now be clear that there is no way of disentangling these roles without reintroducing an expression in which they are once again combined. The conclusion here is that there are not two distinct roles played by predicates; or, as Frege says, the concept is predicative, unsaturated.41

The distinction of levels of generality is of the highest importance to Frege. The first fruits of this distinction is his logical definition of the ancestral of a relation. This definition enables the principle of mathe-matical induction to be deduced from logical laws. Thus, the distinction between generality over objects and generality over concepts is the immediate source of the centerpiece of Frege's logicism: the reduction of "the argument from n to (n + 1), which on the face of it is peculiar to mathematics, to the general laws of logic."42 Small wonder then that the third guiding principle of The Foundations of Arithmetic advises us "never to lose sight of the distinction between concept and object."43

We are now in a position to understand how ontological categories are, for Frege, supervenient on logical ones. The logico-syntactic source of the notion of an object lies in first-level generality. To be an object is to be indefinitely indicated by first-level variables. Our grasp of the notion of an object - simply the notion of an object, not an object of this or that kind - is exhausted by the apprehension of inference patterns and the recognition of the truth of the basic logical laws in which these variables figure.44 We encounter at this juncture the central elucidatory use for the phrase 'means an object' (bedeuten). This phrase is used in contrast with the phrase 'indefinitely indicates an object' (andeuten) to call attention to the inferential difference between state-ments where first-level generality is present and where it is absent, and so to distinguish first-level variables from proper names.

Similar remarks hold for the notion of a concept. The logico-syntactic source of this notion lies in our apprehending basic inference patterns turning on second-level variables. It is worth observing that for con-cepts, or rather functions - for Frege this is the prior and more general notion - there is an expression of this attitude at the outset of Frege's-


paper 'Function and Concept.' Frege there notes that mathematicians had no use for the general notion of a function so long as they were occupied with establishing facts about particular functions. The need for the general notion of a function arose only when mathematical attention turned to abstract generalizations about functions, for instance that any function everywhere differentiable is everywhere continuous, but not vice versa. Frege says:

The first place where a scientific expression appears with a clear-cut meaning is where it is required for the statement of a law. This case arose as regards the function upon the discovery of higher Analysis. Here for the first time it was a matter of setting forth laws holding for functions in general.45

And the introduction of the general notion of a function is signaled not so much by the use of the word 'function' as by the appearance of variables over functions. From the Fregean vantage point, the unclarity surrounding the notion of a function in mid-nineteenth century analysis was the product of an inadequate understanding of variables, including a failure to distinguish sharply first-level from second-level variables, objects from functions.

We have seen how Frege's talk of meaning serves to instruct us in the use of the Begriffsschrift. One aspect of this talk calls for further exami-nation - Frege's notorious doctrine that statements mean truth-values in the same sense in which, say, 'Venus' means a planet. The position appears outlandish. Frege seems to begin with the familiar and pre-theoretical paradigm of the relation between an ordinary proper name and the person, animal, edifice, etc. that bears it. The suspicion is that, however motivated Frege's other extensions of the name-bearer relation are, the analogy snaps when applied to the relation between statements and truth-values.

I suggest that this line of objection to Frege's controversial thesis both misevaluates its significance and gets the connection between Frege's talk of meaning and the name-bearer relation backwards. We saw in Section I how the conception of judgment that emerges from the contrast between the objective and the subjective includes the idea of standards of correctness. But correctness or truth cannot be thought of as a property of the contents of judgments. How then are we to think of it? In 'Thoughts,' Frege says that the meaning of the word 'true' is sui generis.46 In 'On Sense and Meaning,' Frege denies that the relation of a thought to truth is the relation of a subject to a predicate and talks


instead of a difference in level between thoughts and truth-values. He says, "Judgments can be regarded as advances from a thought to a truth value."47 and, "We are therefore driven into accepting the truth value of a sentence as constituting its meaning .... These two objects [the True and the False] are recognized, if only implicitly, by everybody who judges something to be true ... "48 Frege's talk of a difference of level between thoughts and truth-values and of judgment as a movement from one level to the other attempts to introduce a way of thinking about truth in contrast to taking truth to be a property of thoughts. The notorious doctrine should then be undet'stood as an elaboration of this contrast; it is the more appropriate way of speaking to which Frege alludes in 'Thoughts'.49

Nor should we take Frege's thesis to draw on our familiarity with the name-bearer relation for its motivation. If the preceding interpretation of the source of Frege's notion of meaning in the apprehension of inference patterns is correct, the relation between a statement and its truth-value is the paradigm for the meaning relation, a paradigm in terms of which the familiar name-bearer relation is re-interpreted. Frege acknowledges the generalization

If x = y, then Fx if and only if Fy,

to be a logical law. The inferential links between first-level variables and proper names together with the foregoing conception of meaning practi-cally force the identification of a colloquial proper name's having a meaning with its having a specific bearer.

Thus understood, truth is not, properly speaking, ineffable. To treat statements as proper names, and hence truth-values as objects, renders the True and the False as effable as any object. The ineffability that lurks in Frege's discussions of truth should attach instead, I suggest, to the meaning relation thought of as pairing linguistic expressions with the items meant by those expressions. After all, the availability of such a relation would give us the means for converting Frege's remarks about truth-value determination into a genuine theory containing the resources for defining a concept of correctness or truth. It is through his talk of the meaning relation that Frege speaks of truth-value determination for the purpose of instructing us in the Eegriffsschrift. Once this end is achieved, talk of meaning is to drop away; it plays no further role in logic.

There is a final topic that requires at least brief consideration -Frege's distinction between sense and meaning. In Section I it was noted


that some of the ways Frege has of talking about thoughts lend credence to the platonist reading - the ontological status of thoughts as atem-poral abstract objects guarantees the objectivity of judgment. It was claimed, however, that Frege's vocabulary for talking about assertion and judgment - particularly the notion of thought - was introduced in order to redescribe systematically those features of our linguistic practice that fund Frege's conception of logic. It is through Frege's identification of thoughts with the senses of sentences that thoughts obtain the status of objects within Frege's system. The immediate source of Frege's notion of sense lies in the application of his Begriffsschrift to the logical analysis of the observations about judgment, belief, and assertion that are behind his conception of objectivity. There is, after all, no way to direct attention to the distinguishing features of judgment without using locutions such as those of the form 'x asserts that p'. These observations, couched as they are in familiar vocabulary, are clear cases of significant statements; they are also examples of statements that contain the subor-dinate clauses of indirect discourse. Frege takes the familiar apparent failures of Leibniz's law here - including his extension of it to materially equivalent statements - to show that the statements and expressions of indirect discourse do not have the same significance they have in other settings. The sense/meaning distinction provides a delineation of this pervasive ambiguity that logical analysis uncovers. The details of Frege's

about sense are largely the result of his reflections on the char-acter of truth-preserving substitutions within indirect discourse.

Frege has other uses for his notion of sense. But it is the foundational role that observations about assertion and judgment play in Frege's philosophy that suits the notion of sense for those uses. Of course, it is debatable whether the notion of sense, introduced in the manner just sketched, can discharge these other roles; and even Frege's conclusions about truth-preserving substitutions within indirect discourse are con-troversial. The serious problems surrounding Frege's notion of sense lie beyond the scope of this paper.

My present point concerns the status of the senses of proper names, including sentences, as objects. Frege's identification of thoughts as objects is owed entirely to the logical segmentation he discerns within indirect discourse. He takes sentences in indirect discourse to be logically functioning units, proper names. Thus it is that Frege's concep-tion of the objectivity of judgment and his associated view of logical segmentation and meaning eventuate in me claim that thoughts are


objects. We find then talk of thoughts occurring in two different places in Frege's philosophy, places he does not clearly demarcate in his writ-ings. Talk of thoughts occurs at the very beginnings of his philosophy, introduced to redescribe the central features of our practice of assertion. It also occurs subsequently, once logic is in place, in discussing the logical segmentation of indirect discourse. Understood in this way, the objecthood of thoughts, far from explaining the objectivity of judgment, presupposes it.

University of Pennsylvania

NOTES

* The ideas in this essay reflect many conversations about Frege with Burton Dreben. I am deeply and pervasively indebted to Warren Goldfarb, both for instruction received from his 1977 lectures on Frege and for countless ensuing discussions. My interpreta-tion of Frege's anti-psychologism has been influenced by an unpublished paper of Susan Neiman. I have benefited from conversations with Michael Friedman, Peter Hylton, and Joan Weiner. Prof. Weiner, in her dissertation Putting Frege in Perspective (Harvard University, 1982), arrives at an understanding of the point of Frege's talk about Bedeutung similiar to that presented in Section III. 1 Gottlob Frege, The Foundations of Arithmetic, translated by J. L. Austin (Evanston: Northwestern University Press, 1978), p. x. 2 A discussion of Frege's philosophical antecedents that convincingly portrays Frege's work as a kind of rationalist reaction to a naturalistic empiricism is given by Hans D. Sluga in Gottlob Frege (London: Routledge & Kegan Paul, 1980), chapters one and two. I am indebted to Peter Hylton for awakening me to the affinity between post-Kantian Idealism and early Analytic philosophy that lies behind the obvious and important discontinuities. 3 Frege, 'Boole's Logical Calculus and the Concept-Script; in Posthumous Writings,

edited by Hans Hermes et al. (Chicago: The University of Chicago Press, 1979), pp. 15-16. 4 Frege, The Basic Laws of Arithmetic: Exposition of the System, translated and edited by Montgomery Furth (Berkeley and Los Angeles: University of California Press, 1967), p. 14. 5 Basic Laws, p. 13. See also Foundations, p. vi. 6 Basic Laws, p. 14 and p.l5. 7 Basic Laws, p. 17. 8 For instance, see Frege, 'On Sense and Meaning' in Translations from the Philo�

sophical Writings of Gottlob Frege, edited and translated by Peter Geach and Max Black (Oxford: Basil Blackwell, 1952), p. 61; see also Frege, 'Thoughts' in Logicallnvestiga�

tions edited by Peter Geach; translated by Peter Geach and R. H. Stoothoff (New Haven: Yale University Press, 1977), pp. 8-9.


9 Posthumous Writings, p. 142. 10 'Thoughts,' p. 24; but see also the material surrounding the quoted passage, especially Frege's footnote about grasping. 11 Frege, 'Negation' in Logical Investigations, p. 31. See also Posthumous Writings, p. 7. 12 Posthumous Writings, p. 175; see also p. 3. See also Foundations, pp. 3-4, especially the last paragraph of section three. 13 Foundations, p. 102. 14 'Thoughts,' p. 1. Similar remarks are scattered through Posthumous Writings. For example, see p. 2, p. 126, p. 128, p. 253. IS Frege, 'Compound Thoughts,' in Logical Investigations, p. 71. 16 See W. V. Quine, Word and Object (Cambridge: MIT Press, 1960), sec. 56, especially p. 273; see also Quine, Philosophy of Logic (Englewood Cliffs: Prentice-Hall, 1970),pp.l0-13 and pp. 47-50. 17 The point is established in Warren D. Goldfarb, 'Logic in the Twenties,' The Journal of Symbolic Logic 44 (1979), especially pp. 341-44 and in Jean van Heijenoort, 'Logic as Calculus and Logic as Language,' Synthese 17 (1967), especially pp. 325-27. 18 This, I take it, is Michael Dummett's strategy in his masterful interpretation of Frege. At this juncture then, we encounter a consequence of the difference between my understanding of the priority Frege attaches to judgment and Dummett's. 1 sketch in this essay an alternative to Dummett's interpretation of Frege; I have not, either at the level of exegesis or of argument, defended my interpretation against Dummett's. 19 'On Sense and Meaning,' p. 65. 20 This observation, I take it, is behind Frege's remarking that to recognize anything to have any property carries with it, after a fashion, a predication of truth. See 'Thoughts,' p. 5 and Posthumous Writings, p.129. 21 'Thoughts,' p. 4. See also Posthumous Writings, pp. 128-29. 22 'Thoughts,' p. 4. 23 See Frege's letter to Hilbert of Dec. 27, 1899, letter IV/3 in Frege, Philosophical and Mathematical Co"espondence, edited by Gottfried Gabriel et al., abridged from the German edition by Brian McGuinness and translated by Hans Kaal (Chicago: The University of Chicago Press), pp. 36-37. See also Frege, 'On Concept and Object,' in Translations, pp. 42-43. 24 'Thoughts,' p. 6. See Posthumous Writings, p. 251; see also 'On Sense and Meaning,' p.74. 25 Basic Laws, p. 12. 26 Posthumous Writings, p. 128. 27 'Thoughts,' p. 2. 28 Basic Laws, p. 15. 29 Immanuel Kant, Critique of Pure Reason, translated by Norman Kemp Smith (New York: St. Martin's Press, 1929), p. 18 (B ix); see also p. 98 (A 60 = B 85). 30 See Basic Laws Section 48, pp. 105-09 for Frege's formulation 01 the inference rules for the formal system of Basic Laws. Rule 6 on p. 106 is Frege's version of MP. 31 Frege, thanks to his horizontal function that associates every object with a truth-value, in the formal system of Basic Laws formulates truth-functional laws using unrestricted first-level variables. 32 See 'Negation,' p. 34. 33 Posthumous Writings, p. 252.


34 Foundations, p. 103. Frege makes the same point a bit more sharply in 'On Herr Peano's Begriffsschrift and My Own,' translated by V. H. Dudman Australasian Journal

of Philosophy 47 (1979), p. 2. 35 Until the fmal paragraphs of this section, Frege's distinction between sense and meaning will be put to one side; and the term 'statement' will be used with the same ambiguity that Frege uses 'Satz' in his pre-1891 writings. I also follow the growing practice of using 'meaning' and its cognates as the English equivalents of Frege's uses of 'Bedeutung' and its cognates. 36 'On Concept and Object,' p. 45. 37 Foundations, p. x. Austin translates 'Satz' by 'proposition'. 38 Frege can be read as marking this point in his letter to Peano of Sept. 29, 1896, letter XlV /7 , in Correspondence, p. 115. 39 See Foundations, p. 72 and p. 73. 40 See 'On Concept and Object,' p. 54. 41 'On Concept and Object,' p. 43. The argument of the last two paragraphs is in essence the one Frege gives in the penultimate paragraph of this paper; see pp. 54-55. 42 Foundations, p. 93. 43 Foundations, p. x. I am indebted in this paragraph to Warren Goldbarb. The section as a whole bears the imprint of Burton Dreben's repeated insistence on the role of Frege's mathematical training and interests in shaping his philosophy. 44 These remarks about Frege's conception of objecthood should be distinguished from superficially similar things Quine says. Quine puts forward his maxim "To be is to be a value of a variable," in the context of a discussion of ontological commitment. Applica-tion of the standard of ontological commitment it provides is made in a metalanguage into which we may have occasion to semantically ascend in the course of certain investigations and disputes. For present purposes, the important point is that the notion of ontological commitment, and with it this maxim, belong to the theory of reference and thus invoke the semantical perspective Frege eschews. 45 Frege, 'Function and Concept,' in Translation, p. 21. See also p. 41. 46 'Thoughts,' p. 4 and p. 6. 47 'On Sense and Meaning,' p. 65. 48 'On Sense and Meaning,' p. 63. 49 See 'Thoughts,' p. 6; and see above pp. 22-23.

TYLER BURGE

FREGE ON TRUTH

From a natural perspective, Frege's view that sentences denote (bedeuten) objects appears to be an irritating peculiarity. His claim that there are only two objects denoted by sentences and that these are Truth and Falsity has seemed to many to advance from the peculiar to the bizarre. Indeed, a standardized form of philosophical humor has grown up around talk of "naming the True". I think that the natural perspective is sound and that the humor has its point. But understanding Frege's motivations for these views provides insight into the fundamen-tals of his philosophical standpoint and method. Such insight enriches the natural perspective.

The importance of Frege's views on truth values in his system has been appreciated by a number of philosophers. Michael Dummett characterizes Frege's claim that sentences denote objects as "an almost unmitigated disaster" for Frege's later philosophy of language (FPL, 196,643-4).1 Several authors have seen in Frege's writings the skeleton of an a priori argument, later given by Church and Godel, that sentences must denote only Truth or Falsity. And Frege's method of identifying the truth values with certain courses of values has been construed as indicating a non-realistic attitude toward numbers. I think that each of these interpretations is mistaken. But they correctly suggest that Frege's odd-sounding conclusions about truth and falsity should be taken seri-ously as a key to his philosophies of language, logic and mathematics.

My aims here are historical. I shall argue in Section I that Frege's view that sentences denote only truth or falsity has profound and natural motivations, and that his view that truth values are objects is more pragmatically based - and therefore less strange - than has usually been thought. In Section II I criticize Dummett's influential interpretation of Frege's theses on truth values and his evaluation of the effect of those theses on Frege's philosophy of language. I also delineate the development of Frege's views on assertion and truth between Begriffsschrift and Basic Laws. In Section III I argue that Frege's identifi-cation of the truth values with the particular objects he identifies them with undergirds his realism about logical objects, and proceeds from

97

L. Haaparanta and f. Hintikka (eds.), Frege Synthesized, 97 -154. © 1986 by Tyler Burge.

98 TYLER BURGE

some of his deepest philosophical conceptions. In particular, it proceeds from a theory about the nature of logical objects, from a thesis about the aim and ordering of logic, and from his conceptions of assertion and truth.

In order to lay the groundwork for our discussion of Frege's concep-tions of assertion, truth, and logical objects, I will have to go over a fair amount of familiar ground in Section I. Some readers may wish to work through this section quickly in order to concentrate on Sections II and III. I should caution, however, that although many of the doctrines discussed in Section I are well-known, the ways they fit together and the means Frege uses to motivate them are less well recognized. Under-standing these ways and means is critical to a proper appreciation of Frege's use of the notion of truth in his philosophy of logic and mathe-matics - and indeed, to an appreciation of his depth as a philosopher.

Although defective in various ways, Frege's views on truth are richer and more central to his logical theory and much of his philosophy of mathematics than is often realized. One reason why these views are underappreciated is that Frege refused to allow a meta-theoretic seman-tics, as we know it, to be part of his logical theory. Another reason is that Frege's presentation of his views has tended to encourage concen-tration on his philosophy of language or his mathematical work as some-what separate enterprises. The philosophy of language is expounded largely in the great articles of the 1890's and in unpublished writing, with little discussion of its connection to logicism. The mathematical project is spun out in The Basic Laws of Arithmetic, which is cast in the form of a traditional mathematical treatise - its philosophy kept to a minimum. Underlying Frege's work is, however, a remarkably integrated vision. We shall try to layout the central place that Frege's views on truth have in this vision.

It is useful to separate Frege's views on truth values into several theses, although the theses are interrelated and his arguments for them overlap. The relevant theses are

(a) Sentences (when not defective) have denotations (Bedeu�

tungen).

(b) The denotation of a sentence is its truth value.

FREGE ON TRUTH 99

(c) Sentences are of the same logical type as singular terms. (d) The denotation of a sentence is an object.

Frege tends to develop support for the theses in the order in which they are listed. (See Note 6 below for a qualification.)

Frege's arguments often presuppose his distinction between sense and denotation (which he first draws for singular terms). They almost always presuppose or make use of his groundbreaking composition principles:

(1) The denotation of a complex expression is functionally dependent only on the denotations of its logically relevant component expressions.

(2) The sense of a complex expression is functionally dependent only on the senses of its logically relevant component expressions.

(I omit certain qualifications on these principles that are irrelevant to our concerns.) The first principle is the critical one in Frege's thinking; the second makes important but only occasional appearances.

Thesis (a)

Frege argues that the sense of a sentence - its cognitive value, the thought that it expresses - remains the same regardless of whether or not the sentence's component expressions (particularly, the singular terms) have denotations: The sense of a sentence is fixed independently of its components' denotations [G & B, 'S & R', 63/KS 148; Cor 165/

BW 247; PPW 193-4/NS 210.] It follows from (1) and (2) and these considerations that the sense of a sentence cannot be conceived as its denotation.

Frege further argues that one cannot reasonably hold that sentences in general lack a denotation. In at least one passage he draws this conclusion almost directly from the arguments of the preceding para-graph:

It follows that there must be something associated with a sentence that is different from the thought, something for which it is essential whether the parts of the sentence have denotations. This is to be called the denotation of the sentence. (PPW 194/NS 210-11)

This inference clearly relies on the Composition Principle (1). Now one might feel that the inference begs any question one might

have about whether sentences have denotations - about Thesis (a). Why

100 TYLER BURGE

should sentences be included among the complex expressions that have denotations? The last sentence of the passage just cited suggests some-thing wrong with the question. Frege is not using the term 'denotation' with a fixed, meaning in arguing for (a). Rather he is determined to give his Composition Principle (1) a comprehensive role in logical theory, and he is intending to fit the term 'denotation' to the role that the principle might fruitfully play in a logical theory about sentences. "The denotation" of a sentence is whatever is most fruitfully seen as functionally dependent on the denotations of its parts. So far the phrase 'the denotation' has no specific logical grammar or ontological implications. Since the arguments for (a) do not presuppose Thesis (d), there is so far no reason to consider the view that a sentence's denota-tion is an object. One may at this point regard talk about sentence denotation as potentially a de parler for an important semantical aspect of sentences. The ontological import of such talk, if any, is left thoroughly open.

Of course, the term 'denotation' (,Bedeutung') was not devoid of intuitive content in Frege's arguments. 'Bedeutung' is a common word in German, usually translated 'meaning'. In German there is no oddity in saying that sentences have a "Bedeutung". Frege did, however, appro-priate the term for his theoretical uses and introduced it in the essays 'Function and Concept' and 'On Sense and Denotation' through exam-ples of singular terms ('The Evening Star', 'Odysseus' - which lacks a denotation - '1', '2 + 2', 'the capital of England'). The examples suggest that naming or reference - considered as relations between names and their bearers or between a complex singular term and the object it picks out - is one primary sort of "Bedeutung". But since Frege also used his term to apply to a semantical relation between expressions (such as predicates), that he emphatically did not regard as singular terms, on one hand, and non-linguistic entities, on the other, one must view these initial examples with some caution. They are aids in building a theory.

The point I want to press regarding Frege's quick inference to (a) from his composition principle is that the inference is indicative of his pragmatic attitude toward his terminology. As he repeatedly noted, the term 'number' had expanded in its application (from the natural num-bers, to negatives, rationals, reals,' complex numbers) under pressure from the requirements of mathematics. Semantical terminology could be expected to undergo similar stretching in response to the demands of logical theory.

FREGE ON TRUTH 101

Frege provides a closely related, but different argument for (a). This argument occurs in 'On Sense and Denotation' and is repeated in his correspondence with Russell and his posthumously published writings. The following passages suggest the argument:

The fact that we concern ourselves at all about the denotation of a part of the sentence indicates that we generally recognize and expect a denotation for the sentence itself. (G & B, 'S & R' 63/ KS 149)

Now it would be impossible to see why it was of value to us whether or not a word had a denotation if the whole sentence did not have a denotation and if this denotation was of no value to us; for whether or not that is so [whether or not the words have a denotation] does not affect the thought. (Cor 1521 BW235)

[If a sentence had no denotation] the denotation of any part would be a matter of indifference, for, regarding the sense of a sentence, only the sense not the denotation of its parts comes into consideration. (Cor 1581BW 240; cf. also Cor 165, 163nlBW 247,

245n; PPW232INS250-1)

These claims are embedded in discussion of examples of nondenoting names and in an argument for Thesis (b). But in view of the obvious generality of their intent, I think that they are worth isolating.

Frege's argument is that we would not concern ourselves with the denotations of sentence-parts if we were not interested in the denota-tions of whole sentences; we clearly do concern ourselves with the denotations of sentence-parts - we often care whether singular terms denote something; so we are interested in the denotations of whole sentences.

The argument must again be seen in the light of the centrality of the Composition Principle (1) and of Frege's pragmatic use of the term 'Bedeutung'. Denotations of sentences are whatever can be seen as both central to logical theory and functivnally dependent on the denotations of the logically relevant parts of sentences. But this argument adds a further claim. Our interest in the denotations of words is derivative from our interest in the denotations of sentences. That is, word denotation is important because and only because of the importance of some feature of sentences that is central to logical theory and functionally dependent on word denotation.

This further claim appears to be an expression, or outgrowth, of the context principles that Frege had enunciated earlier in The Foundations

of Arithmetic. These enunciations preceded the development of the distinction between sense and denotation, and they took a variety of

102 TYLER BURGE

nonequivalent forms. But they all emphasized that the analysis of the "meaning" of a word (in retrospect, presumably, its sense and its denotation) was to be carried out in the context of an analysis of its role in a sentence. Frege appears to be invoking the primacy of sentences in his argument that sentences have denotations. Our interest in the denotations of words had to be connected in logical and linguistic theory with some feature of sentences. Frege forged the connection by means of the Composition Principle (1), and he called the relevant feature of a sentence its denotation.

It is worth noting that Frege's ,reasoning is prima facie incompatible

with the idea that the notion of the denotation of a term has no other content than that provided by an analysis of the contribution of the term in fixing the denotation (or, truth value) of a sentence.

The argument presupposes that we have a co-equal understanding of and application for the notion of the denotation of a term.2 Indeed it presupposes that the notion of term-denotation is more familiar than that of sentence denotation, though perhaps not more familiar than that of truth value. The argument claims that whether terms have any denotation at all is of importance to us only relative to our interest in relevant semantical properties of sentences. It does not suggest that the notion of term-denotation can be exhaustively defined, or characterized, or reduced by attempting to analyze the relevant semantical properties of sentences in total abstraction from one's ordinary understanding of the notion of term-denotation (reference). The ordinary understanding of term denotation is assumed to be sound. (One could produce numer-ous passages from Frege's opposition to formalism to substantiate this point.) The argument simply demands that such ordinary understanding has to be connected, in one's theory, to the semantical properties of sentences, interest in which motivates interest in the denotations of terms.

Thesis (b)

The role of value and "interest for us" in Frege's argument for (a) needs articulation. Frege saw logic as revealing certain norms governing ideal thought. The sentence was the linguistic correlate of thought. We think, according to Frege, only by means of sentences. So any logical theory had to ground itself in an analysis of the properties of sentences that revealed the relevant norms. Our interest in the denotations of terms,

FREGE ON TRUTH 103

and of functional expressions, was motivated by interest in normative properties governing thinking, normative properties whose laws logic sought to uncover.

This focus on the normative implications of logical theory underlies Frege's primary argument for Thesis (b). Frege specifies what it is about sentences that motivates our interest in the denotations of their parts. The relevant property is the sentence's truth value.

The fact that we concern ourselves at all about the denotation of a part of the sentence indicates that we generally recognize and expect a denotation for the sentence itself. The thought loses value for us as soon as we recognize that the denotation of one of its parts is lacking. We are therefore justified in not being satisfied with the sense of a sentence, and in asking also for its denotation. But why do we then want every proper name to have not only a sense, but also a denotation? Why is the thought not enough for us? Because, and to the extent that, we are concerned with its truth value. This is not always the case. In hearing an epic poem ... we are interested only in the sense of the sentences and the images and feelings thereby aroused. In response to the question of truth we would abandon aesthetic delight and turn to a scientific investigation. Hence also it is a matter of no concern to us whether the name 'Odysseus', for example, has denotation so long as we accept the poem as a work of art. It is the striving for truth that drives us always to advance from sense to denotation. (G & B, 'S & R'63/KS 149)

When we merely want to enjoy the poetry we do not care whether, for example, the name 'Odysseus' has a denotation ... the question first acquires an interest for us when we take a scientific attitude - the moment we ask, 'Is the story true?', that is, when we take an interest in the truth value .... Now it would be impossible to see why it was of value to us whether or not a word had a denotation if the whole sentence did not have a denotation and if this denotation was of no value to us; for whether or not that is so does not affect the thought. And this denotation will be something that will have value for us precisely when we are interested in whether the words have denotation (bedeu-

tungsvoll Sind), therefore when we ask after truth. (Cor 152 BW 235)

... if it is not a matter of indifference to us whether the signs that make up a sentence have a denotation, then it is not just the thought that matters to us, but also the denota-tion of the sentence. And this is the case when and only when we ask after truth. Then and only then does the denotation of the sentence enter into our consideration; it must therefore be most intimately bound up with truth. (Cor 165/BW247)

That the name ... designates is of value to us when and only when we are concerned with truth in the scientific sense. So our sentence will have a denotation when and only when the thought expressed in it is true orfalse. (PPW 232/NS 250-1)

Frege's argument for Thesis (b) clearly presupposes his arguments for Thesis (a). It thus presupposes the primary importance of sentences in logical theory. In fact, the language of the first three passages directly echoes the first statement of context principle in The Foundations of

104 TYLER BURGE

Arithmetic: "never to ask for the Bedeutung of a word in isolation, but only in sentential context". The phrases I have translated "asking for its denotation" and "ask after truth" in these three passages from 'On Sense and Denotation' and the correspondence with Russell use the same phrase 'zu fragen nach der Bedeutung' that occurs in the introduc-tion to Foundations. It is almost inconceivable that Frege did not intend to associate the passages with his slogan. The reason that the denota-tions of words must be "asked for" only in sentential context is that the relevantly related semantical feature of sentences - the denotation of sentences - motivates our interest in word denotation. Our interest in the denotation of words derives from our interest in the truth value of sentences, or of the thoughts that they express. Truth is the relevant norm governing our use of and interest in sentences and thoughts. The point of logical theory should be the analysis of the most general laws governing this norm.

Frege's argument for Thesis (b), the thesis that the denotation of a sentence is its truth value, is not and is not intended as a deductive argu-ment. There. is no attempt to deduce (b) from "first principles". In 'On Sense and Denotation', he twice calls the thesis a conjecture (Vermutung

- conjecture, supposition, surmise) - (G & E, 'S & R' 64, 65/ KS 150,

151). And the remainder of article is presented as a series of "tests" of the conjecture. After considering these tests in detail, he writes at almost the end of the article: 'From this it follows with sufficient probability that the cases where a subordinate clause is not replaceable by another with the same truth value proves nothing against our view that a truth value is the denotation of a sentence whose sense is a thought'. (G & B, 'S & R' 78/KS 162)

There is a closely related argument, proposed by Church, G6del and others, that does take deductive form. Frege has sometimes been constructed as giving an ellipitical, or even invalid, approximation to this argument. I think that such a construal is very poor history. Frege's argument rather has the form: In view of the normative aims of logical theory and in view of the considerations that actually motivate our interest in the denotations of terms, the appropriate feature of sentences to connect with the denotations of the sentence's constituent parts via the composition principle, is the sentence's truth value. We shall discuss the Church-G6del argument shortly.

There is a supplementary argument for Thesis (b). This argument is roughly: a sentence's truth value is dependent on the denotations of its

FREGE ON TRUTH 105

constituent terms in just the way that the Composition Principle (1) requires. So given the way the notion of a sentence's denotation is introduced, truth values are well-suited to be the denotations of sen-tences.

In 'On Sense and Denotation' and elsewhere Frege proposes this argument as a confirmatory consideration or an essential test of the conclusion of the previous argument: 'If our conjecture that the denota-tion of a sentence is its truth value is correct, the latter must remain unchanged when a part of the sentence is replaced by an expression having the same denotation. And this is in fact the case." (G & B, 'S & R', 64/KS 150)

Taking the appeal to the Composition Principle (1) as a confirmation or supplement to the previous argument seems to me to be Frege's most reasonable presentation of the relation between the two arguments for Thesis (b). But sometimes Frege seems to place the appeal to the composition principle in a different light. He asks, "what else but the truth value could be found, that belongs quite generally to every sentence, to which the denotation of its constituent parts is relevant, and that remains unchanged by substitutions of the kind in question?" (G & B, 'S & R', 64/KS 150; ct. also Cor158, 165/BW240, 247)

Although Frege cannot be expected to have foreseen this, his question prompted Russell to open a semantical and metaphysical Pandora's box. One can well imagine Russell turning over in his mind this question, which Frege put to him more than once in their correspondence of 1902-4. For Russell was resisting the view that sentences had truth values as denotations.

A year after the correspondence ended, Russell published his theory of descriptions. The theory opened the possibility of maintaining alle-giance to the Composition Principle (1), yet analyzing the logically relevant parts of a sentence in a very different way from the way Frege regarded as natural and appropriate. The theory simultaneously opened the possibility of assigning a variety of different sorts of denotations to sentential parts, and a variety of different sorts of denotations, other than truth values, to the sentences themselves ("states of affairs", "facts", "propositions" and so forth).

Russell demonstrated that one could do compositional semantics without taking truth values to be the central feature functionally asso-ciated with sentences. But it is no accident that, despite the deep methodological interest of the theory of descriptions, Frege's approach,

106 TYLER BURGE

not Russell's, has been the source of the mainstream development of semantical theory in logic. No doubt one reason for the pre-eminence of Frege's approach lies in the artificiality, from a syntactial or grammatical point of view, of Russell's analysis of sentential constituents. But more profound reasons are suggested by Frege's first argument for Thesis (b). Truth (or some modalized notion of truth, like necessary truth or validity) is the central notion of logical theory. In making truth values the primary functional values of the Composition Principle (1), Frege was simply uniting his formal apparatus with the conception that motivates logical theory.

Russell's theory can, of course, accommodate the representation of truth values, of truth-evaluations; and it maintains allegiance to a notion of logical consequence explained in terms of truth. But formally, the truth values enter through a side door, so to speak. The composition principle yokes words with sentences, but it is not used primarily to relate word-denotation to truth value. The primary semantical feature of sentences is the "fact" they are correlated with. The denotations of words functionally determine a ''fact'' or "proposition" composed of attributes and (perhaps) individuals. The primary semantical feature of a sentence is the ''fact'' that it is correlated with. Thus, from the outset, Russell's formal theory incorporates into its subject matter entities that evince a strong admixture of metaphysical motivation. States of affairs, facts, and the like have a recurring attraction for the metaphysically minded. But they have not obtained general acceptance among logicians, and they have yet to, be shown to be indispensable for the foundations of logic. By contrast, the more abstract notion of truth is firmly entrenched in nearly all logical theories. Formal logical theories that place this latter notion at their center, resting little or no weight on arguably dispensable metaphysical entities semantically correlated with sentences, have formed the mainline development oflogic in this century. Frege may be seen as a certain sort of minimalist in this context. He conceived of the fundamental part of logic - the calculus of truth values and first and second order logic - as having an aim and subject matter that was relatively independent of metaphysical controversy. The laws of logic are fundamentally the laws of truth, not laws about the meta-physical constitution of facts, propositions, or thoughts (Gedanken). (Cf. Kl50SIKS 342; and PPW 1221NS 133.) The metaphysics of thoughts is developed to deal with intensional contexts and with epistemic ques-tions, which are treated only in a heuristic way in Basic Laws.

FREGE ON TRUTH 107

Frege took the notion of truth as a normative primitive.3 He did not leave it unexplicated, and his explications are, as we shall see, highly controversial and involved in metaphysical commitments. But his basic procedure is that of a good scientist in the broadest sense of the term. He created and worked within a theory whose interpretation, for the

fundamental purposes of the science, was largely uncontroversial. Con-troversial views were isolated and confined - to the science's heuristic preliminaries and to its frontiers (the philosophical explication of the notions of truth, sense, and assertion, and the application of the logic to intensional contexts, respectively). Extensional logic, more or less as Frege interpreted it, remains fundamental at least in the sense that it is common ground to all logicians and in the sense that its interpretation expresses, with a minimum of controversial accessories, that notion of logical consequence in terms of truth which has traditionally been seen as the central concept of the discipline.

None of this is to deny that Frege had a controversial metaphysics. His philosophical views about truth (particularly Theses (c) and (d) and the "redundancy" conception), his theory of sense, and his theory of judgment and assertion are widely doubted. Indeed, one might safely count them mistaken.

Frege's philosophical views are not, as such, a set of unfortunate superfluities. I think that a metaphysics - or rather a set of controversial philosophical proposals - in this area can hardly be avoided. There are philosophical questions about truth, meaning, cognitive value, and judg-ment that are genuinely difficult and apparently genuine. Frege responded to - in fact, in some cases introduced - these questions. And in order to deal with problems about informativeness, about the commitments of propositional-attitude discourse, about the mechanisms of word-denotation, and so forth, he postulated certain metaphysical entities (senses, Gedanken) that are no less controversial than Russell's facts or propositions. Russell's own theory is in part an attempt to answer these same questions. So from a certain philosophical standpoint it may seem that until these issues are thrashed through, Frege's position holds no advantages over Russell's. His extensional logic owes debts that must be paid before a balance sheet can be drawn up.

There is surely something to this standpoint. But I think that it overlooks one of Frege's central insights and ignores the cognitive advantages of his pragmatic method. Frege's insight is that the norma-tive notion of truth is the central semantical feature of sentences and the

108 TYLER BURGE

fundamental concept of the science. And his pragmatic method of isolating controversy carries the subject a long way before philosophical issues intrude. That has been the method of all successful sciences, mathematical logic included; and success is perhaps our surest guide to knowledge. These considerations tend to favor Frege's basic approach to logic unless the philosophical issues with which he and Russell grappled were decisively and "scientifically" decided in a way that undermined Frege's extensional starting point. That possibility seems remote.

The Church-Godel Argument

Here is perhaps a good place to enter into a digression on the relation between Frege's argument for Thesis (b) and an argument proposed by Church and G6del that is clearly inspired by Frege. (Church (1943), G6del (1964).) The argument has a number of interesting variants, and it has been put to even more uses. G6del's version is particularly rich in implications. I shall, however, discuss only what has become a stan�dardized form.

The argument is supposed to show that all true sentences denote the same thing; an analogous one would show that all false sentences denote the same thing. The argument first presupposes that sentences have a semantical feature that bears enough of an analogy to the central semantical feature of terms to be given the same expression. (This presupposition is often not made explicit. For convenience we shall, inaccurately, call it a "premise".) Let us dub this feature "denotation" in accord with Frege's Thesis (a). Second, the argument assumes the Composition Principle (1). And third, it assumes that logically equiva�lent expressions have the same denotation. Take any true sentences S and S'; S is logically equivalent with a sentence of the form '(LX)

(X = 0) = (LX) (X = 0 & S)'. So by the third premise, S and this sentence have the same denotation. But the latter sentence yields the sentence '(LX) (X = 0) = (LX) (X = 0 & S')' by substitution of co�denotational terms on the right side of the identity sign. So these two sentences have the same denotation, by the second premise. But the new sentence is logically equivalent with S'. So by the third premise, they have the same denotation. So Sand S' have the same denotation.

Frege accepted not only the conclusion of the argument, but all three premises. But in arguing for the conclusion, in effect Thesis (b), he did

FREGE ON TRUTH 109

not advance this argument. I do not find it plausible to view Frege as giving an elliptical or invalid approximation to this argument. The primary reason for this is the one I have already proposed. Frege invokes the normative foundations of logic and the normative roots of the primacy of sentences in logical theory (and in everyday language use) in arguing for his conclusion. That is, he has a premise about the point of logic; and he connects the notion of sentence denotation both with this point and with his primary analytical tool, the Composition Principle (1). The Church-Godel argument makes no such appeal to the purpose of logic or semantics.

Another reason why Frege's argument is different can be developed by looking ahead in our discussion. One source of plausibility for the third premise of the Church-Godel argument derives from the com-parison of sentences to terms - in effect, Frege's Thesis (c). Clearly, the denotations of logically equivalent terms are the same. Insofar as sentences are terms, or at least designators, they plausibly fall under the same principle. But many of the considerations that led Frege to accept Thesis (c) presuppose a prior commitment to a semantical analysis of sentences in terms of their truth values.

A deeper version of the same sort of point can be made from another angle. It may seem perfectly reasonable to accept the third premise of the Church-Godel argument independently of comparison between the semaI\tics of terms and the semantics of sentences. Suppose that we avoid relying on the view that sentences, like terms, designate or denote entities. The notion of the denotation of a sentence was initially intro-duced as that notion which captured the primary semantical feature of sentences for logical theory that could be linked up, by the Composition Principle (1), with the denotations of terms. Should we not expect, virtually a priori, that logical theory ought to count sentences as being the same with respect to their primary semantical feature if it counts them logically equivalent?

The rhetorical question packs a punch. But it still overlooks how fundamental Frege's starting point is. The sentences that are indicated to be logically equivalent in the Church-Godel argument are so counted under a prior conception of logical equivalence, whether informal or fully articulated. This notion already employs some concept of truth -truth under all interpretations, necessary truth, or the like. From Frege's standpoint, this notion of logical consequence (and logical equivalence) already brings with it a commitment to truth values as the central,

110 TYLER BURGE

logically relevant feature of sentences. So again the third premise of the Church-Godel argument is less fundamental from Frege's standpoint than its conclusion. Frege's syntactical analysis - Thesis (c) - his conception of logical consequence, and the metaphysics of his logical theory, e.g. Thesis (d), all depend on his commitment to logic's being primarily concerned with the normative notion of truth.

It would be absurd, of course, to suggest that Frege's conception of logical consequence in terms of (necessary) truth was somehow arbi-trary, or merely one of many equally suitable choices. The conception lies in the mainline tradition of logic that stretches back to its beginning. Even those conceptions of logic prior to Frege that allowed metaphysi-cal visions to predominate tended to maintain allegiance to the informal conception of logical consequence from which his theory sprang. There have been in this century a few approaches, self-consciously reacting against the main tradition, that have departed from the standard infor-mal conception of logical consequence, following a metaphysical, or more often an epistemological muse. At this point, such approaches must be regarded as secondary developments.

The preceding discussion is not intended to suggest that the Church-Godel argument is circular. (The third premise is not equivalent to the conclusion.) The argument is usually given in a context in which people already have the ordinary notion of logical consequence, and in which the notion of a denotation for sentences is open to determination. The usual way of reading the argument is to give it the flavor of, "If you are willing to concede that there is a notion of sentence denotation that meets these restrictions (those of the argument's second and third premises), I will surprise you with what the denotation of a sentence has to be." The third premise might be bolstered by the argument I gave above [second half, p. 109.] By contrast, Frege already knew exactly how he wanted to use the notion of sentence denotation: it was re-stricted by the Composition Principle (1) (second premise); but it had to accord with the primary aim of logic, as it has traditionally been conceived.

Much of the surprise of the Church-Godel argument derives from implicitly thinking of sentence denotation primarily in terms of the pre-theoretical notion of naming, rather than primarily in terms of a specific conception of its theoretical employment, as Frege did. Once one has taken the dubious step of seeing sentences as names, or at least as desig-nating some entities that are functionally dependent on word-denota-

FREGE ON TRUTH 111

tion, it is intuitively surprising to think of them as designating only one of two entities, and odd to reify truth and falsity. Having come so far, it is perhaps more intuitive to take sentences as "designating" possible states of affairs, or something like that. At least many have thought so. Seen this way, the conclusion of the Church-Godel argument is unap-pealing. It is doubtful, however, whether anyone, except perhaps for Church, has endorsed the argument read in this way.4 The natural and most common response to the argument is to reject.its first "premise": sentences do not name, refer to, or designate any entity.

As 1 have indicated, it is possible to see the argument as using a less determinate notion of denotation that gets around this objection. One can consider the argument without adopting Frege's Theses (c) and (d). Then the oddity of the conclusion disappears - the better for reflecting on the logical relationships that the argument reveals.5

Theses (c) and (d) - Pragmatic Motivations

In my view, the first of Frege's arguments for (b) and both his arguments for (a) are sound. Although (a) and (b) have often been targets of criticism, most of the criticism stems from construing (a) and (b) in the light of Theses (c) and (d). 1 believe that doubts about (c) and (d) are justified. But as 1 shall try to show in the remainder of this section and in Section II, such doubts are less interesting than has sometimes been supposed. The discussion of these theses will serve to introduce back-ground essential to Frege's treatment oflogical objects.6

In a letter to Frege in 1903, Russell challenged Thesis (c), the view that sentences are to be regarded as of the same logical type as singular terms: "I have read your essay on sense and denotation, but 1 am still in doubt over your theory of truth values, only because it seems para-doxical to me. 1 believe that a judgment, or even a thought, is something so completely peculiar that the theory of proper names has no appli-cation to it." (Cor. 155-6IBW238.) The gist of Russell's challenge has been repeated by subsequent generations, and with qualifications to hedge against the overstatement in Russell's phrase "no application," I would echo it. But it is easy to be led by the paradoxical ring of (c) and (d), as 1 think Russell was led, into misunderstanding their import and place in Frege's system.

Frege repeatedly emphasized intra-logical, pragmatic advantages for regarding truth values as objects: "How much simpler and sharper every-

112 TYLER BURGE

thing becomes through the introduction of truth values, only thorough occupation with this book can show. These advantages alone already put a great weight on the balance in favor of my conception, which indeed may seem strange at first glance." (BL 7/ GG x.) In fact, The

Basic Laws of Arithmetic mentions only considerations involving simplification of logical theory as motivations for (c) and (d). These considerations are also dominant in Frege's post-paradox writing.

I should make it clear here that in calling Frege's reasoning "prag-matic" or "intra-logical," I am not suggesting that he took the commit-ments that he based on such reasoning to be less than absolutely serious. Such commitments were not merely practical conveniences or technical artifices. Frege saw himself as making objective discoveries. What I wish to emphasize is the great extent to which Frege tried to develop his positions from his analysis of logical structure and from observations regarding functional analogies between different components of that structure. In his arguments for (a)-(d), considerations that derive from intuitions not firmly entrenched in the actual practice of logic, "meta-physical intuitions," play a secondary role in Frege's argumentation. Whatever conceptions most profoundly clarified and simplified logical theory, whatever language made mathematical practice more rigorous, more comprehensive, more fruitful, and less ad hoc, were seen as providing insight into the most abstract features of the world.

The pragmatic cast of Frege's thought seems to have come naturally. Only rarely did he remark on his methodology in general terms. The effusion from Basic Laws, quoted in the previous paragraph but one, constitutes a relatively unusual example. The following passage from a thrice rejected manuscript of 1880-1 provides another:

All these [mathematical) concepts have been developed in science [Frege terms mathe-matics a science) and have proved themselves fruitful. What we can perceive in them therefore has a far higher claim on our attention thaI'l anything that everyday trains of thought might offer. For fruitfulness is the touchstone of concepts, and the scientific workshop is the real field of observation for logic. (PPW 331NS 36-7; cf also KS 124, 369)

Frege's pragmatic considerations rest on analogies that are quite natural within a formal context. In formulating the propositional calculus, it is natural to quantify t4e letters that stand for sentences in something like the way one quantifies into the places held by singular terms in the first-order functional calculus. The places for sentences, like the places for

FREGE ON TRUTH 113

terms, stand in argument places for functional expressions; they some-times constitute value expressions resulting from functional application; and they never stand for functional or predicate expressions. When one quantifies the letters that stand for sentences, the natural interpretation of the domain of quantification is to take it as consisting of the two truth values - as several generations of logic students have been made aware. (Cf. Cor. 158/BE241; KS 225-6/

Another analogy, which is more debatable, is that between non-denoting names and truth-valueless sentences in natural language. (G & B 'S & R' 62/ KS 148; Cor. 152, BW 235). Both maintain a sense in the absence of denotation. Frege thought that subject-predicate sentences containing nondenoting terms always lacked truth values.

It is difficult to see to what extent he accepted this view on intuitive grounds and to what extent he reasoned to it. If one already sees predicates as denoting functions, then one will see a nondenoting name as providing no argument for such a function. Functions without argument yield no value. And if the denotations of sentences are the values of such functions, and are counted truth values, then sentences involving only the application of predicates to nondenoting terms (and application of functors to sentences so obtained) will lack truth value. This reasoning, of course, assumes the assimilation of predicates to function signs. And this assimilation is tantamount, as we shall see, to accepting Thesis (c). So the reasoning cannot be seen as providing much independent support for Thesis (c).

Very likely, the view that nondenoting terms in subject-predicate sentences yield truth-value-less sentences was also found acceptable by Frege on intuitive grounds. The various examples he gives do elicit in many the intuition that the sentences are neither true nor false. But there are numerous other cases that are at best indecisive witnesses for Frege's defense. Since this issue has been discussed at uncommon length by others, I shall not go into it. I think that Frege's view of the intuitive relation between nondenoting terms and truth-value-less sentences was not very critical to his account of truth values. Since he banned non-denoting terms from his formal theory, he rested little weight on the point.

An analogy of which Frege makes more is that between nonassertive occurrences of declarative sentences (suppositions or occurrences within other sentences) and proper names. (EL 35/GG 7.) Sentences often occur embedded in other sentences (for example, in the antece-

114 TYLER BURGE

dents of conditionals) in such a way as to contribute to semantical structure, without being asserted - like terms. Moreover, whole sen-tences can be put forward merely for consideration without carrying any assertive force - again like terms.

These analogies between sentences and terms are, of course, not very gripping. They take on interest when linked to Frege's larger strategy. One of Frege's most profound contributions was to separate the notions of predication and assertion. More generally, he distinguished the notions of logical structure and pragmatically relevant force. The deeper point of the present analogies is that within a formal theory that attempts to lay bare semantical structure, one can prescind from the primary difference between names and sentences (that only the latter can be used to effect linguistic acts or thoughts, protoypically assertions and judgments). The difference between names and sentences can be taken to lie in their point, their use, not in the form of their contribution to semantical structure. Actually, as we shall see in Section II, Frege's formal theory did make formal distinctions between sentences and terms. But the distinctions do not leap to the eye. Although one might believe (as I do) that form should correspond more closely to use than Frege's logical theory allows, subsequent formal usage has confirmed that Frege's analogy constitutes an insight that affords at least a con-venient alternative in setting up a logical system.

Frege's construal of predicates as functional expressions is perhaps the most obvious and widely appreciated ground for Theses (c) and (d) and for his view that truth values are objects. As far back as the Begriffsschrift in 1879, Frege had interpreted predicates as function signs (B, Section 9). Once he supplemented this initial conception with an explicit semantical analysis, which he arrived at by the early 1890's at latest, he was forced to think of functions as denotations for predicates. He called such denotations "concepts". (We shall limit considerations to 1st-level concepts.) Objects, obviously, served as arguments for (lst-level) concepts. But then there must be values for these functions. These must be the denotations of sentences. Sentences are not themselves functional expressions, so their denotations are not functions. Moreover, the values of prototypical functions, the denotations of prototypical completions of functional expressions (terms), just are objects. (We shall, for now, regard an object as anything that is denoted by what is, under logical analysis, a term.) Taking concepts literally to be functions was tantamount to taking the denotations of sentences to be objects

FREGE ON TRUTH 115

(and the completions of predicates, sentences, to be terms). Since Frege had independent grounds for regarding the denotations of sentences to be truth values, this line of thought entailed that truth values were objects.8

Why did Frege take concepts, denotations of predicates, literally to be functions? One primary and lasting motivation was, yet again, prag-matic. Seeing sentences as created by the application of functional expressions effected a simplification in the understanding of the compo-sition principles. The simplest construal of the Composition Principle (1) is to take 'the denotation of sentence s' to be a singular term, denoting an object. (Cf. Note 8.)

A closely related motive was to provide a simple formal expression of the formal analogies between predicates and function signs. (BL 6,

34-5IGGX, 6-7/PPW 235/243-4/NS 253-4, 263). Like function signs, predicates have open places for terms. The primary role of predicates from the point of view of logic is functional - to take objects into truth values.

Completeness and Incompleteness

Frege's pragmatic motives are, I think, dominant. But the analogy between predicates and function signs is sometimes associated by Frege with remarks that have a darker, more metaphysical hue - remarks about similarity in their ''unsaturatedness'' or "incompleteness". He says that the essence of a function is its making a connection between its arguments and its values, in a specific sort of "need for completion" (BL

33-4/GG 5-6; 'C & 0' G & B 47/KS 171; 'F & C' G & B 24-5/KS

128-9). Moreover, he writes, 'An object is anything that is not a function, so that an expression for it does not contain any empty place. A declarative sentence contains no empty place and on that account its denotation is to be regarded as an object' ('F & 0' GB, 32/KS 134).

Here Frege may appear to be inferring from a metaphysical thesis about incompleteness of functions and completeness of objects and from a thesis about how language must match reality as regards com-pleteness or incompleteness, to the conclusion that truth values are objects. Michael Dummett interprets the characterization of objects as anything that is not a function in this passage (and in an equivalent one in Basic Laws vol. I Section 2) as an ad hoc attempt to induce the reader to accept truth values as objects (IFP, 235n). Neither the metaphysical reading nor Dummett's attribution of desperate improvisation places Frege in a very attractive light.

116

TYLER BURGE

To begin with the latter interpretation, I do not find Dummett's

charge plausible. Frege's characterization of objects is independent of

Thesis (d) and precedes its adoption. The idea that objects can be

recognized as whatever is never the denotation of an incomplete, func-

tional expression goes back at least to The Foundations of Arithmetic

(Cf. pp. x, 77n, 72 - before Frege's adoption of Thesis (d).) In the latter

passage, FA 72, Frege writes that the point of counting number words

as words for objects (or self-subsistent objects) is "only to preclude the

use of such words as predicates or attributes ... " (Cf. PPW 100, 1051

NS 109.) Given Frege's view that truth values are denotations of

complete sentences, and never denotations of predicates, and given this

characterization, truth values fill the bill as objects.

Frege's characterization of an object as the denotation of any ex-

pression other than a predicate or function sign may seem either to

emasculate the notion of object or (perhaps equivalently) to commit one

to objects too easily. In discussing Frege's arguments for Thesis (a) we

attributed to him a notion of sentence denotation that does not carry

genuine ontological commitment. But now, it may seem, we are allowing

Frege to smuggle ontological commitments into his arguments for

Theses (a) and (b) by granting him an excessively liberal criterion for

ontological commitment to objects. I have argued that the relevant

criterion was not fabricated, as Dummett suggests, simply to make

palateable the view that truth values are objects. But it may seem that

Frege made illegitimate use of a criterion that was first developed in a

context in which the denotation of sentences was not an ontological

issue - resorting to a cheap means for ontological gain.

There is something to this worry. I believe, however, that it cannot be

taken at face value. Frege does argue from his characterization of

objects to Thesis (d) (in the paragraph following the relevant charac-

terization of objects in 'Function and Concept'). But he does not take

the characterization as stipulated or ungrounded. In the first place, there

are substantial analogical considerations that undedy his counting

sentences and terms "complete" and predicates and function signs

''incomplete.'' In the second place, Frege seems to have always regarded

the characterization of objects as resting on an antecedent notion of

completeness that he believed he could apply to sentences (and truth

values) as well as to terms and ordinary objects. It is the notion of

completeness that bears the weight, not the bare claim that objects are

the denotations of every sort of symbol other than function signs. The

FREGE ON TRUTH 117

intuitive notion of completeness underlies and motivates the syntactical, semantical, and ontological claims.

Then isn't Frege's engrossment in the completeness-incompleteness distinction simply a metaphysical indulgence? I would not deny that some of Frege's uses of the distinction involve a kind of fixation that is difficult to fathom, much less defend. But his deployment of the distinc-tion to support his view that the denotations of sentences, truth values, are objects, seems to me less problematic than some other uses he makes of the distinction.

Let us lay aside Frege's view that no objects are functions and no functions are objects. I think that this view is extremely doubtful and that it probably does constitute an instance in which Frege allowed his sound conceptions of logical function to harden unnecessarily into a metaphysical doctrine. These matters are, however, intertwined with a suprisingly large number of serious considerations (for some of them, see Burge, 1984). I shall avoid the tangle here.

Let us consider only Frege's views that in using (what were under logical analysis) function signs, one is committed to their denotations, functions; and that in using (what were under logical analysis) terms, one is committed to their denotations, objects. As we have noted, predicates are like function signs in having empty argument places, and in having a functional role in logical theory. Sentences are like terms in not manifesting such formal incompleteness and in not having a func-tional role. In concluding that the denotations of sentences are objects, Frege may be reasonably seen not as drawing a primitively minded inference from some pre-Socratic vision of the world as a mixture of the complete and incomplete - but as simply summing up and embellishing the analogies, within his logical system, between the roles of sentences and terms, and their contrasts with predicates and function signs.

The mapping of objects and functions onto truth values - the central semantical feature of sentences - is the primary formal role of predicate expressions (or concepts) within formal logical theory. The deep differ-ences between predicates and ordinary function signs, and between sentences and terms, were largely shunted off into the theory of force or use. Frege did not lose sight of the differences. But he thought that he could draw ontological conclusions from a semantical theory that abstracted from them. In regarding concepts as functions and truth values as objects on grounds of the "incompleteness" of signs for the latter, Frege was basing ontological commitments on

118 TYLER BURGE

the semantical analysis of the logical forms of sentences in whose truth he believed. Frege's methods, if not his conclusions, seem unexceptionable.

Clarification of Extensions of Concepts

The assimilation of concepts to functions served one other large pur-pose in Frege's system. It provided the key to his attempt to clarify the notions of a concept and of the extension of a concept. As I have tried to show in some detail elsewhere, Frege was unclear about and dissatis-fied with these notions from the time he first introduced the latter in Foundations (Section 68) (1884) up to and through the publication of Basic Laws. (1903). (Cf. Burge, 1984.) The key to the clarifica-tion that he attempted, until Russell unsettled him, was the notion of the course of values of a function. Frege sought to make this notion intuitive by appeal to the graph of a function (G & B, 'F & C' 251KS 129)

which he seemed to think of both algebraically and geometrically. He self-consciously did not interpret the graphs as a set of ordered pairs, for a variety of reasons deriving from his emphasis on the priority of functions over their courses of values. We shall return to these points in Section III.

The notion of a concept had had a long but mathematically barren career in the logical tradition. It was not held in high esteem by mathematicians in Frege's day. By contrast, the notion of a function was well established in mathematics. By assimilating the denotations of predicates to those of function signs - giving them a recognizably mathematical role - Frege hoped to clarify the notion of a concept and burnish its reputation. At the same time, he would be effecting a unifi-cation of the languages of logic and mathematics in accord with his logicist thesis. This motivation is explicit when Frege first introduces the assimilation of concepts to functions:

... for what purpose, then, are the signs '=', '> " , <' admitted into the circle of those that help form a functional expression? It seems that nowadays more and more sup-porters are being won to the view that arithmetic is further-developed logic ... I too am of this opinion, and I base upon it the requirement that the symbolic language of arithmetic must be expanded into logical symbolism (G & B, 'F & c', 30/KS 132.)

The clarification of the notion of a concept was intended to give a firm foundation to those objects logically associated with concepts (their

FREGE ON TRUTH 119

"extensions") with which in Foundations, driven by grammatical con-siderations and his logicist goal, Frege wanted to identify the numbers. The extension of a concept was understood as the course of values or graph, obtained by providing all objects one by one as arguments for the concept (function) and taking the resulting truth values as values. The whole procedure, taken as a completed whole, was what Frege regarded as a logical object. Since the introduction of such objects crucially depended on Axiom V, which led to Russell's paradox, Frege's attempt to clarify the notion of the extension of a concept by assimilating concepts to functions failed.

The Redundancy Conception of Truth and the Notion of Object

I shall conclude our discussion of Frege's reasons for accepting Theses (c) and (d) by considering his redundancy view of truth. In 'On Sense and Denotation' he writes:

One might be tempted to regard the relation of the thought to the True not as that of sense to denotation but rather as that of subject to predicate. One can, indeed, say: ''The thought that 5 is a prime number, is true." But if one observes more closely, one notices that really nothing more is thereby said than in the sentence '5 is a prime number.' ('S & R' G & B 641 KS 150). (Cf. also PPW 128-9, 233-4, 251-2, 255-61 NS 139-140,251-2,271-2,275-6; 'The Thought' Kl. 5141KS 347.)

Frege uses the view to ward off possible doubts about the postulation of the truth values as objects denoted by all sentences, regardless of subject matter. If truth were an attribute of a limited range of entities (thoughts), it would be difficult to motivate the claim that every sentence denotes one of the truth values and his view that (in a sense to be sharpened in Section III) all assertive uses of sentences regardless of subject matter are committed to the object truth.

In 'On Sense and Denotation', two large philosophical ideas emerge in connection with the redundancy conception. One utilizes truth values as objects in an account of assertion and judgment. The other bears on scepticism. We shall consider these themes in turn.

Frege goes on from the passage just cited to argue that the claim or judgment that a thought is true arises not from the predication of 'is true' of a thought, since the sentences 'the thought that 5 is prime is true' and '5 is prime' express the same thought regardless of whether they are used with or without assertive force. Truth claims or judgments depend

120 TYLER BURGE

on the combination of the form of a declarative sentence with its "usual force." Frege thinks that such truth claims are indicative of the real relation between a sentence or thought and its truth value. (G & B 'S & R' 65/ KS 150). Judgments are "advances from thoughts to truth values." Since truth claims and judgments cannot be represented in subject-predicate form, the relation of a sentence or thought to its truth value cannot be regarded as that of subsumption of a thought (sentence) under a property. Frege proposes that the appropriate relation is that of a sentence or thought to its denotation. (Cf. also PPW 128-9, 233-4, 251-2/NS 139-40, 252-3, 271-2.)

One need hardly note that considered as an argument for Thesis (d), this is pretty weak. (It is doubtful that Frege intended it as such.) One could respond that on Frege's own account, two sentences could have the same assertive force - both could count as assertions - while one lacked a truth value and the other had one. So truth values' as objects cannot be essential to the account of assertive force. Even if this reply is not decisive, Frege does not show why it is not.

I think Frege was here again thinking analogically. Normally, the point of using names was to secure a denotation, a bearer, to relate a mode of presentation to an object. Normally, the point of using sen-tences, what "matters to us," is to claim truth for a thought. The object, in the sense of the point or objective, of sentence use was truth. It is illuminating therefore to see truth as an object. There is more than a suggestion of this reasoning when Frege writes:

The designation of truth values as objects may here appear as arbitrary fancy or perhaps a mere play on words, out of which no profound consequences could be drawn. What I call an object can be more exactly articulated only in connection with concept and relation .... But so much should already be clear, that in every judgment, no matter how trivial, the step from the level of thoughts to the level of denotations (the objective) has already been taken. (G & B, 'S & R', 63-41 KS 149)

The parenthetical phrase is the key to the passage. To many this reasoning may seem indeed to rely on a mere pun on

the word 'object'. I think that there is more to it than that. Both the relevant objective of sentence use, truth, and objects that are denoted by terms are for Frege mind-independent. And in some sense they are what sentences and terms are respectively "about." (In fact, as Frege empha-sizes in his arguments for (a) and (b), objects are of interest to us because and only because of the objective of assertion and judgment.)

FREGE ON TRUTH 121

Both points resist simple or quick put-downs. Both mind independence and being the topic of a discourse are involved in traditional explica-tions, stemming from Aristotle, of the notion of object.

I am not suggesting that the analogies between the "objects" of terms and the "object" (objective) of sentence use provide a sound argument for Frege's assimilation. I believe the contrary. In fact, I believe not only that the analogies are not compelling, but also that Frege's redundancy view of truth,which motivates them, is untenable. (Part of the reason for this untenability lies in the semantical paradoxes; cf. Section III.) Rather, what I am suggesting is how Frege might have come to see the analogy as intuitively attractive, given his view that the attribution of truth added nothing to a thought. We shall further articulate Frege's analogy between objects and the objective of assertion, in Section III.

Frege puts Thesis (d) and the redundancy view of truth to use as a weapon against the sceptic about an objective world. Frege writes that the True and the False "are recognized, if only implicitly, by everybody who judges something to be true - and so even by the sceptic" (G & B, 'S & R 63/KS 149). (Frege assumes contrary to the legends about Pyrrho, but probably correctly, that no sceptic suspends all judgments.) The idea is that every act of judgment aims at truth and presupposes some discrimination between truth and falsity. Frege explicates the point by his redundancy thesis: truth values are not a property of thought, where thoughts constitute one subject matter among many: "What distinguishes [truth] from all other predicates is that it is always asserted when anything at all is asserted." (PPW 129/ NS 140.) Since the true is an object logically associated with the truth predicate and so with judgment - a logical, mind-independent object - , judgment itself presupposes an objective world. We shall sharpen Frege's point in Section III. I think that one could probably dispense with the implausibi-lities of the redundancy view to provide the sort of premise needed for joining with (d) in order to defeat the relevant sceptic. If only (d) were true!

Godel remarked that Frege held the view that all true sentences have the same denotation "in an almost metaphysical sense" (Godel 1964, 214). It is true that Frege puts the doctrine to use against the sceptic. There is no question but that he thought of truth as an object. And there are some unfortunate, but qualified and never repeated remarks in 'On Sense and Denotation' about parts of the True (G & B, 'S & R' 65/ KS

122 TYLER BURGE

150-1) - remarks that prompted G6del (inappropriately, I think) to compare Frege's view with Parmenides'. But if the reasons for a view may be seen as an index of its character, Frege's doctrine cannot comfortably be called metaphysical. The brunt of his case for (d) rests on the formal simplifications the view effects within his logical theory and the clarification it was supposed to yield for the notion of the extension of a concept. With the discovery of Russell's paradox, the latter support was undermined, leaving only the former.

It is interesting that in his post-paradox period, Frege cites only considerations of simplification (for example, the congeniality of the view with the Composition Principle (1» in favor of Thesis (d). In his epistolary responses to Russell's doubts he remains doggedly within the elegant confines of his logical theory - repeatedly employing the composition principle and pointing out difficulties in Russell's vague but seminal alternatives. In 1906 at the beginning of the scrap 'What May I regard as the Result of my Work?', he cites "a concept construed as a function" and introduces the citation with the remark, "It is almost all tied up with the Begriffsschrift" (PPW 184/NS 200). In his late writings he gives up on the notion of the extension of a concept, and in 'The Thought' (1918) the arguments against scepticism make no use of truth values.

Once Frege's intra-logical analogies are appreciated, there is, I think, no need other than momentary expositional convenience to treat sen-tences as of the same logical type as names. One may maintain in one's semantical theory a reflection of the large differences in use between sentences and terms. And one may return to the natural view that terms, not including sentences, are the basic avenue of ontological commit-ment.

The primary reason why Frege did not take this more modern view of the matter in his great, pre-paradox writings is that he wanted to use the truth values in his account of logical objects. Logical objects were needed for his logicist project - the project of showing that the mathe-matics of number is reducible to logic. For mathematics was apparently committed to objects, the numbers; and to account for these commit-ments Frege thought he had to generate commitments to appropriate sorts of objects within logic. The truth values were the basic logical objects from which all others were to be generated. (Cf. Section III.)

By roughly 1906, however, Frege seems to have given up logicism'. So the most prominent philosophical motivation for postulating logical

FREGE ON TRUTH 123

objects lapsed. The doctrine that truth values are objects may have become less important to him in his later years. He does not give up the view, however. And I suspect that in addition to analogical or pragmatic considerations, he retained a philosophical motive for holding it. This motive was his desire to explicate the objectivity and informativeness of logic - its "descriptive" as well as normative character. (Cf. Note 3 and Section III.) Although I shall not discuss it here, I think that this motive is profoundly conceived. But after the failure of Frege's logicist project, the attempt to utilize the truth values as means to articulate the motive was deprived of a coherent background theory within which to bring together a conception of truth with a conception of logical objects. So Frege is left without a theory within which he could argue for using Thesis (d) to articulate his thoroughly unKantian view that logic is an informative science of "being" (K150S1 KS 342).

We have briefly touched on Frege's view about logical objects in our discussions of his attempt to clarify the notion of the extension of a concept, his argument against the sceptic, and his conception of the nature of logic. We shall return to them in Section Ill. But first, I want to consider an influential body of thought that seems to me to have placed Frege's views on truth values in the wrong light. This discussion will enable us to develop further Frege's conception of truth, a conception that will dominate our concluding reflections on logical objects.

II

In his two books on Frege, Michael Dummett maintains, as against Theses (c) and (d), that sentences are not names, and truth values are not objects. As is plain, I do not dispute this conclusion. It is the reasoning behind Dummett's rejection of these theses, and the urgency with which he invests that rejection, that constitute, in my opinion, a serious misrepresentation of FregeY

We have already quoted Dummett's statement that Frege's accept-ance of Theses (c) and (d) was an almost unmitigated disaster. For, Dummett writes, ... it obscured the crucial fact that the utterance of a sentence, a complex term ... can be used to effect a linguistic act, to make an assertion, give a command ... the general notion of the sense of a word will now have to be taken to consist in the contribution which that word makes to determining what a complex singular term, in which it may occur, stands for, rather than what are the truth-conditions of a sentence in which it may occur. (FPL, 7)

124 TYLER BURGE

If sentences are merely a special case of complex proper names, ... then, after all, there is nothing unique about sentences: whatever was thought to be special about them should be ascribed, rather, to proper names - complete expressions - in general. This was the most disastrous of the effects of the misbegotten doctrine that sentences are a species of complex name ... : to rob him of the insight that sentences playa unique role, and that the role of almost every other linguistic expression ... consists in its part in forming sentences. (FPL, 196; cf. 643-5)

Dummett takes the adoption of Thesis (c) to underlie the relative inconspicuousness, in Frege's later work, of statements of the context principles, statements which had been so prominent in The Foundations

of Arithmetic (p. x, and Sections 60, 62, 106). Dummett's idea is that since Frege's assimilated sentences to complex singular terms, he "debarred himself from a direct statement of the context principle, since this would have involved acknowledging a difference in logical role, of utmost importance, between sentences and proper names of objects other than truth-values" (IFP, 371). Dummett cites Frege's conclusion to Section 10 of The Basic Laws of Arithmetic (cf. also Sections 29, 31-2) as evidence that only a weakened, generalized analog of the context principle for denotation was still adhered to: The denotations of terms are fixed when it has been determined for every primitive function [whether a concept or not] what the value of the function is to be for the denotations of any terms as argument(s) (IFP, 408ft). In this principle sentences and predication are given no special prominence over terms and ordinary functional application. Dummett goes on to question the coherence of the resulting doctrine.

Now there is much in Dummett's discussion that we cannot take time to go into. The context principles form an exceedingly complex topic. Despite my disagreement on some fundamental matters in this area, I think that Dummett has contributed a great deal to our understanding of the issues. Here I shall concentrate on disagreements that bear most directly on truth values.

There is evidence that Frege did not lose sight of the "crucial fact" that the utterance of a sentence, unlike a term, can be used to make assertion; that he did not draw the unsound inference Dummett does that "if sentences are merely a special case of proper names ... then, after all, there is nothing unique about sentences ... "; and that Frege was never robbed of the insight "that sentences playa unique role."

In the first place, there are a great number of passages throughout his career and especially from the 1890's onward, in which Frege asserts

FREGE ON TRUTH 125

that the aim of logic is to understand the laws of truth (PPW 2-3,

128-9, 149, 197-8, 252, 253/NS 2-3, 139-40, 161, 212-13, 272, 273; Kl 505ff/ KS 342ff). He repeatedly characterizes these laws as normative restrictions on judgment and assertion. Predications of truth are not really distinguishable from the assertoric form of any declarative sentence at all (PPW 129, 2331NS 140,251; cf. our discussion, Section I, of the redundancy theory of truth.) Once he writes that the essence of logic lies in assertoric, or judgmental, force (PPW 252/NS 272). The vehicle of judgment is a thOUght and the vehicle of assertion (the expression of a judgment) is a sentence (PPW 126, 131, 206/NS 157,

142,222-3). Thus the essence and aim of logic is repeatedly associated with sentences and thoughts (the senses of declarative sentences) and their logically relevant uses. The denotations of terms are almost never discussed except in the larger context of this emphasis on the centrality of truth, judgments, thought, assertion, and sentencehood. And in Frege's last years, the denotations of terms receive very little attention at all.

Moreover, there are the passages from the 1890's and later, quoted in Section I, that occur in Frege's arguments for Theses (a) and (b) (G & B, 'S & R' 631KS 149; Cor. 152, 158, 163n, 1651BW235, 240,

245n, 165; PPW232/ NS 250-1). These repeatedly and explicitly make the point that the denotations of terms are of interest to us only because of our interest in the denotations, in fact the truth values, of sentences. Indeed, the remarks constitute a fair approximation to the slogan of Foundations that only in the context of a sentence do words have a Bedeutung.

Further, the implication of the same passages is that our interest and confidence in the truth of sentences that contain terms justifies our interest and confidence in the terms' having the denotations that they are commonly taken to have. This implication appears to echo and perhaps even sharpen the motivation for one of the uses to which Frege put his contextualism in Foundations (Sections 60, 62) - defending ontological commitment to objects (numbers) in the absence of an intuitive, imagistic, or causal relation to them. Only the general prin-ciple underlying this use is suggested in the argument for Theses (a) and (b). But it is clearly indicated: ontological commitment to the denotation of terms is justified insofar as we are justified in acknowledging the truth of sentences that contain them. It is noteworthy that these develop-ments of Frege's contextualist thinking occur in arguments for Theses

126 TYLER BURGE

(a) and (b), which are in turn embedded in arguments for Theses (c) and (d) - the very theses that Dummett holds prevented Frege from main-taining the prominence of sentences in his contextualist principles.

What then are we to say of the considerations Dummett draws from The Basic Laws of Arithmetic to support his view that Theses (c) and (d) undermined Frege's commitment to the centrality of sentences in logical theory? Let us begin with the passage Dummett cites from Section 10 of Basic Laws that states a weakening of the context principle, one that gives no special prominence to sentences. What Frege writes is as follows:

With this we have determined the courses of values so far as is here possible. As soon as there is a further question of introducing a function that is not completely reducible to already familiar functions, we can lay down what value it is to have for courses of values as arguments; and this can then be regarded as much as a determination of the courses of values as of that function.

A similar passage occurs in Section 29:

A proper name has a denotation if the proper name that results from that name's filling the argument places of a denoting name of a first-level function with one argument always has a denotation, and if the name of a first-level function of one argument that results from the relevant proper name's filling the of a denoting name of a first-level function with two arguments always has a denotation, and if the same holds also for the t-argument-places.

These remarks do indeed state a kind of context principle for fixing term denotation - one that does not give prominence to sentences. First-level concepts are not singled out from among the first-level functions. (Part of the reason for this derives from a particular problem that Frege raises in Section 10 about the interpretation of his Axiom V. I shall not go into this point here since it would require substantial stage-setting.)

Although the principles just quoted do not give prominence to predication over functional application, or to sentences over terms, they are unquestionably compatible with the view that ultimately it is the use of a subclass of "terms," the sentences, that counts in justifying interest in term denotation and confidence in identifying the denotations of course-of-value terms. Dummett is right to note that Frege does not explicitly draw this distinction in Basic Laws. He is probably also right in holding that Frege's not doing so is partly explained by his commit-

FREGE ON TRUTH 127

ment to counting sentences as falling in the same syntacial category as terms. But it does not follow that Frege had lost sight of the philosophi-cal motivations underlying the formal system that he repeatedly stated in other writings during the same period. The circumstance bespeaks a lack of perspicuousness in the formal system - the price of the various economies Frege prized. But it does not evince a major philosophical turn away from the centrality of sentences, ultimately judgment, in motivating logical theory.

I think that the main reason Frege gives no special prominence to sentences over terms in Sections 10 and 29 is that to make intelligible the primacy of concepts (or predication) in fixing term denotation, he would have had to have entered on an excursus into his philosophy of language. Such an excursus would have been incongruous in the context of the book as whole, where philosophical discussion was held to a mini-mum. The strategy of Basic Laws is ruthlessly to suppress discussion of philosophical ideas and motivations, except where they are essential to understanding the formal system and the proofs themselves, or where they bear directly on mathematical practice (as in the case of the discus-sions of definition and consistency). Where philosophical ideas intrude, they are presented tersely and in summary fashion. Except for the polemical introduction, the book is steadfastly mathematical.

The chief consideration that Dummett relies upon for holding that (c) and (d) undermined Frege's commitment to the centrality of sentences in logical theory is that "the whole thrust of [Frege's] logical doctrines" was "to recognize no difference in the kind of logical powers that different expressions have save as were explicable by a difference in logical type" (IFP, 371-2). Since by Thesis (c), sentences and terms are of the same logical type, it follows that they can have no difference in logical power. Dummett admits that Frege never states such a principle. But he holds, "it is implicit in his whole procedure; nothing could illustrate it more aptly than the fact that, in the logical system of Grundgesetze, no distinction exists between sentential and individual variables ... " (IFP, 372).

The evidence of the numerous passages that we cited six paragraphs back indicates that this principle must be severely qualified. Although sentences and terms are of the same logical type, according to Frege, some properties in which they differ are of direct and primary impor-tance to logic. Sentences can make assertions and express judgments; terms cannot. The semantical properties of terms are of interest to us

128 TYLER BURGE

only because of our interest in the seman tical properties and use of sentences and thoughts. There is no reason for thinking that Frege wanted to deny or suppress these points in his philosophical writings. As we have seen they are prominent in his post-Foundations work.

In fact, sentences and terms are not everywhere interchangeable even within the formal system Frege presents in Basic Laws. So in a further sense, they do not have the same "logical powers" despite the fact that they are of the same "logical type." Only sentences can follow the vertical judgment stroke in Frege's syntax; ordinary terms cannot. This important point requires detailed explication. I shall develop it by reference to a further consideration that Dummett adduces in favor of his view.

Dummett notes that whereas in the Begriffsschrift there is a restriction in the formation rules against placing the horizontal or content stroke before anything other than an expression with judgeable content -anything other than a sentence - , in Basic Laws this restriction is relaxed. In the latter book, the horizontal may occur before any term (or sentence) yielding "a name of a truth value, of the True if the original expression named the True and of the False in all other cases" (IFP,

371). Dummett does not explain what he takes the significance of this fact to be. But it may suggest to the unwary that Frege's system was set up so as to allow one to "judge" (impossibly) the contents of terms. For example, both '- 5' and 'IT 5' are grammatical expression in Frege's logic, where the shorter vertical line in the latter expression represents negation. (Cf. the end of Basic Laws, Section 6.)

This reasoning would be quite mistaken. (I do not claim that Dummett employs it.) In fact, Frege's use of the horizontal in Basic

Laws constitutes one of the subtleties of the book that suggest that Frege was keeping his philosophical motivations in mind. 1 do not see that the use supports Dummett's view in any way. To begin with, although the horizontal may apply to ,any name, it is itself a concept expression: a function from objects to truth values, as Frege explains (BL, Section 6). Informally, the horizontal means "is the True:" Concept expressions are predicates and concepts are the denotations of senten-tial parts (e.g. PPW 119, 1931NS 129, 210). Thus the expression' - 5' is a sentence, though a false one. It says that 5 is the True. ' IT 5' represents an assertion that 5 is'not the true.

Now the vertical judgment stroke can be applied only to the hori-zontal, content stroke. So it is built into Frege's system, however

FREGE ON TRUTH 129

discretely, that only sentences, not ordinary terms, may be asserted. Only the senses of sentences, thoughts, may be marked as judged. Frege himself makes the point:

I distinguish the judgment from the thought in this way: by a judgment I understand the acknowledgement of the truth of a thought. The presentation in the concept script (begriffsschriftliche Darstellung) of a judgment by use of the sign" f- ", I call a statement (Satz) of the concept script, or briefly a statement .... Of the two signs of which" f-" is composed, only the judgment stroke contains the act of assertion (BL, Section 5).

Judgments acknowledge the truth of a thought, and thoughts are said, over and over again throughout the period and afterward, to be charac-teristically expressed by declarative sentences: 'The proper means of expression of a thought is a sentence' (1897) (PPW 126, 131/NS, 137,

142-3). (Cf. also PPW 129, 138, 167, 174, 197-8,206,216, 243/NS

140,150,182,189,213-14,222-3,234,262; KI'The Tho't' 5111KS

345; G & B'S & R' 64/ KS 150 etc.)IO The result of attaching the judgment stroke to a sentential expression,

begun by the horizontal, asserts something, but it is not a term: "The judgment stroke cannot be used to construct a functional expression; for it does not serve, in conjunction with other signs, to designate an object: , I- 2 + 3 = 5' does not designate anything; it asserts something." (G & B, 'F & C, 34/KS 137) The vertical, judgment stroke is not a function sign, but is the sign of an act - judgment or assertion - , an act that applies only to thoughts or sentences. (This is why one cannot substitute a singular term denoting truth for the sentence beginning with the horizontal in the expression' I- 2 + 3 = 5' (which would yield the ungrammatical 'I the True').) It is here that the distinction between sentences and terms finds its representation within Basic Laws. I I

The change regarding the grammar of the horizontal that Frege makes between Begriffsschrift and Basic Laws is partly motivated by the grammatical assimilation of sentences to terms. But this motivation is less important than one might think. For in one sense the grammatical assimilation of sentences to terms was already present in Begriffsschrift.

Insofar as this is so, the view that adoption of the position effected a major change in Frege's later philosophy of language is rendered further implausible. In Begriffsschrift Section 3, Frege writes:

A language is imaginable in which the sentence 'Archimedes perished at the capture of Syracuse' would be expressed in the following way: 'the violent death of Archimedes at the capture of Syracuse is a fact'. Here one can, if one wishes, distinguish subject and

130 TYLER BURGE

predicate; but the subject contains the whole content, and the only purpose of the predicate is to present this as a judgment. Such a language would have only a single predicate for all judgments, namely 'is a fact' .... Such a language is our Begriffsschrift, and the sign' I- ' is its common predicate for all judgments.

Here Frege is primarily intending to make the point that the subject-predicate distinction of natural language has no comparable importance in his logical theory. But the passage also indicates a more radical point of view that the "content" of the first sentence can be completely captured by the subject, a term, in the second. The sign 'f--' is seen in Begriffsschrift as a predicate that adds nothing to the content of the term to which it applies. This viewpoint contains more than the germ of Frege's later commitments to the grammatical assimilation of sentences to terms and to the redundancy conception of truth.

The changes from this position in Begriffsschrift to his later stance in Basic Laws are fairly easy to separate out. In the first place, Frege more clearly distinguished in the sign 'f--' an element corresponding to jUdgmental force and an element corresponding to the expression 'is a fact' or 'is a truth'. (The running together of force with semantical attribution occurs elsewhere in the Begriffsschrift. Cf. for example the semantics given in Section 5.) Thus, the vertical judgment stroke repre-sents judgmental force, and the horizontal alone comes to represent a semantical predicate, such as 'is a fact' or 'is true'. Presumably this distinction is accompanied by the rejection in 'Function and Concept' of the Begriffsschrift view that the sign' f-- ' is a predicate (G & B, 'F & C' 34/KS 137 - quoted above). On the other hand, the horizontal, taken alone, is a predicate whose meaning is similar to that of 'is a fact'.

Distinguishing force from predication in the sign 'f--' probably made it easier for Frege to relax the Begriffsschrift restriction against following the sign' f-- ' with anything but a judgeable content. What was asserted need not be just what followed the horizontal, it could be the predication of the horizontal onto what followed it. As I have noted earlier, Frege had already in Begriffsschrift come to view predicates as function signs (Section 9). Given that he was also already treating the grammar and "content" of sentences as equivalent to that of terms that nominalize those sentences, it may have seemed a small step to allow the horizontal to be functionally applicable to all terms, simple and complex, clausal and nonclausal. The vertical judgment stroke could still only apply to sentences, the expression of somethingjudgeable.

FREGE ON TRUTH 131

What seems to be the large step in this development, in addition to the distinction between force and predication, is the semantical clarifi-cation that Frege achieved. The semantical standpoint developed in Theses (c)-(d) in effect answered the question of what the arguments and values of the horizontal should be. The redundancy conception of truth is the natural offspring of this semantical development and the viewpoint expressed in Begriffsschrift Section 3. The universal predicate 'is a fact' gives way to the universal predicate 'is the true'. But in neither case does the predicate add to the "content" (sense) of what followsP The predicate does not change the sense (or denotation) of the results of ordinary predication.

Thus the grammatical change in the restrictions on the horizontal between the Begriffsschrift and Basic Laws is not really a change from allowing only sentences to occur to allowing terms to occur. It is from allowing only the occurrence of terms that nominalize sentences to allowing all terms. Thesis (c) played a role in motivating this change. But the developments associated with the changed use of ' 1-' that seem most significant are different. The significant developments are Frege's drawing the semantical consequences of viewing predication as func-tional application (not the mere viewing of predication as functional application, which is already present in Begriffsschrift), and the clear distinction between judgmental force and predication. This latter devel-opment, and the prominence Frege gave to truth and judgment in motivating logical theory, undermine any claim that the grammatical assimilation of sentences to terms deprived him of his insight into the basic role of sentences in logical theory.

III

The claim that truth values are objects inevitably suggested to Frege the question 'Which objects?' A parallel question had arisen in The Foun�

dations of Arithmetic, once it had been concluded that numbers were objects. Frege was sensitive to the initial possibility that the answer to the latter question might be no other than 'why, the numbers - 0, 1, 2 .. .' Similarly, the truth values might tum out to be specifiable only as truth and falsity. But Frege's belief in logicism drove him to seek a different answer in the case of numbers. Similar forces were at work in his views on truth values.

132 TYLER BURGE

In The Basic Laws of Arithmetic the truth values are identified with particular logical objects, particular extensions of concepts. The reason-ing behind this identification is the subject of this final section. Our discussion of this subject must be more conjectural than that of Theses (a)-(d) because Frege wrote very little directly about it. Nevertheless, by piecing together different strands of his views, it is possible, I think, to weave a pattern that has some interest, and even a kind of blemished beauty.

One reason why Frege's reasoning is interesting is that it sheds light on his conception of logical objects. Another is that it is critical to assessing what sort of realism Frege maintained with regard to such objects, and with regard to numbers. Each of these issues is quite difficult and complicated. I shall begin with some very rudimentary background for the realism issue.

A common and straightforward story about Frege's realism goes as follows:

Frege believed that the numbers were genuine, existing abstract objects. He thought, however, that number theory was reducible to logic. He proceeded to try to show this by constructing a logic containing a version of set theory. He gave definitions, within the logic, of the primitive expressions of number theory, and tried to derive the axioms and theorems of number theory within his logic. Since he had a realist attitude toward the ontologies of the languages of both number theory and logic, and since he regarded numbers as particular objects, he thought that there was but one way to construct the definitions of numerical expressions within his version of set theory. As it turned out Frege's set theory is inconsistent; and for any viable set theory there are an infinite number of ways of defining arithmetic within it. So even if his set theory had been consistent and even granting that set theory is logic, Frege's logicism and his realism about the numbers are, if not incompatible, at least deeply at odds.

There is much that is right about this familiar recitation. But it seems to me misleading in some fundamental ways. The first derives from Frege's attitude toward all language other than his own concept script. It is well known that Frege thought that natural language was defective for the purpose of expressing thOUght. But he also thought the same of mathematics itself. Within mathematics, the problem was partly just that the language had not been given logical form. But vagueness was also a problem. Frege repeatedly notes that the content or sense of the term 'number' is not adequately or sharply grasped by even the most com-petent mathematicians. Other fundamental, long-standing arithmetical terms are afflicted by vague usage. In the first section of The

Foundations of Arithmetic, he writes:

FREGE ON TRUTH 133

The concepts of function, continuity, limit and infinity have been shown to stand in need of sharper determination. Negative and irrational numbers, which had long since been admitted into science, have had to undergo closer scrutiny of their credentials. In all directions these same ideals can be seen at work - rigor of proof, precise delimitation of extent of validity, and as a means to this, sharp grasp of concepts.13

In the introduction to the book, referring indirectly to the concept of number, Frege writes, 'Often it is only after immense intellectual effort, which can continue over centuries, that a concept is successfully recog-nized in its purity, stripped of foreign coverings that hid it from the eye of the intellect' (p. vii). (Cf. KS 122.) Clothing, covering, veiling are standard Fregean metaphors for the interferences of language in thought. This theme runs throughout the book. Indeed, the book may be fruitfully read as an attempt to remedy the inadequacies of language (primarily mathematical language) for ideally rational conceptualization and thought. If one substitutes 'perception' for 'language', one has the schema for the traditional rationalist program for freeing the intellect from non-rational factors.

Any number of senses and denotations were compatible with the conventional significance of vague terms. It is clear that Frege thought that conventional mathematical usage left mathematical terms vague. That is, what a conventionally competent speaker masters does not fix a definite sense or denotation. Frege thought that his logicist program was required to uncover the senses and denotations of number words. So strictly speaking, defining arithmetical terminology is not a matter of capturirig linguistic meaning as we commoly understand it, but of uncovering supplementations of such meaning so that the terminology can be seen to have a definite sense and denotation. Sense lay beyond or beneath conventional significance. (These points are discussed and substantiated in some detail in Burge, 1984.)

It would be a mistake to infer from this point that Frege held that defining numerical terminology involved stipulating meanings for it. The definitions had to respect mathematical practice. Moreover, I think it plausible that Frege thought that only one set of definitions (his) respected all relevant philosophical considerations. Frege's point is that by merely understanding the linguistic meaning of ordinary mathemati-cal language, by being a competent participant in the conventions governing it, one did not thereby attain a completely adequate grasp of numerical concepts; one did not thereby secure completely definite denotations for number words.

134 TYLER BURGE

Thus if Frege did think that there was a single, correct set of logicist definitions, it was not because the ordinary conventional meaning of mathematical language and standard mathematical practice allowed only one such set. Since mathematical usage was vague, it admitted of many sharpenings. Rather the unique propriety of a set of logicist definitions would have to depend partly on philosophical considera-tions, considerations attendent on the logicist program.

Thus Frege's realism about the ontology of ordinary mathematics is more subtle and qualified than the familiar narrative spun above suggests. His realist attitude toward the language of mathematics is tied to and supplemented by the assumption that his logical theory gives a proper account of the objects and Junctions in the mathematical world.

Even with respect to his logical language, his realism is not com-pletely unqualified. Not every logical constituent of the language corre-sponds to an item in reality. For example, the function sign negation does not in general correspond to a thought component. (PPW 149-50,

185, 1981NS 161-2, 201, 214; Cf., however, G & B "N" 131-21KS

374-5). Nevertheless, Frege thought of his logic as a tool for discovering the

nature of the mind-independent world, at least that portion of the world with which mathematics was concerned: " ... The mathematician cannot create something at will, any more than a geographer can; he too can only discover what is there and name it" (FA 107-8). The theme runs through Foundations, his correspondence with Hilbert, and his attack on the formalists; it emerges in the introduction to Basic Laws, and it recurs in his late writings. Functions, thoughts, and (at least until the despair over Russell's paradox) courses of values are among the charter members of the mind-independent world. Although this traditional "realist" interpretation has been questioned now and again, I think it fundamentally secure and will not argue for it in general terms here.14

A second way in which the familiar account of Frege's realism that I recited above is misleading concerns the references to set theory. I will not go into this complicated and somewhat obscure matter here. (I have discussed it in Burge, 1984.) I will just say bluntly that it is a mistake to think of Frege's theories of courses of values and extensions of concepts primarily in terms of what we now know as set theory. This is not because Frege's theory turned out inconsistent. It is because-he consciously and repeatedly argued against the basic intuitions that underlie the iterative conception of sets, and because the fundamental

FREGE ON TRUTH 135

intuitions underlying his own theory have only scattered echoes in mainstream set theory and even in the various nonstandard versions. The question of whether and in what sense Frege's logicism and his realism about numbers are affected by the multiplicity of models of arithmetic within set-theory is clearly bound up with these matters. This is a question that I shall not attempt to settle here.

If Frege's extensions of concepts were thought of set-theoretically, the axiom of foundation would be everywhere violated. This is a prime reason for feeling queasy about the set-theoretic explication. This reason is a corollary of a more fundamental one - the primacy for Frege of concepts over classes, or courses of values. It is better to think of extensions of concepts visually in terms of a geometrically represented graph, or yet better as the total (abstract) event of matching each of the arguments with their truth values, one by one. Needless to say, these heuristics give one only a vague start at the notion. (As logical objects, extensions of concepts were not supposed to be dependent for their conception on intuition or vision.) Frege never achieved a dear con-ception of extensions of concepts that accorded with his philosophical and mathematical preconceptions. It is arguable that no one else has either. So we must be willing sometimes to grope along in the dim afterglow of Russell's devastating paradox if we are to follow the course of his reasoning.

The common thrust of the two main caveats that we have entered in the familiar story about Frege's realism is that we need to be sensitive to the role of his philosophical considerations, beyond what we now think of as standard mathematical (arithmetical and set-theoretic) practice, in assessing the character and plausibility of Frege's realism about mathe-matical and logical objects.

How do truth values figure in all of this? They are fundamental in Frege's notion of the extensions of concepts, a subset of which consti-tutes the primary logical objects. Since concepts are functions from arguments to truth values, and extensions of concepts are courses of such values, the truth values chart the courses. Since the numbers are certain extensions of concepts and since truth values thus figure essen-tially in the ontology of the numbers, consideration of them is insepar-able from consideration of Frege's realism about numbers.

But there is a more specific reason for ontological interest in truth values. Frege identifies not only the numbers but the truth values themselves with courses of values, extensions of concepts. Frege indi-

136 TYLER BURGE

cates that his identification of the truth values with the specific courses of values he chooses in Section 10 of Basic Laws is arbitrary relative to the axioms of his logical theory. Any other choice would have been equally consistent with those axioms. Getting straight what Frege means in this passage, which we shall scrutinize shortly, is critical for under�standing his whole philosophical standpoint.

On its face, the passage in Section 10 of Basic Laws suggests that Frege's theory of truth had a large stipulative component. It also suggests that Frege's ontology of the numbers contains a massive dose of arbitrariness, and that he was aware of this. If these suggestions are correct, then the traditional view of Frege's realism and of the intentions governing his logicist project must suffer substantial qualification. For different choices as to how to identify the truth values with extensions of concepts would yield different accounts of which objects the numbers are.1S

I believe that these initially plausible suggestions are mistaken. Although I shall stop short of a general discussion of how Frege regarded his definitions of the numbers, I shall argue that his identifica�tions of the truth values were, and were known to be, not in the least arbitrary, but supported by reasons. To understand these reasons, we must consider the philosophical context in which Frege conceived his logicist program. The reasons are not narrowly mathematical. Their failure to appear in Basic Laws accords with the predominantly mathe�matical emphasis of the book.

In Section 10 of Basic Laws Frege correctly argues, first, that whether or not one or both truth values are courses of values and, second, which courses of values they are, granted that they are courses of values, is left completely undecided by the axioms of his logical system (in particular by Axiom V). He concludes:

Thus without contradicting our setting 'tq,(e) = t'P(E)' equal [in denotation] to '(x)

(q,(x) = 'II (x»,' it is always possible to stipulate th,!t an arbitrary course of values is to be the True and an arbitrary different one, the False. Accordingly, let us lay down that t (-e) is to be the True and that t(e = - (x) (x =' x» is to be the False.

't4>(e) = l'¥(e)' is read 'the course of values of the concept 4> is identical with the course of values of the concept lJI'; 't( -e), is read 'the course of values of the concept is the True'; 't(e = - (x) (x = x))' is read 'the course of values of the concept being identical with the truth

value of not all objects' being self-identical'. This passage certainly appears to support the view that Frege's choice is a matter of stipulation.

FREGE ON TRUTH 137

And, of course, it is in two ways. The choice involves "stipulation" relative to the commitments of natural language. As Frege often notes, ordinary uses of 'true' and 'false' do not explicitly commit themselves to truth values as objects at all. The commitment is promoted by logical theory. Thus relative to natural language use, the identification of any object as one of the truth values is arbitrary. Frege's choice also involves stipulation relative to considerations of consistency with Axiom V, and the other axioms of his system. Within a context in which mathematical consistency is all that matters, the choice is arbitrary. But let us broaden the context.

On numerous occasions outside of Basic Laws Frege holds that consistency does not suffice for truth. Frege repeatedly defends this view in opposition to early expressions of the model-theoretic viewpoint toward mathematics. Frege's best known defenses of the view occur in his correspondence with Hilbert (1895-1900) and in 'On the Foun-dations of Geometry, I' (1903) (e.g. Cor 48/BW 75; FG, 25-37/KS

264-72). But the view is already quite explicit in 'On Formal Theories of Arithmetic' (1885) (KS, 110). And it clearly guides the criticism of formalism in the closing pages of Foundations (pp. 104-119).

These latter passages are particularly relevant to our theme.16 Frege notes that the denotation of 'the square root of -1' was not fixed by mathematical usage prior to the advent in mathematics of complex numbers (pp. 106-7, 11 On). He then ridicules the view that one can simply introduce a denotation for the term by mere stipulation or definition (pp. 107-8). One reason Frege gives is that even granted that the purported definition is consistent, one is not thereby guaranteed that there exists an object that satisfies the concepts used in the definition (pp. 108ff.): in effect, consistency does not entail truth. A second reason is that even if one succeeds in attaching the term to an object and even if one stipulates meanings for the usual mathematical function signs in application to this object that are compatible with those mathematical principles that had been established prior to the introduction of com-plex numbers, there might be philosophical considerations that militate against the definition.

Let us elaborate the second reason in more detail since it bears directly on the treatment of truth values in Basic Laws. Frege notes that simultaneously with the introduction of new numbers, the meanings of functional words like 'sum' and 'product' are extended. Suppose we choose some object as the denotation of 'the square root of -1', say, the Moon. So the moon multiplied by itself is -1: This explication seems to

138 TYLER BURGE

be permitted because the [denotation] of such a product does not at all arise from the erstwhile denotation [Bedeutung] of multiplication, and therefore in extending this erstwhile denotation it [the denotation of the product] can be arbitrarily determined' (p. 110).17

Frege goes on to consider multiplication and addition as applied to imaginary numbers, and in so doing capriciously takes the time interval of one second, instead of the Moon, as the denotation of 'the square root of -1'. He summarizes by saying, 'one is tempted to conclude: Thus it is quite immaterial whether i denotes a second or a millimeter or anything else, provided only that our laws of addition and multiplication hold good; that alone is what matters; the rest need not trouble us' (p. 111).

Frege does not accept this position. One point he makes against it, less interesting for our proposes, is that a contradiction may lurk between the definitions and the rest of mathematical theory. There is no evidence that Frege thought that the relevant definitions in fact lead to contradiction. Frege's other objection is philosophical. By letting the interval of a second be the denotation of 'the square root of -1',

we are bringing something quite foreign, time, into arithmetic. The second stands in

absolutely no intrinsic relation to the real numbers. Propositions proved with the aid of complex numbers would be a posteriori judgments, or at least synthetic, unless we could find some other proof for them, or some other [denotation) for i. We must at any rate first make the attempt to show that all propositions of arithmetic are analytic (p. 112).

For two sections Frege develops the theme of not importing anything foreign into arithmetic. And he ends the book by recapitulating it: by explicitly defining the numbers as extensions of concepts, one can avoid importing physical objects or geometrical intuitions into arithmetic (p. 119).

These sections of Foundations provide an initial clue to understand-ing Frege's remarks about truth values in Section 10 of Basic Laws. Like complex numbers, truth values are seen by Frege as introduced (recog-nized) for theoretical reasons. Their introduction also extends previous mathematical and natural-language usage. And a variety of ontological choices are compatible with that usage. In showing that Axiom V does not fix the denotation of the course of value notation (or of the expres-sions 'the True', 'the False'), Frege is indicating, as he does in his attack on Hilbert and the formalists, that the (partially interpreted) axioms of a theory do not by themselves fix the objects of the theory (or the senses

FREGE ON TRUTH 139

of its terms). One must have an understanding of the senses and denota�tions of the terms used in the axioms that is not reducible to mere commitment to the truth of the axioms taken as linguistic objects. (Cf. Note 2 and the accompanying text in Section I.) In stating that the stipulations as to the identity of the truth values are arbitrary relative to previous usage and previously stated axioms, however, Frege is not stating that his stipulations are arbitrary, absolutely speaking.

What we need now is to understand the reasons for the particular identifications Frege proposes in Section 10 of Basic Laws. I shall approach his position by a process of elimination. The Foundations

passages already make it clear why the truth values could not be identified with physical, mental, or geometrical objects. The domain of logic is universal, whereas these objects are the topics of special sciences, and thus their natures are explicated in synthetic propositions (Cf. Foundations, Section 3.) Moreover, physical and mental objects exist contingently and are known by a posteriori methods. Logic encom�passes the necessary and is fundamentally a priori.

Similar considerations seem to rule out identifying truth values with any course of values associated with a function or concept denoted by a term that is a primitive of, or is definable with primitives of, one of the special sciences. Thus the extension of the concept is a cat (is an image,

is a line) is inappropriate. Truth values could not be identified with senses because of the

arguments of Thesis (b) that we considered in Section I. A corollary of these arguments is that such identification would be inappropriate because senses are denoted primarily in oblique contexts, whereas a truth value is denoted in the expression of any thought. Since, by Thesis (d) truth values are objects, and since Frege thought no function (or concept) is an object, truth values could not be identified with functions.

These considerations leave Frege either with identifying truth values with the course of values of some logical function or with not identifyinK them with any courses of values, taking them rather as primitive, "independent" logical objects.

We have been ruling out possible identifications by appeal to what is foreign to logic. In order to proceed further, we need to recall what Frege saw as essential to logic. Logic essentially concerns itself with the laws of truth. As we have seen, Frege sharpened this claim by stating that the laws of truth yielded those norms governing ideal assertions or judgments (PPW 2521NS 272; Kl 507-SIKS 343). Truth was the

140

TYLER BURGE

"objective" of judgment; the most general laws governing this objective

formed the subject matter of logic. Thus the true is in a sense the most

basic logical object. We shall return to this point and sharpen it.

The conception of truth as the aim of logic informed Frege's view

that logic had an internal ordering. Logic is founded on the proposi-

tional calculus, or the calculus of truth values. Frege repeatedly empha-

sizes that one of his seminal contributions is to begin in logic with the

sentence and to derive an analysis of sentential parts and their various

semantical functions only in the context of a semantical analysis that

already features logical relations among sentences: the functional calculi

are built upon the calculus of truth values. We have cited various

passages to this effect in his arguments for the view that truth values are

the denotations of sentences (Theses (a) and (b». Frege also makes the

point in his earliest and latest writings (PPW 17, 253/NS 18-19,273).

The sense of an ordering within logic is perhaps most clearly enunciated

in a footnote Frege wrote in 1910 to Jourdain's chapter on Frege in a

history of mathematical logic:

To found the 'calculus of judgments' on the 'calculus of concepts' ... is to reverse the

correct order of things; for classes are something derived, and can only be obtained from

concepts (in my sense). But concepts are something primitive that cannot be dispensed

with in logic .... And the calculation with concepts is itself founded on the calculation

with truth-values (which is better than saying 'calculus of judgments') (Cor 192n!NS

287n).

The truth values are on the ground floor of logic - in the ontology of

the propositional calculus. They are a "subject matter" for all parts of

logic. In a sense to be explicated, truth is even more basic than falsity.

The laws of logic were for Frege "nothing other than an unfolding of the

content of the word 'true'" (PPW 3/NS 3; cf. Kl 507/KS 343).!Now

courses of values were supposed by Frege to be logical companions of

functions, and functions were denoted in all parts of logic. Prior to

discovery of the paradox, each function was thought to be accompanied

by its associated course of values. Logical objects such as courses of

values could be canonically specified only through denoting the

associated functions. Frege repeatedly emphasizes that (denoting) a

course of values is derivative from (denoting) a function. A function

sign denoted a function, but its use determined an associated course of

values. Frege seems to have regarded sentences containing function

signs as ontologically committed to their associated courses of values.

FREGE ON TRUTH 141

(See G & B 27, 49-50/ KS 130, 173; the point is also suggested in BL 4/001,7 and PPW123/NS 134.)

Since Frege saw logic as having a fixed order, his taking truth values as part of the ontology of the propositional calculus meant that his specification within logic of these objects could not depend on concepts whose specification was conceptually derivative.

These considerations suggest an argument for taking truth, or the True, to be specifiable in terms of a concept that is primitive within the propositional calculus, assuming that truth is a course of values. Since all logic is concerned with truth and is in fact the unfolding of the laws of truth, and since truth is an object, truth must be ia the ontology of all parts of logic - in particular the most fundamental part, the proposi-tional calculus. For a course of values to be in the ontology of the propositional calculus it is necessary and sufficient that it be specifiable in terms of functions denoted in the propositional calculus. So assuming that truth is a course of values, it must be specifiable in terms of functions within the propositional calculus. The only such functions are concepts.

One might worry about the argument, both as a reconstruction of Frege and as a substantive proposal, by concentrating on the second premise. Let us assume with Frege that to be specifiable at all, a course of values must be specifiable in terms of its associated functions. But why iS'it necessary, in order to be in the ontology of the propositional calculus for a course of values to be specifiable in terms of functions denoted by expressions in the propositional calculus? And is denoting certain functions really sufficient for being onto logically committed to their associated courses of values?

Of course, with the hindsight that we have gained from the semantical and set-theoretic paradoxes, both questions can be pressed. And I would not wish to defend a Fregean answer to either. It is arguable that sometimes the ontological commitments of a language (say, those involved in giving a semantical theory for it) are specifiable only in a stronger metalanguage. And since Frege's Axiom V is inconsistent, it is sometimes the case that commitment to a given function is not sufficient for commitment to an associated extension (a course of values). This problem makes it plausible to deny that in using function expressions one has dual commitments, to a function and a course of values, even in cases where there is an associated course of values. One can make one's commitments separately. But these problems are bound up with larger

142 TYLER BURGE

problems concerning Frege's conceptions of truth and courses of values - particularly with the inconsistency of his system. They do not under-mine the argument we gave as an interpretation of Frege.

How would Frege answer the two questions about the second premise? To the first, I think that he would reply that a language that lacked the concepts (or could not denote the concepts) needed to specify its ontology would be logically deficient. Such a language could certainly not serve to express the fundamental part of logic. A logical language that could not specify its own ontology would be dependent on some other language to specify its primary subject matter, truth. Thus it could not be the fundamental language for expressing the laws of truth. But Frege regarded logic as an ideal language of thought; the fundamen-tal part of that language should be complete for its own purposes.

This point should be seen in the light of Frege's redundancy concep-tion of truth. Frege believed that semantical discourse about a language could add nothing to what could already be said in the language itself. It is obvious that he did not anticipate the sorts of considerations that lead us to be cautious about claims that a language must be able to specify its own ontology.

The second question about the second premise of our argument is less interesting insofar as it bears on the interpretation of Frege. The truth values are in fact in the ontology of his propositional calculus and they are in fact specifiable in terms of functions (concepts) that are denoted in the propositional calculus. Since the truth values are courses of values in his view, and since he thought courses of values could be specified only in terms of their associated functions, it is hard to see any ground for denying that he thought that being so specifiable was suffi-cient for being in the ontology of the language. (For a discussion of texts that suggest that Frege thought that language always carried dual commitments to concepts and their associated extensions, see Burge, 1984, Sections III-IV.)

I have argued that Frege conceived truth as the subject matter of the most basic part of logic, and that truth had to be specifiable by means of the primitive predicates in the propositional calculus. Frege's con-ception of an ordering within logic motivates a corollary restriction. Truth should not be the extension of a concept (or course of values of a function) whose specification is in any way conceptually deriva-tive. Truth must be specifiable in terms of a concept whose own specifi-

FREGE ON TRUTH 143

cation need not presuppose other types of concepts. This consideration would seem to rule out identifying truth any course of values of a second-level function (including second-level concepts). For the second-level functions are introduced in logic only after, and only in terms of, first-level functions. The consideration also seems to rule out iden-tifying truth with any course of values of a function, of any level, whose canonical explication presupposes specification of functions.

The effect of these exclusions is substantial. Two large categories of logical objects are barred as candidates for being identified with the truth values. First, the courses of values that are not extensions of concepts are excluded. There is only one purely logical, primitive first-level function sign that is not a concept sign, or predicate, in Frege's logic. This is the description operator, and its explication presupposes specification of the course-of-values operator, which is second-level (at least!). Of course, the course of values associated with the course-of -values function sign 'f' is excluded. For the function it denotes is (at least) second level. A second category of logical objects that is excluded consists of the extensions of concepts with which the numbers are identified. For these are extensions of second-level concepts. In fact, the definition of 'equinumerous', which is essential to specification of the numbers, depends on second-order quantification - i.e. third level concepts.

So, if the truth values are to be identified with courses of values at all, and if Frege's philosophy of logic is to be respected, it appears that they must be identified with the extensions of logical concepts that are first-level. Such concepts ought to be denoted in the propositional calculus.

What underlies the path of exclusion that we have so far followed, and indeed what guides us to our destination, is Frege's conception of truth. The argument we have just given is rather speculative, considered as an interpretation of Frege's own reasoning. The considerations that follow are much less so.

Frege identifies truth (the True) with fe-e). '-', the horizontal, denotes the concept that maps truth onto truth and everything else onto falsity. Thus the horizontal denotes the concept under which only truth falls. Truth is the extension, or course of values, of this concept. Falsity is the extension of the concept under which only falsity falls.

Now this identification can seem to be a piece of artifice if one thinks of courses of values purely in set-theoretic terms. From this persp'ective

144 TYLER BURGE

truth (falsity) is identified with its own unit class. (Cf. Dummett, IFP, 404.) What could be more typical of a mere technical convenience? But the set-theoretic gloss misrepresents Frege's view.

Frege's identification should be seen as the result of drawing out the implications of the two ideas that we have discussed so far, and supple-menting them with his redundancy conception of truth. The first idea was that the truth values are logical objects. Their specification should not "import anything foreign into logic". Further, as logical objects, their specification must be derivative from the specification (or denotation) of logical concepts. The second idea is that logic is an ordered unfolding of the laws of truth, where truth is the aim of sentence use within logic. Truth must somehow be the objective and subject matter (object) of all parts of logic, including the most primitive part. In fact, it must some-how be implicated in the aim of every sentence of logic. As we have seen in earlier sections, Frege interprets the logically relevant aim of sentence use in terms of our "striving after truth". This aim is revealed in assertion and judgment. So truth must somehow be implicated in the assertive use of every sentence of logic. Putting the two ideas together, we seek a concept in terms of which we can specify truth as a logical object, a concept that is present in the assertive use of every sentence of logic.

It might appear that we have an approximation to the idea that truth must be implicated in the assertive use of every sentence of logic, in Frege's doctrine that every sentence of logic denotes truth or falsity. As we have seen, in Section I, there was a connection in Frege's mind between the point of sentence use and the denotation of sentences. But truth is the aim of logic; falsity, strange to say, is not. This aim is revealed in assertion, not simply in the grammatical form of sentences. The concept in terms of which truth is specified is present in every assertive use of a sentence, whereas the counterpart concept for falsity is not. Thus the specification of truth is philosophically primary. The specification of falsity will present itself as natural once we have under-stood Frege's specification of truth.

The only concept that fits the requirement of being present in the assertive use of every sentence of Frege's logic is his concept of truth, the concept denoted by the horizontal. In unpublished writing contem-poraneous with Basic Laws, Frege is quite explicit about the point. He held that what distinguishes 'true' from all other predicates, and what fits it to indicating the a'im of logic, is that "it is asserted when anything at all

FREGE ON TRUTH 145

is asserted" (PPW 129/NS 140). In accord with his redundancy concep-tion, Frege held that 'it is true that _', when filled by a declarative sentence, expresses the same sense as '_' when filled by the same sentence. But this neutrality of sense in predications of 'true' is collateral with the predicate's omnipresence in assertions and judgments. As we have seen in Section II, this idea of the omnipresence of the truth predicate traces all the way back to the Begriffsschrift (B, 3).

The horizontal expresses the notion of truth in Frege's system. It means 'is the True' or 'is the truth' or 'is truth'.18 It is present in the formulation of every assertion. It may accompany any declarative sentence without adding to its sense. The concept denoted by the horizontal is the only one within Frege's logic that meets the condition set by his redundancy conception of truth. Frege alludes to this condi-tion without fanfare in Section 5 of Basic Laws, where he notes the equivalence:

/:l. = (-/:l.)

where '/:l.' varies over truth values. The import of the condition comes clear if one sees sentences as substituting for '/:l.', reads '=' (as in such cases one may in Frege's system) as the material biconditional, and reads the horizontal as the truth predicate. The equivalence is the analog within Frege's system of Tarki's truth schema.

Given his redundancy conception, Frege regarded the two sides of the equivalence as having the same sense when a sentence is substituted for'S. (Cf. G & B 63-4/KS 149-150.) This suggests that the concept of truth may in a sense be implicated in the ontology of every (declara-tive eternal) sentence, whether the sentence is asserted or not, and regardless of whether a truth predicate explicitly occurs in the sentence. This conclusion needs the assumption that a sentence is committed to the existence of the extension of a concept C if C is a denotation deter-mined by (a component of) the sense that the sentence expresses. Since every sentence has the same sense as a sentence in which the concept of truth is denoted, every sentence would by this reasoning be committed to the existence of the extension of this concept - the truth value truth. Although I think that there is some reason to believe that Frege considered and accepted the relevant assumption, he did not explicitly assert it. (Cf. G & B 27, 49-50/KS 130, 173; BL 4/GGJ 7; PPW 123/ NS 134; the numerous passages where he speaks of the senses of sentences as being decomposable into component senses; and Burge,

146 TYLER BURGE

1984.) I shall not defend the attribution of the assumption, however, since I regard it as somewhat speculative.

What is indisputable and important is that Frege forces the truth concept to be explicitly denoted by a truth predicate (the horizontal) when a sentence is assertively used in his logic. Thus it is through assertion that the aim and ultimate subject matter of logic are revealed.

The truth concept also expresses Frege's conception of the order within logic. It is the concept in terms of which all others are explicated and understood. In this regard, it is prior to any other concept canoni-cally denoted in the propositional calculus.

In specifying truth as the extension of his truth predicate, Frege is "deriving" a logical object from a logical concept. Obviously, it is intuitively natural to derive the object truth from the concept of truth (assuming that one wants such an object and that such an object must be derived from a concept). Frege is in effect nominalizing the truth pre-dicate by generalizing on his truth schema (in the light of his construal of the schema in terms of Theses (b) and (d».19 But Frege's specification of the object truth is not merely intuitively natural, granted his assump-tion that it is a logical object and must be derived from a concept. It also expresses his conception of the point, subject matter and order of logic, and flows from his redundancy conception of truth. It is the only identification that is consonant with his philosophical views.

The evidence we have been considering suggests reconsideration of a pair of long-standing criticisms of Frege's view of truth values as objects. One is that once truth becomes one object among others, it is difficult to explain what it is about it that makes us want to strive after it, assert it, acknowledge it, and so forth. (Cf. Furth, BL, pp. lii-liii. The point is also made by Dummett in various places in FPL.) What is so terrific about the relevant object? Frege seems to be inviting us to join a kind of secular religion without explaining the attributes of its god that merit our worship.

We may begin to appreciate the weakness of this criticism by recalling Frege's contextualist arguments for Theses (a) and (b). Denota-tion of objects with terms is of interest only because of our interest in semantical features of sentences. Interest in sentences derives from interest in norms governing their use in making assertions and express-ing judgments. So the practice of using sentences to denote truth values derived its interest from the role of assertion and judgment in our lives. From Frege's point of view, the idea that we must explain this role in

FREGE ON TRUTH 147

terms of the features that certain objects, the truth values, have, would be to put the horse behind the cart.

Frege underlines this point by specifying the object truth as what every assertive use of a sentence (and every judgment of a thought) is committed to. We know what the object truth is and what it is like only through reflecting on the fact that assertion and judgment aim at it. To be sure, Frege thinks that the laws of truth, which are laws of "being" (Sein, Kl 507-SIKS 342-3), generate norms for assertion and judg-ment. (Cf. Note 3.) This is because Frege assumes it as obvious that we ought to judge in accordance with logical laws, and because he con-strues these laws as governing an objective world of objects. But it is not part of his view that we should be able to explain the interest for us of the object truth, or the way that we think about it, independently from consideration of the point of assertion and judgment - to "strive after truth".

Frege thinks truth and judgmental force are primitive ideas. And he does not try to explicate or philosophize about the value of "striving after truth." But it is foreign to his system, and thus not a pseudo-question concocted by it, to ask what it is about the object truth that engenders our interest in it. The activity of jUdging and the practice of assertion are primary.

The second longstanding criticism of Frege is closely related to the first. Dummett states it in his thought provoking article 'Truth':

... it is part of the concept of truth that we aim at making true statements; and Frege's theory of truth and falsity as the references of sentences leaves this feature of the concept of truth quite out of account. Frege indeed tried to bring it in afterwards in his theory of assertion - but too late; for the sense of the sentence· is not given in advance of our going in for the activity of asserting, since, otherwise there could be people who expressed the same thoughts but went instead for denying them. (TOE, pp. 2-3)

It is certainly true that Frege said less about the roles of jUdging and asserting in our lives than one might want in a post-Wittgensteinean climate. But it is not true that Frege leaves our aim at making true statements "quite out of account" in his exposition of the concept truth. It is this aim, he says in 'On Sense and Denotation', that motivates our asking for the denotations of terms and sentences. Late in life he writes: " ... 'true' only makes an abortive attempt to indicate the essence of logic, since what logic is really concerned with does not at all lie in the word 'true' [since by the redundancy view, it contributes nothing new to

148 TYLER BURGE

the sense of whole sentences in which it occurs as predicate] but in the assertoric force with which the sentence is uttered" (PPW252/ NS 272).

The last main clause of the passage cited from Dummett's criticism is difficult to interpret. But it does not seem relevant to Frege's view. Frege nowhere, to my knowledge, writes or implicates that the sense of a sentence is given "in advance" of our going in for the activity of assertion. Denotation is motivated and justified by Frege in terms of our "striving after truth." Sense is postulated as the way denotations are presented to us in thought. Thoughts, the senses of sentences, are truth conditions; and we are interested in truth conditions because we are interested in arriving at truth. So our attaching senses and denotations to sentences and sentential parts is motivated by their roles in judgment and assertion. Frege's thesis that sentences denote objects is proposed in the context of these motivations, not in contradiction to them.

We should now consider the possibility of taking truth and falsity as "new" logical objects, not identical with any course of values. Frege would not have introduced a new name for truth, an individual constant, into his system since doing so would have created an inelegant logical connection between his specification of truth and his truth predicate. Moreover, such a move would controvert his view that logical objects are derivative from logical concepts, a view bound up with his con-textualist defense of the existence of abstract objects. But suppose that one identified truth with \f(-e) (where the slash is the description operator - cf. Section 11 Basic Laws) - the unique object that falls under the concept - (is the true).

I think that Frege would answer this suggestion by asking rhetorically what purpose a distinction between truth and the extension of the concept is the true would serve. He would take \f(-e) to be fe-e). (Compare the way Frege argues against postulating negative thoughts (PPW 149-50,185, 198/NS 161-2,201,214).) Prior to the discovery of the paradox, there were no evident logical advantages to the distinc-tion. This point is implicated in Note 17 in Section 10 of Basic Laws.

It is arguable that Frege accepted the intuitive or metaphysical point that a physical, mental, or geometrical object is distinct from any course of values - on the ground that they are obviously distinct categories of objects. (The argument would have to develop out of the central sections of Foundations.) But such a point is evidently not applicable to the relation between truth and courses of values. Truth is a necessary, logical object, unlike physical, mental, and geometrical objects; and

FREGE ON TRUTH 149

since it is introducted on predominantly pragmatic grounds, against the grain of prior intuition, there is no force in the claim that we make an intuitive distinction between the object truth and courses of values. From Frege's pre-paradox point of view, I see no purpose to the distinction and so, for him, no point in drawing it.

I believe that the urge to draw such a distinction derives from thinking that has already been informed by the set-theoretic and seman-tical paradoxes. One wants to insist on a difference in "level" between basic individuals and higher-order objects. Of course, merely taking truth to be an individual will not suffice to deal with the semantical paradoxes. Frege's redundancy theory of truth and his truth predicate ('is the True') are not capable of representing common uses of the notion of truth, much less to explicating and representing the derivative, indexical, and schematic aspects of the notion, aspects revealed by the paradoxes. (Cf. Burge, 1979.) If there is not some pervasive provision for levels of just the sort that Frege ignored, there can be no adequate theory. The notion of truth cannot be adequately represented in terms of a truth predicate that lacks any sort of stratification.

These remarks are, however, anachronistic. They presuppose knowl-edge that Frege lacked when he wrote Basic Laws. Given his redun-dancy conception of truth, his notion of course-of-values, his view of truth as an object, and his logicist ambitions, the identification of truth values that he proposed in Basic Laws Section 10 must have seemed uniquely appropriate.

Truth is the basic logical object in Frege's system. It is the object on which the purely logical, first level functions (including concepts) are initially construed as operating (BL, Section 31). The logical objects with which the numbers are identified are derivative from second and third level concepts that operate on the first level functions. So the numbers are less fundamental than the truth values (though all are, of course, seen by Frege as necessarily existent). The reason truth is basic is that it is the object (objective) of assertion and judgment. It is through these activities that the point and the ontology of logic are revealed.

As we have noted, the truth values formed, at one time, the basis for an argument against scepticism. They also formed the basis for Frege's answer to Kant's dictum that there could be no knowledge of objects without intuitions. Frege held that commitment to mind-independent objects was inseparable from the very act of judging something to be

150 TYLER BURGE

true. The idea is surely the quintessential distillation of the ambitions of rationalist epistemology.

Even apart from these ambitions, the profundity and breadth of Frege's conception must be seen as admirable. What enables Frege's views on truth value to be of enduring importance is that they are so largely grounded in fruitful and highly articulated insights into the anatomy and function of logic, mathematics, language, and thought. I think it unlikely that we have fully harvested Frege's insights.

University of California,

Los Angeles

NOTES

I Abbreviations of titles listed in the bibliography, together with page numbers, will be included in the text. Where German editions and translations of Frege's works differ in pagination, both occurrences will by cited, separated by a slash. Responsibility for the translations of all quotations from Frege is mine, although frequently the translations are similar to and benefit from already published translations. 2 The fact that Frege's notion of term-denotation cannot be entirely separated from the "name-bearer" relation has been appropriately emphasized by Michael Dummett in FPL, Chapter 12; and in IFP, Chapter 7. For some views that proceed on the assump-tion of the primacy of sentences (or their truth-values) and on the view that the notion of the denotation of a term has no content other than that which is derivative from an analysis of how the term functionally determines truth-value, see Quine (1960, Chapters 1-2; 1969, Chapters 1-2); Wallace, (1977); Davidson, (1977); Putnam (1981, Chap-ters 1-2). Several of these authors explicitly invoke Frege's inspiration. I find the views not only uncongenial to Frege (though unquestionably inspired by part of his doctrine), but unpersuasive. But I shall not be able to go into these points here. 3 In calling Frege's notion of truth "normative", I am glossing over a very interesting set of views that he held regarding the normative and descriptive aspects of logic. From the beginning to the end of his career, Frege regarded logic as being descriptive of the laws of logical objects, in particular those of truth. (PPW 31NS 3; K1507-81KS 342-3.) In fact, Frege seems to have believed that in a sense logic was fundamentally "descriptive," fundamentally a science of "being." Normative restrictions on assertion and judgment derived from "the way things are" regarding the laws of truth. (Kl 5081KS 342.) To many this view of logic will seem quaint at best. I think that stripped of the particular metaphysics with which Frege endowed it, and supplemented by Quinean and other considerations, it can be made very powerful. In emphasizing that truth is a normative notion, I am not ignoring the "descriptive" elements in his view. I am simply highlighting a feature of Frege's methodology. Frege attempts to arrive at the laws of truth not by invoking metaphysical assumptions but by concentrating on our practices of assertion, judgment, and deductive inference and by developing his science of logic through reflecting on the "oughts" of good intellectual practice.

FREGE ON TRUTH 151

4 Even he enters qualifications, (1956), p. 25 Note 66. 5 Prior to the argument's formulation, as early as 1903, Russell rejected the first premise. He insisted that the relevant semantical notions for sentences are quite different from those for terms. It is a bit open to question how seriously this rejection is to be taken. Russell refused to call the relation between a sentence and a fact or pro-position the relation of "naming" or "denoting." But he treats propositions or facts as complexes made up entirely of entities that have traditionally been thought of as referred to by words - properties, relations, individuals and the like. And he regards these complexes as playing a central role in his semantical theory of sentences. Thus at times Russell's point seems to come to little more than that sentences are not ordinarily

speaking names, a point with which Frege might well agree. Actually, of course, the issue in Russell is quite complex. (Cf. e.g. the first lecture in 'Logical Atomism'.)

Russell remains at odds with the Church-Go del argument even in its less titillating form - even after sentence denotation is construed as the yet-to-be-determined seman-tical feature that is connected to the denotation of terms by means of the Composition Principle (1). As I mentioned earlier, Russell may be interpreted as rejecting the third premise of the argument. He accepted more or less the traditional conception of logical equivalence and judged logical consequence in terms of the traditional modalized notion of truth (e.g. PM Section A*I). But he took the primary semantical correlates of sentences to be what he called "propositions" and sometimes "facts." If one interprets Russell as accepting the first "premise" of the Church-Godel argument by granting it the liberal conception of sentence-denotation that involves no commitment to Thesis (c) (so sentence denotation is merely the central semantical feature of sentences in one's formal semantics), then one must see Russell as rejecting the argument's third premise. For then facts or "propositions" are sentence denotations; and logically equivalent facts could, on Russell's view, differ.

What allows this position to remain compatible with the principle that exchange of co-denotational expressions preserves truth value is, of course, the theory of descrip-tions. This theory by itself blocks the Church-Godel argument by depriving one's language (artifically, I think) of the definite description operator, or any comparable device for forming complex singular terms that have denotations.

A discussion of the Church-Godel argument that is Russellian in its metaphysical cast occurs in Barwise and Perry (1981). Unfortunately, the paper contains much that is misleading. Frege's arguments are dismissed in two paragraphs. One paragraph charac-terizes Frege's rhetorical question 'What else besides truth value is compatible with the composition principle?' as a metaphysical oversight. This dismissal would perhaps be fair if it did not ignore Frege's normative motivations and methodology. Frege's argument from our primary interest in sentences, glossed in one sentence, is countered by an irrelevant appeal to embedded sentences (irrelevant because the reason sentences are interesting is that they are the vehicles of assertion and judgment). The Church-Godel argument is discussed only on the unquestioned (but widely rejected) assumption that sentences as wholes "designate" some entity. And it is resisted as if it had been presented on this unquestioned assumption, and widely accepted, as a "proof' from "virtually apriori" first principles. The incompatibility of Russell's system with the argument's conclusion has been widely recognized. No one has tried to utilize the argument to refute Russell, least of all Church and Godel, who were, of course, thoroughly familiar with Russell's system. The role of the argument in the history of

152 TYLER BURGE

semantics is more subtle than treating it as a proof from purportedly obvious first principles could suggest. 6 The relation between Theses (c) and (d) is a complicated and subtle matter. Ontologi-cally, of course, logical objects, such as truth values, and functions on these objects are prior for Frege to singular terms and function signs. They exist before the signs existed, and would have existed regardless of whether the signs did. On the other hand, Frege held, on several occasions, that one could not engage in reasonably sophisticated thought except by means of language. The analysis of thought relied heavily on analysis of linguistic structure. But even in the analysis of thought, the analysis of language was not prior in any simple sense. For thinking could and did correct the deficiencies of language. In the light of all this, there is no simple answer to the question of whether Frege reasoned from Thesis (c) to Thesis (d) or vice-versa. Part of why he concluded that numbers are objects and numerals are terms was that he was able to give explicit definitions which amounted to a criterion of identity for the numbers. On the other hand, much of his reasoning to this conclusion was based on observations regarding the structure of mathematical language. Similarly, his reasoning about Theses (c) and (d) is a mixture of considerations regarding the role of objects in logic and the anatomy of the language of logic (properly construed). I shall therefore treat Theses (c) and (d) more or less together, without trying to sort out the various relations of relative priority that obtain between them. 7 There are variants of the analogy between terms and sentences as regards their sense and denotation that Frege mentions, but which I shall skim over. For example, he cites his theory of indirect discourse as tending to confirm the introduction of truth values. (BL 7/ GGX). The idea seems to be that, in indirect discourse, just as terms shift from denoting their customary denotations to denoting their customary senses, so sentences shift from denoting their truth values to denoting the thoughts they customarily express. On Frege's conception, there need be no shift in grammatical category between the occurrence of a sentence standing alone, and its occurrance in indirect discourse, which is clearly a singular term position. S One might be tempted to think that Thesis (a), or at any rate (b), already commits Frege to taking truth values to be objects. For by our stipulation, an object is anything that is denoted by a term. But the phrase 'the denotation of Sentence 5' (d. Thesis (b» is a term. So by Thesis (b), truth values are objects. This line of reasoning misses the fact that the phrase 'the denotation of a sentence' need not be a term under logical analysis.

Similarly, for the phrase that begins Principle (1). 'Denotation' as applied to sentences in the initial construal of Thesis (b) is guided by' the compositional method, loosely

expressed in Principle (1). As far as Frege's arguments for (a) and (b) are concerned, there need be no entity ·that could be called under logical analysis "the denotation" of the sentence. Of course, once Frege has committed himself to Theses (c) and (d), he can consider Theses (a) and (b) to have proper construals, under logical analysis, that commit him to taking truth values to be objects. 9 I do think that Dummett neglects to convey the. richness and inter-related nature of the theoretical considerations supporting Frege's theses. His remark about Thesis (c) that it is a "ludicrous deviation" from the forms of natural language and a "gratuitous blunder" (FPL, 184) is, at best, immoderate. Incidentally, in FPL, Dummett spends two thirds of his chapter 'Truth Value and Reference' on Frege's view that a sentence with a non-denoting name thereby lacks a truth value. As I mentioned in Section I, I think that Frege rested little weight on this view in defending Thesis (d). I think Frege regarded his view as a consequence of the rest of his doctrine. Since the consequence accorded with his intuitions, it had some value for him in confirming the doctrine.

FREGE ON TRUTH 153

10 It is true that Frege writes in Basic Laws Section 2, 'The sense of a name of a truth value I call a thought'. This might seem to be intended to include all terms (not just sentences and their nominalizations), as long as the term denotes a truth value. I think that there is no independent evidence that this was Frege's intention. The remark is illustrated only by sentences. As we have just seen, three sections later, Frege says that thoughts are what are judged. And his system allows the act of assertion or judgment to apply only to sentences or what they express. Thus I think that the remark in Section 2 is a slip encouraged, to be sure, by the formal assimilation of sentences to terms. 11 Furth errs in calling (rather than the vertical alone) the judgment stroke, a distinction critical to the points we have been making. But he gives an excellent account of the role of the notion of assertion in Basic Laws (Cf. BL, pp. xlviii-lii). 12 With one exception. When in Basic Laws the horizontal applies to expressions like '5' or 'the course of values such that ... ', which are not sentences, the sense of the result of the application is different from the sense of the argument expression. The former is a thought; the latter is not. There is no analog in Begriffsschrift since the horizontal only applied to judgeable contents. 13 The words 'determination' (for 'Bestimmung') and 'grasp' (for 'zu fassen') in this translation replace two occurrences of the word 'definition' in J. L. Austin's otherwise good translation of this passage from FA. Austin's choices may obscure the fact that Frege thought not only that we needed a reduction of arithmetical terminology to other terms, but that we needed a better grasp of the notions that such terminology expressed. 14 For a discussion that rebuts recent attacks on this interpretation, and with which I am in broad agreement, see Michael Dumrnett, IFP.

15 This interpretation has been urged in an article, containing many interesting secon-dary points, by Paul Benacerraf (1981). Benacerraf cites not only Section lOaf Basic

Laws but some passages in Foundations. I have discussed these latter in Burge, 1984. I believe that I have shown them not to support the view. Here I shall concentrate on Basic Laws Section 10. 16 I am indebted to Mary Dant for bringing home to me the kinship of these sections of Foundations to the writings on Hilbert. 17 The bracketed substitution of 'denotation' (,Bedeutung') for 'sense ('Sinn') is specif-ically suggested by Frege, in the light of his subsequent sense-denotation distinction, in a letter to Husser! of May 24,1891. 18 Almost needless to say, Frege's representation of the truth predicate is not intended exactly to reproduce ordinary language. In particular, '-t(-c)' turns out true but has no analog in ordinary uses of 'is true'. Cf. also Note 12. 1" Frege's method of specifying or "defining" truth (in the extensional, mathematical sense) is a primitive ancestor of Tarski's set-theoretic methods of attaining the same objective. (Frege differs from Tarski in having no ambition to explicate, or provide a vindication for, the concept of truth.) Both authors may be seen as summing up or generalizing from their respective versions of the truth schema. Frege's identification of falsity as the extension of the concept is the false is the natural counterpart of his specifi-cation of truth.

BIBLIOGRAPHY

Barwise, Jon and J. Perry: 'Semantic Innocence and Uncompromising Situations' Mid�

west Studies 6, 1981. Benacerraf, Paul: 'Frege: the Last Logicist' Midwest Studies 6, 1981.

154 TYLER BURGE

Burge, Tyler: 'Semantical Paradox' The Journal of Philosophy 76 (1979), 169-198. Burge, Tyler: 'Frege on Extensions of Concepts, From 1884 to 1903', Philosophical

Review 93 (1984), 3-34. Church, Alonzo: 'Carnap's Introduction to Semantics' Philosophical Review 52 (1943),

298-305. Church, Alonzo: lntroduction to Mathematical Logic (1956) (Princeton University

Press, Princeton, 1956). Davidson, Donald: 'Reality without Reference' Dialectica 31 (1977),246-258. Dummett, Michael: Frege: Philosophy of Language (FPL) (Duckworth, London, 1973). Dummett, Michael: Truth and Other Enigmas (TOE) (Harvard University Press, Cam-

bridge, Mass.; 1978). Dummett, Michael: The Interpretation of Frege's Philosophy (IFP), (Harvard University

Press: Cambridge, Mass.; 1981). Frege, Gottlob: Begriffsschrift und Andere Aufsiitze (B) Angelelli (ed.) (Georg OIms,

Hildesheim, 1964). Frege, GottIob: Foundations of Arithmetic (FA), Austin trans. (Northwestern University

Press, Evanston, 1968). (The volume contains both the German and the English. Pagination is the same as in the original.)

Frege, Gottlob: Kleine Schriften (KS), AngeIelli (ed.) (Georg OIms, Hildesheim, 1967). Frege, Gottlob: Translations from the Philosophical Writings of Gottlob Frege (G & B)

Geach and Black (eds.), (Basil Blackwell, Oxford, 1966) 2nd edition. Obvious abbreviations of relevant articles are also used in the text.)

Frege, Gottlob: Grundgesetze der Arithmetik (GG), (Georg Olms, Hildesheim, 1962). Frege, GottIob: The Basic Laws of Arithmetic (BL), Furth trans. and ed. (University of

California Press, Berkeley, 1967). Frege, Gottlob: On the Foundations of Geometry and Formal Theories of Arithmetic

(FG), Kluge ed. (Yale University Press, New Haven, 1971). Frege, GottIob: Nachgelassene Schriften (NS) Hermes, Kambartel, Kaulbach (eds.)

(Felix Meiner, Hamburg, 1969). Frege, Gottlob: Posthumous Writings (PPW), Long and White trans., (University of

Chicago Press, Chicago, 1979). Frege, GottIob: Briefwechsel (BW), Gabriel, Hermes, Kambartel, Thiel, Veraart eds.,

(Felix Meiner, Hamburg, 1976). Frege, Gottlob: Philosophical and Mathematical Correspondence (Cor), McGuinness

and Kaal (eds.) (University of Chicago Press, Chicago, 1980). G6del, Kurt: 'Russell's Mathematical Logic' in Philosophy of Mathematics, Benacerraf

and Putnam (eds.) (prentice Hall: Englewood Cliffs, New Jersey; 1964). Klemke, E. D. (ed.): Essays on Frege (Kl) (University of Illinois Press, Urbana, 1968). Putnam, Hilary: Reason, Truth, and History (Cambridge University Press; Cambridge,

England; 1981). Quine, W. V.: Word and Object (MIT Press, Cambridge, Mass.; 1960). Quine, W. V.: Ontological Relativity (Columbia University Press, New York, 1969). Russell, Bertrand and A. N. Whitehead: Principia Mathematica (PM) (Cambridge

University Press, New York, vol. 1,1910). Wallace, John: 'Only in the Context of a Sentence Do Words Have Meaning' Midwest

Studies 2, 1977.

LEILA HAAPARANTA

FREGE ON EXISTENCE*

1. INTRODUCTION

In his philosophy of language Gottlob Frege strives to present the basic structure of language which is supposed to correspond to the structure of what is referred to. He makes a distinction between proper names, which refer to objects, and function-names, which refer to functions. Function-names include concept-words and relation-words, which stand for concepts and relations, respectively. Frege also assumes that, besides a reference (Bedeutung), each name has a sense (Sinn), through which the name is directed to its reference. l

In his monumental work on Frege's philosophy of language Michael Dummett lists ten theses of Frege's concerning sense and reference (Dummett, 1981, pp. 152-203). One central principle is missing, however, and it has likewise been ignored by most of the other Frege scholars. This is the thesis that the word 'is' is ambiguous in a certain way. Ignacio Angelelli comes close to attending to it when he makes some remarks on identity and predication, and Matthias Schirn puts special emphasis on the role of the thesis in Frege's work? but the great majority of Frege scholars have completely overlooked the ambiguity doctrine. Frege and Russell proposed this thesis and made it one of the basic ingredients of modern logic. Likewise, in the Tractatus Ludwig Wittgenstein emphasized the ambiguity of the verb 'to be' and stressed the importance of constructing a language which prevents confusions between different meanings of 'is'. Wittgenstein remarked that Frege's and Russell's conceptual notation was purported to be such a language although it did not succeed in excluding all mistakes (TLP, 3.323-3.325). Some philosophers and linguists, including Jaakko Hintikka, Charles Kahn, and Benson Mates, have recently discussed the ambiguity doctrine and raised criticism against it.3

The roots of the ambiguity thesis do not reach farther than the beginning of the nineteenth century. Charles Kahn (1973a, 1973b, 1985) has argued that in the nineteenth century there was curious interaction between the views of linguists and philosophers concerning

155

L. Haaparanta and J; Hintikka (eds.), Frege Synthesized, 155-174. © 1986 by D. Reidel Publishing Company.

156 LEILA HAAPARANTA

the verb 'to be', particularly concerning the notions of existence and copula. Linguists and philologists misunderstood the ancient use of the Greek verb einai and based their account of the verb on a mistaken philosophical exegesis of ancient theories of being, while philosophers relied on the work of linguists and philologists, when developing their theories of being. In 1801 Gottfried Hermann, a German philologist, proposed a rule which attached different accents to different meanings of einai, thereby in effect attributing the ambiguity between existence and copula to ancient Greek (Hermann, 1801, pp. 84-85).

Among the early opponents of the ambiguity thesis as applied to Greek philosophy, G. E. L. Owen (1960) hinted that Aristotle's view of being had been misinterpreted. Michael Frede (1967), for his part, questioned the possibility of reading any sharp distinction between existence and copula out of Plato's texts (Frede, 1967, p. 37). R. M. Dancy (1975, 1983) has explicitly argued against efforts to apply the ambiguity thesis to the verb einai, especially to Aristotle's verb einai,

and Jaakko Hintikka (1983, 1985) has discussed Aristotle's doctrine of being in detail and maintained that the ambiguity thesis is completely anachronistic when applied to Aristotle. Benson Mates (1979) has criticized the view that Plato made a semantical distinction between the 'is' of identity and the 'is' of predication.

According to Aristotle, the realm of beings falls into different cate-gories. Being itself is not a genus, and no single category exhausts all beings (Met. B 3, 998b22-27, An. Post. II 7, 92b14; cf. Owen, 1965, p. 77). Aristotle assumes that to be is always to be either a substance of a certain sort, or a qUality of a certain sort, or a quantity of a certain sort, etc. (An. Post. I 22, 83bI3-17). However, this does not mean that Aristotle takes 'to be' to have a completely different meaning for different kinds of subjects. Instead, he argues:

There are many senses in which a thing may be said to 'be', but all that 'is' is related to one point, one definite kind of thing, and is not said to be by a mere ambiguity. (Met. r 2, lO03a33-36.)

Aristotle makes a distinction between homonymy and multiplicity of uses. In the beginning of the Categories he states that things are homonymous if they have only a common name but completely differ-ent definitions. This is not what he assumes all existing things to be, but he argues that the different uses of 'being' in the different categories have the same focal meaning. This amounts to saying that the different

FREGE ON EXISTENCE 157

applications of 'being' are not homonymous for Aristotle but 'is' has, on his view, only a multiplicity ofuses.4

The distinction between different Aristotelian categories is, however, quite different from the Frege-Russell distinction between different meanings of 'is'. Fregean logic distinguishes between the following meanings of 'is':

(1) the 'is' of identity (e.g., Phosphorus is Hesperus; a = b), (2) the 'is' of predication, i.e., the copula (e.g., Plato is a man;

(3) the 'is' of existence,

and

(i) expressed by means of the existential quantifier and the symbol for identity (e.g., God is; (3x) (g = x»,

or (ii) expressed by means of the existential quantifier and the

symbol for predication (e.g., There are human beings/ There is at least one human being; (3x)H(x»,

(4) the 'is' of class-inclusion (e.g., A horse is a four-legged animal; (x) (P(x):::> Q(x»).

Frt(ge's conceptual notation expresses these meanings as follows:

(1) I-- (A =B)

(2) I-- <P(A)

(3) (i) a)

(ii)

(4) X(a)

These formulas <?f Frege's language are judgements (Urteile), since they are provided with the symbol' f-', which consists of the content stroke (Inhaltsstrich) '-' and the judgement stroke (Urteilsstrich) 'I'. The vertical stroke which connects the two horizontal ones Frege calls the conditional stroke (Bedingungsstrich) (BS, §5). Negation is expressed in his symbolism by a small vertical stroke which is attached under the content stroke (ibid., §7). In the Begriffsschrift Frege uses three parallel strokes as a sign for identity, but in the Grundgesetze he adopts the

158 LEILA HAAP ARANT A

usual sign for identity with two strokes (GGA I, 'Vorwort', p. IX). Generality is expressed by a concavity containing a German letter plus the same German letter holding the place of the argument (BS, § 11). Frege does not pay attention to (3) (i) in his formalism, even if he discusses it in detail in his informal articles, such as in 'Dialog mit Piinjer tiber Existenz' (NS, pp. 60-75). Nor has he any separate symbol for existence, but he expresses it by means of the symbol for generality and two negation signs.

Kahn, Hintikka, and others have mainly been worried about earlier writers' efforts to derive the Fregean ambiguity thesis from Aristotle's words. Since, for Aristotle, being is not a genus and to be is thus always to be something or other, it cannot be claimed that he believed in any pure ambiguity between existence and predication. Whether Aristotle hinted at any of the Fregean ambiguities or not, it is at least obvious that he gave these suggestions a minor position in his thought. This is shown by the fact that he did not recognize any need for writing down these different meanings of 'is' in any specific language. He was satisfied with our natural language which does not provide us with any such distinctions as that between identity, predication, existence, and class-inclusion. Accordingly, even if Aristotle believed in such forms of being as his categories, which he also found in natural language, he did not believe in those forms of being which are shown by Frege's distinction. Unlike Frege, Aristotle did not think that there are such relevant forms of being as identity, predication, existence, and class-inclusion.

Frege's doctrine concerning the words 'to be' and 'is' can be gathered from various sources, mainly from the article 'Dialog mit Piinjer tiber Existenz', written before 1884 and published posthumously, from Die Grundlagen der Arithmetik (1884), and from the article 'Uber Begriff und Gegenstand' (1892). In these works Frege deals with the difference between predication and existence, on the one hand, and the difference between predication and identity, on the other. The distinction between predication and class-inclusion is discussed by Frege as early as in the Begriffsschrift (1879), where he also introduces the rest of the distinc-tions but does not comment on them in detail. Frege's thesis not only bears upon the different uses of the words 'to be' and 'is', but it also concerns the different concepts that those words stand for. Frege argues that our natural language is deficient, since it offers us one single word for these various purposes. What we are in need of, then, is a language which correctly reflects the distinctions between the different concepts


of being. Frege regards it as a philosopher's task to show where natural language leads us to see things in the wrong perspective (NS, pp. 74-75 and p. 289). As Frege himself states in the BegriffsschriJt, his conceptual notation is meant to be a language of pure thought, which is free from ambiguities.

It is true that Frege's principal aim was to realize his logicist program. To carry out the program, Frege had to define arithmetical concepts by means of logical concepts and to prove that arithmetical truths are derivable from the axioms of logic by means of logical deduction. Frege developed the logical implements for the derivations in the Begriffs�schriJt, and by doing so he became the foremost pioneer of modern logic. However, it was not Frege's sole purpose to present the rules of logical inference. Indeed, his conceptual notation was purported to be a Leibnizian lingua characterica, from which all ambiguities were elimin-ated and which was thus meant to be the correct semantic representa-tion of natural language.s Frege's paradigm of first-order language was thus essentially semantically determined. However, he did not himself present it semantically, for, as Jean van Heijenoort (1967) and Jaakko Hintikka (1979a, 1981b, 1981c) have argued, he believed in the ineffa-bility of semantics. This means that he did not take it to be possible for us to step outside the limits of language in order to consider the relations between language and the world, because, in his view, all talk already presupposes these semantical relations. Frege explained the different uses of the word 'is' simply by describing his notation for a first-order language, which was for him the only correct linguistic representation of our concepts.

It is true many twentieth-century logicians have adopted the idea of language as a calculus which can be freely reinterpreted in a fixed domain of individuals.6 Some logicians have even completely rejected the claim that a Fregean first-order language - suitably supplemented, for instance, by adding to it some higher-order logic, as Frege does - is a universal medium of communication in the Fregean sense; meaning both the sense that the interpretation of its names and predicates do not vary in a fixed domain of individuals and the sense that it speaks about one fixed domain. None the less, logicians have accepted the Fregean quantification theory, where the ambiguity doctrine is firmly codified. The simple reason why they have accepted this doctrine merely by accepting the quantification theory is that the meanings of the logical constants of the quantification theory, including those that stand for

160 LEILA HAAP ARAN

various kinds of being, remain unchanged, even though the classes of individuals the quantifiers range over may change. Thus adding new elements to the basic Fregean quantification theory or relativizing the ranges of his quantifiers does not remove a logician's commitment to the ambiguity of 'is'.

I have here claimed that Frege has several concepts of being, without carefully attending to Frege's own terminology, according to which concepts are references of concept-words. We can try to avoid the distinction between senses and references by saying that, in Frege's logic, the word 'is' not only has a number of uses but it has various meanings. In this paper, I shall not deal with the possibility of applying the distinction between senses and references to such an auxiliary verb as 'to be'.7

This paper focuses on Frege's doctrine of existence. One of the innovations of Frege's logical theory was to construe existence as a second-order concept, i.e., as a property of concepts.8 This paper is, however, an attempt to elucidate some lesser known aspects of Frege's view of existence. I shall argue that Frege regards existence both as a proper second-order concept and as an empty first-order concept and that the distinction between the two references of 'exists' is motivated by his metaphysical and epistemological tenets. In constructing these tenets, we must be satisfied with Frege's brief remarks and hints. Thus, Frege not only assumes that the word 'is' is ambiguous but he also considers the verb 'to exist' and hence also the 'is' of existence an equivocal word. Frege's concept of Wirklichkeit, which likewise turns out to be a kind of concept of existence, will not be discussed in this paper.9

I have spoken of Frege's logical language as a first-order language. Admittedly, Frege did deal with higher-order quantifiers. However, for reasons which Will not be discussed here the higher-order component of Frege's lingua characterica may be considered inessential, and it is in any case largely irrelevant for the purpose of this paper.IO

2. THE EQUIVOCITY OF 'EXISTS'

The concept of existence is discussed by Frege in detail in his 'Dialog mit Piinjer iiber Existenz', which has been published in Frege's Nachge-

lassene Schriften. The paper was written before 1884, the year Frege completed his Grundlagen. In the Grundlagen Frege presents his doc-trine of existence in a mature form. The argumentation put forward in


'Dialog mit Piinjer iiber Existenz' supplements the Grundlagen and is very instructive if we are interested in the different aspects of Frege's concept of existence. I have discussed the dialogue in my paper 'On Frege's Concept of Being', where I tried to shed some light on how Kant's views on existence influenced Frege's ideas. In what follows, I shall present the main points that Frege makes in the dialogue and give my suggestions concerning the implicit motivation of his view.

In 'Dialog mit Piinjer iiber Existenz' Frege supports the claim that such sentences as 'Leo Sachse is' and 'Leo Sachse exists' are self-evident (selbstverstandich), while Piinjer suggests that the word 'is' carries the same meaning as 'is something that can be experienced' (ist erfahrbar).

For Piinjer, the set of objects of experience (Gegenstande der Erfahrung)

is a subset of the set of objects of ideas (Gegenstande der Vorstellungen).

Frege argues that Piinjer's account results in contradiction: If 'A is not' means the same as 'A is not an object of experience', then the statement 'There is something that is not an object of experience' means the same as 'There is an object of experience which is not an object of experience' (NS, pp. 71-72). In an afterword to the dialogue Frege continues his argumentation and states that if the sentence 'A is' were not self-evident, its negation could be true in some circumstances. He concludes that if the sentence 'There are entities which do not have the property of being' means the same as 'Something that has being falls under the concept of not-being (der Begriff des Nichtseienden)', it is a contradic-tory sentence, and if the sentence 'There are B's' is equivalent in meaning to the sentence 'Something that has being is B', the concept of being is self-evident.

Frege seems to be driven to denying the meaningfulness of sentences like 'A is' or 'A exists' because of his conceptions of language and the world. He cannot say that the sentence 'Something that has being is not' means that something for which it is possible to exist does not exist in the actual world, since he is committed to the view that there is only one world and that his conceptual notation is a universal language which speaks about that world. He does not even divide his universe into various sorts. That is indicated by his principle of completeness (Grundsatz der

VoNstandigkeit), according to which any function must be defined for all objects (GGA II, §§56-65).11 Due to his one-world view, he concludes that the concept of being is not a determination of an object, i.e., it does not help us to distinguish between any two objects (NS, p. 73).

We may put the same point as follows: Because of Frege's one-world


view, there can be quantifiers of one kind only, namely, quantifiers ranging over all actually existing objects. For this reason, Frege cannot escape the threatening inconsistency by assuming that we have two different ranges of quantifiers in sentences like 'Something that has being is not', which otherwise might have seemed a plausible way out for someone who distinguished from each other the different meanings of 'is'.12

After having rejected the idea that existence is a real property of an object, Frege makes an effort to convert existential statements into the form of particular statements (NS, p. 70).'3 For instance, he turns the sentence 'There are men' into the sentence 'Some living beings are rational'. If the concept that occurs in a given sentence cannot be defined by means of two concepts, Frege resorts to the concept of being identical with itself (sich selbst gleich sein), which he takes to be the most general concept of the hierarchy of concepts. He identifies this concept with the concept of being. Hence, he can turn the sentence 'There are men' also into the sentence 'Something that has being is a man', or 'Something that is identical with itself is a man' (NS, p. 71).

Consequently, Frege holds the view that we are forced to regard being (in the sense of existence) as a concept which is superordinate to every concept. What Frege shows here is that, according to his doctrine of being, existence can be used as a first-order concept if one i's willing to pay the price of its becoming an empty concept.14 The concept of being that we are here interested in is such that we predicate it of every object of which we predicate anything. By saying that a is X, we say that a is and that a is X Here the copula is purported to posit the object a in the sense that it is the part of the predication which makes the predication carry with itself the claim for existence. Copula is for Frege a concept which applies to entities in this one world of ours as well as any other concept, even if more generally.

The idea that existence is included in every predication apparently brings Frege quite close to Aristotle. Frege seems to repeat Aristotle's view that the expression 'existent man' says nothing more than 'man', i.e., 'existent' is an empty and hence redundant word in any context whatever (Met. r 2, 1003b27-30,I 3, 1054a16-18). What is even more, as far as we consider Frege's concept of being as a first-order concept, Frege does not believe in the analogy of 'is' in the sense in which Aristotle does. For Frege, being is a 'genus'; it is an empty concept, which has an infinitely wide extension and no comprehension.


If we limited our consideration of Frege's concept of existence to what we have found out up to now, we could conclude that Frege's concept of being is a univocal concept.

Why does Frege regard being as a univocal concept in this limited sense? The reason is, again, that Frege does not divide his universe into various sorts. Unlike that of Aristotle's, Frege's realm of beings does not

fall into different categories. In his article 'Kritische Beleuchtung einiger Punkte in E. Schroders

Vorlesungen iiber die Algebra der Logik' Frege suggests that the sentences 'A is' and 'A exists' could be rendered as the metalinguistic sentence 'the name "A" has a reference' (KS, p. 208). But if Frege is consistent in his view that we cannot step outside the limits of language, he must consider such statements as illegitimate talk about the expres-sions of our language.15

Frege requires that in the language of science all proper names have to be taken to be nonempty (NS, p. 135). He also assumes that if we talk about an object we already presuppose the existence of that object ('Uber Sinn und Bedeutung', KS, p. 147). None the less, Frege admits that we speak meaningfully about entities which do not exist. On Frege's view, a sentence lacks only a truth-value - but not a sense - if it contains a name which has no reference ('Uber Sinn und Bedeutung, KS, p. 148).

Since Frege regards being as a characteristic of every concept, it might be suggested that if we attach any concept-word to an empty proper name, the concept brings it about that the name has a reference, after all. That this is not the case becomes obvious if we consider Frege's concept of concept. According to Frege, a concept-word - and likewise a concept to which it refers - is ungesiittigt. It has a 'gap', which can be filled with a complete expression, i.e., with a proper name (GGA I, pp. 5-8). If we fill the gap of the concept-word with a proper we also aim at filling the gap of the corresponding concept. If we succeed in filling the gap of the concept, it is also true that the proper name has a reference. If the gap of the concept is not filled, we do not attach existence to anything, since we do not succeed in predicating anything. If we succeed in predicating something, we succeed in predi-cating existence at the same time. This is so because, for Frege, to be an object implies existence. For if a is an object, then the proper name 'a'

has a reference, which amounts to saying that a exists. Consequently, Frege's idea that we presuppose the existence of the objects we talk


about concerns only the nature of our speech acts and the pragmatical aspects of our language, and it has nothing to do with the semantical relation between the sentences of our language and the objects and the functions of the world. Frege simply wants to remark that, when we say something about an object, we do not add that the object exists.

In the Tractatus Wittgenstein argues that we arrive at nonsense if we treat such concepts as those of object, function, number, and concept as ordinary concepts. According to Wittgenstein, in a consistent and perspicuous language, that something is an object, a function, etc. could only be shown but could not be said (fLP, 4.126). David Bell (1979) claims that Frege supported this view as far as functions are concerned, for, on his view, that something is a function is shown by the incom-pleteness of the sign which is used to refer to it, but it is not possible to say that something is a function ('Uber Begriff und Gegenstand' , KS, p. 170). Bell assumes that Frege never extended this doctrine to include the expression '() is an object' (Bell, 1979, p. 47). Frege's treatment of existence is, however, a kind of extension, since, on his view, we do not say that something is an object because, in using the name, we already presuppose that there is an object to which the name refers.

There is one important and manifest reservation that Frege makes in his argumentation in 'Dialog mit Piinjer tiber Existenz'. He concludes there that if the sentence 'There are B's' is equivalent in meaning to the sentence 'Something that has being is B', then the concept of being is self-evident. His formulation suggests an alternative way of handling the problem, in which the mentioned equiValence is denied and which thus hints at the equivocity of 'being'. But if the expressions 'x has being' and 'there is an x' differ in meaning, Frege's argumentation for the view that 'A is' is self-evident collapses.

At the end of 'Dialog mit Piinjer tiber Existenz' Frege introduces the doctrine that existence is a property of a concept (NS, p. 75). Frege is inclined to maintain that existence used as a first-order concept is an empty concept, but he insists on preserving the meaningfulness of existence used as a second-order concept. That conviction is conspicu-ous in his criticism of the idea that every concept is abstracted from a number of objects. He remarks that if all concepts were elicited from existing objects, existential statements would lose all content; once we had a concept, we could infer that there is an object which exemplifies the concept (GLA, § 49).16

FREGE ON EXISTENCE

3. THE PHILOSOPHICAL BACKGROUND OF FREGE'S EQUIVOCITY VIEW

165

The above discussion shows that Frege assumes that 'exists' and the 'is' of existence have two readings. They may refer either to an empty first-order concept or to a meaningful second-order concept. In the former case the existential statement becomes meaningful if it is trans-formed into a metalinguistic statement which expresses that a given proper name has a reference. In the latter case the statement tells us that a concept is instantiated, i.e., that there is an object which has a given property. First-order existence is formalized by means of the existential quantifier and the symbol for identity, while second-order existence is expressed by means of the existential quantifier and the symbol for predication. In order to find out the philosophical motivation for Frege's view of existeace, let us first consider the distinction between identity and predication.

Frege discusses the problem of interpreting the concept of identity already in the Begrijfsschrift, where he states that in an identity state-ment a name seems to represent itself. He adds, however, that an identity statement does not concern names only but it expresses that two signs have the same content (Inhalt), which is determined in two different ways by the two signs (BS, § 8). He reformulates that idea in 'Uber Sinn und Bedeutung' by saying that an identity statement expresses that two names have the same reference but different senses. The sense of a name is the way in which the reference of the name is presented (KS, pp. 143-144).

Frege thinks that the symbols which occur on the different sides of the identity symbol can be replaced by each other in any context whatsoever, and he also assumes that two objects are identical if and only if they fall under the same o:mceptsP As we saw in the preceding section, Frege also considers identity to be a relation of an object to itself. These renderings of the concept of identity are subject to Wittgenstein's criticism, according to which to say of two objects that they are identical, is nonsense, and to say of one object that it is identical with itself, is to say nothing at all (TLP, 5.5303). The details of Frege's doctrine of identity and the possible changes in his view will not be discussed in this paper.IS It is, however, worth mentioning that Frege does not regard it as possible to define the identity of objects by the sameness of their properties or by any other means, since any definition


is itself an identity ('Rezension von: E. G. Husserl, Philosophie der

Arithmetik 1', KS, p. 184). In his article 'Uber Begriff und Gegenstand' Frege emphasizes that, in

order to keep objects and concepts apart, we must make a sharp distinction between identity and predication (KS, p. 168). The principle according to which objects must be clearly distinguished from concepts also manifests itself in that, unlike traditional grammatical analysis, Fregean analysis of sentences distinguishes the relation between two concepts of the same order from predication, which, for its part, concerns the relation between an individual and a concept or a relation between two concepts of two different orders (NS, p. 207 and p. 210). In the Grundlagen the distinctioJ) between objects and concepts occurs in the list of Frege's basic principles (GLA, p. X). He also stresses the distinction in the Grundgesetze (GGA I, p, X and p. XIV), in 'Uber die Begriffsschrift des Herrn Peano und meine eigene' (KS, p. 233), and in 'Uber die Grundlagen der Geometrie II' (KS, p. 270).

Why does Frege stress the distinction between objects and concepts? Frege rejects the grammatical analysis of sentences and substitutes objects and concepts (and other functions) for subjects and predicates, and thereby he changes the structure of universal and particular sentences. As far as singular sentences are concerned, he does not accept the identification of individuals with their essential properties. For Frege, the sentence 'Plato is a man' contains the 'is' of predication, which must be sharply distinguished from the 'is' of identity ('Logik in der Mathematik' (1914), NS, pp. 230-231).19 This means that Frege does not regard it as possible for us to have knowledge of what an object is in itself by means of our concepts. For him, all properties are on a par, whether they are called essential or accidental in traditional philosophical literature.

Surprisingly enough, Frege's much debated distinction between Sinn

and Bedeutung witnesses the same epistemological view. Disregarding Frege's views on the sense and the reference of a sentence, which Frege also labels a proper name, we can present Frege's doctrine of the sense and the reference of a proper name as follows: The sense which a proper name expresses and which is a way of presenting the object to which that proper name refers belongs to the object. Moreover, we should have complete knowledge of the object, only if we knew all its senses, which is not possible for us ('Uber Sinn und Bedeutung', KS, pp. 144-147). Accordingly, Fregean senses of an object seem to be


complexes of the concepts under which the object falls. This interpreta-tion of Frege's concept of Sinn is supported by his examples, according to which 'the Evening Star' and 'the Morning Star' are senses of Venus and 'the teacher of Alexander the Great' arid 'the pupil of Plato' are senses of Aristotle. Frege also argues that a proper name is related to an object via a sense and only via a sense and each proper name must express at least one sense (NS, p. 135). Hence, according to Frege, it is not possible to talk meaningfully of an object without thinking of the object as falling under some concept. Frege's remarks (;oncerning senses and references thus give us more support for the hypothesis that Frege believes in the universality of language. They may even hint at the view that there are no properties which belong to objects before there is a conceptual system which attaches senses to objects.

Frege's theory of sense and reference shows that Frege not only takes it to be impossible to find out any essential properties of objects, which would be identical with objects themselves, but he also considers the forming of complete concepts of objects as being beyond the abilities of a finite human being. Frege echoes Leibniz's thOUght that a human being is only able to form partial concepts of individuals, while God sees in the concept of an individual all that can be predicated of that individual (Leibniz, 'Discourse on Metaphysics', sec. 8 and sec. 9). For Frege, an object is neither identical with any essential property nor with any combination of the concepts under we can know the object to fall. By stressing the distinction between objects and properties, or concepts, Frege in effect draws the limits of human knowledge.

What has been said above shows why Frege calls special attention to the distinction between identity and predication. Frege's conceptual notation, which is meant to be a universal language, allows us to speak about objects only through different configurations formed by concept-words and other function-names. We cannot step outside these con-figurations in order to consider the relations between our language and objects themselves. An identity statement can only tell us that two names have the same Bedeutung but, according to Frege, we cannot say what this Bedeutung is. If Frege were consistent, we could not even accept a metalinguistic sentence which tells something about the relation between names and references. An identity statement tries to say some-thing that cannot be said, while predication is precisely the way in which our reason is capable of handling objects.

Even if Frege subscribed to the principle that one and only one


linguistic symbol or distinction should correspond to each reference or distinction of the universe, he did not eliminate the identity symbol from his language. Wittgenstein was more consistent in this respect, for he regarded an identity statement only as a rule, which concerns the substitutivity of names in different contexts. He assumed that we can give up the identity symbol, when we have realized the idea of universal language so that there are no more two different names for anyone single object in our language. According to Wittgenstein, the identity symbol is not an essential part of the conceptual notation (fLP, 5.53).

Frege considered the role of the identity symbol from a completely different perspective. As he mentioned already in the Begriffsschrift, he did not regard the identity statement only as a rule which concerns names. At the same time, he insisted on the principle that his language speaks about something and that each distinction and each symbol in language must correspond to a unique sense, a unique reference and a unique distinction in the universe. Hence, we must try to find out what the counterpart in the world is in the case of identity statements. What Frege found in the world was a distinction between senses and references, which follows from the distinction between objects in themselves and objects as we know them. Frege wanted to make a distinction between objects as metaphysical entities and objects as we know them, and he also wanted that this distinction is visible in his universal language. For this reason, he distinguished the 'is' of identity from the 'is' of predication in his conceptual notation and hence incorporated the identity symbol in his language. Frege did not notice that he ought to have excluded identity statements from his language precisely because they aim - at saying something about objects as metaphysical entities, that is, in the way in which we cannot speak about objects in language, according to Frege.

In the previous section I argued that Frege's view of language and the world influenced his doctrine of existence. The discussion above con�cerning identity and predication serves to clarify the details of Frege's view that the word 'exists' is equivocal. It is not only the case that Frege regards 'exists' and the 'is' of existence as equivocal in the sense that there are two concepts of existence apart from all linguistic contexts but that each context determines to which concept the words refer in each case. What is more, Fregean analysis has the additional consequence that 'exists' and the "is' of existence preserve their equivocity in some contexts. That is what happens if we attach them to proper names. I


shall clarify this point in what follows on the basis of what I argued above concerning identity and predication. That also yields an answer to the question concerning the philosophical background of Frege's view of existence.

If Frege's Sinne are complexes of properties of objects, the sentence 'a exists' expresses the thought that there is an object which is P, Q, R, etc. Since the sentence 'There is a P' means, for Frege, the same as the sentence The concept P is instantiated', likewise, the sentence 'a exists' means that a certain bundle of concepts is instantiated. Here existence turns out to be something that is asserted of a bundle of properties. Frege nowhere explicitly draws this conclusion from his premisses, but his suggestions are, however, evident. In 'Uber Sinn und Bedeutung' Frege undertakes to show that identity statements can be meaningful even if they seem to be either vacuously true or self-contradictory. The solution he offers is that one may associate a different sense with 'a' and with 'b' even if 'a = b' is true. If Frege supports this kind of analysis, he must also admit that 'a exists' makes sense. That is because one can, of course, attach a sense to 'a' without knowing that a exists, as easily as one can attach a sense to 'a' and 'b' without knowing that 'a= b' is true. Existence is not included in the Sinn expressed by a proper name. If we take the individual properties expressed by the proper name apart and form a judgment of each, the sentences which express the judgments may either be true or lack a truth-value. For Frege, forming a concept or a bundle of concepts is independent of the instantiation of that concept or bundle of concepts.

Like the distinction between identity and predication, the thesis of the equivocity of 'exists' and the 'is' of existence is motivated by epistemological considerations concerning the limits of human knowl-edge. My suggestion for construing Frege's doctrine of existence in the case of sentences like 'a exists' or 'a is' is hence the following: If we say that a exists and if someone asks us what it is that exists, we are not able to answer the question in any other way than by mentioning some of the concepts under which that objects falls. We can say that the sentence 'a exists' means that there is an object which has the properties P, Q, R, etc. Existence turns out to be a second-order concept, which means instantiation of a bundle of properties. But since we cannot say what the object a is as abstracted from our concepts, our answer to the question concerning what a is in itself comes down to saying that a is a, which is an empty statement. On Frege's view, we can say that an object is what it


is, that it is identical with itself, which is an empty statement, but we cannot say what it is, i.e., what it is identical with. Therefore, in the sentences 'a exists' and 'a is' words 'exists' and 'is' can be read either as expressions of a meaningful second-order concept or as expressions of an empty first-order concept. If we interpret the words as referring to the first-order concept, the corresponding sentence can be transformed into the sentence 'The name "a" has a reference', but that is, of course, of no help for us in finding out what the reference is.

Thus, Frege's distinction between the two concepts of existence ensues from his endeavour to distinguish objects in themselves from objects considered through the descriptions that we can attach to them. As I concluded above, Frege wants to make a distinction between objects as metaphysical units and objects as we know them, and he also wants that this distinction is visible in his universal language. But again, if Frege were to be consistent, he should eliminate the existence expressed by the existential quantifier and the symbol for identity from his language, for it tries to say something that, on Frege's view, cannot be said in language.

Academy of Finland

NOTES

* I am very grateful to Professor Jaakko Hintikka for valuable suggestions and comments.

My thanks are due to the Philosophical Society of Finland, which has given me the permission to use extracts from my monograph Frege's Doctrine of Being (Acta

Philosophica Fennica 39, 1985) in this article. 1 See 'Uber Sinn und Bedeutung', KS, pp. 144-147, GGA I, p. X and § 26, and 'Uber die Grundlagen der Geometrie I-Ill', KS, p. 285. 2 See AngeleIli (1967), pp. 253-254, and Schirn I (1976), pp. 20-21. 3 See Hintikka (1979b, 1981a). See also Kahn (1973a, 1973b) and Mates (1979). 4 See Hintikka (1973,1983,1985) and Knuuttila (1985). 5 See, e.g., 'Uber den Zweck der Begriffsschrift', BS, 1964, p. 98, 'Uber die Begriffs-schrift des Herrn Peano und meine eigene', KS, p. 227, and 'Anmerkungen Freges zu: Philip E. B. Jourdain, 'The development of the theories of mathematical logic and the principles of mathematics', KS, p. 341. 6 See van Heijenoort (1967), Goldfarb (1979), and Hintikka (1979a, 1981b, 1981c). 7 For further discussion of this problem, see Haaparanta (1985a), pp. 56-58. B See, e.g., GLA, § 53. 9 For Frege's concept of Wirklichkeit, see Haaparanta (1985a).


10 Note that Frege tried to treat higher-order entities by correlating with each of them a Werthverlauf, which is always a first-order entity, for instance, a value of first-order quantifiers. See, e.g., GGA I, § 3 and § 10. II See van Heijenoort (this volume). 12 This was suggested to me by laakko Hintikka. 13 Frege makes the same suggestion in 'Aufzeichnungen fiir Ludwig Darmstaedter', NS, pp.274-275. 14 This is also pointed out by R. Stuhlmann-Laeisz (Thiel, 1975, p. 126). 15 However, Frege anticipates the distinction between object-language and metalan-guage in the article 'Logische Allgemeinheit', where he makes the distinction between Hilfssprache and Darlegungssprache. See NS, p. 280. Cf. also Haaparanta (1985a), p. 35. 16 I do not want to argue that if existence is considered as a second-order concept, it is no more a problematic concept. The expression 'There is a - ' is far from clarified by the expression 'The concept - is instantiated'. Cf. Williams (1981), pp. 59-60. 17 See BS, § 8, GLA, § 65, and 'Uber Sinn und Bedeutung', KS, p. 150. See also GGA I, § 20, and NS, p. 131. 18 See Schirn (1976), Band II. See also Haaparanta (1985a). 19 Cf., e.g., Aristotle, Met. Z 6. M. J. Woods (1975) argues for the interpretation that, according to Aristotle, the sentence 'Socrates is a man' means that Socrates is identical with a man. See also Hintikka (1985). For further discussion, see Haaparanta (1985a), pp.94-96.

REFERENCES

Frege, G.: 1879, Begriffsschrift, eine der arithmetischen nachgebildete FormeZsprache des reinen Denkens. Verlag von L. Nebert, Halle a. S.; repr. in G. Frege (1964), pp. 1-88. (Referred to as BS.)

Frege, G.: 1883, 'Uber den Zweck der Begriffsschrift', in Sitzungsberichte der lenaischen Gesellschaft flir Medizin und Naturwissenschaft fUr das lahr 1882, Verlag von G. Fischer, lena, pp. 1-10; repr. in Frege (1964), pp. 97-106.

Frege, G.: 1884, Die GrundZagen der Arithmetik: eine Zogisch mathematische Untersu-chung tiber den Begrijf der ZahZ, Verlag von W. Koebner, Breslau; repro and trans!' by l. L. Austin in The Foundations of Arithmetic! Die Grundlagen der Arithmetik, Basil Blackwell, Oxford, 1968. (Referred to as GLA.)

Frege, G.: 1891, Funktion und Begriff, Vortrag, gehalten in der Sitzung vom 9. Januar 1891 der lenaischen Gesellschaft fiir Medizin und Naturwissenschaft, H. Pohle, Jena; repr, in KS, pp. 125-142.

Frege, G.: 1892, 'Uber Sinn und Bedeutung', Zeitschrift fUr Philosophie und philoso-phische Kritik 100, 25-50; repr, in KS, pp. 143-162.

Frege, G.: 1892, 'Uber Begriff und Gegenstand', Vierteljahrschrift flir wissenschaftliche Philosophie 16,192-205; repr. in KS, pp. 167-178.

Frege, G.: 1893, Grundgesetze der Arithmetik, begrijfsschriftlich abgeleitet, 1. Band, Verlag von H. Pohle, lena. (Referred to as GGA I.)

Frege, G.: 1894, 'Rezension von: E. G. Husser!, Philosophie der Arithmetik 1', Zeit-


schrift [iir Philosophie und philosophische Kritik 103, 313-332; repr, in KS, pp. 179-192.

Frege, G.: 1895, Kritische Beleuchtung einiger Punkte in E. Schroders Vorlesungen iiber die Algebra der Logik', Archiv [iir systematische Philosophie 1,433-456; repro inKS,pp.193-21O.

Frege, G.: 1896, 'Uber die Begriffsschrift des Herrn Peano und meine eigene', in Berichte uber die Verhandlungen der koniglich siichsischen Gesellschaft der Wissen-schaften zu Leipzig, Mathematisch-Physische Klasse, 48. Band, pp. 361-378; repr. in KS, pp. 220-233.

Frege, G.: 1903, Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, II. Band, Verlag von H. Pohle, Jena. (Referred to as GGA II.)

Frege, G.: 1903, 'Uber die Grundlagen der Geometrie II', Jahresberichte der Deutschen Mathematiker- Vereinigung, 12. Band, pp. 368-375; repr. in KS, pp. 267-272.

Frege, G.: 1906, 'Uber die Grundlagen der Geometrie I-III', in Jahresberichte der Deutschen Mathematiker- Vereinigung, 15. Band, pp. 293-309, 377-403, 423-430; repr. in KS, pp. 281-323.

Frege, G.: 1912, 'Anmerkungen zu: Philip E. B. Jourdain, The development of the theories of mathematical logic and the principles of mathematics', The Quarterly Journal of Pure and Applied Mathematics 43,237-269; repr, in KS, pp. 334-341.

Frege, G.: 1952, Translations from the Philosophical Writings of Gottlob Frege, ed. by P. Geach and M. Black, Basil Blackwell, Oxford.

Frege, G.: 1964, Begriffsschrift und andere Aufsiitze, ed. by I. Angelelli, Georg Olms, Hildesheim.

Frege, G.: 1964, The Basic Laws of Arithmetic, Exposition of the System, trans!. and ed. by M. Furth, University of California Press, Berkeley and Los Angeles.

Frege, G.: 1967, Kleine Schriften, ed. by I. Angelelli, Wissenschaftliche Buchgesellschaft, Darmstadt, and Georg Olms, Hildesheim. (Referred to as KS.)

Frege, G.: 1969, Nachgelassene Schriften, ed. by H. Hermes, F. Kambartel, and F. Kaulbach, Felix Meiner Verlag, Hamburg. (Referred to as NS.)

Frege, G.: 1972, Conceptual Notation and Related Articles, trans!' and ed. by T. W. Bynum, Clarendon Press, Oxford.

Frege, G.: 1979, Posthumous Writings, trans!. by P. Long and R White, Basil Blackwell, Oxford.

Angelelli, I.: 1967, Studies On Gottlob Frege and Traditional Philosophy, D. Reidel, Dordrecht.

Aristotle: 1928, Analytica Posteriora, in The Works of Aristotle, Vol. I, ed. by W. D. Ross, Oxford University Press, London.

Aristotle: 1928, Metaphysica, in The Works of Aristotle, Vol. VIII, ed. by W. D. Ross, The Clarendon Press, Oxford.

Bell, D.: 1979, Frege's Theory of Judgement, The Clarendon Press, Oxford. Dancy, R: 1975, Sense and Contradiction: A Study in Aristotle, D. Reidel, Dordrecht. Dancy, R: 1983, 'Aristotle and Existence', Synthese 54,409-442. Dummett, M.: 1981, Frege: Philosophy of Language, 2nd ed., Duckworth, London.

(First published in 1973.) Frede, M.: 1967, Priidikation und Existenzaussage, Hypomnemata 18, Vandenhoeck &

Ruprecht, Gottingen.


Goldfarb, W. D.: 1979, 'Logic in the Twenties: the Nature of the Quantifier', The Journal of Symbolic Logic 44, 351-368.

Haaparanta, L.: 1985a, Frege's Doctrine of Being, Acta Philosophica Fennica 39. Haaparanta, L.: 1985b, 'On Frege's Concept of Being', in Knuuttila and Hintikka

(1985). van Heijenoort, J.: 1967, 'Logic as Calculus and Logic as Language', Synthese 17,

324-330. van Heijenoort, J.: 1985, 'Frege and Vagueness', in this volume. Hermann, G.: 1801, De emendanda ratione graecae grammaticae, Gerhard Fleischer,

Leipzig. Hintikka, J.: 1973, 'Aristotle and the Ambiguity of Ambiguity', in J. Hintikka, Time and

Necessity: Studies in Aristotle's Theory of Modality, The Clarendon Press, Oxford, pp.1-26.

Hintikka, J.: 1979a, 'Frege's Hidden Semantics', Revue Internationale de Philosophie 33,716-722.

Hintikka, J.: 1979b, '''Is''.. Semantical Games, and Semantical Ralativity', Journal of Philosophical Logic 8, 433-468.

Hintikka, J.: 1981a, 'Kant on Existence, Predication, and the Ontological Argument', Dialectica 35,127-146.

Hintikka, J.: 1981b, 'Semantics: A Revolt Against Frege', in G. Fl0istad (ed.), Contem-porary Philosophy, Vol. 1, Martinus Nijhoff, The Hague, pp. 57-82.

Hintikka, J.: 1981c, 'Wittgenstein's Semantical Kantianism', in E. Morscher and R Stranzinger (eds.), Ethics, Proceedings of the Fifth International Wittgenstein Sym-posium, HOlder-Pichler-Tempsky, Vienna, pp. 375-390.

Hintikka, J.: 1983, 'Semantical Games, the Alleged Ambiguity of "Is", and Aristotelian Categories', Synthese 54, 443-468.

Hintikka, J.: 1985, 'The Varieties of Being in Aristotle', in Knuuttila and Hintikka (1985).

Kahn, {:.: 1973a, 'On the Theory of the Verb "To Be"', in M. K. Munitz (ed.), Logic and Ontology, New York University Press, New York, pp. 1-20.

Kahn, c.: 1973b, The Verb 'Be' in Ancient Greek, D. Reidel, Dordrecht. Kahn, C.: 1985, 'Retrospect on the Verb 'to be' and the Concept of Being', in Knuuttila

and Hintikka (1985). Knuuttila, S.: 1985, 'Being qua Being in Thomas Aquinas and John Duns Scotus', in

Knuuttila and Hintikka (1985). Knuuttila, S. and Hintikka, J. (eds.): 1985, The Logic of Being: Historical and Critical

Studies, D. Reidel, Dordrecht. Leibniz, G. W.: 1969, Philosophical Papers and Letters, 2nd ed., ed. by L. E. Loemker,

D. Reidel, Dordrecht. Mates, B.: 1979, 'Identity and Predication in Plato', Phronesis 24, 211-229. Owen, G. E. L.: 1965, 'Aristotle on the Snares of Ontology', in R Bambrough (ed.), New

Essays on Plato and Aristotle, Routledge and Kegan Paul, London, pp. 69-95. Schirn, M. (ed.): 1976, Studien zu FregelStudies on Frege I-III, Frommann-Holzboog,

Stuttgart - Bad Cannstatt. Stuhlmann-Laeisz, R: 1975, 'Freges Auseinandersetzung mit der Auffassung von

'Existenz' als ein Priidikat der ersten Stufe und Kants Argumentation gegen den ontologischen Gottesbeweis', in C. Thiel (ed.), Frege und die moderne Grundlagen-forschung, Anton Hain, Meisenheim am Glan, pp. 119-133.


Williams, C. J. F.: 1981, What is Existence?, Clarendon Press, Oxford. Wittgenstein, L.: 1961, Tractatus Logico-Philosophicus, The German text of Ludwig

Wittgenstein's Logisch-philosophische Abhandlung with a new Translation by D. F. Pears and B. F. McGuinness, Routledge & Kegan Paul, London. (Referred to as TLP.)

Woods, M. J.: 1975, 'Substance and Essence in Aristotle', Proceedings of the Aristotel-

ian Society 75,167-180.

PART III

LOGICAL THEORY

MICHAEL D. RESNIK

FREGE'S PROOF OF REFERENTIALITY

A logically perfect language (Begriffsschrift) should satisfy the conditions, that every expression grammatically well constructed as a proper name out of signs already introduced shall in fact designate an object, and that no new sign shall be introduced as a proper name without being secured a reference. The logic books contain warnings against logical mistakes arising from the ambiguity of expressions. I regard as no less pertinent a warning against apparent proper names having no reference. The history of mathematics supplies errors which have arisen in this way. ([7), p. 41; trns., [10], p. 70)

Here again we likewise see that the laws of logic presuppose concepts with sharp boundaries, and therefore also complete definitions for names of functions ... In Vol. 1 we expressed this as follows: every function-name must have a reference. ([4), p. 78; trans., [10), p. 170)

After reaching the end in this way, one may reread the Exposition of the Begriffsschrift as a connected whole, keeping in mind that the stipulations that are not made use of later, and hence seem superfluous, serve to carry out the basic principle that every correctly-formed name is to denote something, a principle that is essential for full rigor. ([3), p. 12; trans., [9), p. 9)

No methodological principle was more important to Frege than the one at stake in these passages: in a properly constructed scientific language every name (including function-names as well as object-names) must have a reference. In his eyes the repeated failures of his fellow mathe-maticians to be certain of satisfying this tenet was one of their most grievous errors. Thus it was entirely in keeping with this that he proved that every name in his own system has a reference. The foundations for the proof appear in Sections 28-30 of the Grundgesetze (13]) with the proof itself taking place in Section 31. Evidently, Frege thought that he need only show that his primitive names were referential since the referentiality of names composed from them would follow by an easy induction.! Accordingly, the proof focused almost exclusively on the "basis" case.

Frege's proof anticipates contemporary proofs that a given valuation for a language assigns a unique value to each sentence of the language. Since many authors regard those proofs 'as too obvious to state, it might be hard to imagine any difficulty with Frege's proof. There is agreement among Frege scholars that serious problems exist with his proof but that

177


178 MICHAEL D. RESNIK

is as far as the agreement goes. Some have claimed that there is no way to assign references uniquely to all sentences of Frege's language, because doing so would establish the consistency of his system.2 Others have claimed that the proof is circular; while most recently it has been claimed that the proof is perfectly valid but rests upon a contradictory assumption ([11], pp. 157-158). I scarcely exaggerate in saying that each person interested in the proof has thought that it involves one big mistake, and the dispute concerns the location of that mistake. I will give here still another account of the proof and show that the proof rests upon not one, but several, big mistakes whose significance has not been previously appreciated.

Some of the problems with Frege's proof involve simple oversights; others result from deficiencies in the semantical formalism Frege used; still others are due to conflating substitutional and referential inter-pretations of logical operators. Finally, one is due to Frege's use of the semantical version of Axiom V as a premise. As it turns out, semantical formalisms adequate for Frege's proof were not available until several generations later. They also took forms which would not have been acceptable to Frege.

These formal considerations bear upon the interpretation of Frege's philosophy. Several scholars have wondered whether Frege's proof and the sections of the Grundgesetze related to it reflect the signs of much broader semantical and ontological doctrines - such as a contextual theory of abstract entities. ([2], pp. 139-141; [16].) Certainly, those sections demand interpretation in the light of Frege's wider doctrines. But, as will emerge below, the nature of Frege's mistakes and his inability to give a fully satisfactory proof attenuate conclusions about his wider doctrines drawn from the proof itself or its attendant seman-tical formalism. For Frege's proof may have reflected no ontological or semantical doctrines at all; it may simply have been the best solution he found for the mathematical problem of proving the referentiality of his system.

I am not fully prepared to take such a conservative stand. I, too, suspect that deep issues may have been at stake in the proof. For, as we will see, there is a simpler and logically more satisfactory version of \"loth Frege's semantics and his proof which it is hard to believe that he could have simply overlooked. In my opinion, Frege rejected this alter-native because he thought that it would not properly represent classes as dependent upon concepts.

FREGE'S PROOF OF REFERENTIALITY 179

METATHEORY IN THE GRUNDGESETZE

1. Formation rules

The first step towards proving that all well-formed names are referential is to delineate the class of well-formed names. Frege does this with formation rules which are totally foreign to contemporary logicians. Instead of constructing "(x) (y) (x = y)" by first forming "x = y" and applying the quantifiers "(x)" and "(y)" to it, Frege starts with the first level function name "; = and an object name A. He then forms the first level function name "A = ;" which he combines with the second level function name "(y)o(y)" to yield "(y) (A = y)". Deleting the "A" from this, he obtains the function name "(y) (; = y)", which is, in turn, combined with "(x)o(x)" to yield "(x) (y) (x = y)". Frege calls the method by which two names are combined to yield a new one the first way of forming names. The method which forms a new name by deleting a well-formed part of a name he calls the second way. The closest Frege comes to an inductive definition of the class of well-formed names is to follow his presentation of the two ways of forming names with the statement "all correctly-formed names are formed in this manner".3 There is no problem in giving such an inductive definition, however ([14J, pp. 76-78).

Although Frege never discusses alternative formation rules, it seems clear that the rules he does give are chosen to reflect the way in which objects and functions of the various levels may be combined. Forming "(x) (y) (x = y)" from "(x)" and "(y) (x = y)" by juxtaposition would belie the incomplete nature of quantification qua second level function. Frege often warned against misrepresenting the unsaturated nature of functions by using names or variables for them which did not, them-selves, exhibit unsaturatedness.4 He would hardly have violated these strictures for metamathematical convenience. This passage makes it quite plain:

For the same reason many of Peano's designations in which a function-letter occurs without an argument are to be rejected. They contradict the very essence of a function, ... Such designations, which belie the real situation, may indeed seem at first sight convenient, but in the end they always lead into a morass .... But the designations that promise best to survive are those which adapt themselves most readily to diverse requirements and can be applied over the most extensive domain, and this because they fit the subject-matter best. We shall not be able to discover such designations if we are merely concerned to cope with the case in hand and do not pursue our reflections


further, but only if we attain the deepest possible insight into the nature of the subject-matter ([6], p. 156).

2. Semantical rules

Just as Frege's ontological views preclude the use of concatenation-style formation rules, they also exclude the Tarski approach to semantics. According to Frege, the denotation of a compound name is yielded by application of the functions denoted by some of its components to the arguments denoted by its other components. The universal quantifier symbol, for instance, denotes a second level function, and Frege's explanation

["(x)F(x)"] is to denote the True if for every argument the value of the function is the True, and otherwise is to denote the False" ([3], p. 12, trans. [9], p. 42).

tells us which function this is. A Tarski-style rule, such as (x)Sx is true if Sx is true for every assignment of values to x; and is false otherwise, (where a domain, interpretation and assignment to the free variables in S is presumed fixed), can be taken as defining a function denoted by the universal quantifier symbol. But, and this is the important point, it can also be taken as merely giving conditions for a universal quantification to be true. Furthermore, the Tarski approach extends to sentences containing free variables in a way that a Fregean approach should not. For Frege did not regard open sentences as names and reduced their content to that of their universal closures ([3], Sections 17 and 32). Thus the denotation of the universal quantification, "(x) (x = x)", which is a name, cannot be construed as a function of the denotation of the "x = x"; for the latter has a denotation only in the derivative sense of being stipulated to be coreferential with the former. As we shall see, Frege's non-Tarskian approach precludes the use of a simple inductive proof to show that if his simple names denote then so do all names which are correctly formed from them; although that very proof wouid seem altogether obvious to us today. (Frege's semantical conditions are like Tarski's in being recursive. Yet Frege's entire conception is unlike Tarski's in not relativizing interpretations to domains or to assignments to nonlogical symbols. The idea that a sentence could be true for one interpretation and false for an other, while familiar to Frege, was totally inimical to him. For him logic deals with one domain of each logical type - the universal domain of that type.)5


Frege explains the truth-function and quantifier symbols by directly specifying the functions which they denote. However, the explanation of the abstraction symbol does not follow suit. Instead, it is based on the stipulation "that the combination of signs 'iF(x) = yG(y)' has the same denotation as '(x) (F(x) = G(x))'" ([3], p. 16; trans. [9], p. 45). This does not assign denotations to abstracts; it only interprets identities between them, and leaves open questions concerning identities between abstracts and other terms. This odd feature of Frege's method, which has puzzled and provoked most Frege scholars, is responsible for several of the difficulties with his proof.

It is quite possible that Frege's unusual semantical stipulation was largely responsible for his undertaking his proof in the first place. For in Section 10 he attempts to deal with referential indeterminancies engen-dered by the stipUlation and he refers to this in Section 31. It is also possible that he regarded his stipUlation and the proof as justifying the introduction of classes. One can hardly fail to believe that some deep motive grounded his procedure, since Frege almost certainly considered the alternative of interpreting the abstraction operator by stipulating that it designates a second level function which maps each first level function onto its course-of-values. In fact his words,

If I say generally that "iF( x)" denotes the course-of-values of the function F( ;),

this requires a supplementation ... ([3], p. 15, trans., [9], p. 44),

virtually amount to such a stipulation. (Be that as it may, only the first explanation figures in either Frege's semantical discussions or his proof.) I will postpone further speculation on Frege's motives until later in this paper.

3. The Criteria of Referentiality

Frege Qpens Section 32 of the Grundgesetze with these words:

In this way it is shown that our eight primitive names have denotation, and thereby that the same holds good for all names correctly compounded out of these. However, not only a denotation, but also a sense, appertains to all names correctly formed from our signs. Every such name of a truth-value expresses a sense, a thought. Namely, by our stipulations it is determined under what conditions the name denotes the True. The sense of this name - the thought - is the thought that these conditions are fulfilled ([3], p. 50-51; trans. [9], pp. 89-90).


(This passage is the source of the famous truth-condition theory of meaning attributed to Frege by Wittgenstein and others.) As the passage shows, Frege thought that his proof of referentiality not only established that each well-formed name has a unique reference but also determined the identity of each name's reference. (This is not to say that their references were effectively determined or that the proof was construc-tive.) It is quite clear that Frege thought that the proof showed not only that "(x) (y) (x = y)", for example, denotes a truth-value but also that it denotes the True just in case every object x is identical with every object y. Had Frege not believed this he would have hardly claimed that his referential stipulations also sufficed to determine truth conditions for the sentences of his system.

The proof thus had two tasks: 1) to establish that each name has a (unique) reference and 2) to fix the identity of that reference. Unfortu-nately, Frege's semantical formalism was not equal to the second task.6

Frege needed a semantical formalism which specifies the reference of each compound name in terms of the reference of its component names. Only in this way can the reference of each compound name be deter-mined inductively from the reference of simple names. Now one might say, "What could be easier? '(x) 0 (x)' refers to the universal quantifier function; '- .;' refers to the horizontal function; so '(x) (- x)' refers to the True in case every object is the True, and refers to the False other-wise." Yes, there is no problem here; but consider instead the case of "(y) (x) (x = y)". This can be formed from the universal quantifier functor and the first level function-name "(x) (x = ';)". Hence we should be able to derive its reference from those of these components. But then we must already have determined a reference for "(x)

(x = .;)". However, "(x) (x = .;)" must be formed using the "second way", that is, by forming the name "(x) (x = A)", where A is an object name, and then dropping the occurence of A. But there is no analogous method for obtaining the reference of the name "(x) (x) (x = .;)". We cannot start with the object (x) (x = A) and "knock out" the object A in analogy to Frege's second way of forming names. For the same object can be the value of many functions for many different arguments; and, thus, there is no inverse operation to map a pair of objects into a unique function. Instead the reference for "(x) (x = .;)" must be obtained by applying the universal quantifier function to the identity relation. This would either require universal quantifiers which can apply to two place functions, ones which can apply to three place ones, etc, or it would

FREGE'S PROOF OF REFERENTlALITY 183

require an operation which applies to functions of any number n of places and carries them to functions of n-l places in accordance with the behavior of the universal quantifier. The latter alternative would violate Frege's theory of types and the former would require having infinitely many primitives. The abstraction operator poses similar prob-lems. (To the modern logician the quickest solution would be to add functional abstraction to Frege's metalanguage. Yet Frege could not accept this; for him functional abstracts do not designate functions, they designate objects!)

In Grundgesetze, Section 29, Frege "solved" this problem by intro-ducing criteria of referentiality which give conditions for a name to be referential but which fail to determine what the name's reference is. We read, for example, that "a name of a first-level function of one argument has a denotation (denotes something, succeeds in denoting) if the proper name that results from this function-name by its argument places being filled by a proper name always has a denotation if the name substituted denotes something" ([3], p. 46; trans. [9], p. 84). As far as these criteria go, "has a denotation" and "denotes something" can be one place predicates which ascribe to a name no more connection to objects, functions, and truth-values than does the predicate "is a quantifier".

This hints that all there is to an expression having a reference is that it always yields a referential expression when properly combined with referential expressions. This in turn is akin to the view that an expres-sion is referential so long as its occurence in a sentence does not disqualify the sentence itself from having a truth-value. So, perhaps, a contextual theory of reference is implicit in the Grundgesetze. Despite the suggestiveness of both these criteria and the very title of Section 29 ("When does a name denote something?") and the clear need to account for Frege's procedure, we should resist attributing a contextual view of reference to Section 29 of the Grundgesetze. Such an interpretation is virtually scotched by the opening lines of section 30, where we read:

The foregoing provisions are not to be regarded as definitions of the phrases "have a denotation" or "denotes something", because their application always presupposes that we have already recognized some names as denoting. They can serve only in the extension step by step of the sphere of such names. From them it follows that every name formed out of denoting names does denote something ([3], p. 46; trans. [9], p. 85).

First, as the passage clearly states, the criteria do not give the entire account of reference; they neither define what it is for an expression to


refer nor do they suffice for establishing the referentiality of each expression of Frege's system. They are to be used to show that names formed from names already known to denote also denote. Plainly, on pain of an infinite regress, the referentiality of some names must be established by other means. The remainder of Section 30 discusses the two ways in which new names may be formed from the names already at hand. Section 31 then opens with the remark that given what has been said before, one need only show that the simple names of the system denote in order to show all the names of the system do, and then turns to the proof that the simple names do denote. Thus it is clear that the primary purpose - and probably the sole purpose - of the criteria of Section 29 is to enable Frege to carry out his proof. To clinch the case, remember that simple names, with the notable exception of the abstrac-tion function name, are directly assigned references.

One might object that all this is compatible with a quasicontextual theory of reference. According to such a theory (some) simple names are referential because they refer to specific functions or objects, while other simple names and all compound ones are referential only in the sense of meeting the criteria of Section 29. But the idea that there might be such a distinction between names of the same syntactic category seems utterly unFregean. This is, perhaps, too strong; there is a passage in which Frege discusses, though in the most tentative terms, an idea similar to the contextual account. The text occurs in the appendix to the second volume of the Grundgesetze as part of his reaction to the Russell Paradox ([4], pp. 225-256; trans. [9], pp. 129-130). Frege says that if we are unwilling to recognize classes as "proper objects", that is, "as admissible arguments for every first level function" then we must regard class names as syncategorematic and without denotation. He continues, 'They would in this case have to be regarded as parts to signs that had denotations only as wholes." Now this sentence supports my contention that Frege would mark no distinction between the ways in which names of the same syntactic category can refer; names that do not refer standardly are not really names at all. (Frege proceeds to reject the syncategorematic approach, because it excludes classes and thereby numbers, from the range of object quantifiers).

There is a curious footnote to the sentence last quoted that I find difficult to reconcile with my interpretation. The footnote refers the reader to Vol. 1, Section 29, where the criteria of referentiality are given. Since the footnote is appended to the sentence stating that class


names refer only in the sense of occurring in wholes which have denotation, it seems to imply that the criteria of referentiality in Section 29 do not give sufficient conditions for genuine reference. In a sense, this does support my view, since it says that contextual reference is not reference at all. It also demolishes my view that Frege thought that the criteria of referentiality afforded a means for proving that certain names have reference in the full sense. There is a way out. The sentence prior to the one footnoted says that a proper object must be admissible as an argument for every first-level function. But a corollary of this is that a proper name refers to an object if and only if substituting it into the argument place of any first level function name which refers to a function yields a name which denotes the value of that function for the object as argument. This, of course, resembles the criterion of referen-tiality of proper names. Thus it is quite possible that the otherwise puzzling footnote was intended for the sentence prior to the one to which it was in fact affixed. If we" accept this explanation then there are no problems with my reading of the text in question. We can safely conclude that the criteria of Section 29 were intended neither as a contextual nor as a quasicontextual theory of reference.

It is worth recalling at this point that there are other explanations for the form taken by the criteria of Section 29, which have nothing to do with contextual views of reference. The first is that Frege may have known no other way to carry out the inductive part of his proof. The second is that he may have realized that approaches of the sort we considered above would conflict with his fundamental principles. I accept the first explanation. As noted earlier, Frege was apparently unaware of the failure of his proof to determine the identity of the reference of each name. Furthermore, another problem with Frege's proof - his use of what I will call the substitutional-referential approach to quantification and abstraction - indicates that he may have been unaware of the subtle differences between a name being referential in the sense of his criteria and referring to some particular entity. Finally, the semantical formalisms (the predicate functor logics and algebraic semantics) which measure up to a Fregean proof of referentiality have only been recently developed.

Do the criteria of referentiality succeed in at least establishing that every name is referential given that the simple names are? Yes, but they establish too much. For they state, in essence, that a name of a given type is referential provided it yields a referential name whenever


properly combined with other referential names.7 By appealing to them one can introduce names into Frege's system which the criteria will mark as referential but which fail to refer according to Frege's more familar canons of referentiality. For instance, let us suppose that all the names in the system refer in the usual sense of designating an entity of the type appropriate to their syntactic category. Now let us introduce a new first-level function name, "S", with the stipulation that "S(E)"

refers to the reference of E if E has one. Here E may be any proper name, including one which has not yet been introduced. Then the criteria of Section 29 will deem" S" as a referential function name, because whenever it is combined with a referential proper name the resulting compound name has a denotation, namely, that of the proper name. But plainly "s (';)" does not denote a totally defined function. Consequently, our introduction of it violates Frege's principle of com-pleteness ([4], sects. 56-65). Yet once again I cannot believe that Frege was aware of this consequence of his criteria or the difficulties it raises for his proof.

4. The Proofin Section 31

In this section Frege tries to prove that his simple names have refer-ences, assuming that once this basis case is established the referentiality of his compound names will follow automatically from the criteria of Section 29. He takes the referentiality of the truth-functors to be imme-diate and pauses briefly over the quantifiers. Here he uses the substitu-tional-referential approach. He shows that "(X)0(X)" is referential by showing that "(x)F(x)" is, for every referential function name, F(';). In other words, he shows that every referential substitution for "0" yields a referential compound instead of showing "(x) 0 (x)" denotes a function which has a value for every first level function as argument. He does that in tum by noting that "F(A)" denotes the True for every referen-tial object name A or it denotes something else (again, rather than by arguing that the function denoted by "F(';)" has a value for each argument). As we have already seen, this form of reasoning would lead Frege to other consequences which would be totally unacceptable to him. The major concern in Section 31, however, is with the abstraction operator.

Due to its unusual semantical stipulation, Frege approaches the abstraction sign differently. He argues that he can restrict himself to


occurences of abstracts within identity contexts, and notes that by using the criteria of referentiality he need only show that if F is referential then so is the function name "iF(x) = ';". To do so he need only show that "iF(x) = A" is referential for each referential object name A. But in Section 10 he had identified the truth-values with two value-ranges. Therefore, he restricts himself to those A which are referential abstracts of the form "yG(y)". Thus everything reduces to showing that "iF(x) = yG(y)" is referential. And by the semantical stipulation for abstracts this is if "(x) (F(x) = G(x»" is. Of course, it is referential, given that F and G are.

There are several problems with this argument. First, Frege has failed to provide for all the contexts in which an abstract may occur. One of these contexts is "(g) (g(';»)", and the proof becomes circular with respect to it. The context is a referential first level function name. Hence, following the model of the rest of Frege's proof, the way to show that an abstract A is a referential abstract is to show that "(g) (g(A»)" is referential (in addition to showing that A = .; is). However that is shown by showing that F(A) is referential for every referential first-level function name F(';), and "(g) (g(';»)" is one of these; so we have gone in a circle.s (Note that we do not have this problem with showing that "(g)

(g( ';»)" is referential; for we can restrict ourselves to function names already known to be referential and use the semantical condition for the universal quantifier.)

Bartlett, who was the first to see that Frege's proof was faulty,9 observed that it is circular in another sense. He noted that the proof shows only that if F(';) is referential then so is "iF(x)", and that in admitting the abstraction operator we can form first-level function names in which the abstraction operator itself occurs. Thus to apply the italicized principle non-circularly we must order the abstracts appropri-ately, and the presence of impredicative abstracts is incompatible with the ordering required. Parsons follows Bartlett in making a similar criticism. 1O Of course, as I have just remarked, even the proof of the italicized principle fails.

On the other hand, Martin sees no problem at all with the features of the proof considered so far, and thinks that its only flaw is that it rests upon a contradictory assumption - the semantical embodiment of Axiom V. I find his assessment dubious. The Bartlett-Parsons criticism points out that Frege tried to run an inductive proof while confusing the basis and inductive cases. An inductive proof should furnish a scheme


for establishing the theorem's assertion with respect to any finite initial sequence of its instances, and that is exactly what Frege's proof fails to do. Martin apparently thinks that there is another procedure, which is at least implicit in Frege's proof, and that it can handle with ease the impredicative abstracts, such as "y(j) (x/(x) = y)".

Here is how the Martin method works. We can prove that

(1) yif) (Xf(x) = y)

is referential by first applying the method of Section 31 (the basis case) to reduce the referentiality of (1) to that of

(2) yif) (x/(x) = y) = xF(x),

where F( s) is a referential function-name. The referentiality of (2) then reduces to that of

(3) (z) [if) (x/(x) = z) = F(z)].

Then we apply the inductive step, taking the universal quantifier to be referential, and reduce the referentiality of (3) to that of

(4) if) (x/(x) = B) = F(B),

where B is any referential object-name. Since F(s) and B are referential and the identity symbol is already known to be referential, the referen-tiality of (4) reduces to that of

(5) if) (x/(x) = B),

and since the universal quantifier symbols are already known to be referential we can reduce this case to

(6) xG(x) = B,

where G( s) is referential function-name. Then returning to the method of the basis case we replace B by an abstract formed from a referential function-name, say, xF(x), and reduce the question of the referentiality of (6) to that of

(7) (x) (G(x) = F(x»,

which is referential since F and G are. Undoubtedly, this method can be applied to any wellformed abstract of Frege's system.

Thus if we assume 1) the criteria of referentiality, 2) the semantical version of Axiom Y, and 3) the referentiality of the truth-function and


quantifier symbols, then we can argue validly that each wellformed name is referential. For in a finite number of steps Martin's procedure will reduce the question of the referentiality of any name to that of a name composed of the quantifier and truth-function symbols. Thus it could be argued that Frege had another "proof' of referentiality in mind; to wit, one which first establishes that all names constructed from the truth-functions and quantifiers are referential and then uses this to show that the abstraction operator is referential.

Is this what Frege had in mind? I think not. For this account fails to respect Frege's division of his proof into a basis case (all simple names are referential) and an inductive case (if N is a name correctly formed from referential names then N is referential too). Martin's method "proves" that the abstraction functor (a simple function name) is refer-ential by using the results of the inductive case. In addition, Frege's remark that the abstraction operator greatly complicates matters sug-gests that he did not think that a separate proof of referentiality for the truth-function and quantifier symbols could be carried over intact once the abstraction symbol was added.

I must confess some uneasiness with this criticism, since it is possible that Frege attempted to give an induction within the basis case, and confused this with an argument like Martin's. Whether this was Frege's procedure or not, it is not generally sound. Consider a language con-taining a supply of predicates and the description operator. Suppose that no vacuous descriptions are constructable in this language. Then suppose that we add a new predicate F, which is referential considered in itself and together with the other predicates of the language. It is still entirely possible that the definite description, ''the F", will fail to refer. Then what should we the blame for the failure of reference? Surely, the problem lies with the whole system of names and is not a fault of the predicates or of the description operator taken singly. The proof considered above suffers from the analogous mistake of assuming that if the truth-function and quantifier symbols are referential then they auto-matically remain so in the presence of the abstraction symbol. This assumption, in tum, can be traced to the criteria of referentiality, whose other faults we have already noted.

CONSISTENCY AND REFERENTIALITY

A common opinion is that no sound proof of referentiiility is possible


for Frege's system because it would amount to a proof of its consistency. Here is how Sluga has recently put it:

Since he treats sentences as names of truth-values, the proof that every name has exactly one reference amounts to a proof that no sentence refers to both the True and the False. If Frege's argument were successful it would in effect give us a consistency proof for the system of the Grundgesetze ([15), p. 167).11

But without some qualification that is clearly mistaken. The set of sentences of a formal language and the set of its theorems are not necessarily identical. It is quite easy to give examples of interpreted formal languages with sentences having unique truth-values and incon-sistent sets of theorems. Thus referentiality does not imply consistency. Furthermore, it is also easy to construct consistent formal languages which have nonreferring terms, and are thus not referential.

The independence of referentiality from consistency can be demon-strated for the symbolism of the Grundgesetze too. By using a standard interpretation (or Frege's) for the quantifier and truth-function symbols and by assigning each abstract the True as its denotation, one can secure referentiality while falsifying Frege's Axiom V. This together with Russell's contradiction establishes referentiality and inconsistency. On the other hand, by using a partial valuation for Frege's language we can verify a subset of his axioms. Then by taking these as the full set we obtain a consistent but not fully referential system (this is analogous to Quine's mixing of virtual and real classes).

But we cannot give a sound interpretation of Frege's system in which abstracts denote value-ranges construed as objects that satisfy Axiom V. The Russell paradox shows that there are no such objects. This means that no sound repair of Frege's original proof is possible (perhaps, this is all that Sluga and others have meant). Despite this there is no reason to think that we cannot interpret abstracts as denoting objects like value-ranges, so long as we lay down sufficiently weak axioms governing them.

There is also an indirect connection between consistency proofs and Frege's particular proof. That proof could be seen as an attempt to use Axiom V, or its semantical correlate, to eliminate abstracts in favour of formulas of second order logic. If all occurrences of abstracts were eliminable in favor of second order logic, we would have a consistency proof of Frege's system relative to second order logic. Thus Martin is right; there would be no way to eliminate abstracts entirely from Frege's system even if the referentiality proof were correct. I find little reason to

FREGE'S PROOF OF REFERENTIALlTY 191

believe that Frege had the elimination of abstracts in mind when he gave his proof. When Russell proposed the no-class theory to Frege, he used a definition quite similar to Axiom V. But Frege was only concerned with Russell's failure to represent functions as incomplete, and, though he did compare Russell's abstracts with his, he did not show much interest or enthusiasm for the no-class idea itself ([51, pp. 160-161).

We have just seen that referentiality and consistency are independent. Does referentiality bear on other major methodological issues? At the beginning of this paper I emphasized that there was a strong connection for Frege between referentiality and good methodology. Some of the passages I cited were portions of Frege's criticisms of the faulty defini-tions, such as contextual definitions, implicit definitions, conditional definitions or creative definitions, which many of his contemporaries used. It is thus ironic that a proof of referentiality for the primitives of a system in no way insures against improper definitions. Frege seems to have seen this too, and laid down rules of definition which can be shown to satisfy Lesniewski's criteria of eliminability and noncreativity.12 Whether or not the primitives of a system are referential, definitions in it which satisfy these criteria will be formally satisfactory in the sense of giving rise to a conservative extension of the system. If the primitives are referential besides, then such definitions will guarantee that the defined expressions are too.

However, there is no methodological impropriety in presenting, developing and studying uninterpreted formal systems. Hence Frege's requirement of referentiality is not a requirement for rigorous work with formal systems - at least not as such work is done today. Furthermore, even if rigor did demand that formal systems be interpreted, referen-tiality need not be a requirement of rigor. For some rigorously inter-preted systems might contain nondenoting terms. Referentiality is not an uncontroversial requirement of rigor, despite the emphatic protests by Frege with which I opened this paper.

Granted, Frege believed that a formal system of logic cannot be applied unless it is fully referential; thus he portrayed formalism as a grievous methodological error. Many formalists criticized by Frege did employ faulty definitions and committed other methodological sins; however, formalism itself can be given rigorous foundations. (Indeed, Frege saw how to do it. (Cf. [13], pp. 54-62.» Formalism is at worst a philosophical error and not a methodological one. Frege would like us to think that referentiality is a requirement of rigor; and it is, given his


views on logic and inference. But we must recognize that those have never been uncontroversial views. From the point of view of rigorous mathematical practice, both of today and of Frege's time, referentiality proofs are merely embellishments.

CONCLUSION

I have argued that Frege's proof contains a number of· mistakes and confusions, that it would not prove what he wanted even if it were correct, and, finally, that the demands of rigor do not even require it. Yet we still have no explanation for Frege's giving the proof in the first place. As I noted earlier, several possibilities come to mind. Frege did interpret his language referentially, and, thus, the proof was entirely appropriate for a rigorous presentation of his language. Perhaps that was the sole motive for Frege's proof. Another possibility, suggested by the criteria of referentiality, is that Frege used the proof to buttress a contextual theory of reference. I argued against that interpretation earlier. Possibly Frege felt that both the demands of rigor and the peculiarity of his semantics for abstraction necessitated giving the proof. This explanation seems the most plausible to me.

But this just raises another question. Why did Frege use the strange semantical stipulation for abstraction? Was it in order to guarantee that Axiom V would be true by virtue of the meanings of its terms? Suppose that he had introduced value-ranges in his metalanguage (as he in fact did), with the stipulation that he would use the words "the function has the same value-range as the function as having the same denotation as the phrase "the functions and have identical values for identical arguments" (Cf. [3], section 3). But then suppose that he went on (as he did not), to stipulate that the abstraction operator is a second-level function which maps first-level functions onto their value-ranges. His proof would have been greatly simplified and Axiom V, being a formalization of his meaning condition, would remain true in virtue of the meaning of its terms.

However, there is a serious flaw in Frege's metalanguage, and it would persist even if Frege made the changes just mentioned. Frege introduced the term ''value-range'' by means of a contextual stipulation. It is dangerously similar to definitions which he strongly castigatedY Value-ranges themselves are not necessarily at hand prior to Frege's stipulation nor are they introduced by an explicit existential postulate.


Frege could have avoided this situation by insisting instead that value-ranges exist and that it must be recognized as a fundamental logical law that there is a one to one association between them and first-level functions.

Then why didn't he do that? One is again tempted to see Frege as resisting the recognition of value-ranges in their own right, to see them as epiphenomenal, or along the lines of Russell's no-class theory or the results of a contextual view. But, as emerges in his discussion of the Russell paradox, Frege clearly thought of value-ranges as full-fledged objects and would reject any resolution of the paradoxes which treated them as less than SUCh. I4 The explanation for Frege's treatment of value-ranges seems to me to be connected with his insistence that the concept of class is a derivative concept of logic and that classes must be conceived of as extensions of concepts - the more fundamental logical entities. Is If we introduced value ranges, and classes, by means of an axiom that read,

With each function there is associated in a one to one fashion an object, its value-range,

we would fail to emphasize the "derivative" status of classes. Indeed, Frege might have supposed that since his readers might identify value-ranges with one of their own pet entities, e.g., with classes in Schroeder's sense, the logical self-evidence of the axiom introducing them would not be assured. I think that this explanation is as good an interpretation of the very puzzling treatment of value-ranges in the Grundgesetze as we currently have.

My account of Frege's attempt to prove his system referential por-trays him as confused and groping. I will neither moderate nor apologize for it. But Frege's greatness demands that it be put into perspective. Prior to Frege no one had clearly grasped the notion of a formal system, no one had even formulated the distinction between a system and its interpretation;nor had anyone tried to state or prove theorems relating them. Frege was the first to do all of these. Moreover, in assessing Frege's failure we must recall that no satisfactory general semantical theories for formal languages existed until Tarski's. If, in trying to rank Frege against other great figures of early mathematical logic, we compare his attempt with Hilbert's early effort (1904) to prove the consistency of arithmetic, the result is an embarrassment to Hilbertp6

The University of North Carolina Chapel Hill


NOTES

1 The criteria of referentiality given in Section 29 state, roughly, that a name is referential in case it yields referential names whenever combined properly with other referential names. The opening paragraph of Section 31 contains the sentence: "By what has been said, it is necessary to this end only to demonstrate of our primitive names that they denote something". 2 This view, which has been advanced by Bartlett and Sluga, is related to Frege's own remark to Russell (in the letter of 22.6.1902) that his "explanations to Sect. 31 do not suffice". But note that this remark is made with reference to the specific semantical stipulations for abstracts which he used in Section 31. See Note 9 below. It does not preclude proofs based upon seman tical stipulations other than those used by Frege. See [1], p. 75; [15J, p. 167. 3 [3], p. 48; trans., [9], p. 86. 4 For two examples see: [8], pp. 663-665; trans., [10], pp. 113-115; [4], p. 148, Note 2; trans., [10], p. 180. 5 See the Frege-Hilbert and Frege-Peano exchanges. 6 Martin was, to my knowledge, the first to remark on this problem with Frege's proof ([11], p. 163). 7 In fact Frege fails to lay down criteria which cover each way of combining names in his system. S This is not the circle discussed by Bartlett or by Parsons, but it has been noted by Thiel ([17], p. 82). 9 In the letter to Russell giving his first reactions to the contradiction Frege wrote:

my Law Y (Sect. 20, p. 36) is false, and ... my explanations in Sect. 31 do not suffice to secure a meaning for my combinations of signs in all cases ([5], p. 132).

It is thus clear that Frege realized that something was wrong with his proof, but given that he connected the problem to Axiom Y, it is most likely that he saw his proof was based upon an incorrect semantical stipulation for abstraction and was unaware of the other problems with the proof. 10 See [1], pp. 72-75; [12], p. 190. II See also [I],p. 75. 12 Lesniewski's notion of creative definitions is not the same as Frege's. To Frege a creative definitiol'l amounts to mislabeling, as a definition, an existence postulate or theorem. Frege discusses "creative definitions" in [4], Sect. 143; his principles of definition are given in [3], Section. 33. 13 This did not escape Frege's notice. In [4], pp. 148-149 Frege argues that since he introduces value-ranges by means of an axiom rather than a definition his procedure is not open to the criticisms he had directed at his colleagues. My claim, however, is that his metalanguage uses such a definition. 14 The question of why Frege introduced value-ranges in his metalanguage by means of a' contextual stipulation is related to, but distinct from, the question of why he was willing to settle by stipulation the question of whether, say, Julius Caesar is a value-range.


15 This is especially clear in [5], pp. 191-192. See [13], pp. 206-207 for further discussion. 16 I would like to thank Catherine Elign, Susan Hale, Philip Kitcher and Andrew Rein for their help with this paper.

REFERENCES

[1] Bartlett, J.: 1961, Funktion und Gegenstand, Dissertation, Ludwig-Maximilians-Universitiit, M. Weiss, Munich.

[2] Dummett, M.: 1981, The Interpretation of Frege's Philosophy, Harvard University Press, Cambridge.

[3] Frege, G.: 1893, Grundgesetze der Arithmetik, Vol. I, H. Pohle, Jena. [4] Frege, G.: 1903, Grundgesetze der Arithmetik, Vol. II, H. Pohle, Jena. [5] Frege, G.: 1980, Philosophical and Mathematical Correspondence, edited by H.

Hermes, F. Kambartel, C. Thiel and A. Veraat, abridged from the German by B. McGuinness and translated by H. Kaal, University of Chicago Press, Chicago.

[6] Frege, G.: 1979, Posthumous Writings, edited by H. Hermes, F. Kambartel and F. Kaulbach, translated by P. Long and R. White, University of Chicago Press, Chicago.

[7] Frege, G.: 1892, 'Uber Sinn und Bedeutung', ZeitschriJt fUr Philosophie und philosophische Kritik 100, 25-50.

[8] Frege, G.: 1904, 'Was ist eine FunktionT, in Festschrift fUr Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20. Februar 1904, A. Barth, Leipzig.

[9] Furth, M.: 1964, The Basic Laws of Arithmetic: Exposition of the System, a partial translation of [3] , University of California Press, Los Angeles.

[10] Geach, P. and Black, M.: 1952, Translations from the Philosophical Writings of Gottlob Frege, Blackwell, Oxford.

[11] Martin, E.: 1982, 'Referentiality in Frege's Grundgesetze', History and Philosophy of Logic 3, 151-164.

[12] Parsons, c.: 1965, 'Frege's Theory of Number', in Philosophy in America, edited by M. Black, Cornell University Press, Ithaca.

[13] Resnik, M.: 1980, Frege and the Philosophy of Mathematics, Cornell University Press, Ithaca.

[14] Resnik, M.: 1963, Frege's Methodology, Dissertation, Harvard University. [15] Sluga, H.: 1980, Frege, Routledge and Kegan Paul, London. [16] Snapper, J.: 1974, 'Contextual Definition: What Frege Might Have Meant but

Probably Didn't', NoUsS, 259-272. [17] Thiel, c.: 1965, Sinn und Bedeutung in der Logik Gottlob Freges, Verlag Anton

Hain, Meisenheim am Glan.

NINO B. COCCHIARELLA

FREGE, RUSSELL AND LOGICISM:

A LOGICAL RECONSTRUCTION

Logicism by the end of the 19th century was a philosophical doctrine whose time had come, and it is Gottlob Frege to whom we owe its arrival. "Often," Frege once wrote, "it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, in stripping off the irrelevant accretions which veil it from the eyes of the mind" ([Fd] , p. xix). Prior to Frege, logicism was just such a concept whose pure form was obscured by irrelevant accretions; and in his life's work it was Frege who first presented this concept to humanity in its pure form and developed it as a doctrine of the first rank.

That form, unfortunately, has become obscured once again. For today, as we approach the end of the 20th century, logicism, as a philosophical doctrine, is said to be dead, and even worse, to be impossible. Frege's logicism, or the specific presentation he gave of it in [Gg], fell to Russell's paradox, and, we are told, it cannot be resurrected. Russell's own subsequent form of logicism presented in [PM], moreover, in effect gives up the doctrine; for in overcoming his paradox Russell was unable to reduce classical mathematics to logic without making at least two assumptions which are not logically true; namely, his assump�tion of the axiom of reducibility and his assumption of an axiom of infinity regarding the existence of infinitely many concrete or non�abstract individuals.

Contrary to popular opinion, however, logicism is not dead beyond redemption; that is, if logicism is dead, then it can be easily resurrected. This is not to say that as philosophical doctrines go logicism is true, but only that it can be logically reconstructed and defended or advocated in essentially the same philosophical context in which it was originally formulated. This is true especially of Frege's form of logicism, as we shall see, and in fact, by turning to his correspondence with Russell and his discussion of Russell's paradox, we are able to fonnulate not only one but two alternative reconstructions of his form of logicism, both of which are consistent (relative to weak Zermelo set theory).

In regard to Russell's form of logicism, on the other hand, our

197


198 NINO B. COCCHIARELLA

resurrection will not apply directly to the form he adopted in [PM] but rather to the form he was implicity advocating in his correspondence with Frege shortly after the completion of [POM]. In this regard, though we shall have occasion to refer to certain features of his later form of logicism, especially in our concluding section where a counterpart to the axiom of reducibility comes into the picture, it is Russell's early form of logicism which we shall reconstruct and be concerned with here.

Though Frege's and Russell's early form of logicism are not the same, incidentally, they are closely related; and one of our goals will be to reconstruct or resurrect these forms with their similarity in mind. In particular, it is our contention that both are to be reconstructed as second order predicate logics in which nominalized predicates are allowed to occur as abstract singular terms. Their important differences, as we shall see, will then consist in the sort of object each takes nominalized predicates to denote and in whether the theory of predica�tion upon which the laws of logic are to be based is to be extensional or intensional.

1. LOGICISM AND THE PREDICA TIVE NATURE OF CONCEPTS

The doctrine of logicism can be succinctly stated in the following two�fold claim: (1) that all of the concepts of classical mathematics are explicitly definable in terms of purely logical concepts; and (2) that all of the theorems of classical mathematics can be derived from the laws of logic through purely logical deductions (cf. [Camap]). This is not a doctrine about the reducibility of mathematics to set theory, it should be noted, but about the reducibility of classical mathematics to the con�cepts and laws of logic. In other words, it is not a doctrine about the reducibility of mathematics to a theory of membership (in a class or set) but about the reducibility of mathematics to a theory of predication, and in particular to a theory about the concepts which predicates stand for in their role as predicates. For this reason both Frege and Russell maintain that the logistic framework within which classical mathematics is to be represented must consist at least of a second order predicate logic where quantification is not only with respect to the role of singular terms but to that of predicates as well. Indeed, it was Frege himself who first formulated and developed standard second order predicate logic, and he did so precisely as a framework within which classical mathe�matics was to be reduced to logic.

FREGE, RUSSELL AND LOGICISM 199

The distinction between predicates and singulars terms, it must be emphasized, is fundamental to both Frege's and Russell's forms of logicism; and to fail to attend to this distinction is to fail to understand the nature of predication in either framework. In particular, the attempt to characterize predication in terms of a first order theory of exemplifi�cation (or a first order theory of membership without an axiom of extensionality) is a nonstarter as far as either form of logicism is concerned. Predicates are not singular terms or what Frege called "proper names" in his extended sense, and the role of concepts in predication is not that of objects or individuals to which other objects or individuals stand in a relation of exemplification.

Now this does not mean that predicates cannot be nominalized and transformed into singular terms the way that 'human', 'triangular', 'wise', etc. can be transformed into 'humanity', 'triangularity', 'wisdom', etc.; or to use Frege's example, the way 'horse' can be transformed into 'the concept horse'. Rather, the point is that any account of nominalized predicates as abstract singular terms presupposes an account of their role as predicates; and in particular any relational predicate such as 'exemplifies' or 'falls under', as in 'Socrates exemplifies humanity' and 'Bucephalus falls under the concept horse', is to be viewed as derived

from an account of predication in which predicates do not have such nominalized forms. Indeed, the priority of the role of a predicate as a predicate over the corresponding role of its nominalization as an abstract singular term is in fact one of the ways we are to understand Frege's famous context principle (cf. [Fd], p. xxii). For it is only in the context of a sentence that a predicate can occur as a predicate, and it is only throught a correlation with such occurrences that we are to under�stand the role of a nominalized predicate as an abstract singular term. This sort of correlation is fundalnental not only to Frege's logicism but also to the reconstruction we shall propose.

What distinguishes a predicate from its nominalization, according to Frege, is that a predicate has an unsaturated nature which is essential to its role in predication. That is, whereas a nominalized predicate is a complete, saturated expression in its own right, the predicate itself is in need of supplementation, and it is this unsaturatedness or need of supplemenatation which is the basis of its predicative nature. Moreover, because this unsaturatedness is essential to its role in predication, the concept which a predicate stands for (bedeutet) is said by Frege to have a corresponding unsaturated nature as well. Indeed, it is precisely the


corresponding unsaturatedness of a concept which Frege identifies as its predicative nature (cf. [G & Bj, p. 47, [PW], p. 177).

Now just as the predicative nature of a predicate excludes its being a singular term, so too, according to Frege, the predicative or unsaturated nature of a concept excludes it from being the dena tatum (Bedeutung)

of a singular term; and therefore what a nominalized predicate denotes, according to Frege, is not the concept which the predicate otherwise stands for in its role as a predicate. Instead, according to Frege, the concept "must first be converted into an object, or, speaking more precisely, represented by an object" ([G & Bj, p. 46), and it will be this object, or concept-correlate as Frege also calls it, which is the real denotatum of the nominalized predicate. Thus in particular, what 'the concept horse' denotes, according to Frege, is not a concept but an object, that is, a concept-correlate.

This is not the conclusion Russell comes to in [POMj and thereafter, it should be noted; and in fact, as already indicated, Russell's different view in this regard is one of the principal differences between his and Frege's form of logicism. Russell did hold something rather similar to Frege's view in an unpublished manuscript of 1898, it might be noted; but in [pOM], it is clear, he explicitly rejects the earlier view. Thus (keeping in mind that terms and predicates for Russell are individuals and concepts, respectively, and not expressions) in 1898 he wrote that "the peculiarity of predicates is that they are meanings," and "although it is impossible to speak of meanings without making them subjects ... , yet meanings as such are the antithesis of subjects, are destitute of being, and incapable of plurality. When I say 'Socrates is human,' human as used in this judgment does not have being, and is not a logical subject. I am, in a word, not asserting a relation between two subjects. As soon as I make human a term, ... I have added something, namely being, one-ness, and diversity of being from other terms which human as predicate did not not possess" ([AMRj, book I, p. 10).

Russell briefly restates this view in [pOMj, observing that "it might be thought that a distinction ought to be made between a concept as such and a concept as used as a term" (p. 45); but he quickly rejects it, claiming that "inextricable difficulties will envelop us if we allow such a view" (ibid.). The difficulties in question, it turns out, are those con-nected with such claims as that the concept horse is not a concept, which Russell thinks is false and leads to a contradiction. His conclusion is that "terms which are concepts differ from those which are not, not in


respect of self-subsistence, but in virtue of the fact that, in certain true or false propositions, they occur in a manner which is different in an indefinable way from the manner in which subjects or terms of relations occur" (ibid., p. 46). In other words, concepts, including relations, are individuals (since 'term' for Russell is synonymous with 'individual' - cf. [POM], p. 43), and they are denoted as such by nominalized predicates occurring as abstract singular terms. Yet, predicate expressions are not singular terms and the concepts which predicate expressions stand for in their role as predicates do not occur in propositions the way that individuals denoted by singular terms do. That is, even though concepts, according to Russell, do not have an unsaturated nature, nevertheless they do have a predicative nature in virtue of which they can occur in propositions in "a manner which is different in an indefinable way" from that in which individuals denoted by singular terms occur.

Now the "indefinable way" in which a concept can occur in a proposition as a concept and not as a term, according to Russell, is also a feature of what he calls a propositional function; and in fact in [pOM] a propositional function can occur in a proposition only in this "indefin-able way". That is, unlike concepts, propositional functions in [POM] are not individuals. To be sure, to each concept there corresponds a unique propositional function in the propositional values of which the concept occurs as a concept; but upon discovering his paradox, Russell was led in [POMI to doubt that every propositional function is either itself a concept or that it has a concept corresponding to it. Of course, "apart from the contradiction in question," Russell observes, "this point might appear to be merely verbal: 'being an x such that fjJx,' it might be said, may always be taken as a predicate [concept]. But in view of our contradiction," Russell continues, "all remarks on this subject must be viewed with caution" (ibid., p. 88). In particular, the contradiction is avoided for Russell in [POM) "by the recognition that the functional part of a propositional function is not an independent entity" (ibid.); and in fact it was precisely for this reason that Russell was led in his commentary on Frege (in Appendix A of [POM]) to claim that "the word Begrijf is used by Frege to mean nearly the same thing as propositional function" (p. 507).

In his later form of logicism in [PM], Russell was able to avoid his paradox while also claiming that propositional functions are single entities after all; i.e., that propositional functions are individuals, albeit of a higher order/type than concrete individuals, a fact which is often


missed or ignored by philosophers and logicians alike since, contrary to his earlier practice, Russell chose to use 'individual' in [PM] only to refer to concrete individuals. There is no point in distinguishing concepts from propositional functions in such a framework, needless to say, and in fact we shall not assume any such distinction even in our resurrection of the form of logicism which is implicit in Russell's correspondence with Frege shortly after [POM] was written. That is, having noted the difference between concepts and propositional functions which Russell thought he was committed to in [POM] as a result of his paradox, we shall nevertheless assume that concepts, including relations, are none other than propositional functions, since that in fact was what Russell originally thought and returned to in his later form of logicism. We return in this way to the original context of Russell's paradox as it applies to Russell's early form of logicism no less so than as it applies to Frege's form of logicism. In that regard, in other words, we may assume that the only important difference between Frege's and Russell's early form of logicism which need concern us at this point is that whereas according to Frege nominalized predicates denote concept-correlates, i.e., objects somehow correlated with concepts, for Russell nominalized predicates denote the same concepts or propositional functions which the predicates in question otherwise stand for in their role as predicates. What is common to both forms of logicism, on the other hand, is that the concepts assumed in each form have a predicative nature, and that it is this predicative nature which is the basis of the laws of logic.

2. PREDICATION VS. FUNCTIONALITY

Standard second order predicate logic with identity, we have already noted, was first formulated by Frege as a framework in which to carry out the reduction of classical mathematics to logic. The reduction is not forthcoming, to be sure, without a logistic treatment of nominalized predicates or some such equivalent device, but still it is a framework in which the laws of logic have their basic form prior to any assumption about how nominalized predicates are to be interpreted. In this regard it is a system of the laws of logic which are common to both Frege's and Russell's early form of logicism; and for that reason we shall tum to its formulation first.

Accordingly, let us use 'x', 'y', and 'z', with or without numerical subscripts, to refer (in the metalanguage) to individual variables, and

similarly let us use 'pn" 'Gn' and 'Rn' to refer to n-place predicate variables. (JVe shall usually delete the superscript when the context makes clear the number of subject or argument positions that go with a predicate variable or constant; and we shall only use 'Rn' when n > 1.) As primitive logical constants, let us take -> (the material conditional sign), - (the negation sign), = (the identity sign), and 'if (the universal quantifier sign). As usual, we understand the juxtaposition of signs to represent their concatenation. We shall also use parentheses and brackets as auxiliary signs.

Ignoring the introduction of predicate and individual constants for the special applications of logic, the basic or atomic formulas are all of the expressions of the forms x = y and pn(XI> ... , x

n

), where n is a natural number. Well-formed formulas, or wjfs, are then defined as the members of the smallest set K containing the atomic formulas and such that - <jJ, (<jJ -> 'IjJ), ('if x)<jJ, ('if P)<jJ are in K whenever <jJ, 'IjJ are in K and x

and P are an individual and an n-place predicate variable, respectively, for all natural numbers n. For convenience, we shall use '<jJ' and ''IjJ' to refer (in the metalanguage) to wffs (with and without predicate and individual constants as well). We assume the usual notions of bondage and freedom of variables, of one variable or constant being free for another in a given wff, and of the proper substitution of a wff for an n�

place predicate variable (relative to n individual variables occurring free in that wff as subject-position indicators).

No function symbols other than predicate variables (and constants) have been introduced into our present logical syntax, it should be noted; and in this regard, it might be said, our formalism is more in line with Russell's form of logicism than with Frege's. For whereas Frege explained predication in terms of the mathematical notion of function-ality (cf. [G & B], p. 47), Russell explained mathematical functionality in terms of predication, i.e., in terms of the predicative nature of proposi-tional functions. That is, according to Russell, "the sort of function which is fundamental in logic is the propositional function; and the functions customary in mathematics are defined by means of this" «(EA], p. 261). Thus "if f(x) is not a propositional function, its value for a given value of x ... is the term y satisfying, for the given value of x, some relational proposition; this relational proposition is involved in the definition of f(x), and some such propositional function is required in the definition of any function which is not propositional" ([POM], p. 508).


All functions other than propositional functions, in other words, can be identified with many-one relations (cf. [POM], p. 83); and this, from either a philosophical or a logical point of view, is not an unimportant observation. Stated in this way, however, the observation in no way runs counter to Frege's form of logicism. For just as Frege analyzed all truth-functions in terms of those for negation and the material conditional, so too he could have analyzed all functions in terms of those which he calls concepts and relations, i.e., in terms of functions from objects to truth-values. (Frege never spoke of relations as concepts; i.e., concepts were always unary functions from objects to truth-values for Frege. For convenience, however, we shall speak here of relations as relational concepts; and whether a concept is unary or relational, we shall in either case refer to the saturated object corresponding to that concept as a concept-correlate.)

There is something in Russell's way of making the above observation, incidentally, which should not be overlooked; namely, that functionality presupposes predication in that it depends essentially on the unity of a proposition. Curiously, though Frege himself did not explain function-ality in terms of predication, nevertheless precisely this sort of con-sideration, whether applied to the unity of a sentence or to the thought (Gedanke) expressed by that sentence, seems to be fundamental to his notion of unsaturatedness, which of course is the basis of his notion of functionality. Thus in regard to the unsaturated nature of a predicate as the predicative component of a sentence, Frege argued that "this unsaturatedness ... is necessary, since otherwise the parts [of the sentence) do not hold together" ([PW), p. 177); and, similarly, "not all the parts of a thought can be complete; at least one must be 'unsaturated', or predicative; otherwise they would not hold together" ([G & B), p. 54). In other words, though Frege explained predication in terms of his mathematical notion of functionality, ultimately his only argument for the unsaturatedness of functions is his argument for the unsaturatedness of the functions involved in predication, whether that predication be syntactical or otherwise. In this regard our present for-malism neither distorts nor runs counter to Frege's form of logicism; and indeed, if anything, it rather emphasizes what is really fundamental in Frege's view of logicism.

3. EXISTENTIAL POSITS AND THE LAWS OF LOGIC

The laws of logic, according to Frege, must be universal, but only in the


sense that they must be applicable to any objects whatsoever. Frege does not mean to deny, in other words, that there are any existential posits among the laws of logic. To be sure, none of Frege's axioms in his Begriffsschrift are other than universal in form, but this does not mean that no existential posits are provable on the basis of these axioms. In particular, the (impredicative) comprehension principle,

(CP) (3P) (Vx) . .. (Vxn) [F(x), . .. ,xn

) ... ¢j,

where ¢ is a wff (pure or applied) in which P does not occur free and XI' •.. , xn are pairwise distinct individual variables occurring free in ¢, is easily seen to be provable on the basis of Frege's basic law (lIb).

Besides being provable, however, (CP) can be taken as an axiom schema and Frege's basic law (lIb) derived instead. That is, together with the remaining axioms and rules of standard second order predicate logic with identity, Frege's basic law (lIb), which we can formulate as follows,

(UI2) (VP)'IjJ -> 'IjJ[¢/F(xI"" ,x

n

)],

is provable on the basis of (CP) (cf. [Henkin]). Now although there is something to be said in favor of having the basic laws of logic all be universal in form, there is also something to be said for putting one's existential posits up front. In addition, there is something appropriate about avoiding the notion of proper substitution in stating the basic laws of logic, especially such a complex notion as the substitution of a wff for a predicate variable (relative to certain individual variables free in that wff as subject-position indicators).

Such an observation applies on the first order level as well, inciden-tally, especially if we are to adhere to Frege's view that "correctly-formed

names must always denote something" ([Gg], vol. 1, §28); for of course that view amounts to the assumption that logic is not free of existential presuppositions regarding singular terms. That is, where a is a singular term in which X does not occur (free), the fact that a denotes something (as a value of the bound individual variables), i.e., (3x) (a =

x), is provable on the basis of Frege's basic law (IIa):

(UII ) ('V x)'IjJ -> 'IjJ( a/ x).

But, conversely, given the remaining axioms and rules, (UI

I

) is also provable on the basis of (:Ix) (a = x) (cf. [K & MD.

A substitution free axiom set for standard second order predicate logic with identity can be described, accordingly, as follows. As axioms


(or basic laws of logic) we need take only all wffs (pure or applied) which are either tautologous or of one of the following forms:

(AI) (Vu) [¢ -+ 1/J]

-+ [(Vu)¢ -+ (Vu)1/J],

(A2) ¢ -+ (Vu)¢,

(A3) (3x) (a = x),

(LL) (a= b) -+ (¢ - 1/J),

(CP) (3P) (V Xl) ... (V xn)

[F(XI , ... , xn) - ¢j,

where u is an individual or a pre-dicate variable,

where u is an individual or pre-dicate variable not occurring free in ¢'

where a is a singular term in which x does not occur free,

where a, b are singular terms and 1/J comes from ¢ by replacing one or more free occurrences of b by free occurrences of a,

where Fn does not occur free in ¢, and Xl' ... , Xn are among the distinct individual variables oc-curing free in ¢.

As inference rules we need take only modus ponens and universal generalization:

(MP) if f- ¢ and f- (¢ -+ 1/J), then f- ¢;

(UG) if f- ¢, and u is an individual or a predicate variable, then f-(Vu)¢.

Unlike Frege, incidentally, Russell does not take (Ull ) and (UIz) as basic laws, but instead he takes their contrapositives, i.e., their corresponding form as existential generalizations (cf. [PM], *9.1):

(EG1) 1/J(a/x) -+ (3x)1/J,

(EG2) 1/J[¢/F(XI"'" xn)] -+ (3P)1/J.

Of course, Russell does not distinguish these two laws and expresses (EG2) in the form of (EG I); but that is because his "individual" variables are really metalinguistic variables subject to systematic ambiguity. The substitution of wffs for predicate variables which is involved in (EG2) is only implicit in Russell, moreover, and (EG2) is really effected through his use of the cap-notation, ¢(Xj, ... , xn)' for the representation of propositional functions. Such a use of the cap-notation, needless to say,


amounts to applying a syntactical operation for the generation of com-plex predicates. An alternative notation which we shall adopt here is Alonzo Church's A-operator for functional abstraction. That is, instead of Russell's notation ... , xn) we shall use [AXI ... for the expression of a complex predicate. Russell's "primitive proposition" (EG2) can then be restated as follows:

1Jl([A..

X

i· .. P) (3P)1Jl.

According to Russell, incidentally, "the above primitive proposition gives the only method of proving 'existence-theorems'" ([PM], p. 131). That is, "in order to prove such theorems, it is necessary (and sufficient) to find some instance in which an object possesses the property in question. If we were to assume 'existence-axioms', i.e., axioms stating (:lz) . fjJz for some particular fjJ, these axioms would give other methods of proving existence. Instances of such axioms are the multiplicative axiom (*88) and the axiom of infinity .... But we have not assumed any such axioms in the present work" (ibid.).

This is somewhat misleading, needless to say, since it suggests that in logic "existence-theorems" are only conditional theorems, and this we know from our discussion of (CP) is false. Thus, in particular, where ai' ... , an are singular terms which are free, respectively, for Xl' ... , Xn

in fjJ, the basic law regarding complex predicates corresponding to Russell's of his cap-notation is the principle of A-conversion:

(A-Conv)

Generalized, this principle can be stated as follows:

(V/A-Conv) (Vxl ) ... (VXn)([AXI ... XnfjJ] (Xl" .. , Xn) .... fjJ);

and of course from this, (CP) follows by (EG2).

It is historically noteworthy, incidentally, that the first explicit use of (CP) as a basic law of logic occurs in [Tarski] where it is credited to Lesniewski (cf. [Henkin], p. 203). Lesniewski apparently, referred to special cases of (CP), each of which is really an instance of (V / A-Conv), as "pseudo-definitions", which of course is quite appropriate since (CP) is the logical basis of all explicit definitions of predicate constants. That is, it is on the basis of (CP) that any such definition can be shown to be noncreative and that the predicate constant so defined is eliminable in principle. It is in terms of these "pseudo-definitions", in other words, that the reduction of classical mathematics to logic is to be effected.


4. WERTVERLAuFE AS CONCEPT-CORRELATES

No reduction of mathematics is forthcoming, however, without a logistic treatment of nominalized predicates, or some such equivalent syntactic device such as Frege's notation for value-ranges (Wertverliiufe). Value-ranges, it should be noted, are not sets or classes in the sense of being composed of their members; rather, they are the saturated logical objects which Frege also informally called concept-correlates. That is, value ranges are for Frege the denotata of norninalized predicates - or at least that is what we claim in our reconstruction and shall attempt to show in what follows. If we are right in this claim, then we may in effect identify Frege's implicit logic of nominalized predicates, that is, the logic which is implicit in his informal remarks, with his explicit logic of value-ranges. Indeed, this way of viewing Frege's theory of value-ranges, we claim, will not only provide the essential rationale for his basic Law V regarding value-ranges but it will also explain why his theory of value-ranges is not really a second-order set theory.

In justifying our basic claim that value-ranges are concept-correlates, accordingly, let us turn first to 'On Concept and Object', where Frege explicitly states that an expression of the form 'the concept F' denotes not a concept but an object which is somehow correlated with that concept. In particular, in response to Benno Kerry's objection that concepts are objects, and, moreover, objects other than their extensions, Frege explains that "in my way of speaking expressions like 'the concept F' designate not concepts but objects" ([G & Bj, p. 48). That is, "if he [Kerry] thinks that I have identified concept and extension of concept, he is mistaken; I merely expressed my view that in the expression 'the number that applies to the concept F is the extension of the concept like-numbered to the concept F' the words 'extension of the concept' could be replaced by 'concept'. Notice carefully that here the word 'concept' is combined with the definite article" (ibid.). In other words, according to Frege, 'the extension of the concept like-numbered to the concept F' denotes the same object as that which is denoted by 'the concept like-numbered to the concept F'. In a footnote of his original draft of this passage, Frege writes that "the question whether one should simply put 'the concept' for 'the extension of the concept' is in my view one of expediency" ([PW], p. 106). Of course, by the extension of a concept Frege means none other than a value-range (ct. [Gg], vol 1, §3).

The above explanatory remark is about as explicit as Frege gets in identifying concept-correlates with value-ranges. More indirect, but also more important, evidence for this identification can be found in the connection Frege implicitly makes between second level concepts and concept-correlates on the one hand and that which he explicitly makes between second level concepts and value-ranges on the other. A second level concept, we should explain, is essentially what a variable binding operator on wffs to wffs (such as a quantifier) stands for, or, equiva-lently, it is a concept corresponding to an open wff whi<:h may be used in a third order comprehension principle to specify such a variable binding operator, such as the wff (Vx) [F(x) -+ G(x)j which specifies the second level relational concept of the subordination of one first level concept to another (cf. [NBC-I]). A first level concept, i.e., one which a predicate stands for, is said by Frege to fall within a second level concept in a way analogous to (but still not the same as) the way that an object is said to fall under a first level concept.

Now the connection between second level concepts and concept-correlates which is implicit in "On Concept and Object" is the thesis that corresponding to each second level concept there is a special first level concept such that a first level concept, say, G, falls within that second level concept if, and only if, the object correlated with G, i.e., the

concept G, falls under the corresponding special first level concept; or in symbols (in the monadic case), where subject-position occurrences of a predicate variable are nominalized occurrences of that variable:

(VQx) (3F) (VG) [(Qx)G(x) ++ F(G)j.

Thus, for example, corresponding to the second level concept of (objectual) existence, i.e., the second level concept that first order existential quantifier phrases stand for, there is the special first level concept of being realized; and the correspondence, according to Frege, is so tight that even the same thought is expressed by 'there is a square root of 4' and 'the concept square root of 4 is realized' (cf. [G & Bj, p. 49f). A concept-correlate is realized, in other words, if, and only if, there exists an object which falls under the concept in question.

Needless to say, but if value-ranges really are concept-correlates, then Frege's thesis connecting second level concepts with concept-correlates should connect second level concepts with value-ranges; and indeed Frege explicitly states this to be the case in [Ggj, vol. 1, §25. That


is, according to Frege, "second level functions can be represented in a certain manner by first level functions, whereby the functions that appear as arguments of the former are represented by their value-ranges" (op. cit.); or in symbols (in the case of unary concepts):

(VQx) (3F) (VG) [(Qx)G(x) - F(tG(E»].

The singular term tG(E) is of course Frege's notation for the extension (value-range) of the concept G; and the above thesis, it should be noted, amounts in effect to a restatement of Frege's context principle regarding such expressions. That is, just as it is only in the context of a sentence (or of a wff in general) that a predicate can occur as a predicate, it is only through a correlation with such occurrences of a predicate that we are to understand the role of the name for a value-range; and of course it is precisely the same thesis, and therefore the same restatement of his context principle, which Frege implicitly gives for nominalized predi-cates, i.e., for abstract singular terms of the form 'the concept G'. We may read 'tG(E)', in other words, either as 'the concept G' or as 'the extension of the concept G', which, as already noted, is exactly what Frege said in the footnote referred to above ([PW] , p. 106).

For convenience, we shall hereafter refer to the above thesis (in either form) as Frege's double correlation thesis. This is because Frege assumes as part of his thesis both a one-to-one correlation between second level concepts and certain special first level concepts on the one hand, and a one-to-one correlation between first level concepts and certain special objects called concept-correlates or value-ranges on the other. Our two alternative reconstructions or resurrections of Frege's logicism, as we shall see, will turn precisely on a minor modification of one or the other of the correlations involved in this thesis.

5. FREGE'S DOUBLE CORRELATION THESIS AND HIS BASIC LAWV

There is at least one other place in Frege's wntmgs which clearly indicates that value-ranges are concept-correlates, and which, given Frege's extensional view of concepts (as functions from objects to truth-values) provides the essential rationale for his Basic Law V. Both the identification and the rationale, it should be noted, are again based on Frege's double correlation thesis, only now applied to second level


relational concepts, and in particular to the second level relation of mutual subordination.

Indeed, Frege's Basic Law V, viz.,

tF(E) = tG(E) ++ (Vx) [F(x) ++ G(x)j,

should be viewed precisely as a special instance of his double correla-tion thesis. For what is indicated on the right hand side of this law is none other than the second level relation of material equivalence or mutual subordination of two first level concepts; and on Frege's exten-sional view of concepts (as functions from objects to truth-values) such an equivalence amounts in effect to their "identity". That is, Frege's Basic Law V amounts to correlating the first level relation of identity with his second level relation of mutual subordination. Such a correla-tion is needed, Frege observes, since "to construe mutual subordination simply as equality is forbidden by the basic difference between first and second level relations. Concepts cannot stand in a first level relation. That wouldn't be false, it would be nonsense. Only in the case of objects can there by any question of equality (identity). And so the said transfor�

mation [from mutual subordination to identity] can only occur by

concepts being correlated with the same object. It is all, so to speak, moved down a level" ([PW], p. 182, italics added). Of course, if concept-correlates are identical when the concepts in question are mutually subordinate, then concept-correlates are none other than the extensions of the correlated concepts; i.e., then concept-correlates are value-ranges.

The identification of value-ranges as concept-correlates also explains, it should be noted, why Frege's theory of value-ranges is really not a second order set theory. For as a concept-correlate, Frege observes, a value-range "simply has its being in the concept, not in the objects which belong to it" (ibid., p. 183). That is, unlike sets whose existence or being is constituted by their members, concept-correlates, and therefore value-ranges, are "logical objects" whose sole determination is given by Frege's double correlation thesis, which, as already noted, is really a restatement of his context principle applied specifically to nominalized predicates. It is this thesis, in other words, which explains how "by means of our logical faculties we lay hold upon the extension of a concept, by starting out from the concept" (ibid., p. 181); for as also already noted, a value-range, and therefore a concept-correlate, is none other than the extension of a concept. In this regard, accordingly, the confusion sometimes made of Frege's theory of value-ranges with a


second order set theory might best be obviated by directly describing his theory of value-ranges as a theory of concept-correlates; that is, by describing his logicism as a second order predicate logic with nominal-ized predicates.

6. RUSSELL AND FREGE ON NOMINALIZED PREDICATES

Unlike Frege for whom the extension of a concept "has its being in the concept, not in the objects which belong to it," Russell originally took the extension of a concept to be a class, or rather he took it to be what he called a class as many as opposed to a class as one; and a class as many, according to Russell, is essentially many, i.e., it is essentially composed of its members (cf. [POM), chap. VI). Being many, however, a class as many is not a single object, according to Russell, and therefore it could not occur in a proposition as a term, which in effect defeated the whole point of Russell's original form of logicism.

Of course, prior to the discovery of his paradox, Russell assumed that a class as one always existed corresponding to a class as many (cf. [pOM), p. 104); and as so determined, a class as one was also composed of its members. But since a class as one is an individual, according to Russell, then it, unlike its corresponding class as many, can occur in a proposition as a term; and therefore the fact that a class as many, that is, the extension of a concept, was not a single object was without any real effect in Russell's original form of logicism.

With the discovery of his paradox, however, Russell gave up the assumption that a class as one always existed corresponding to a class as many. That is, Russell came to believe that it was this assumption which was "the source of the contradiction" (ibid.). In particular, in regard to his paradox of the class of classes that are not members of themselves, Russell found that he could only conclude that "the classes which as ones are not members of themselves as many do not form a class - or rather, that they do not form a class as one, for the argument cannot show that they do not form a class as many" (ibid., p. 102). Having given up this assumption, however, Russell in effect was forced to give up his original form of logicism.

Shortly after completing [POM) , Russell gave up not only the assumption that a class as one always existed corresponding to a class as many but he gave up assuming that there are any classes at all - i.e., classes in the sense of objects that are essentially composed of their


members. Thus, in his May 24, 1903 letter to Frege we find Russell writing that "I believe I have discovered that classes are entirely super-fluous. Your designation e¢(E) can be used for ¢ itself, and x n e¢(E) for ¢(x)" ([PMq, p. 158). (Frege's notation 'x n ecp(E)' in effect amounts to 'x E ecp(e),' as Russell observed in [POMI, p. 512.) The important suggestion here, it should be noted, is that propositional func-tions are individuals after all, and therefore they can occur in proposi-tions as terms instead of the classes as ones Russell originally assumed to correspond to their extensions. That is, instead of assuming that there are any classes, whether as "ones" or as "manys", Russell is now proposing what later he called his "no classes" theory; i.e., the theory that propositional functions are single entities (individuals) after all, and that all talk of classes is to be reduced to talk of propositional functions.

Frege's response to Russell's letter is of course predictable. "I cannot regard your attempt to make classes entirely dispensable as successful, the reason being that you use function letters in isolation" ([PMq, p. 160). In other words, according to Frege, "to use a function sign in isolation is to contradict the nature of a function, which consists in its unsaturatedness" (ibid.). Russell's reply in turn is of course also predict-able by now, for he writes that "it is not dear to rile that it is never permissible to use to function letter in isolation" (ibid., 166). That is, Russell simply refuses to accept the unsaturated nature of concepts.

Despite the stalemate on this point, however, the exchange is instruc-tive for our present purposes. For to do what Russell suggests, Frege (ibid., p. 161) observes that "we would first have to transform all [unary] function names in such a way that there was only one argument place and that was on the right-hand side. Thus, we would have to transform, e.g., 'x 2 = l' into '£(e2 = l)x', and 'x(x - 1) (x + 1) = 0' into '£[(e - 1) (e + 1) = Olx', so that we could write

£(e 2 = 1)x::? £[e(e-l) (e+ 1) = O]x

and for this, according to your [Russell's] definition,

£(e 2 = 1) c £[e(e- 1) (e+ 1) = OJ.''

Now what should be especially noted here is that Frege uses the sipritus asper or rough breathing operator to generate a complex predicate as opposed to his use of the spiritus lenis or smooth breathing operator to generate a complex singular term (for a value-range). Stated in terms of our own notation, and using the A-operator to generate complex predi-


cates, Frege's point is that in transforming a wff into a complex predicate, [AX¢(X)], we must keep in mind that [A.x¢(x)] really has a pair of parentheses (and commas as well in the case of a relational predicate) accompanying it. Thus whereas

(Vy) ([AX¢(X)] (y) [AX1jJ(X)] (y»

is a well-formed formula in which two A-abstracts occur as complex predicates, the expression

R([AX¢(X)], [AX1jJ(X)]),

where R is a 2-place predicate constant, cannot be well-formed or meaningful as far as Frege is concerned, since as a complex predicate a A-abstract "would be defined only in connection with an argument sign following it, and it would nevertheless be used without one; it would be defined as a function sign and used as a proper name, which will not do" (ibid.,p.161t).

In his reply on this point, Russell is undaunted, insisting, as already noted, that an expression for a propositional function can at least in some cases be used as a singular term. This will not do as it stands, of course, but we can reconstruct Russell's position here by distinguishing between two transformations of a wff ¢(x) where Frege has acknowl-edged only one. That is, we can first transform ¢(x) into the complex predicate [AX¢(X)] () which does have an accompanying argument or subject position (indicated by the last pair of parentheses); and then we can transform this complex predicate into the singular term [A.x¢(x)]

which does not have an accompanying argument or subject position. A confusion might arise here if we allow ourselves to speak of [AX¢(X)] as both a predicate and a singular term; but so long as we understand that when used as a predicate it must be accompanied by a pair of paren-theses (and commas in the case of a relational predicate), no confusion should arise if these parentheses (and commas) are informally dropped for abbreviatory purposes.

Frege, incidentally, is not unaware of this second transformation of a predicate into a singular term; for in applying Russell's suggested nota-tion in just this way, he notes that "we would have 'i( e2 = 1) = i[( e - 1) (e + 1) = 0]" which does not differ essentially from my 'f(e

2 = 1) = e[e e - 1) (e + 1) = 0]'" (ibid., p. 162). But in that case, Frege observes, Russell's suggested notation for nominalized predicates "would lead to the same difficulties as my value-range notation" (ibid., p. 161). That is,


other than assuming that nominalized complex predicates denote con-cepts or propositional functions as single entities, Russell's suggested use of such singular terms in 1903 was no less immune from his paradox than was Frege's use of the same singular terms to denote value-ranges.

7. RUSSELL'S PARADOX REVISITED

In turning to Russell's paradox and its resolution in our reconstruction of Frege's and Russell's early form of logicism, i.e., the form of logicism Russell was implicitly advocating in his correspondence with Frege, let us first give a more explicit formulation of both the logical grammar and the logical principles involved in the above exchange. As indicated, instead of using both the spiritus asper or rough breathing operator for the generation of complex predicates from wffs and the spiritus lenis or smooth breathing operator for the generation of complex singular terms from wffs, we shall follow the reconstructed Russellian strategy sug-gested above and use only the A-operator for both purposes. Also, for convenience of expression we shall informally drop parentheses and commas when referring to predicates. It should be noted in this regard, however, that using one operator and adopting an informal convention of dropping parentheses and commas when referring to predicate expressions in no way prejudges the case in favor of Russell's view that concepts have a saturated or individual nature. Aside from simplicity and economy of notation, in other words, the convention allows us to formulate a logical grammar which is common to both forms of logic-ism. (We will later introduce an intensional operator for Russell's form of logicism as well, but this will not affect the grammar which is common to both forms of logicism.)

In describing our logical grammar we shall for convenience of exposi-tion identify the diffferent types of meaningful expressions by associat-ing them with different natural numbers, where 0 is understood to represent the type of all singular terms, 1 the type of all wffs or proposi-tional forms, and n + 1, for n > 1, the type of all n-place predicate expressions. Individual variables, accordingly, are of type 0, proposi-tional variables are of type 1, and n-place predicate variables are of type n + 1. We continue to ignore the introduction of special individual and predicate constants, and, for nEro, we recursively define the meaning�

ful expressions of type n, in symbols, MEn> as follows:

216

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

NINO B. COCCHIARELLA

every individual variable (or constant) is in MEa, and every n-place predicate variable (or constant) is in both MEn+ I and MEo;

if a, b E MEo, then (a= b) E MEl;

if :n E MEn+I' and aI' ... , an E MEo, then :n(al , ... , an) E

MEl;

if E MEl, and Xl' ... , xn are pairwise distinct individual variables, then [AXI ... E MEn+ l ;

if E MEl, then - E MEl;

if and ljJ and in MEl> then ..... ljJ) E MEl;

if E MEl' and a is an individual or a predicate variable, then ('V E MEl;

if E MEl, then E MEo; and

if n > 1, then MEn MEo.

Singular terms, which we shall also refer to simply as terms, are now understood to be all the members of MEo; and for n > 0, we under-stand the members of MEn+1 to be n-place predicate expressions. We are in general to think of each n-place predicate expression as having n argument or subject positions associated with it, and, as in clause (3) above, these are all understood to occur within parentheses and to be separated from one another by commas. Wffs or propositional forms are of course all the members of MEl. Note that whereas by clause (9) every predicate expression is a term, not every wff is a term. We differ in this regard from what Frege would allow; but our difference is negligible since by clause (4), where n = 0, is a wff if is a wff, and by clause (8) is a term. In other words, besides O-place predicate variables (and constants), wffs are terms only when prefixed by the A-operator. We shall in general read as 'that f when it occurs in a wff as a term, i.e., when it occurs in one of the arguments or subject positions of a predicate expression.

It is clear of course that predicate expression occurring in the argu-ment or subject positions of other predicates, or of themselves as well, are intended to represent the nominalized predicates that occur in natu-rallanguage. For this reason, we shall refer to such occurrences of a


predicate as nominalized occurrences of that predicate, acknowledging thereby that it has been transformed into an abstract singular term. Note that adding such suffixes as '-ity', '-ness', or '-hood' to nominalized occurrences of predicates would be completely superfluous here since such occurrences are already formally identified as subject position

occurrences. The same observation applies, needless to say, to the use of such related phrases as 'the concept P or 'being an P. Such phrases and suffixes are important in transformational grammar. no doubt, since they serve to mark derived nominal expressions in the surface grammar of English; and in that regard we shall ourselves use such expressions when translating or verbally stating certain theses in English. Neverthe-less, it is sheer sophistry to insist that such surface grammatical features of English either must or should occur in our "deep structure" logical forms, as though a logical error were being committed otherwise.

As logical principles regarding this grammar and with respect to which Russell's paradox is to be derived, we assume exactly the same axioms and inference rules of standard second order predicate logic with identity already described in §3, but understood now to apply to wffs containing nominalized predicates as well. We assume in this regard (but avoid going into the details here) the obvious definitions of bondage and freedom of terms and predicate expressions in wffs and A-abstracts, and also when one such expression can be properly substi-tuted for another of the same type. In general we shall use a *-label in referring to axioms and other theses so as to remind ourselves that we are now dealing with wffs which may contain nominalized predicates as singular terms. Thus, e.g., Axiom (A3), which is now referred to as (A3*), has not only

(3y) (Fn-= y)

but also

(3y) ([AXI ... xn

¢] = y)

as an instance. Similarly, Leibniz' Law, (LL), which is now referred to as (LL *), has not only

but also

pn = Gn -. (V'YI) ... (V'Yn) [F(YI' ... ,Yn) ... G(YI'···' Yn)]

[AXl •.. x

n

¢] = [AXI •.. xntp] -. (V'Yl) ... (V'Yn) ([AXl

x

n

¢] (Yt> ... ,Yn) - [Axl · .. xntp] (YI' ... ,Yn»


as an instance. Together with (A-Conv*), moreover, this last instance of (LL *) has the following generalized form of Frege's basic law (Vb) as an instance:

[AX

j

. .. = [AXj ... Xn1fJ] -+ (V'Xj) . .. (V'xn) ++ 1fJ).

Note incidentally that the comprehension principle (CP), now referred to as (CP*), does not have the following standard formulation of Rus-sell's paradox as an instance:

(3F) (VG) [F(G) ++ - G(G)].

This is because concepts are posited by (CP*) only by means of condi-tions that apply to all individuals or object, whether those individuals be abstract or concrete. It is for exactly the same reason, moreover, that one cannot define in this context a predicate, say, 'Impredicable', as fol-lows:

(V' G) [Impredicable( G) ++ - G( G)].

For, as already indicated, an explicit definition of a predicate constant must be based upon the comprehension principle (CP*) in the sense of being one of its existential instantiations, and the above fails in this regard for the same reason that the preceding wff is not an instance of (CP*).

Russell does give another version of his paradox, however, in terms of ''what seems like a complex relation, namely the combination of non-predicability with identity" ([POMj, p. 97); and this version is an instance of (CP*):

(3F) (V'x) (F(x) ++ (3G) [x= G& - G(x)]).

A contradiction is derivable from this instance of (CP*), it should be noted, only because

(UIf) -+

is derivable from (A3*) and (LL*). (By (LL*),

F= x -+ -+

and therefore by (UG), (A1 *), (A2*) and tautologous transformations,

(3x) (F= x) -+ -+

(UIf) then follows, . accordingly, by (A3*) and modus ponens.) Indeed,


by weakening either (A3*) or (LL*) in certain obvious ways it can be shown that the resulting second order logic of nominalized predicates is consistent, not inconsistent. The weakened version of (LL *) is appro-priate, it should be noted, only when identity is defined (or replaced) by indiscernibility. (Cf. [NBC-4], §4.6 for a consistency proof when (LL *) is weakened to its version for indiscernibility; and also §4.1 0 for a consistency proof when (A3*) is weakened instead.) We do not con-template rejecting or modifying (LL *) here, however, and although (A3*) will be weakened in our second alternative reconstruction of Frege's logicism, we shall retain it in our first and more fundamental reconstruction, since, together with (LL *), it implies that part of Frege's double correlation thesis, namely that every concept has a unique saturated concept-correlate, which will remain intact in our first recon-struction of Frege's logicism. Of course, since this correlation for Russell is really an identity, then (A3*) cannot be weakened in any reconstruction of Russell's early form of logicism.

Now in his correspondence with Frege, it should be noted, Russell does suggest weakening one other principle, namely, the comprehension principle (CP*). Thus, in his December 12, 1904 letter to Frege, we find Russell writing that "I believe the contradiction does not arise from the nature of a class, but from the fact that certain expressions of the form

(¢) . F(x, ¢x, ¢£)

... do not represent [propositional] functions of x. That is, we have

f- ::(:3F):: -'(3!):.(x):fx= (¢). F(x,¢x,¢£).

This is easy to prove ill the case of

x = (¢£). :::> • - ¢x .

¢ , For this proposition denies fj£(f£)} for any f" ([PMq, p. 167). Re-stated in our own present notation, Russell's particular example amounts to

(VG) [x = [AyG(y)] -> - G(x)],

and given the identity

G= [AyG(y)]

as a law of the logic of norninalized predicates, Russell's particular instance of (CP*) which he wants to reject is


(3F) (VX) (F(x) - (VG) [X= G .... - G(X)]),

which in the present context is also easily seen to lead to a contradic-tion. The editors of [PMC] erroneously claim there is no contradiction here (cp. p. 168), incidentally, and their explanation suggests that they are ignoring Russell's original proposal that f= But even without this particular identity, it is clear that a contradiction is derivable on the basis of (LL*), or what amounts to Frege's basic law (Vb), and the following instance of (CP*) which is clearly intended by Russell in the letter in question:

(3F) (V x) (F(x) - (VG) [x = [AyG(y)] .... - G(x)]).

Now it is clear of course that Russell is not proposing that we are to reject all instances of (CP*) in his letter to Frege, but only those of a certain form. What is not clear, on the other hand, is the precise delimitation of the excluded forms in question; that is, it is not clear what restricted form (CP*) is to have according to Russell. What we shall suggest in our reconstruction of this early form of Russell's logicism is that (CP*) is to be restricted in accordance with the theory of simple types Russell described in Appendix B of [pOM), though applied now to propositional functions as. individuals rather than to classes as many as the extensions of propositional functions. Such a theory, in other words, seems to be what Russell had in mind in his correspond-ence with Frege shortly after the completion of [POM).

Before turning to the specifics of Russell's proposal, however, let us note that independently of its contradictory instances (CP*) is a conse-quence of a still simpler form of comprehension principle, namely,

(Cpn (3P) ([AXI ... = F),

where P does not occur free in For by (LL *) and (V / A-Conv*),

[AX

I

••• = F .... (Vx

l

) ••• (Vxn) [F(x

l

, ••• , x

n

) -

from which (CP*) follows by (UG), (Al*), tautologous transformations and (Cpn. A restricted form of (Cpn, needless to say, will imply only a restricted form of (CP*); and for this reason we shall apply our recon-struction of Russell's proposal to (Cpn instead.

FREGE, RUSSELL AND LOGICISM

8. FREGE'S REJECTION OF SCHRODER'S HIERARCHY OF INDIVIDUALS

221

Four months prior to his completion of [POM] in Dec. 1902, Russell wrote Frege suggesting that ''the contradiction could be resolved with the help of the assumption that ranges of values are not objects of the ordinary kind; i.e., that needs to be completed (except in special circumstances) either by an object or by a range of values of objects or by a range of values of ranges of values, etc. ''This theory," Russell observed, "is analogous to your theory about functions of the first, second, etc. levels. In x ("\ u it would be necessary that u was the range of values of objects of the same degree as x: x ("\ x would therefore be nonsense. This view would also be useful in the theory of relations" ([PMC] , p. 144). This suggestion was subsequently described as the simple theory of types which appears in Appendix B of [POM].

In his reply to Russell and the suggestion "that we are to conceive of ranges of values and hence also of classes as a special kind of objects whose names cannot appear in all argument places of the first kind" (ibid., p. 145), Frege noted that "a class would not then be an object in the full sense of the word, but - so to speak - an improper object for which the law of excluded middle did not hold because there would be predicates that could be neither truly affirmed nor truly denied of it. Numbers would then be improper objects" (ibid., italics added). In his discussion of Russell's paradox in the appendix to Volume 2 of [Ggj, Frege reiterated this objection, nothing that if "classes were proper objects, the law of excluded middle would have to hold for them" ([G & Bj, p. 235).

Now it is noteworthy that in 1895 Frege had already considered and rejected the restriction on the laws of logic which would be necessary for the kind of hierarchy of classes (and later of propositional functions as individuals) Russell was suggesting as a way of avoiding his paradox. The circumstances of this rejection had nothing to do with Russell's paradox, needless to say, since Frege was at that time unaware of the paradox; but rather they had to do with E. Schroder's conceptual difficulties with the empty set as an extension consisting of nothing, on the one hand, and with his notion of a singleton as an extension which was identical with its only member, on the other (cf. [G & Bj, pp. 86-106). Be that as it may, Schroder, as Alonzo Church has observed, anticipated the theory of types when he took ''the universal class 1


which appears in his algebra, not as an absolute universal class, but as composed of all the elements of a certain domain fixed in advance" ([Church), p. 150). Once such a universal class or "manifold" was given, moreover, a second may be obtained (to which the algebra is to be applied in turn) "by taking the subsets of the first to be the individuals of the second" (ibid.); and by continuing in this way a "hierarchy of reine Mannigfaltigkeiten may be extended to infinity" (ibid.). The crucial restriction Schroder imposed on this hierarchy was that no subset of the domain of "individuals" considered at any stage of the hierarchy was to be among the individuals of that stage, and that consequently the laws of logic, as the laws of his algebra, were to be restricted in any given application only to the individuals of the stage in question.

It was precisely this sort of restriction of the "field for our logical activities" ([G & B), p. 92), in other words, which Frege criticized and rejected in his 1895 review of Schroder's book. For ''whereas elsewhere logic may claim to have laws of unrestricted validity, we are here required to begin by delimiting a manifold with careful tests, and it is only then that we can move around inside it" (ibid.).

Russell was on the mark, it should be noted, when he observed that such a restriction on the laws of logic was already imposed by Frege on his functions or concepts of first, second, etc. levels. But then for Russell, it must be remembered, these functions or concepts do not have an unsaturated nature; and as far as Russell is concerned if the laws of logic can be restricted when applied to concepts of different levels, then they can also be restricted when applied to the extensions of these concepts as well. Indeed, once classes as the extensions of concepts are eliminated from logic altogether, then as far as Russell is concerned the restrictions in question are essentially the restrictions already imposed by Frege on the concepts of different levels.

Now the crucial point ill Russell's view of Frege's hierarchy, it must be emphasized, is that concepts do not have an unsaturated nature; i.e., that despite their predicative nature (as functions from individuals to propositions), concepts for Russell are abstract individuals. This is essential to Russell's interpretation, in other words, since if concepts really do have an unsaturated nature as Frege claims, then they cannot be construed as objects or abstract (higher order) individuals; and as a theory of different types of individuals, the theory of types, whether simple or ramified, would then not be applicable to Frege's concepts of different levels. Indeed, it is precisely because concepts, and functions in


general, have an unsaturated nature according to Frege that Church rejects "the claim sometimes made on behalf of Frege that his Stufen ... constitute an anticipation of the simple theory of types" ([Churchl, p. 151). And in his rejection of classes as "improper objects" Frege himself points out that "there is nothing 'unsaturated' or predicative about classes that would characterize them as functions, concepts, or rela-tions" ([G & BI, p. 235). Numbers in particular are objects, according to Frege, not concepts or "improper objects" as they would have to be on Russell's proposal.

9. FREGE'S DOUBLE CORRELATION THESIS AND THE THEORY OF SIMPLE LOGICAL TYPES

There is a way, it turns out, of reconstructing Russell's proposal while still agreeing with Frege that if nominalized predicates denote objects as individuals "in the full sense," then the laws of logic, and the law of excluded middle in particular, must not be restricted when applied to such objects. We shall do so, moreover, by applying the notion of a simple logical type not to a description of Frege's Stufen or levels of concepts as "improper objects" the way Russell suggests but rather as a description of the conditions determined by Frege's double correlation thesis for positing first level concepts and their corresponding concept-correlate:;.

Now the point of our reconstruction is that if second level concepts can be correlated with certain first level concepts, then third level concepts can be similarly correlated with second level concepts, and therefore, by the product of these correlations, third level concepts can in effect also be correlated with first level concepts. Similarly, fourth level concepts can be correlated with third level concepts and therefore with first level concepts as well. In general, in other words, all concepts of whatever level can in effect be correlated with first level concepts, and these in turn can be correlated with certain (saturated) objects called concept-correlates. Thus instead of speaking of a higher level concept Q falling within a concept M of one level higher we shall instead generalize Frege's double correlation thesis and speak of the concept-correlate of the first level concept corresponding to Q falling under the first level concept corresponding to M. Doing so, however, requires that the conditions for specifying the first level concepts and concept-correlates in question must be stratified in a way corresponding

to the stratification of the higher level concepts to which these first level concepts and their concept-correlates correspond. The comprehension principle (CPt), in other words, must be restricted in a way that corre-sponds to the stratification of the unsaturated concepts of Frege's hierarchy.

We shall actually need a more stringent form of stratification than Frege allows, it turns out for reasons shown below, and specifically one which assumes that there are no unequal higher level relations (or at least none that are involved in Frege's double correlation thesis); that is, we shall be required to assume, on pain otherwise of generating Russell's paradox after all, that higher level relations are only homo-geneously stratified. This in fact will be the only modification of Frege's original form of logicism which we shall assume in our first reconstruc-tion of Frege's logic.

Returning to the logical grammar of §7, accordingly, we shall say that a formula or A-abstract of that grammar is homogeneously stratified if, and only if, there is an assignment t of natural numbers to the set of terms occurring in (including itself if is a A-abstract) such that (1) for all terms a and b, if (a = b) occurs in then tea) = t(b); (2) for all n

1, all n-place predicate expressions n and all terms ai' ... , an> if n(a

l

, ... , an) is a wff occurring in then (i) tea;) = teak)' for 1 j, k

n, and (ii) ten) = teal) + 1; and (3) for all mEw, all individual variables x, ... , X

m

, and all wffs tfJ, if [AXI ... xmtfJ] occurs in then (iii) t(x;) = t(x

k

), for 1 j, k m, and (iv) t([AXI ... XmtfJ]) = t(Xl) + 1. If clauses (i) and (iii) are dropped and clauses (ii) and (iv) are replaced by the weaker requirement that ten) = 1 + max [t(a

l

), ... , t(an)] and t([AXI

... xmtfJ]) = 1 + max [t(xl

), ... , t(xm

)], then we shall say that is hetero�

geneously stratified, or simply stratified; and if clause (1) as well as clauses (i) and (iii) are dropped and clauses (ii) and (iv) are replaced by the still weaker requirement that max [t(a

l

), ... , t(an)] < ten) and max[t(xI)' ... , t(Xm)] < t([AXI ... XmtfJ]), then we shall say they is cumulatively stratified.

We include the idea of cumulative stratification here because it is a natural generalization of the idea of a stratified hierarchy of concepts. Involved in a such a generalization, in other words, is L1.e suggestion that we should restrict (CPt) as little as possible in our representation of Frege's double correlation thesis; that is, that we should replace (CPt) by the cumulatively stratified comprehension principle, (CSCPt), which is exactly like (CPI) except for the added constraint that the A-abstract


in question must be cumulatively stratified. Unfortunately, however, this is too much of a generalization in our present context, since the A-abstract [AX (3 G) (x = G & - G(x)] involved in Russell's paradox is easily seen to be cumulatively stratified. That is, (CSCpn is an insuffi-cient restriction of (Cpn since it still leads to Russell's paradox.

Now although [Ax(3G) (x = G & - G(x»] is cumulatively stratified, it is not heterogeneously stratified, and therefore it will not fulfill the conditions for the (heterogeneously) stratified comprehej1sion principle;

(SCPI), which is exactly like (CPt) except for the constraint that the A-abstract in question must be (heterogeneously) stratified. This princi-ple, it should be observed, is in accordance both with the simple theory of types briefly described in Appendix B of [POM] and with Frege's hierarchy of unsaturated concepts. That is, although unsaturated concepts cannot be cumulatively stratified, higher level relational con-cepts can be inhomogeneously stratified. Such higher level relations are referred to by Frege as unequal leveled relations. Thus, for example, the second level relation of an object to a concept under which that object falls is said by Frege to be an uneq1lal second level relation because it has as arguments both a saturated object and an unsaturated first level concept ([Gg], vol 1, §22). Needless to say, but (SCPt) posits a first level relation corresponding to tJ:>Js second level unequal relation of sub-sumption; that is,

(3R

2

) ([Axy(3G) (x= G& G(y»] = R)

is an instance of (SCpn; and in that regard (SCPt) is easily seen to be in full accordance with Frege's double correlation thesis.

Unfortunately, however, precisely because predication stands for a relation according to (SCPt), then being impredicable with respect to this relation is also specifiable in terms of (SCPt). That is, since [Axy(3G) (x = G & G(y»] is heterogeneously stratified, then so is [AZ- [Axy(3G) (x = G & G(y)] (z, z)]; and therefore despite its com-plexity of expression in terms of A-abstracts Russell's paradox of the concept which is predicable of itself if, and only if, it is not predicable of itself is derivable on the basis of (SCPt) after all.! It follows, accordingly, that if Frege's double correlation thesis is to apply to all higher level unsaturated relations, then we must assume that no such relations are inhomogeneously stratified, i.e., that there are no unequal higher level relations. In particular, we must not assume that there can be such an unequal second level relation as the sUbsumption of an object under a


concept; for it is fundamental to Russell's paradox that predication cannot stand for a relation between an object and a concept-correlate.

10. THE THEORY OF HOMOGENEOUS SIMPLE TYPES AS A SECOND ORDER LOGIC

If unsaturated higher level concepts, including relations, are not to be inhomogeneously stratified, then the appropriate restriction of (CPt), needless to say, is the homogeneously stratified comprehension principle, (HSCP!), which is exactly like (CPt) except for the added constraint that the A-abstract in question must be homogeneously stratified. Indeed, in the second order logic of nominalized predicates in question, which hereafter we shall refer to as AHST*, the only A-abstracts recognized as well-formed are those which are homogeneously stratified. This does not mean, it should be noted, that every wff of AHST* must be homo-geneously stratified, but only that the A-abstracts occurring in such wffs are. That is, in general, where n is a natural number, a meaningful expression of type n (as defined in §7) is a meaningful expression of type n in AHST* if, and only if, every A-abstract occurring in that expression is homogeneously stratified.

It follows, accordingly, that if F is a I-place predicate variable (or constant), then F(F) and - F(F) are both wffs of AHST*; and similarly if [Ax¢] is homogeneously stratified, then both [AX¢] ([AX¢]) and its nega-tion are well-formed formulas of AHST*. In other words, the laws of logic, and the principle of excluded middle in particular, apply in AHST* to concept-correlates as the denotata of nominalized predicates no less so than they apply to objects or individuals in general; and in that regard, therefore, concept-correlates may be said to be objects or individuals "in the full sense" as far as AHST* is concerned.

The basic laws of logic according to AHST*, it should be emphasized, are exactly those of standard second order predicate logic with identity already described in §3, but extended now to include homogeneously stratified A-abstracts and nominalized predicates. In other words, by an axiom of AHST* we understand any wff of AHST* which is either tautologous or of one of the following forms:

(AI*)

(A2*)

(Vu) [¢ --+ 'IJI] --+ [(Vu)¢

--+ (V u) 'IJI] ,

¢ --+ (Vu)¢,

where u is an individual or predicate variable,

where u is an individual or predicate variable not occur-ring free in ¢'


(A3*) (3x)(a= x),

(LL*) (a= b) -+ ... 'I/J],

(A-Conv*) [Ax

1

• •• (a

1

,· •• , an)

.... .. , ajx

n

),

where a is a singular term of ,1.HST* in which x does not occur free,

where a, b are singular terms of AHST* and 'I/J comes from

by replacing one or more free occurrences of b by free occurrences of a,

where a

1

, ••• , an are singular terms of AHST* and each a

i

is free for Xi in

(IdA) [Ax

1

• •• xn

P(x

1

, . .. , xn)] = P, where P is an n-place predi-

(HSCPA) (3P) ([Ax1 ... = P),

cate variable or constant,

where Fn does not occur free in and [Ax1 ... is homogeneously stratified.

Modus ponens and universal generalization (of an individual or pre-dicate variable) are still the only inference rules, and theoremhood and derivability from premises are defined in the usual way.2

Now it is important to note that if is a A-free wff containing no nominalized occurrences of predicates, then [Ax1 ... is homo-geneously stratified, and therefore by (HSCPA), the comprehension principle (CP) of standard second order predicate logic with identity is easily seen to be provable in AHST*. In other words, any A-free wff containing no nominalized occurrences of predicates which is a theorem of standard second order predicate logic with identity is a theorem of AHST* as well. In this regard, AHST* goes beyond the laws of standard second order predicate logic only in its recognition of homogeneously stratified A-abstracts as complex predicates and in its logistic treatment of nominalized predicates, whether simple or complex, as singular terms. Both of these features are incorporated in the comprehension principle (HSCPI), which, as already indicated, is in full accordance with Frege's double correlation thesis, so long as we assume that unsaturated higher level concepts are only homogeneously stratified.

In regard to the question of the consistency of AHST*, let us note first that the full AHST* system can be shown to be consistent relative to monadic AHST* (i.e., AHST* restricted to monadic predicates only) in


essentially the same way that the simple theory of types can be shown to be consistent relative to the simple monadic theory of types. That is, it can be shown that if monadic AHST* is consistent, then the full AHST* system is also consistent (cf. [NBC-4], §4.8). Secondly, by interpreting monadic predication as membership, monadic AHST* can be readily shown to be consistent relative to R. Jensen's system NFU ("New Foundations with Urelements"); and therefore if NFU is consistent, then so is the full AHST* system. In [Jensen], however, Jensen has shown that NFU is consistent relative to weak Zermelo set theory;3 and therefore, by putting these results together, we can make the following consistency claim regarding the full AHST* system (cf. [NBC-4], §4.9).

THEOREM: If weak Zermelo set theory is consistent, then so is AHST*.

11. FREGE AND THE PRINCIPLE OF EXTENSIONALITY

We observed in §7 that (LL*) and (A-Conv*) together yield the follow-ing generalized form of Frege's basic law (Vb):

[AXI · .. = [AXI ... xntP] -+ ('VxI)· .. ('Vxn) ++ tP)·

It was this law, it will be remembered, which together with (A3*) and the unrestricted comprehension principle (CPt) led to Russell's paradox. Given the restriction of (CPt) to (HSCPt), however, Russell's paradox is no longer derivable (if weak Zermelo set theory is consistent), and we are still able to maintain Frege's basic law (Vb) as a law of logic.

Of course, this is only one direction of Frege's Basic Law V. The other direction, i.e., Frege's basic law (Va), is the following principle of extensionality:

(Ext*) ('VxI )··· ('Vxn) ++ tP) -+ [AxI · .. = [AXI • •• xn tP]·

This principle, needless to say, is not provable in AHST*, and in that regard, it might be claimed, it is not a "law of logic". For Frege, however, it is a law of logic because concepts and relations are (unsaturated) functions from objects to truth-values. That is, according to Frege, ''what two concept-words mean [bedeuten] is the same if and only if the extensions of the corresponding concepts coincide" ([PW], p. 122). In making this claim, Frege is aware that he has made "an impor-tant concession to the extensionalist logicians. They are right," he claims, ''when they show by their preference for the extension, as against


the intension, of a concept that they regard the meaning [Bedeutung] and not the sense of words as the essential thing for logic. The inten-sionalist logicians are only too happy not to go beyond the sense; for what they call the intension, if it is not an idea, is nothing other than the sense. They forget that logic is not concerned with how thoughts, regardless of truth-value, follow from thoughts, that the step from thought to truth-value - more generally, the step from sense to meaning has to be taken in. They forget that the laws of logic are first and foremost laws in the realm of meanings [Bedeutungen] and only relate indirectly to sense" (ibid.). In other words, as far as the laws of logic are concerned, "concepts differ only so far as their extensions are different" (ibid., p. 118); and "therefore just as proper names can replace one another salva veritate, so too can concept-words, if their extension is the same" (ibid.).

Regardless of his commitment to the principle of extensionality as a law of logic, however, it must not be overlooked here that Frege still maintains "that the concept is logically prior to its extension" ([G & B], p. 106), and that he regards "as futile the attempt to take the extension of a concept as class, and make it rest, not on the concept, but on single things" (ibid.). In other words, despite his commitment to the principle of extensionality, Frege's second order logic of value-ranges as concept-correlates is not a second order set theory in the sense in which sets are essentially constituted or composed of their members. That this is so is especially brought out by the reconstruction and identification of Frege's form of logicism with the system AHST* + (Ext*) as a second order logic of nominalized predicates.

12. NFU-."SETS" AS CONCEPT-CORRELATES

Our first reconstruction of Frege's form of logicism, accordingly, is its reconstruction as the system AHST* + (Ext*). The question whether this system is consistent, and, if consistent, whether it suffices for the reduc-tion of classical mathematics is answered, it turns out, by its relation to Jensen's modification of Quine's well-known "set" theory NF; i.e., by its relation to NFU. In particular, the concept-correlates of monadic AHST* + (Ext*), it turns out, are none other than the "sets" of NFU; and therefore classical mathematics is reducible to AHST* + (Ext*) at least to the same extent that it is reducible to NFU. Moreover, by representing concepts by their extensions, monadic AHST* + (Ext*) can


be easily seen to be equiconsistent with NFU; and therefore since monadic AHST* + (Ext*) is already equiconsistent with the full AHST* + (Ext*) system, then AHST* + (Ext*) is equiconsistent with NFU.

In other words, by defining membership as follows,

XE y=df(3F) [y=F&F(x)],

Jensen's "set" theory NFU can be shown to be contained within monadic AHST* + (Ext*); and, similarly, by interpreting monadic predication as membership in NFU, monadic AHST* + (Ext*) can be translated into NFU so that the translation of a theorem of monadic AHST* + (Ext*) is a theorem of NFU (cf. [NBC-4], §4.9).4 But monadic AHST* + (Ext*), for reasons already indicated, is equiconsistent with the full AHST* + (Ext*) system (cf. [NBC-4], §4.8); and therefore we are able to establish the following result.

THEOREM: AHST* + (Ext*) is consistent if, and only if, Jensen's "set" theory NFU is consistent; and therefore AHST* + (Ext*) is consistent if weak Zermelo set theory is consistent.

This theorem is perhaps not surprising, it might be said, since both AHST* + (Ext*) and NFU are constructed as type-free counterparts of the theory of simple types (with an axiom of extensionality for each type greater than 0). This is somewhat misleading, however, since AHST* + (Ext*) is really a reconstruction of standard second order predicate logic with nominalized predicates and not, strictly speaking, a recon-struction of the theory of simple types as a theory of "improper objects". The guiding principle of our reconstruction is indeed in accordance with a homogeneously stratified hierarchy of unsaturated concepts since that principle is none other than a generalized form of Frege's double correlation thesis (minus unequal leveled relations); but that is not quite the same as its being a reconstruction of the theory of simple types (with axioms of extensionality for each type greater than 0) as a theory of "improper objects". This is especially so, moreover, insofar as the latter is intended as a model of all of the finite "stages" at which classes are generated from concrete individuals (as objects of type 0) in accord-ance with the power set axiom of set theory. For as a theory of classes which are essentially constituted or composed of their members, the number of classes of any given type in the theory of simple types will be less than the number of classes of the next succeeding type; and this,


needless to say, runs directly counter to Frege's double correlation thesis. Thus, in particular, if second level concepts are to be correlated with certain first level concepts and first level concepts are to be correlated in turn with their saturated concept-correlates, then there can be neither more second level concepts than first level concepts nor more first level concepts than objects. In this regard, since AHST* + (Ext*) was constructed in accordance with Frege's double correlation thesis (minus unequal leveled relations), it is misleading to describe it as a type-free reconstruction of the theory of simple types.

Now Quine's "set" theory NF, on the other hand, was originally constructed precisely as a first order counterpart of the theory of simple types as a theory of classes which are essentially constituted or com-posed of their members (d. [Quine], §§40-42). Its failure to be what it purports, however, is precisely what makes NF so controversial as a theory of sets. For as a first order theory NF is committed to construing all individuals as "sets", since by the axiom of extensionality every individual in NF either has a member or is identical with the empty "set". That is, unlike the situation in AHST* + (Ext*), NF cannot "recognize" the concrete individuals of the theory of simple types without first transforming them into "sets" (e.g., by identifying them with their singletons). In this regard, NF fails to capture an important feature of the theory of simple types as a theory of classes which are composed of their me.mbers, since, as such, classes are ultimately founded on concrete individuals or urelements which are not themselves classes. NFU, needless to say, was designed to overcome precisely this failure of NF.

In both NF and NFU, however, there is a universal "set", a fact which runs counter to the way classes are generated by "stages" which never end in the theory of simple types, thoagh not of course counter to the Fregean view that the universal concept of self-identity should have an extension as its concept-correlate. Furthermore, in both NF and NFU every "set" has an absolute complement, which also runs counter to the strictly relative complements a class has at each "stage" in the theory of simple types, but not counter to the absolute complements which concept-correlates have as the extensions of unsaturated concepts. There is a symmetry between the small and the large in both NF and NFU, in other words, which runs counter to the limitation of size doctrine which is incorporated in the theory of simple types as a theory of classes that are essentially constituted or composed of their members. The same


symmetry obtains, of course, between the extensions of concepts in 2HST* + (Ext*), since as concept-correlates these extensions have their being in the concepts whose extensions they are, and not in the objects which are their members. That is, there can be no asymmetry between the small and the large among the extensions of concepts in 2HST* + (Ext*), since by the laws of logic regarding concepts the extensions of concepts must satisfy the conditions of a Boolean algebra; and this is so, moreover, precisely because "the extension of a concept simply has its being in the concept, not in the objects which belong to it" ([PW), p. 183). In this regard, it makes more sense to identify NFU-"sets" with the concept-correlates of monadic 2HST* + (Ext*) than to attempt to construe them as sets which are essentially constituted or composed of their members.

Now there is a result of Ernst Specker's which shows that NF is equiconsistent with the theory of simple types as a theory of classes if we add to the latter the assumption that all of the classes of anyone type can be correlated one-to-one with the classes of the next succeeding type and that all of the classes of urelements can be correlated one-to-one with these urelements, an assumption which Specker calls "com-plete typical ambiguity" (cf. [TA], p. 118). Such an assumption, needless to say, runs directly counter to the idea of classes as composed of their members, and therefore it fails to explain in what sense NF is to be viewed as a theory of sets. It does not run counter to the idea of NF-"sets" as concept-correlates, on the other hand, and indeed, given the assumption that every object is a concept-correlate it conforms perfectly to our generalized form of Frege's double correlation thesis for unsaturated (unary) concepts. This assumption, formalized as (\Ix)

(3pl)(X = F), was in fact briefly considered by Frege himself in a footnote to [Gg) , vol. 1, §1O; and, indeed, it is easily seen that NF is contained in mST* + (Ext*) + (\Ix) (3P) (x = P) in precisely the same way that NFU is contained is 2HST* + (Ext*). In this regard, we maintain, it is more appropriate to identify NF-"sets" with the concept-correlates of monadic AHST* + (Ext*) + (\Ix) (3P) (x = F) as a recon-struction of Frege's form of logicism than to construe either NF or 2HST* + (Ext*) + (\fX)(3Pl)(X = F), as type-free reconstructions of the theory of simple types, even when Specker's axiom of "complete typical ambiguity" is added to the latter. A similar observation applies, needless to say, to the identification of NFU-"sets" with the concept-correlates of monadic HST* + (Ext*). In other words, despite the


original motivation or purpose for its construction, NFU makes more sense as a partial description of the concept-correlates of AHST* + (Ext*) as a reconstruction of Frege's form of logicism than it does as a reconstruction of the theory of simple types as a theory of classes which are composed of their members.

The appropriate definition of the natural numbers in NFU, inciden-tally, is precisely the well-known Frege-Russell definition and not either von Neumann's or Zermelo's purely set-theoretic definitions. Given the Frege-Russell definition of the natural numbers, moreover, we can go on to construct all of the integers, rational numbers and real numbers using only the resources already available in NFU, and therefore in AHST* + (Ext*) as well. In other words, except for the assumption of an axiom of infinity the reduction of classical mathematics to logic, and especially of arithmetic to logic, as originally conceived by Frege is fully realized in our reconstruction of Frege's form of logicism as AHST* + (Ext*).

Finally, in regard to the only "existence-axioms" described as such by Russell, viz., the axioms of choice and infinity, we should take note of Specker's result in [AC] that the axiom of choice is disprovable in NF; and therefore, since the axiom of choice is provable for finite sets, the existence of an infinite "set" is provable in NF. But since NF is contained in AHST* + (Ext*) + ('Ix) (3P) (x = F), then Specker's results apply here as well. That is, in AHST* + (Ext*) + ('Ix) (3F) (x = F), the axiom of choice is disprovable and the axiom of infinity is provable. Of course, this does not mean (as it did in Russell's later form of logicism) that there are infinitely many concrete individuals, but only that the total number of objects, whether abstract or concrete, is infinite. However, by another argument of Specker's regarding NF, we can prove in AHST* + (Ext*) + ('Ix) (3F) (x = F), that there are infinitely many natural numbers in the sense in which these are defined in the Frege-Russell manner (cf. [Quine], p. 299); and that, needless to say, is exactly what Frege thought held in his form of logicism. Specker's proofs do not apply to NFU, however, and indeed Jensen (op. cit.) has shown that the axiom of infinity is not provable in NFU. The same observation applies, needless to say, to AHST* + (Ext*).

13. RUSSELL AND THE PRINCIPLE OF RIGIDITY

The only fundamental difference between Frege's and Russell's early form of logicism so far emphasized is that for Russell concepts are their


own concept-correlates. That is, Russell refuses to accept the unsaturated nature of concepts, and he assumes accordingly that nominalized pre-dicates denote the same concepts which these predicates otherwise stand for in their role as predicates. The prdicative nature of a concept for Russell, in other words, does not consist in its being an unsaturated function from objects to truth-values, but rather only in its being a function whose values are propositions. That is, concepts for Russell are none other than propositional functions, and propositional functions are themselves individuals.

As functions whose values are propositions rather than truth-values, however, propositional functions are intensional and not extensional entities; and this in fact is another fundamental difference between Frege's and Russell's early form of logicism. For when applied to AHST*, this difference is reflected in the acceptance of (Ext*) by Frege as a law of logic as opposed to its rejection as such by Russell. That is, besides assuming that concepts have an individual nature, Russell also assumes that in general they are only intensionally individuated.

What exactly Russell means by the intensionality of a proposition, and thereby of a propositional function as well, he never says. Neverthe-less, in our reconstruction of his early form of logicism, we shall assume that propositions can be represented by (or rather correlated with) functions from possible words to truth-values. This is not what a proposition really is according to Russell, needless to say, especially since possible worlds would themselves be constructed in Russell's early framework in terms of propositions about merely possible as well as actual individuals;5 but it does serve as an intuitive guide regarding the intensional individuation of propositional functions. Thus, instead of Frege's principle of extensionality, (Ext*), being a law of logic, Russell (or so we shall assume) would have as a law of logic the corresponding principle of intensionality.

(DExt*) o (V'Xj) ... (V'xn) (rp ... 'I/J) - [.hj ... xnrp] = [AXj ... Xn'I/J].

This means that we shall need to take 0 as a new primitive logical constant of our logical grammar, and that we shall need to add the following clause in the definition of a meariingful expression given in §7:

(10) ifrp E MEl' then 0 rp E MEl'

We shall retain all of the axioms and inference rules of AHST*, needless to say, except that now these axioms and rules are understood to apply


to wffs containing occurrences of 0 as well. In addition, we shall also assume the axiom schemas of the S5 modal propositional logic and the rule of modal generalization; i.e., the rule that if ¢ is provable, then so is O¢. For convenience, we shall refer to the resulting system as OAHST*. Our initial proposal, accordingly, is that Russell's early form of logicism is to be reconstructed as OAHST* + (OExt*).6

Unfortunately, however, OAHST* + (OExt*) does not suffice for Russell's view of classes as analyzable in terms of concepts. In particu-lar, assuming that classes are to be represented in terms of concepts (as individuals), we shall need an account of how this representation is to be given; and assuming the adequacy of that account we shall need to establish the following thesis as one of its consequences:

(itF)(3G)(Cls(G) & (Yx) [F(x) +> G(x)]).

Now without going into the details here, we shall only say that Russell's own later contextual analysis in [PM] of 'CIs', or of expressions for particular classes as "incomplete symbols", will not suffice in our present context, since as reconstructed here, Russell's contextual analy-sis would have the inappropriate consequence that all concepts are "classes"; i.e., that concepts are extensionally, not intensionally, indi-viduated after all. Instead of Russell's analysis, however, we can begin with the notion of a rigid (n-ary) propositional function as a concept which has the same extension in every possible world. A class, on this analysis, will then simply be a rigid concept. Indeed, if concepts are themselves individuals, as Russell claimed, there would be little or no point in distinguishing a rigid concept from the class which is the extension of that concept, at least not if classes are really "superfluous" in Russell's form of logicism? Accordingly, where definitions' of predi-cate constants are given in terms of homogeneously stratified A-abstracts, we define the notion of rigidityn as follows:

Rigidn =df [Ax(3P) (x = P & (ity!) ... (itYn) [OF(Yl' ... , Yn) V 0 - F(y), ... ,Yn)])]'

Thus, an n-ary relation-in-extension, on this analysis, is simply a rigidn

relation (in intension), and a class is simply a rigid l concept:

Cls=df [Ax(3Fl) (x= F & Rigidl(F»].

Now the fundamental new assumption which we need as a "law of logic" in Russell's intensional form of logicism is not just the above thesis that


every concept is extensionally equivalent to a rigid concept (i.e., that the extension of a concept exists), but rather the following more general principle of rigidity:

(PR) (V p)(3Gn)(Rigid

n

(G) & (Vx/) . .. (Vxn)[F(x/, . .. ,xn

)

- G( xl> . . . , xn)]).

Our proposal, accordingly, is that Russell's early form of logicism is to be reconstructed as the system DAHST* + (DExt*) + (PR).

It is clear, of course, that by interpreting D as double negation, DAHST* + (DExt*) + (PR) collapses to just AHST* + (Ext*). That is, on our reconstruction, Russell's (early) intensional form of logicism is equiconsistent with Frege's extensional form; and therefore by the equiconsistency of the latter with Jensen's "set" theory NFU, we have the following result (cf. [NBC-4], §6.3).

THEOREM: DAHST* + (DExt*) + (PR) is equiconsistent with both AHST* + (Ext*) and Jensen's "set" theory NFU.

Classical mathematics, it is clear, is reducible to our reconstruction of Russell's intensional form of logicism no less so than it is to our recon-struction of Frege's extensional form. The identification of NFU-"sets" with Frege's value-ranges as concept-correlates is perhaps more plau-sible than their identification with rigid concepts; but then their identi-fication with concepts to begin with should also more readily obviate their confusion with sets as essentially constituted or'composed of their inembers.

14. FREGE'S CONTEMPLATION OF THE ABELARDIAN VIEW

In reconstructing Frege's logicism, it should be emphasized, we have made only one relatively minor change in his overall view. We have assumed, in particular, that all higher level unsaturated concepts are only homogeneously stratified; that is, that there are no unequal higher level relations. Only on the basis of this assumption, in other words, can we maintain that (CPt) is to be restricted to (HSCPt) in accordance with Frege's double correlation thesis.

There is an alternative to this assumption, however, which allows us to retain both (CPt) and to avoid Russell's paradox. This alternative, is implicit in Frege's discussion of Russell's paradox and is contained in his


suggestion that we might "suppose there are cases where an unexcep-tional concept has no class answering to it as its extension" ([G & B], p. 235). In other words, instead of maintaining for each singular term the dubious existential presupposition that that singular term actually denotes (a value of the bound individual variables), we are to allow on this suggestion that some singular terms, and in particular some nomi-nalized predicates, are denotationless.

Given the standard second order predicate logic with nominalized predicates described in §7 - that is, the logistic context in which Russell's paradox was originally formulated - the proposal in question, accordingly, amounts to replacing the dubious axiom (A3*), which explicitly expresses all such existential presuppositions, by the following weaker, but also unexceptionable, law of logic:

(A3**) (\Ix) (:Jy) (x= y).

A little more is actually needed on this proposal, however, since without (A3*) the identity wff (a = a), also an unexceptionable law of logic, is no longer derivable. In replacing (A3*) by (A3**), in other words, we shall also need to add (a = a) as an axiom, where a is any singular term. Similarly, in dropping all existential presuppositions regarding singular terms, we shall need to replace (A-Conv*) by the presupposition free form of A-conversion:

(:J/A-Conv*) [AXI ••• Xn

¢] (al , ... ,an) ++ (3x

l

) . .. (:JXn)

(al = Xl & ... & an = Xn & ¢),

where no Xi is free in any aj, for all i, j such that 1 i, j n. Now with the replacement of (A3*) by (A3**) and (a = a), and of

(A-Conv*) by (:J/A-Conv*), the principle of universal instantiation (VIi)

regarding singular terms is no longer provable, it turns out, except in the following qualified form:

(:J/VIi) (:Jy) (a= y) -+ [(\lx)¢ -- ¢(a/x)],

where a is any singular term which is free for X in ¢ and in which y has no free occurrences. With this qualification, however, what follows from Russell's paradox as described in §7 is not a contradiction but only that the complex predicate [AX(:JG) (x = G & - G(x»] is denotationless in its occurrences as a singular term: i.e., instead of a contradiction,

- (3y) ([AxC:lG) (x== G& - G(x») = y)


is provable. In other words, even though (CP!) posits the existence of an unsaturated concept corresponding to [Ax(3G) (x = G & - G(x»] as a predicate, nevertheless by Russell's argument it is provable that there can be no saturated object corresponding to [A.x(3G) (x = G & - G(x»] as a singular term; or in Frege's words here is a case ''where an unexceptional concept has no class answering to it as its exten-sion"(ibid.).

Unfortunately, however, mathematics, and arithmetic in particular, is not reducible to the system resulting from the above changes; for unlike the situation in AHST* where (A3*) remains in force, but where only A-abstracts which are homogeneously stratified are recognized as well-formed (and where (CPt), accordingly, is replaced by (HSCPl», we can no longer prove that there are any objects denoted by nominalized predicates, or, equivalently, that there are any concept-correlates at all. (The logical grammar of the system in question, it should be empha-sized, is just the full unrestricted grammar described in §7, and the axioms are (Al*), (A2*), (A3**), (a = a), (LL*), (Idi), (3/A-Conv*) and (CPi).)

That there might be no concept-correlates at all, incidentally, was not a possibility Frege refused to consider in his discussion of Russell's paradox. For in commenting on the proposal in question, Frege seemed to interpret this alternative as one in which we can only "regard class names as sham proper names, which would thus not really have any reference" (ibid. p. 236). We agree, it should be noted, that for Frege this apparently meant that "class" names "would have to be regarded as part of signs that had reference only as wholes" (ibid.), as, for example, names of natural numbers might occur as parts of quantifier expres-sions, but in our present framework this can also be formulated as the Abelardian thesis that no concept is a thing (cf. [NBC-4], §4.1):

('VF') - (3x)(F'=x).

In other words, instead of maintaining only that some nominalized predicates must be denotationless, Frege generalized this alternative into the Abelardian view that all nominalized occurrences of predicates are denotationless. Abelard, incidentally, apparently believed that the same universal can be shared by different individuals, but he refused to grant that what individuals have in common is a ''thing''; that is, he denied that universals are individuals. In this regard, he might be said to have anticipated Frege's view of concepts as unsaturated functions. Be


that as it may, nevertheless it was Frege's view throughout most of his career that nominalized predicates, and abstract singular terms in general, denoted objects even though these objects could not themselves be unsaturated concepts. For Abelard, on the other hand, at least on our interpretation of his view here, nominalized predicates were singular terms which simply failed to refer to any individual at all. Such a view, needless to say, is obviously "safe" from Russell's paradox, and indeed the result of adding the Abelardian thesis to the system in question can be shown to be consistent (cf. [NBC-4], §4.10).

Though consistent, however, such a view does not lead to a recon-struction of Frege's form of logicism; and in fact, it leads directly away from it, a direction which Frege was not disinclined to take in 1924/5 at the end of his long and brilliant career in defense of logicism. Thus, in speaking of "the formation of a proper name after the pattern of 'the extension of the concept a'" ([PW), p. 269), Frege not only noted that "because of the definite article, this expression appears to designate an object" (ibid.), but he went on to suggest that "there is no object for which this phrase could be a linguistically appropriate designation" (ibid.). Weare misled by language here, he suggests, and "from this has arisen the paradoxes of set theory which have dealt the death blow to set theory itself' (ibid.).

These final thoughts of a great logician are unfortunate, however, for the paradoxes, and Russell's paradox in particular, do not affect set theory as a theory of classes which are composed of their members, but rather affect only a theory of classes as the extensions of concepts; and whereas set theory has continued to flourish throughout the 20th century, it is the theory of classes as the extensions of concepts, and thereby logicism itself, which has fallen into disrepute. Such disrepute is not deserved, however, since logicism, and Frege's form of logicism in particular, can be easily reconstructed in such a way as to avoid the paradoxes. In this regard, we have not only the reconstruction of Frege's logicism already given, but the following alternative reconstruction as well.

15. A SECOND RECONSTRUCTION OF FREGE'S LOGICISM

The proof that some nominalized predicates must be denotationless in the system presently under consideration, it should be noted, does not show that all must be denotationless; i.e., the Abelardian thesis is not


provable in our modified second order logic with nominalized predi-cates as described above in §14. We may assume, in other words, that some concepts do have concept-correlates after all.

Such an assumption, needless to say, can be given in a number of different ways, such as those corresponding to the existence conditions for sets in different set theories. The latter, however, will result only in a second order theory of sets in the sense in which sets are essentially constituted or composed of their members; and dropping the axiom of extensionality so as to avoid calling such objects sets is really pointless, since their existence conditions are in accordance with the limitation of size doctrine, which in tum is based on the notion of sets as composed of their members. Such an assumption, in other words, will not result in a coherent form of logicism since as objects which are essentially consti-tuted or composed of their members sets are mathematical and not logical objects. The relevant assumption, in this regard, must be based on Frege's double correlation thesis and the way it pertains to the positing of concept-correlates as logical objects and not on conditions that pertain to the existence of sets as composed of their members.

Now in returning to Frege's double correlation thesis in our present framework where A-abstracts need not be homogeneously stratified, we should note that there is a difference in applying the thesis in the positing of first level concepts from applying it in the positing of con-cept-correlates. In partiCUlar, since the full unrestricted comprehension principle, (CPr), is to remain as an axiom schema, we in effect retain the unrestricted form of Frege's double correlation thesis as it applies to the positing of first level concepts; i.e., the form in which it posits first level concepts corresponding to unequal leveled or inhomogeneous higher level relations no less so than to homogeneous or equal leveled relations. Thus, corresponding to Frege's unequal second level relation of subsumption or predication, there is a first level relation posited by (CPt); that is,

(3R2) ([Axy(3G) (x= G& G(y»] = R)

is provable in the system in question. Of course, by Russell's argument, this same predicate when nominalized simply fails to denote (a value of the bound individual variables). In other words, as applied to concept-correlates, Frege's double correlation thesis must be restricted; and in particular it is not to apply in general to inhomogeneous higher level relations. The assumption we shall make here is that it is to apply at


least to all higher level unsaturated concepts which are homogeneously stratified; that is, to all of the concepts which have concept-correlates in AHST*.

In order to formulate our assumption as an axiom schema, we shall say that a meaningful expression (as defined in terms of the logical grammar of §7) is bound to individuals if, and only if, for all natural numbers n, all n-place predicate variables P, and all ¢, if (V P)¢ is a wff occurring in then for some individual variable x and some wff 1/J, ¢ is the wff [(3x) (P = x) -> 1/J]. In other words, to be bound to individuals, every predicate quantifier occurring in must refer only to those concepts posited by (CPr) which have corresponding concept-correlates. Our assumption may now be stated as the following axiom schema:

(3/HSCPr) (3y) (a1 = y) & ... & (3y) (ak = y) -> (3y) ([AXI ... x

n

¢] = y),

where [AXI ... xn¢] is homogeneously stratified, ¢ is bound to individ-uals, y is an individual variable not occuring in ¢' and aI' ... , ak are all ofthe variables or non-logical constants occurring free in [AXI ... xn¢]'

The axiom schemas of our present system, accordingly, are (Al*) , (A2*), (A3**), (a = a), where a is any singular term, (LL*), (Idr), (3/A-Conv*), (CPr) and now (3/HSCPI) as well.8 (Modus ponens and universal generalization are its only inference rules.) Because of its relation to our earlier system, we shall refer to this system hereafter as HSTi·

We must again emphasize that unlike the situation in AHST*, A-abstracts are not required to be homogeneously stratified in HSTi. That is, the meaningful expressions of HSTi are just those described in the logical grammar of §7. Also, unlike AHST*, the system HSTi is free of existential presuppositions regarding singular terms, including of course nominalized predicates as abstract singular terms. It follows, accord-ingly, that HSTi is not a conservative extension of AHST*, since whereas, by (A3*), (V P) (3x) (F = x) is provable in AHST*, this same wff is actually disprovable in HST!. Nevertheless, since every wff of AHST* is provably equivalent, again by (A3*), to a wff which is bound to individuals, HSTi may be said to contain AHST* in the sense of the following lemma (which is easily seen to hold).

LEMMA: If ¢ is a wff of AHST* which is bound to individuals, y is an individual variable not occurring in ¢, and aI' ... , ak are all the


variables or non-logical constants occurring free in rp, then IAHST* rp only ifIHsn (3y) (al = y) & ... & (3y) (ak = y) -> rp.

Restricting ourselves to pure wffs, i.e., wffs in which no predicate or individual constants occur, if follows by the above lemma that every sentence (or wff with no free variables) of AHST* which is bound to individuals and provable in AHST* is therefore provable in HST;.,* as well. In addition, by (3IHSCPI), every object which is a concept-correlate in AHST* is also a concept-correlate in HSTf, and therefore, for the same reason, every object which is a concept-correlate in AHST* + (Ext*) is also a concept-correlate in HST! + (Ext*). In other words, since Jensen's "set" theory NFU is contained in AHST* + (Ext*), then in the sense of the above lemma NFU is also contained in HST! + (Ext*); and in that regard classical mathematics, and arithmetic in particular, is reducible to HST! + (Ext*) at least to the same extent that it is reduci-ble to NFV. (Actually, HST! + (Ext*) is an improvement over AHST* + (Ext*), and therefore over NFU as well, for the same reason that Quine's "set" theory ML is an improvement over NF; in particular, in both HST! + (Ext*) and ML, mathematical induction is provable in an unrestricted form (cf. [Quine], §42).) Our proposal that HST! + (Ext*) be taken as an alternative, and perhaps a preferred, reconstruction of Frege's form of logicism is in that case quite in order.

Finally, it should be noted that just as Hao Wang was able to prove the consistency of Quine's set theory ML relative to NF, we are able to prove the consistency of HST! + (Ext*) relative to AHST* + (Ext*), and therefore relative to NFU as well (cf. [NBC-4], §6.4).

THEOREM: If AHST* + (Ext*) is consistent, then so is HST! + (Ext*).

16. AN INTENSIONALIZED FORM OF FREGE'S LOGICISM

By the intensional counterpart of HST! we shall understand the system HST!o which is developed from HST! in the same way that DAHST* was developed from AHST*. That is, the axiom schemas of HST!o are just those of HST!, extended now to apply to wffs containing D as well, plus those of the SS modal propositional logic. The inference rules are of course the same as those for DAHST*, viz., modus ponens, universal generalization and modal generalization.


Now it is perhaps noteworthy that insofar as (DExt*) is to be assumed as well, then we need not take 0 as a primitive logical constant since it can in that case be contextually defined as follows:

=df = [A(\fX) (x= x)].

That is, since the biconditional,

.... = [A (\fx)(x = x)]

is provable in HSTio + (DExt*), then we need not take 0 as a primitive logical constant of this system since it can be introduced by means of the above contextual definition. (Some, but not all, of the modal axioms will then become redundant.) This definition is not available in DAHST* + (DExt*), our reconstruction of Russell's early from of logicism, since it cannot be used in that context to explain occurrences of 0 in wffs which are not homogeneously stratified. That is, the above definiens, as a wff of AHST*, would restrict the application of 0 to wffs which are homogeneously stratified, since only these wffs can occur within the A-abstracts of AHST*. But since there are wffs of AHST*, including tau-tologous wffs, which are not homogeneously stratified, defining 0 as above would in that case fail to validate the rule of modal generalization when applied to those axioms of AHST* which are not homogeneously stratified. It is only in HSTi, in other words, where A-abstracts are not required to be homogeneously stratified that the above definition of 0 will suffice.

In addition to the principle of intensionality, (DExt*), we shall also need to assume the principle of rigidity, (PR), if we are to represent classes in this alternative the way they are represented in DAHST* + (!;JExt*), + (fR\.our reconstruction of Russell'searlvJorm_ofJolicisltL. Of course, with the addition of (PR) we can bring about the same reduction of classical mathematics to logic as is already achieved in HSTi + (Ext*), which contains all NFU-"sets" as concept-correlates. Moreover, we can prove the relative consistency of HSTio + (DExt*) + (fR) in essentially the same way as we proved the relative consistency of HSTi + (Ext*).

THEOREM: If DAHST* + (DExt*) + (PR) is consistent, then so is HSTio + (DExt*) + (PR).


Despite its relative consistency, however, HSTl'o + (Ext*) + (PR) cannot be construed as a reconstruction of Russell's early form of logicism. For as an abstract singular term a nominalized predicate, according to Russell, denotes the same concept as an individual which the predicate in question otherwise stands for in its role as a predicate; and therefore whatever is the value of a bound predicate variable, according to early Russell, is also a value of the bound individual variables. That is, in Russell's early form of logicism no predicate can stand for a concept in its role as a predicate and yet fail to denote an individual when transformed into an abstract singular term. But since there are predicate expressions satisfying just these conditions in HSTAo, and therefore in HSTAo + (DExt*) + (PR) as well, then neither HSTl'o nor any of its extensions can be the basis of a reconstruction of Russell's early form of logicism. In other words, unlike Frege's form of logicism, we have no way in Russell's framework by which to explain why some concepts have individuals as concept-correlates while others do not, since on Russell's account concepts are their own concept-correlates.

We might adopt a mixed strategy here, however, whereby concepts are unsaturated functions in Frege's sense but still have propositions as values in Russell's sense. Concepts are then not individuals after all, and there is nothing odd or contradictory in the idea of some concepts having a concept-correlate while others do not. Of course, if in addition concept-correlates are none other than the extensions which these concepts have in different possible worlds, then the resulting framework may be described more as an intensionalized form of Frege's logicism than as a counterpart to Russell's. In this regard, we would not adopt (DExt*) but (Ext*) instead, since even intensional concepts which are materially equivalent will have the same extension, and therefore con-cepts which are materially equivalent in a given possible world will have the same concept-correlate in that world. In such a framework, needless to say, the principle of rigidity, (PR), would be completely superfluous.

There is a problem here with HSTAo as the basis for such an intensionalized form of Frege's logicism, however, since by (LL *),

(VP) ('iGn) (F= G -> DF= G)

is provable in HSTl'o, and therefore all concepts, even when intensional-ized, will always have the same extension. We can of course modify (LL *) so that it remains valid only for extensional contexts, that is, only for wffs in which D does not occur; for in that case concepts having the

same extension in one possible world need not necessarily have the same extension in every possible world. That is, in that case,

(3P) (3Gn) (F= G& <> F G)

will be consistent and not inconsistent (cf. [NBC-4], §6.5). Thus, if is the system resulting from HSTTo by weakening (LL *) in this

way, but also by adding the commutative law,

(V - 0 (V

as an axiom schema (since concepts themselves remain the "same" from world to world), then we can identify this intensionalized form of Frege's logicism with + (Ext*). The extensional system HST! + (Ext*) described earlier, needless to say, is a subsystem of + (Ext*), and by interpreting 0 as double negation the two systems collapse into one, thereby showing the consistency of + (Ext*) relative to HST! + (Ext*).

17. RUSSELL'S LOGICISM AS CONCEPTUAL PLATONISM

The mixed strategy described above favors Frege's form of logicism over Russell's, it should be noted, since unsaturated concepts, even when intensionalized, were assigned their extensions as their concept-correlates in the different possible worlds. There is another mixed strategy, however, which is more in line with Russell's form of logicism than with Frege's, and in fact which may be taken as a counterpart even to Russell's later form oflogicism in (PM].

On this mixed strategy, the predicative nature of a concept consists in its having both an intensional and an unsaturated nature, though the latter is understood in even a more radical sense than Frege would have allowed. In particular, on this account all predicable concepts are cognitive capacities to identify (in a classificatory sense), characterize or relate objects in various ways, and the purely dispositional or non-occurrent nature of such a capacity is none other than its unsaturated nature. That is, on this account, concepts are intersubjectively realizable cognitive abilities which may be exercised by the same person at different times as well as by different persons at the same time; and in that regard concepts are neither ideas nor mental images in the sense of particular mental occurrences. The exercise (or saturation) of a concept on this account does indeed result in a mental event; i.e., a mental act,


and, if overtly expressed, a speech act as well; but the concept is not itself the mental or speech act as an event but rather what accounts for that act's predicable nature. Second level concepts are really referential concepts on this account, incidentally, and it is their joint exercise or mutual saturation with predicable concepts (in a kind of mental chem�istry) which is the basis of our particular thoughts and acts of communi�cation (cf. [NBC-4], chap. 2, and [NBC-3], §§ 11-15).

Predicable concepts, on this version of conceptualism, are not independently real properties or relations, it must be emphasized; and, unlike referential concepts, their primary role in thought and communi�cation is not referential but predicative. Yet, by a curious development of the interplay between language and thought, predicable concepts as cognitive capacities can be transformed into secondary or derived abilities which enable us to apply predicable concepts in a denotative manner corresponding to the use of nominalized predicates in natural language. It is by means of such a secondary or derived application, moreover, that we purport to refer to independently real platonic forms as the denotata of nominalized predicates. Thus, for example, not only do we predicate of a shape that it is triangular, and of a person that he is wise by applying a predicate concept in each case, but, in addition, we also purport to denote the properties of triangularity and wisdom, respectively, by applying these same concepts denotatively.

Purporting to denote and actually denoting are of course not the same, and in fact, despite all our purportings, there may be no inde�pendently real properties or relations at all which are actually denoted by any nominalized predicates; or at least that is a view which is compatible with our present account of concepts as unsaturated cogni�tive capacities. Adopting the Abelardian thesis here, however, will not result in a counterpart to Russell's form of logicism. In other words, we shall consider here only the alternative platonic view according to which most, even if not all, of the predicable concepts which we can form and articulate actually do denote an independently real property or relation when applied denotatively. That some predicable concepts which we can form and articulate must fail to denote such a property or relation will, on such a view; be an interesting but hardly problematic fact. What would be problematic, of course, is the different platonic view that concepts are themselves the independently real properties and relations they purport to denote when applied denotatively while admitting that only some of them are individuals while others are not.


Now it might objected that if concepts are cognitive capacities, then none of them can be impredicatively formed in the Russellian sense in which they can be specified only be means of a wff which contains a quantifier ranging over all concepts. In other words, as based on the capacities of the human intellect, the laws of compositionality regarding concept-formation in the sense intended here exclude the possibility of validating the full impredicative comprehension principle (CPt). In this regard, only a conceptualism which is "constructive" in a sense corre-sponding to the theory of ramified types will be permitted.

While such a constructive conceptualism can be formalized as a logic of nominalized predicates without resorting to Russell's treatment of concepts as "improper objects" for which the law of excluded middle is to be restricted, we shall, for reasons of space, forego such a presenta-tion here; for our concern is with the validation, not the invalidation, of (CPt). In this regard, it should be noted, there is a form of concep-tualism, which we shall call holistic conceptualism, which agrees that the initial, and perhaps most important, stages in concept-formation are "constructive" in the above sense, but which nevertheless maintains that there is also a stage of concept-formation (usually occuring only in post-adolescence) at which so-called impredicative concept-formation becomes possible. Such a stage is realized, moreover, only through our capacity for language, and in particular through our capacity to use language for the expression of "constructible" concepts. Impredicative concept-formation, in other words, is a mediated process, and language and the linguistic ability to use predicate expressions is the means used to master and direct such a process. Holistic conceptUalism, in this regard, presupposes constructive conceptualism as an antecedent stage of conceptual development, and in particular as a stage which is subse-quently reconstructed through a certain reiterable pattern of reflective abstraction which proceeds through the ramified hierarchy of predicable concepts and which is finally completed or closed only by an idealized transition to a limit at which impredicative concept-formation becomes possible (cf. [NBC-4], chap. 2).

According to holistic conceptualism, in other words, the impredica-tive comprehension principle (CP) of standard second order predicate logic is valid as a description of the laws of compositionality for concept-formation, as long as the predicate quantifiers occurring therein are taken as ranging over all predicable concepts, including those that are only impredicatively specifiable. A holistic conceptualism which also


adopts the platonic assumption described above regarding the denota�tive application of predicable concepts will in that case also validate (CP1) as well. In other words, HSTio can serve as the basis for a formulation of Conceptual Platonism as described above. Moreover, if thoughts are assumed to be at least in principle always overtly express�ible in language, and 0 is interpreted as ranging over all possible contexts of use of language, that is, in the sense of pragmatics, then the principle of intensionality, (DExt*), as a principle regarding the condi�tions under which predicable concepts are understood to denote the same property or relation when applied denotatively, would seem also to be valid in Conceptual Platonism.

HST!o + (DExt*) does not itself suffice for the reduction of classical mathematics to logic, however; and in particular the axiom of infinity is not provable in HSTio + (DExt*) just as it is not provable in either HST! + (Ext*) or HST]j + (Ext*). That there are infinitely many properties and relations as platonic forms is no doubt plausible, if we are assuming that there are platonic forms as individuals to begin with; but it is not in any case a consequence of Conceptual Platonism in the sense in which the latter assumes only that predicable concepts denote properties or relations when applied denotatively. In this regard, Conceptual Platonism is very much like Russell's later form of logicism in [PM], since an axiom of infinity is required in that framework as well. In [PM], however, the axiom of infinity requires that there be infinitely many concrete individuals, and that assumption is less plausible than one regarding properties and relations as abstract individuals. In particular, an axiom of infinity in the form:

(Inf*) (VE"') (T/Gk)(F 'f G), where n 'f k,

is especially natural, since it only assumes that different properties or relations are denoted by predicable concepts of different degrees (adicity).

The situation is more problematic for an account of classes as the extensions of concepts in HST!o + (DExt*), however, though such an account is forthcoming if we also assume the principle of rigidity, (PR). But (PR), unfortunately, cannot be validated in Conceptual Platonism, since as a principle about concepts as cognitive capacities it amounts to an unwarranted (and in fact contraindicated) reducibility axiom to the effect that every predicable concept is materially equivalent to one having the same extension (or rigid property or relation when applied


denotatively) in every possible context of use. (PR), in other words, can be validated only by the full platonic view which identifies concepts as being themselves the independently existing properties and relations they otherwise purport to respresent, which is to say that (PR) can be validated only in Russell's early form of logicism reconstructed as DAHST* + (DExt*) + (PR).

Russell's own reducibility axiom in his theory of ramified types is also said to be unwarranted (and contraindicated) in his later form of logicism, incidentally. If it is, then this can only be- so because the concepts or propositional functions assumed in the theory of ramified types are not indepedently real properties and relations, as Russell in fact described them to be, but are rather really concepts as cognitive capacities in the sense intended here. In that case, however, they cannot also be the logical objects in terms of which classical mathematics is to be explained, since in fact they are not objects at all. Russell's later form of logicisim, in other words, is better understood as a form of Conceptual Platonism, though apparently one based only on construc-tive rather than holistic conceptualism as well.

The upshot is that although HSTto + (DExt*) + (Inf*) , but not HSTto + (DExt*) + (Inf*) + (PR) , can be taken as a counterpart to Russell's later form of logicism, nevertheless neither HSTto + (DExt*) + (Inf*) nor Russell's theory of ramified types suffices as an adequate framework for logicism. This still leaves our reconstruction of Russell's early form of logicism intact, however, and it in no way affects either of our reconstructions of Frege's form of logicism. In other words, as philosophical doctrines go, logicism, whether in Frege's or Russell's early form, is alive and well.

Indiana University

NOTES

I I am indebted to Edmund Gettier for bringing this fact about (SCpn to my attention. 2 The rewrite law

[AXI ... = [AYI ... .. , Yn1xn)],

where no Yi occurs in is derivable in AHST* on the basis of the principle of exten-sionality, (Ext*), i.e., Frege's basic law (Va), or its intensional counterpart, (OExt*), described in §§11 and 13 below. If neither of these principles are assumed, then the rewrite law must be taken as an additional axiom schema of AHST*.


3 Weak Zermelo set theory is the restriction of Zermelo set theory to those instances of the Aussonderungsaxiom in which all quantifiers in the comprehension clause are limited or restricted; that is, in which all quantifiers have the form (''Ix E y)¢ or (3x E

y)¢.

4 Strictly speaking the translation proceeds through an intermediate system NFU' which is obtained from NFU by adding A as an individual constant together with the axiom - (3x) (x E 1\) and the modification of NF's comprehension principle for NFU to the following:

(CP-NF')(3y)((y= A V (3x)(x E y)] & (lfx) [x E y .... IPI)

where IP is a stratified wff of NFU' in which y does not occur free. Similarly, instead of AHST* + (Ext*) we translate the A-free wffs of its A-free counterpart HST* + (Ext*) into the first order wffs of NFU'. (AHST* + (Ext*) is a conservative extension of HST* +

(Ext*) since every wffs of AHST* is provable equivalent in AHST* + (Ext*) to a A-free wff.) The important clause in the translation, besides that translating monadic predica-tion into membership, is the clause for predicate quantifiers:

trs«lfFI)IP) = (ifF) [F=A V (3X)(XE p) - trs(IP)].

where '-' (bar) is a one-to-one mapping of predicate and individual variables onto the individual variables. 5 Cf. [NBC-2] for a description of Russell's early framework as a form of possibilism. 6 Note that by (LL*),

(a= b) - D (a= b)

is provable in DAHST* + (DExt*), and therefore so are

(3x)D(a==x)

and (3F)D([Ax

I

... xn¢] = F),

where x does not occur free in a, and Fn does not occur free in [Axl ... xn

¢], and [Axl

... xn¢] is homogeneously stratified. From these principles, the following commutative laws follow:

(lfx)D91 .... D(lfx)91

(lfP)D91 .... D(lfP)91.

As applied to concrete individuals, the first of these commutative laws makes sense only in Russell's early possibilist framework (cf. [NBC-2]). 7 This approach towards the representation of classes and relations-in-extension was first given by Montague in [FP], p. 132. It also occurs in [Gallin], p. 77, where the principle of extensional comprehension is the type-theoretical counterpart of the prin-ciple of rigidity described below. 8 As indicated for AHST*, the rewrite law,

[Axl ... xn

91] = [AYI·· .. Yn91(Y/x

l

, ••• , Yn!xn)],

where no Yi occurs in 91, is provable in this system on the basis of either (Ext*) or (DExt*). If neither of thege principles are to be assumed, however, then the rewrite law must be added as an axiom schema.


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Specker, Ernst [TAJ Typical Ambiguity', in Logic, Methodology and Philosophy of Science,

eds. E. Nagel et al., Stanford University Press, 1962: 116-124. [AC] The Axiom of Choice in Quine's New Foundations for Mathematical

Logic', "Froc. of the Nat. Acad. of Sciences 39 (1953): 972-975. Tarski, Alfred

[TarskiJ 'Der Wahrheitsbegriff in Formalisierten Sprachen', Studia Philosophica 1 (1936): 261-405.

ROBERT B. BRANDOM

FREGE'S TECHNICAL CONCEPTS:

SOME RECENT DEVELOPMENTS

INTRODUCTION

Today we find ourselves at the outset of a golden age in the interpreta-tion of Frege's philosophical writings. Judged by the number of articles, books, and seminars addressing his thought, interest in Frege is at an all-time high. More importantly, as Frege has come out of the shadow of Russell and Wittgenstein into the fun light of critical attention, the degree of sophistication of discussion has achieved a quantum improve-ment. Many factors conspired to bring about this result, but two events may be singled out as having madt;: contributions both to the resurgence of interest in and to our greater understanding of Frege's work.

First is the publication, more than sixty years after his death, of that part of his Nachgelassene Schriften which survived the vicissitudes of the intervening years. These papers appeared in German in 1969 and in English in 1979.1 Some of the contents are rough in form, though not without value. We are offered, for example, tables of contents and partial drafts of a textbook on logic and its philosophy which Frege made starts on at various crucial periods of his life. Even draft frag-ments of this sort permit important inferences from the order of presentation and different emphases given various topics to conclusions about the explanatory priorities Frege associated with his central technical concepts. But not all of the selections represent rough cuts or abandoned projects. Included are some fully polished articles, dealing with Frege's most central technical concepts - fine examples of his concise, sometimes lapidary mathematician's prose - which he had tried unsuccessfully to publish. In a number of cases, these additional texts permit the resolution of exegetical disputes occasioned by what can now be seen to be accidental lacunae and merely apparent emphases in the canonical published corpus.

The other landmark event is the publication in 1973 of Dummett's monumental and long-awaited full-length treatment of Frege's philoso-phy of language.2 It would be difficult to overestimate the significance of

253


254 ROBERT B. BRAND OM

this classic work. Anyone interested in the interpretation of Frege must give it the same close attention owed to the primary texts. Its clarity of thought, patient rehearsal of considerations, and exercise of the best critical judgement in final appraisal will not be soon equalled. This essay will not offer a systematic account of Dummett's views, since the most important of these are so intimately tied up with the development of powerful novel approaches to contemporary philosophy of language as to defy brief characterization, even by their author. The original volume has now been supplemented with another containing many valuable amplifications and clarifications.3 The result is a 1300 page corpus which, Dummett's complaints4 to the contrary notwithstanding, by now deserves to be considered as setting out the canonical reading of Frege. It is so considered by the authors discussed below, and forms the background against which their own accounts are set out.

Two examples will serve to indicate the sort of interpretive advance signalled by these events. First, it was widely believed in the 1950's and 1960's that Frege did not intend the distinction between sense and reference to apply to functional expressions such as predicates, but only to complete expressions such as terms and sentences.s Although the famous essay on sense and reference does not discuss such an applica-tion of that distinction, the Nachlass makes clear that this is only because that discussion was reserved for a further article which is quite explicit in its endorsement of that application, but which was repeatedly rejected for publication until Frege abandoned the attempt. Several other passages reprinted in [19] decisively refute the interpretation which would restrict the distinction to complete expressions. A some-what less important mistake may also be mentioned as indicative, which was done in as much by Dummett's arguments as by the unearthing of further evidence. In 'On Sense and Reference' Frege says "One might also say that judgements are distinctions of pans within truth-values," and that "the reference of the word is part of the reference of the sentence."6 These remarks have sparked the attribution of a variety of bizarre ontological views to Frege, centering on the notion of the True as representing the whole world, sometimes conceived as a Tractarian world of facts, sometimes as comp'osed of objects (and what about the False?). The remarks stem from a hasty assimilation, soon explicitly rejected, of the relation between the argument of a function and the value it determines to the relation of part and whole. For although the function 'capital of .. .' takes the value Stockholm when Sweden is taken

FREGE'S TECHNICAL CONCEPTS 255

as argument, Sweden is not part of Stockholm. Dummett's discussion of this issue has permanently disposed of the temptation to take these remarks seriously as interpretive constraints. We shall see below, how-ever, that there remain genuine controversies which are not so easily disposed of (concerning the senses and referents of functional expres-sions) which may be regarded as successors to these two mistaken lines of thought.

Dummett has shown that Frege should be treated as a modern thinker in the sense that one can think about contemporary philosophi-cal issues of considerable significance by thinking about his concepts and their explanatory deployment, and that one cannot think about those concepts and their principles of deployment without thinking about such contemporary issues. In what follows those concepts are approached from three different directions. First, an attempt to interpret and develop Frege's technical scheme in light of contemporary discus-sions of the issues he was addressing is considered. Then attention is turned to an argument to the effect that ignoring the historical context in which Frege developed his theories, treating him we might say merely as a contemporary, leads to substantive misinterpretation of those theories. ,Finally, following one strand of the account of the path by which Frege developed and defended some of his central concepts, leads to a novel diagnosis of the status of those concepts.

SECTION I

One important recent offering is David Bell's book Frege's Theory of

Judgment.

7 This is a clear and well-written work. The issues it raises and the form in which they are addressed merit the attention of anyone interested in the significance for current inquiry of Frege's strategic deployment of a battery of technical concepts to explain various aspects of linguistic practice. Its title is worthy of some consideration. It is a measure of the degree of sophistication of contemporary Frege commen-tary that a controversy exists even over how one should describe the topic which his philosophical work addresses. Of course no one disputes his concern with the foundations of mathematical reasoning and knowl-edge, expressed above all in his three books, the Begriffsschrift, the Grundlagen der Arithmetik, and the Grundgesetze der Arithmetik But the more general conceptual framework he found it necessary to

256 ROBERT B. BRANDOM

elaborate in order to express clearly and precisely his claims about the nature of mathematics and its objects cannot easily be characterized without prejudging substantial issues of interpretation. It may seem obvious that Frege was pursuing a project in the philosophy of lan-guage.8 But such a description is misleading in the context of Frege's own insistence on the priority of thoughts (though not of thinkings) to their linguistic expression. For he was interested in natural languages only insofar as they permitted rough formulation of objective and language-independent thoughts, and he crafted artificial languages only as more adequate means for their expression. It would be inappropriate to build into the description of the subject-matter at the outset a post-Wittgensteinian conviction of the wrong-headedness of such an approach, by assimilating his concerns to contemporary investigations under the rubric "philosophy of language". One of the major theses of Sluga'S book, discussed below, is that such Whiggish presuppositions of continuity of concern have consistently led Frege's readers to overlook important strands of his thought. Dummett has also suggested "theory of meaning" as a general characterization, but this seems to apply better to his own enterprise than to Frege's. For 'meaning' is correlative to 'understanding', and Frege's concern lay at least equally with reference, which is not in general grasped when one understands a claim, as with the sense which must be grasped in that case.

In his discussion of the book,9 Dummett objects that Bell has misdescribed his topic, in that Frege's treatment of the act of asserting is the topic of only one chapter, while the rest of the book talks about the notions of sense and reference. This seems unfair, for the heading "theory of judgement" ought to entitle Bell to offer an account of the contents which are judged as well as of the acts which are the judgings of those contents. It has the advantage of placing Frege's concerns in appropriate historical and philosophical context. Bell's denomination of Frege's topic as judgement displays his recognition of the importance Frege, in company with Kant and Wittgenstein, placed on inverting the traditional order of explanation which took concepts as primary and sought to account for judgments in terms of them. At least until 1891, Frege clearly regarded the claim that concepts can only be understood as the products of analysis of judgements as one of his most central insights. Although Bell does not say so, it is equally clear in the Begriffsschrift that Frege completes the inversion of the classical priority of concepts to judgements and judgements to syllogisms by taking the


contents of sentences (judgement in the sense of what is judged rather than the judging of it) to be defined in terms of the inferences they are involved in.1O Concepts are to be abstracted from such judgements by considering invariance of inferential role (which pertain only to judg-ments) under various substitutions for discriminable (possibly non-judgemental) components of the judgement. Both in the introduction to BGS and in his essay on "Boole's logical Calculus and the BGS",ll the virtue of the purely formal perspicuous language of inference in non-formal contexts is described as its permitting for the first time the scientific formation and expression of concepts. Although it is for this reason that Frege called his first work a "concept script", he later came to believe this phrase misleading precisely because it obscured his doctrine of the primacy of judgements. On the other hand, it would be equally misleading to describe Frege simply as a theorist of inference, in spite of the explanatory priority he accorded to it. For his primary theoretical focus always lay on the sentential and thence sub-sentential contents attributable to different expressions in virtue of the roles they played in inference, as revealed by their behavior under substitution. So "judgement", which is (a translation of) an expression Frege himself used pretheoretically to describe the object of his theorizing, seems a good choice to delimit his subject matter.

Like any other choice, however, it does prejudge some controversial issues of interpretation, for instance that concerning the persistence in Frege's thought of the so-called "context principle". It is often unclear exactly what this principle means, but the canonical statement of it is the Grundlagen claim that "only in the context of a sentence does a word have any significance". (I use 'significance' here for Frege's 'Bedeutung' because in 1884 he had not yet distinguished Sinn from Bedeutung, and the undifferentiated term should be marked.) It is often claimed,12 even by those such as Dummett who take the putative change in view to be a serious mistake, that when Frege achieved his mature views in 1891 with the formulation of that crucial distinction he discarded the context principle. If that is so, then Bell's choice of "theory of judgement" to describe the topic of the mature semantic views he discusses would be misleading or simply incorrect. As we shall see below, Sluga argues that Frege never relinguishes the context principle. Bell does not argue this, however, nor does he even claim it. He is simply silent on this issue, as on others concerning detailed questions about the attribution of various views to Frege based on textual evidence.


Bell's enterprise lies in a different direction entirely. He is concerned to look closely at the explanatory roles played by Frege's various concepts and at the ways in which Frege takes them to be related, in order to refine and reconstruct a broadly Fregean account of the nature of judgement. In keeping with this aim, he is not engaged in the exegesis of Fregean texts, and freely discards from his reconstruction a number of doctrines which Frege clearly held, in favor of incompatible princi-ples (for instance, in Bell's reconstruction functional expressions are assigned senses but not referents). His project is to salvage from Frege's account those insights which can be put together to form a workable theory of judgement. The result is broadly Fregean in endorsing the following "major strands" of Frege's theory:13

I. There is the methodological principle that 'we can distinguish parts in the thought corresponding to the parts of a sentence, so that the structure of the sentence serves as a model of the structure of the thought'. II. A thought is (a) objective, (b) the sense of an indicative sentence .... III. A thought must have at least one 'unsaturated' or functional element, otherwise its elements would fail to coalesce and would remain merely disparate atoms. IV. In a thought the complete elements refer (if at all) to objects.

The nature of this enterprise makes it hard to evaluate its success. There are many issues one would think to be central to any attempt to offer a theory of judgement which Bell nevertheless does not address. For instance, although he argues that it would be wrong to require an account of judgement to restrict itself to the form of an account of the propositional attitude constructions used to attribute judgements to others, he does not justify the books failure to present any such account as a proper part of such a theory. Again, althought it has been suggested above that Bell was not obliged to restrict his attention to the notion of assertoric force (the analysis of the act of judging), one would certainly like a fuller and more satisfactory account of that notion than the cursory sketch we are offered.14 The book does its work in a sort of methodological no-man's land between textual exegesis and theory construction owing allegiance only to the phenomena it seeks to theorize about.

This is not to say that the analysis is not enlightening, however. Bell is at his best when dissecting the explanatory role assigned by Frege to his


technical concepts. When he succeeds we learn both about Frege and about the phenomena. Consider for instance the notion of Bedeutung.

Bell tells us that: 15

... Frege had not one, but two notions of reference. These notions hang together so well in the case of singular terms that they are hard to distinguish in this context. In the case of predicates, however, they are not only distinguishable, they are different to reconcile. One notion is this: the reference 01 an expression is that extra-linguistic entity with which the expression has been correlated or which it picks out. The other notion of reference is that it is a property which an expression must possess if that expression is to be truth-valuable (to coin a phrase). By truth-valuable I mean such that it either possesses a truth-value, or is capable of being used (and not just mentioned) in a sentence which possesses a truth-value.

Bell claims that although in the case of singular terms one notion can play both of these roles, since for them to be truth-valuable just is to be correlated with an object, in the case of sentence and functions the two notions diverge. All that Frege ever offers in the way of evidence for the application of the notion of reference to expressions in these categories is considerations showing them to be truth-valuable. Since he does not distinguish the two different notions of reference which he has in play, he feels entitled to conclude that they possess reference in the first sense as well. But this is a non sequitur, or at any rate a transition which must be justified, and not simply assumed on the basis of the conflation of the two different senses of Bedeutung. Thus Bell rejects the notion of truth-values as objects, and of functions as the references of functional expressions, as excess conceptual baggage mistakenly mixed in with the second notion of reference, which is the only one doing any explanatory work for these categories.

This analysis is clear-headed and valuable but can be faulted on two grounds, each of which amounts to a request for further analysis. First, as Dummett points out,16 the characterization of the second notion of reference does not seem right. For as Bell has described it, reference is a property which an expression either has or lacks, depending upon whether sentences containing it can have or always lack truth values. But Frege's notion is that in addition to having or lacking reference, expressions which have reference can have different references, accord-ingly as they make different contributions to the truth-values of sentences containing them. The test is always substitutional - two expressions which have reference have different references iff in some context the substitution of one for the other changes a true sentence into one

260 ROBERT B. BRAND OM

which is not true. Others who have noticed the distinction Bell is after have put things better. For instance, Tugendhat17 (who seems to have introduced this line of thought) calls this non-relational sense of refer-ence "truth-value potential" and in effect identifies the truth-value potential of a sub-sentential expression with the equivalence class of expressions intersubstitutable salva veritate.

The sharpening of Bell's distinction (which makes it similar to that between 'referent' and 'reference' which Dummett uses throughout [8]) does not affect his criticism of the inference from possession of refer-ence in this non-relational sense to possession of reference in the relational sense, of course. But it does affect a further use he wants to make of the distinction, to argue that it is incorrect to think of predicate expressions as having a reference at all, even in the non-relational sense. For here Bell argues that Frege incorrectly takes as a necessary and sufficient condition for the truth-valuability (in Bell's sense) of predi-cates that they have sharp boundaries. He accordingly takes it that the assignment of reference to predicates is motivated only by this require-ment, and so that showing the untenability of such a requirement is sufficient to show the inappropriateness of assiging reference to pre-dicate expression at all. This line of argument is undercut by seeing that there is more to the second notion of reference than truth-valuability. Since the denial of the cogency of the application of the notion of reference to predicates (or function expressions generally) is one of the main innovations of Bell's analysis, his failure adequately to characterize that part of Frege's notion of reference which remains when one takes away correlation with an extra-linguistic object has serious conse-quences for the subsequent course of his argument.

Dummett, however, rejects not only Bell's characterization of the second notion of reference, but also the claim that there are two notions of reference. He claims that the relational and the nonrelational senses represent "two ingredients of one notion". The second "tells us what Frege wanted the notion of reference for, and the oher tells us how he thought that it applied to the various categories of expression".18 It may be granted that the explanatory work Frege wanted the notion of reference for is its truth-value potential or contribution to the truth-conditions of sentences, and that he thought that the intersubstitutability equivalence class of equipollent expressions was determined by the correlation of all and only its members with the same extra-linguistic entity. But it would still remain to be asked, for instance, whether the


identity of the correlated object and the nature of the correlation can be inferred from the semantic equivalence class of expressions they deter-mine, as Frege's arguments concerning the reference of sentences and functional expressions would seem to require. Such a question is in no way made less urgent or easier to answer by rephrasing it in terms of two ingredients of one notion rather than in terms of the relations of two notions. In the final section of this paper it will be argued that this difficulty is one instance of a quite general definitional failure of Frege's part, one which in another context he tried unsuccessfully to resolve in a purely technical way.

Putting the issue in these terms raises the second source of dissatis-faction with Bell's argument. For the sort of question just raised seems no less important or difficult for the paradigmatic case of singular terms than for the parts of speech Bell finds problematic. The basic substitu-tional/inferential methodology which yields the sense of reference as an equivalence class of expressions vastly underdetermines the correlated objects and mode of correlation invoked by the relational sense even for proper names. Tugendhat, having formulated the non-relational notion of reference, takes it to be the notion of reference, discarding correlation with an object as a realistic confusion best extruded from Frege's thought. Sluga follows Tugendhat in this regard. The reason in each case is that all that Frege's analysis of the use of expressions seems to require is the sorting of expressions according to the non-relational sense of substitutional role. The semantic analysis he developed is a method for the perspicuous codification of inferences. Truth is what is preserved by good inferences, and sub sentential expres-sions can be grouped into co-reference classes accordingly as inter-substitution within the classes preserves such good-iriference potentials. Such an approach can give rise to specification of the conditions under which two expressions have the same reference, but how can it warrant a claim that the shared reference is to be identified with some object (among all those which in one way or another could be taken to deter-mine the same coreference classes) specified otherwise than as the reference of an expression? The answer seems to be that Frege's arguments for this identification are straightforwardly substitutional ones, in particular that for any singular term t we can always substitute (saving the inferential potentials) the. term the object referred to by

the singular term 't'. The expressions which license intersubstitution of expressions are identity locutions (as Frege had argued in the


Grundlagen), and so we are correct to say that the object referred to by the singular term 'Julius Caesar' is Julius Caesar. Whether this fact has the significance Frege thought it had is another matter. I 9

One of the most important discoveries of the early 1970's, both from the point of view of the interpretation of Frege and of the philosophy of language generally (for once, made independently of Dummett) con-cerns the need to distinguish two different explanatory roles which are conflated in Frege's technical concepts of sense. Kripke and Putnam independently argued 20 that the cognitive notion of the sense of an expression, what one who has mastered the use of that expression may thereby be taken to understand, and the semantic notion of the sense of that expression, what determines the reference of the expression, cannot in general be taken to coincide. In particular, in the case of proper names no knowledge or practical capacity which can plausibly be attributed to an ordinary competent user of the name will suffice to determine the object of which it is a proper name. A similar point can be made about the use of natural kind sortals. Since Frege had required that his notion of the sense of an expression play both the cognitive and the semantic role, and since for an essential range of expressions no single notion can do so, it is apparent that his concept must be refined by dividing it into two distinct sense-concepts, whose interrelations it then becomes urgent to investigate.

A further distinction within the semantic notion of sense has been urged by a number of writers, on the basis of the consideration of the behavior of indexical or token-reflextive expression.21 In Kaplan's idiom, we must distinguish for such expressions between their character,

which is associated with the expression type, and the content associated with each contextually situated token(ing) of that type. The distinction in question is evident in the following dialogue:

A: I am anxious to get started. B: No, it is possible that you are eager, but I am the anxious one.

We are concerned with the semantic notion of the sense of an expres-sion, that is, with the way in which its reference is determined. In one sense both tokens of "I" have their reference determined in the same way, for in each case it is the speaker responsible for the tokening who is referred to. These expressions share a character. But in another sense A's token of "I" and B's token of "you" have their reference determined in the different ways (e.g. for the purpose of tracking the referent through


the other possible worlds which must be considered to evaluate the model qualifications in B's remark). The referents of these tokenings will coincide in every possible world relevant to the evaluation of these utterances, in virtue of the identity of their contents. The characters of these expressions, together with the context in which they are uttered, determine a content which in tum determine a referent in every possible world. It is this latter task with which the semantic notion of sense is charged for nonindexical expressions. Such expressions may accordingly be thought of as those whose character determines a content without needing to be supplemented by a context. The point is that as we ask about what would be true in other worlds of the individual picked out by B's indexical utterance there is a double relativity to possible worlds, accordingly as those worlds can be relevant to the two different stages in the determination of a referent. First, since B's remark could have been addressed to someone other than A, we must consult the world-context in order to determine what content is fixed by the character of the exression when uttered in that context. The individual concept so determined as a content can then be tracked through various possible worlds and assigned referents in each, so that model claims can be evaluated.

Without referring to either of these antecedents, Bell distinguishes two notions of expression sense in a way which partakes of some of the features of each of the other distinctions. He calls his two notions "input sense" and "output sense", and introduces them by reference to two Fregean principles:22

PSI: The sense of a sentence is determined by the senses of its com-ponent parts,

and

PRI: The truth-value of a sentence is determined by its sense. (And, of course, how things stand.)

His claim is that although the ''two principles depend for their plausi-bility and usefulness on there being a sense of 'sense' which remains constant throughout", in fact they demand different ones. Input sense is that notion of which principle PS I holds, and output sense is that notion of sense of which PRI holds. Input sense is that which is preserved by correct translations and that for which synonymy claims assert identities of sense. Sub-sentential expressions have input senses (''meanings''), and


these combine to determine the input senses of sentences containing them. Output senses are defined as what is common to claims such as ''Today I ate plum pudding," and "Yesterday you ate plum pudding". The input senses of sentences together with a context of utterance determine such output senses. The output senses of sentences are what can meaningfully be described as true or false, as per principle PR2.

As described so far, Bell's distinction amounts to the claim that the cognitive/semantic and character/content partitions of the notion of sense ought to be seen as coinciding. For the compositionality of sense is a postulate required for the explanation of the possibility of under-standing complex expressions, so that it must be input senses which are in the first instance grasped cognitively. Semantic senses, determining truth values of sentences, are in turn identified with output senses. But since the latter are determined by the former together with a context of utterance and the distinction is enforced by attention to indexical expressions, the character/content distinction is likewise subsumed by the difference between input and output senses.

Such an identification is clearly subject to a number of objections, as consideration of the quite different motives and functions of the con-flated distinctions indicates. But these difficulties may not be insur-mountable. Perhaps a useful view could be elaborated based on the assimilation of the sense in which the referent of a proper name token is determined not by what its utterer understands by it, but only by this together with a causal, historical, and social context in which the token is embedded, on the one hand, and the sense in which the reference-determining sense of a token of "yesterday" is given not just by what one can understand as the meaning associated with the expression type, but only this together with a concrete context of use. But Bell does not attempt to develop such an account. In part this is because he has nothing whatever to say about what "contexts" are, or how these together with input senses determine output senses. And it is just here that all the detailed work is involved in making out either half of such an assimilation, and hence in justifying their conflation. But Bell is precluded from addressing such a task by other, less defensible features of his view.

For Bell denies that sub-sentential expressions have output senses at all, claiming that "output sense is essentially sentential",23 No argument or even motivation for this position is presented. It is suggested that for


sentences the distinction between input senses and output senses corre-sponds to that between sentences and the statements they can be used to make, and that it is better to think of the former not as possessing truth-values which change, by contrast to statements whose truth-values do not, but rather to think of the former as not the kind of thing which can have truth values at all. But no reason is given for not extending this distinction to sub-sentential expressions. The distinction hetween the two varieties of sense is introduced, as indicated above, in terms of two Fregean principles. PR2, the 'sense determines reference' principle, is quoted at this portion of the argument as restricted to sentences and truth values. But of course the principle Frege uses is not so restricted. Indeed, when Bell first introduces it some sixty pages earlier it is in unrestricted form. He has just been discussing the principle he calls PRI, that the reference of complex expressions is determined by the references of their components (which Bell discards because as we have seen he does not attribute reference of any kind to functions). He says:24

Elsewhere in his writings, however, he seems to invoke a quite different principle which we can call PR2. It is this: (a) the reference of any expression is determined by its sense, (b) the sense of a complex expression is determined by the senses of its component parts.

Two features of this definition deserve comment. First, part (b) of principle PR2 as here stated is what he later calls PS I and is concerned precisely to distinguish from PR2. Second, part (a) of this original state-ment differs from the later version in not being restricted to sentences. Neither of these substantial changes in the significance of his expression "PR2" is announced, acknowledged, or motivated in the intervening text. Such carelessness in specifying a central interpretive principle which one has taken the trouble to name for clarity of reference is bad enough under any circumstances. It is unforgivable when essential features of one's own claims and their justifications depend precisely on the matters obscured by the sloppiness. As things stand, the reader is left with no idea why in using the two principles PR2 and PSI (= PR2(b) in earlier statement) to distinguish two notions of sense one should employ the later version of PR2 rather than PR2( a) from the earlier version, which is the principle Frege endorsed. Apart from the invocation of PR2, output senses are specified as what is common to the two "plum-pudding" sentences quoted above. As our sketch of the


character/content distinction show, it is not at all obvious why this characterization should not extend to what is common to 'today' and 'yesterday', on the one hand, and'!' and 'you' on the other.

Bell does, however, employ the restriction of output senses to sentences to argue for a further point. For he claims that the "context principle" of the Grundlagen may be understood in terms of the fact that terms only have input senses, which together with the input senses of other expressions determine sentential input senses, which in context determine a truth-value. Since the reference of terms matters only in determining truth-values, it is "only in the context of a sentence that a term have a reference". Clearly nothing can be made of this line of thought in the absence of a rationale for its basic premises.

These difficulties with the distinction between input senses and out-put senses also make it difficult to evaluate another novel interpretive suggestion which Bell offers. He concludes his discussion of the senses of proper names with the claim25 "The sense of a proper name, then, is that it purports to refer to a determinate object of a given sort with which it has been conventionally correlated." The sense of a proper name is here taken as "that which one understands when one is able to use it correctly".26 As indicated above in the discussion of the relation of the cognitive notion of sense to Bell's notions, this must be the input sense, for sub sentential expressions aren't supposed to have output senses. It is accordingly obscure what the connection is supposed to be between the senses Bell is offering a theory of here and the determina-tion of referents for the proper names they are senses of. What then are the criteria of adequacy for an account of what a name user must be taken to understand? Bell examines the conditions under which we would want to deny that someone had mastered the use of a name, and concludes that in addition to using it as a singular term one must at least know some sortal under which the referent is taken to fall in order to be judged a competent user. This is useful as a necessary condition, but much less plausible as a sufficient condition to be taken to be using an expression as a proper name. For a sufficient condition would seem to require that one be appropriately connected to a community of users of the name, perhaps an historically extended one, whose joint use does

determine a referent, though no individual's use need do so. It is not obvious that merely believing that some conventional correlation has been established with an object of the right sort is sufficient to be appropriately connected with the community of users of that name. In


any case, to argue for such a principle would require looking at how input senses and various specific sorts of context can together determine output senses and eventually referents for the names in question, and this Bell does not undertake.

Bell wants his notion of proper name sense in order to develop an appropriate account of the senses of functional expressions. This latter task is made especially urgent by the confrontation between his denial that the referents of functions have any explanatory value, on the one hand, with the undeniable importance in Frege's scheme of functions and concepts understood as functions, on the other. Bell's reconstruc�tion reconciles these ideas by interpreting concepts and functions as the senses rather than the references of functional expressions. A concept, accordingly, is to be understood as a function which can take as arguments proper name senses of the sort he has described, and yield thoughts, the senses of sentences. While this identification of concepts must be seen as a revision rather than an interpretation of Frege's thought, it might seem that, setting that identification aside, at least the account of the senses of functional expressions as functions from the senses of argument expressions to the senses of value expressions ought to be uncontroversial. It is not, and it is instructive to see why not.

As Bell has pointed out in his discussion of senses generally, the concept of sense is required to play two distinguishable roles. First, the sense of a component of a complex expression must contribute to the determination of the sense of that complex. But also, the sense of the component must determine a reference for that component. This gives us two different ways to think about the senses of functional expressions such as predicates. On the one hand they must combine with the senses of terms to yield the senses of sentences. On the other hand they must be the way in which a function from objects to truth values is deter�mined or given. It is not obvious that these two jobs can be done by one notion. In particular, Dummett has argued27 that "once the proper name has specified the way in which the object is given, then it has made its contribution to the sense of the sentence; if it had not, then it would be impossible to see how its sense could both contribute to the sense of the sentence and consist in the way in which the object is given." That is, maintaining the coincidence of the two roles of sense in the case of proper names (presumably where our grasp is firmest) commits us not only to their divergence for functional expressions, but also to which half we give up, namely the identification of their senses with sense


functions. Geach has objected to this doctrine of Dummett's,28 and it is instructive to examine Dummett's response.29

It is not disputed that once a sense has been assigned to a predicate, a function from the senses of proper names to thoughts is determined. For according to Dummett the predicate sense is the way in which a function from objects to truth values is given. Hence, when that function is supplemented by an object, it determines a way in which a truth-value is given, that is, a thought. But since a term sense will determine such a supplementing object (according to the second role of senses mentioned above), the predicate sense will induce indirectly a function from term senses to sentences senses. As Dummett says, "the question is whether the sense of the predicate just is that function."

To argue that it is not, Dummett appeals to a further thesis of Frege's about senses, namely that the senses of component expressions are parts

of the senses of the complex expressions in which they occur. We have seen that it is a mistake to think of functions or their arguments as parts of the values they generate, as Frege's retraction of his careless claim that objects are parts of truth values shows. But since Frege did hold that predicate senses are parts of thoughts, we would be committing precisely this howler if we identified those senses with functions taking term senses into thoughts. This is an ingenious counter-argument, but it cannot be considered decisive. For while it would be a howler to treat functions and their arguments generally as parts of the values they determine (as in the combination of Sweden and the function the capital

of ... to yield Stockholm), this consideration does not show that particular functions and kinds of function cannot have values which contain the functions or their arguments as parts. Stockholm is part of the value of the function the country of which . . . is the capital. And mathematical examples of function-values which contain functions as parts in the set-theoretic sense are easy to come by. (One thinks of the story of the oracle who offered to answer a single question, and upon being asked "What is the ordered pair whose first element is the best question I could ask you, and whose second element is its answer?" replied (falsely, I suppose): "the ordered pair whose first element is your question and whose second element is this answer.")

Insulated from this dispute about sense functions by his distinction between input senses and output senses, Bell backs up his commitment to treating the senses of functional expressions as functions by citing a number of passages, both published and from the posthumous works, in


which Frege unequivocally describes such senses as "unsaturated", "incomplete", and "in need of supplementation", going so far in fact as to say that "The words 'unsaturated' and 'predicative' seem more suited to the sense than to the reference".3o To motivate his identification of concepts with sense functions, Bell argues as follows.31 The only reason Frege had for believing in concepts as predicate referents was the need to deal with a situation in which predicates have a sense and so determine a thought, but lack a reference, and so determine a thought which has no truth-value. The only case where this can happen which does not reduce to the failure of a term to have a reference is where the predicate is not defined for the sort of argument to which it is applied. But this sort of case can be much more plausibly excluded by considera-tions concerning predicate senses. For such cross-categorial predications (such as "Julius Casesar is the sum of two prime numbers") ought properly to be seen as not s\:lcceeding in expressing thoughts at all. Bell's solution accordingly is to see predicates as having sortal restrictions associated with their argument places, which together with the 'sortal physiognomy' he has already assigned to proper name senses yields the result he desires. One of the benefits which might be derived from such a radical reconstruction should be made manifest by the discussion to be given below to the difficulties ensuing from Frege's insistence that functions be defined for all arguments whatsoever. However, as before, the evaluation of this thesis about senses must await some resolution of the general questions Bell has left open concerning his distinction between input and output senses.

SECTION II

Hans Sluga's new book on Frege in the "Arguments of the Philoso-phers" series32 represents an approach complementary to Bell's in almost every regard. It's central aim is to reread Frege's work in the light of that of his precursors and contemporaries, rather than by reference to his successors in the analytic tradition, as has been traditional. Although Frege's unprecedented innovations in symbolic logic have made it natural to think of him exclusively in the role of the founder of a tradition - as a man without a past - Sluga argues that we ignore at our peril his intellectual climate and .the influences which conditioned various aspects of his technical concepts and of the explanatory tasks he set for them. Sluga's task is not purely historical, however. For he is also


concerned to set out and justify novel readings of some of Frege's purely philosophical doctrines, readings which are suggested and motivated by the historical recontextualization he recommends. The result is a stimu�lating new picture of Frege's thought which will be of interest even to those who are not in the end persuaded in detail by it. Furthermore, since the narrative strategy employed is to trace the development of Frege's ideas chronologically (starting, as it were, before he was born) and surveying all of his important writings seriatim, this book is excel�lently constructed to serve as an introduction to these ideas (as Bell's or Dummett's books, for instance, could not) as well as to challenge specialists.

The book's historical orientation, then, is not adopted only for its own sake, but also in order to guard against blinding ourselves to inter�pretively significant features of Frege's work by the importation of anachronistic prejudices. Accordingly, it is primarily in terms of the philosophical illumination they provide for our appreciation of Frege's concepts and claims that we must evaluate the success of Sluga's various invocations of historical influence. The claimed influences may be considered under four headings. First, a view is presented about who Frege took to be his philosophical opponents. Next, Leibniz is identified as a precursor. Third, claims are made about the influence of two logicians of the generation preceding Frege's, Lotze and Trendelenburg. Finally and most significantly, it is claimed that overlooking the intellec�tual debt which Frege owes to Kant has most seriously distorted our understanding. We will consider these claims in this order.

In his first chapter, Sluga is concerned to refute the claim that "In a history of philosophy Frege would have to be classified as a member of the realist revolt against Hegelian idealism, a revolt which occurred some three decades earlier in Germany than in Britain."33 In this aim he succeeds unequivocally. Hegelianism had ceased to be dominant or even popular in German philosophical circles some years before Frege was born. The view against which Frege was reacting is the scientific natural�ism which Sluga claims was held by the physiologists turned philoso�phers Vogt, Moleschott, Buchner, and Czolbe, popularized during Frege's lifetime by Haeckel, and shared with some reservations by Gruppe. Ontologically this view is a reductive materialism, and epistem�ologically it is an empiricist psychologism. Sensations are viewed as material processes of the brain. Concepts, and hence the thoughts constructed from them are taken to be reflections of such sensations.


Logic is seen as the study of the laws of thought, that is, as an empirical investigation seeking to establish the natural laws governing the associa-tion of concepts in judgment and of judgments in inference. It is this psychologism which Frege so vigorously opposed, and on those rela-tively few occasions when he describes his opponents as 'idealists' it is clearly this school which he has in mind.

This is a point of no small moment, especially in the context of an evaluation of Frege's role as progenitor of the analytic tradition. For his over-arching objection to the naturalists is their failure appropriately to distinguish between the normative and ideal order of correct inference and justification on the one hand, and the descriptive and actual order of causation and empirical processes on the other. Their concommitant confusion of features of cognitive acts with features of the contents of those acts is merely the expression of this original sin. And in his insistence on the centrality of this basic distinction Frege is at one with Kant and the post-Kantian idealists, and at odds with the primarily physicalist and empiricist tradition in Anglo-American philosophy which he fathered, and in the context of which it has been natural for us to read him.34

Throughout his book Sluga talks about Leibniz' influence on Frege, but when he specifies the details of this influence his claims turn out to be quite weak. Like Leibniz (and Kant), "Frege is interested in the study of logic and the foundations of mathematics because they allow one to ask in a p'recise form what can be known through reason alone."35 Aside from this general rationalist commitment to the possibility of a priori

formal knowledge, the only Leibnizian doctrine which is attributed to Frege is the endorsement of the project of the universal characteristic. Frege explicitly describes the motivation for his Begrijfsschrift in this way. That at this level of generality Frege owes a debt to Leibniz is hardly a novel or surprising claim, however. Sluga also discusses the influence of Trendelenburg, but in the end the claims seem to come to little more than that he was the conduit through which Frege became familiar with Leibniz' ideas.

It is otherwise with the connection discerned between Frege and the logician Lotze. The suggestion of influence here has specifically been denied as "a remarkable piece of misapplied history".36 Yet in this case Sluga shows sufficiently striking similiarities to make the hypothesis of influence persuasive. It is known that Frege read Lotze. Indeed it has been argued that the theory of judgement in opposition to which he


presents his innovation in the Begriffsschrift just is Lotze's formulation.37

The essay immediately preceding 'The Thought' in the journal in which it was originally published, which Sluga takes to have been intended by the editors as an introduction to Frege's essay, mentions Frege in the context of an exposition of Lotze which highlights several Fregean doctrines.38 From Sluga's account of Lotze's views (as presented in the Logik of 1874 and an earlier work of 1843) one can extract eight points of similarity with Frege.

First, Lotze inveighs against psychologism and indeed is the figure Frege's contemporaries would probably have identified as leading the battle against the dominant naturalism of the day and in favor of a more Kantian position. Next, Lotze was a logicist about mathematics, although there is no hint in his works that he took the detailed working out of such a reduction to logic as part of what would required to justify this view. Third, Lotze insists, against empiricistic sensationalism, upon the distinction between the objects of our knowledge and our recogni-tion of such objects, in much the same terms that Frege did. Fourth, Lotze emphasized and developed the Kantian strategy of explaining concepts as functions (though of course he does not have the notion of functions as unsaturated which Frege derived from his own substitu-tional method of assigning contents to sub-sentential expressions). Fifth, Lotze attacks the empiricists with a distinction between the causal conditions of the acquisition of concepts and the capacity to use such concepts in correct reasoning which mastery of the concepts consists in (see note 34 above). Next, Lotze offers a theory of identity statements according to which the two terms share a content, but differ in form. This is the Begriffsschrift view, and the language survives into the opening paragraphs of 'Uber Sinn und Bedeutung.' Seventh, Lotze endorses the Kantian principle of the priority in the order of explana-tion of judgements to concepts which Frege endorses in the Grundlagen.

Lotze does not succeed in being entirely consistent on this point, since he also is committed to atomistic principles which are not obviously compatible with the view on the priority of judgements. Although Sluga does not say so, those who take Frege not to have discarded the context principle in the post-1890 writings must find a similar tension in some of the procedures of the Grundgesetze. Finally, Lotze is committed to the objectivity of sentential contents, and treats them as neither mental nor physical just as Frege does. Lotze, however, specifically denies that this objectivity is grounded in the correlation of sentences with objects


such as Frege's thoughts appear to be, taking a more Kantian position. Sluga, as we shall see below, argues that despite apparent statements to the contrary we should understand this to be Frege's view as well.

This is a suggestive set of similarities to find in a prominent near-contemporary logician with whose work Frege was familiar. Recogniz-ing them as important need not commit one to minimizing the signifi-cant, perhaps dominant, differences in outlook which remain between Lotze's revived Kantianism and Frege's philosophical e1aboration of his semantic methodology (although Sluga does on occasion succumb to the temptation to treat Frege's agreement with Lotze on one point as evidence that he probably agreed with him on others). Only according to the crudest notion of what philosophical originality consists in is there any incompatibility between finding enlightenment in the demon-stration that these general principles were in the air and so came complete with a history and a tradition, on the one hand, and the appreciation of the genius shown in the use such adopted and adapted raw materials were put to in service of quite a different explanatory project on the other.

Sluga's most important and sustained argument, however, concerns the influence of Kant on Frege. He claims that Frege should not be thought of as a dogmatic realist about physical objects nor as a Platonist about abstract objects, as he almost universally has been thought of. He should be seen rather as a Kantian whose realistic remarks are to be interpreted as expressing that merely empirical realism which is one feature of transcendental idealism. This is certainly a radical reinterpre-tation. What evidence can be adduced for it? Sluga's considerations may be assembled as five distinct arguments.

First it is pointed 'out that Frege joined a philosophical society whose manifesto is explicitly idealist and Kantian, and that he published in their journal. By itself, this shows little, for Frege had so much trouble getting his work into print and finding others willing to discuss it that we cannot be sure how much he would have put up with to secure such opportunities. The rationale Sluga suggests39 is that ''what tied him to the idealists was primarily his opposition to the various forms of naturalism". Specifically, Frege and the idealists (a) were anti-psycho-logistic, (b) endorsed an objectivist epistemology (taking the contents of judgements to be independent of their entertainment by thinkers), and (c) endorsed a rationalistic a priorism about mathematics. These points are well taken, but the views involved are all consistent with Platonism


and realism generally as well as with transcendental idealism. Indeed Sluga admits that "one can read much of Frege and not raise the question of transcendentalism". So we must look elsewhere for a warrant for such an attribution.

The second argument concerns Frege's attitude towards the truths of geometry.40 It is remarked to begin with that in his Habilitationsschrift Frege held a Kantian view on this topic, saying that geometry rests "on axioms that derive their validity from the nature of our capacity for intuition (Anschauungsvermogen)". Furthermore, throughout his career Frege describes geometrical knowledge as synthetic a priori, and on this basis rejects non-Euclidean geometry as false. From this fact Sluga concludes: "Frege held a Kantian view of space and hence a transcen-dentally sUbjective view of the objects that occupy it." The only elucidation offered of this crucial 'hence' is the later statement that "Frege's view must be close to Kant's: Empirical objects are in space and time, but space and time are a priori forms of sensibility. That seems to imply that for Frege empirical objects can only be empirically real, but must be transcendentally ideal." That Kant believed the two views to be linked in this way falls far short of showing that Frege did so. Certainly such an argument cannot be taken to undermine an inter-pretation which takes Frege's realistic remarks about physical objects at face value, and admits that his views are inconsistent to the extent that be never confronted these latter with his views about geometry with an eye to reconciling them. On the other hand some interpretive cost is clearly associated with attributing such an inconsistency to Frege.

The next two arguments must be judged less satisfactory.41 First, Sluga argues that in the context of Kantian transcendentalism (as just discussed) Platonic realism looks like dogmatic metaphysics. So Frege should have been expected to argue that views (a) through (c) above, on which he argees with the idealists, cannot in fact be warranted tran-scendentally. But Frege nowhere argues this. The trouble with this argument is that there is no evidence that Frege did not, as most of his contemporaries did, read Kant's transcendentalism as a form of psy-chologism. If he had done so, he would have dismissed it and so not felt the force of the demand in question. Sluga next argues that every claim of Frege's that can be taken as evidence of Frege's realism can be matched by a passage in Lotze, who had a Kantian idealistic theory of validity. This argument seems to do no more than restate the point that (a) through (c) are consistent with either position. For it is a criterion of adequacy of anyone's transcendentally idealistic position that it have


room for all of the claims the realist wants to make, suitably reinter-preted. Further, Frege does insist that thoughts are independent, not just of this thinker or that, but of the very existence or even possibility of thinkers at all. This seems to contradict Lotze's account of objectivity as rule-governed intersubjectivity.

Sluga'S final argument is weightier and involves more interpretive work, both in construction and evaluation. The basic claim is that "there are strewn through Frege's writings statements that appear irreconcil-able with Platonic realism. In particular the central role of the Fregean belief in the primacy of judgements over concepts would seem to be explicable only in the context of a Kantian point of view."42 Arguing in this way obviously commits Sluga to showing that Frege does not discard the context principle when he arrives at the distinction between sense and reference. We will see below that he contributes significant new considerations to that debate in furtherance of this aim. But the incompatibility of realism with the recognition of the primacy of judge-ments must also be shown. The latter view is 'Kantian', but it does not obviously entail transcendental idealism, which is the view in question. Sluga takes the principle of the primacy of judgements to serve the purpose for Kant43 of refuting any atomistic attempt to construct concepts and judgements out of simple components, and in particular to resist the empiricist sensationalist atomism of Hume. Such a view is indeed incompatible with the reism of Kotarbinski (to which Tarski's recursive semantics owes so much), which sees the world as an arrange-ment of objects out of which concepts and judgements must be con-structed set-theoretically.44 But the Kantian principle need not be taken to be incompatible with Platonic realism about abstract entities such as thoughts which are the contents of judgements. Given that the context principle does not show that Frege was a transcendental idealist about thoughts, it seems also open to him to hold some form of realism about other objects, provided thoughts retain an appropriate priinacy (as, given the very special status of truth in the late works, even those who see the context principle as discarded are committed to granting) even if he has not discarded that principle. So if the case for the persistence of the context principle can be made out, it should be taken as showing

Frege was a Kantian in the sense of holding the context principle, not in the sense of being a transcendental idealist.

Still, this point is worth establishing for its own sake. Sluga correctly sees the Begriffsschrift as the confluence of three lines of thought: (1) that judgements, as involved in inference, are the original bearers of


semantic significance, so that it is only by analyzing such judgements according to the procedure of "nothing invariance under substitution" that such significance can be attributed to sub-sentential expressions ('the primacy of judgements'), (2) the Leibnizian idea of a perfect language, and (3) the idea of reducing mathematics of logic. Assuming the context principle was thus "anchored deeply in Frege's thought, it is implausible to conclude with Dummett that in his later years Frege simply let it slip from his mind."45 Sluga advances five arguments for the persistence of the principle, and along the way addresses two commit-ments of Frege which have been taken to be incompatible with such persistence.

First, Sluga offers an important consideration which has not pre-viously been put forward in the extensive literature discussing this question. The first of the 1891-2 essays which Frege wrote is a seldom read review of L. Lange's The Historical Development of the Concept of Motion and its Foreseeable End Result called 'The Principle of Inertia'. In it Frege argues at some length that the concepts of a theory are not given prior to and independent of that theory. Rather those concepts can be arrived at only by analyzing the contents which the judgements constituing the theory are given by the inferences concerning them which that theory endorses. This is a significant new piece of evidence supporting Sluga'S view. The only question which might be raised about it is that since this semi-popular piece does not deploy the full-blown apparatus of sense and reference it may be wondered whether the views there expressed were confronted by Frege with that apparatus, or whether the essay might not be seen as merely the latest of his early works. But to take such a line would be to concede a lot, and future claims that the context principle was discarded will have to confront this argument of Sluga's in detail.

Next Sluga offers a novel reading of the essay on the distinction between sense and reference which denies that, as has often been claimed, that distinction as there presented applies primarily to singular terms and their relations to the objects which are their referents, and hence commits Frege to an assimilation of sentences to terms which is incompatible with the context principle. The strategy here is, in effect, to deny that 'Bedeutung' as Frege uses it ever has the relational sense which indicates correlation with an object. Relying on the Tugendhat essay mentioned above in connection with Bell, Sluga understands 'Bedeutung' as a nonrelational semantic potential defined paradigmati-


cally for sentences, in virtue of their role in inference. The introduction of this notion in the context of the consideration of identities involving singular terms is seen as a rhetorical device of presentational signifi-cance only. In the final theory subsentential expressions are taken to inherit indirect inferential significances in virtue of their substitutional behavior in sentences, which alone are directly inferentially and hence semantically significant. Thus 'Bedeutung' is paradigmatically a senten-tial notion.

To this analysis is conjoined an account of 'Sinn' as a cognitive notion, as what matters for knowledge. But again, the units of knowledge are judgements, and sub sentential expressions can become relevant only insofar as they can be put together to form sentences which can express judgements. So sense also should be seen as primarily a sentential notion, which applies to sub sentential expressions only in a derivative way. This line of thOUght concerning senses is then combined with that concerning reference in a subtle and sensitive account of the puzzling relations between the Lotzean rendering of identity locutions offered in the Begriffsschrift and its successor in 'Uber Sinn und Bedeutung'.

The previous discussion of Bell's interpretation suggests that these readings leave something to be desired. Sluga does not acknowledge the existence of any passage or considerations indicating that Frege does have a relational notion of reference in play. Yet such passages and considerations do exist, and merely elaborating the nonrelational ver-sion of Frege's concept, as Sluga does, does not obviate the necessity of investigating the relations between the two notions and the possibilities for reconciling them. Similarly, Sluga pushes his discussion of the notion of sense no farther than the discrimination of the cognitive role played by that concept. He has nothing to say about the semantic notion of sense, or accordingly about how senses are to be understood as deter-mining references, even nonrelational references. On these points Sluga's analytic net does not have as fine a mesh as Bell's. As a result, his ingenious interpretation of sense and reference will require further filling-in before its eventual promise can be assessed.

The overall interpretation which results from all of these arguments, however, is challenging and powerful. The primary objections . to the persistence of the context principle are that Frege nowhere explicitly endorses that principle after the 1884 Grundlagen formulation, and that the principle is incompatible with two central doctrines of the 1891-92 essays, namely the semantic assimilation of sentences to terms and the


account of concepts as functions from objects to truth-values. Sluga claims that his readings of the "Inertia" essay and of USB meet these objections. He does not say in detail how the doctrine about functions is to be reconciled with the context principle, but does argue that the "Inertia" essay justifies us in attributing Bedeutung to any expression which makes an appropriate contribution to the possession of truth values by sentences containing it. Thus function-expressions may be assigned (nonrelational) reference on this account. Using intersubsti-tution equivalence classes to move from Tugendhat's nonrelational sentential semantic significances to those of sub-sentential expressions does indeed justify such an attribution. But in the "Inertia" essay, Frege seems to be using "concept" in the ordinary sense rather than his technical one, that is, to refer to the senses of predicate expressions rather than their references. This being the case, it is not clear how the envisaged reconciliation of the context principle with the view of concepts as functions from objects to truth values is to be achieved.

Besides the evidence of the essay on inertia, Sluga offers two further reasons to deny that the later Frege is silent on the topic of the context principle. First, he mentions in several places the posthumously published 'Notes for Ludwig Darmstaedter' (of 1919) as showing that Frege continued to endorse the principle. He does not say what pas-sages he has in mind, but he presumably intends the following: 46 "What is distinctive about my conception of logic is that I begin by giving pride of place to the content of the word 'true', and then immediately go on to introduce a thought as that to which the question 'Is it true?' is in principle applicable. So I do not begin with concepts and put them together to form a thought or judgement; I come by the parts of a thought by analyzing the thought." Such a passage does show that sentences play a special explanatory role for the late Frege, but that much is not in question. At most such claims would show that a version of the context principle held for senses, confirming Sluga's claim that the cognitive origins of the concept of sense require that priority be given to sentences. No version of the context principle for referential signifi-cances follows from these claims. Unfortunately, Sluga never says what exactly he takes the context principle to be, whether a doctrine about senses, references, or both. Frege's original formulation, of course, preceded his making this distinction. So perhaps the best conclusion is that Sluga takes the principle to persist as applying to senses, that is that it is only in the context of a thought that a term or other sub-sentential


expression expresses a sense. This seems to be something Frege indeed did not surrender. Such a reading has the additional advantage that the doctrine that concepts are functions from the references of singular terms to truth-values is not incompatible with it.

The final argument fares less well. It is claimed that Frege's late treatment of real numbers shows that his practice is still in accord with the context principle.47 Here the point seems to be that the real numbers are given contextual definitions. Such an argument would be relevant to a context principle applying to reference rather than senses, since Frege does not pretend to specify the senses of numerical expressions in his formal definitions. But the definition of real numbers he offers is of just the same form as the Grundgesetze definition of natural numbers. If this style of definition does exhibit commitment to a form of the context principle, that case should be argued for the more central and important case of natural numbers. It is not clear how such an argument would go.

SECTION III

One of the themes around which Sluga usefully arranges his presenta-tion of Frege's development is that of the pursuit of the definition of purely logical objects. The reason offered for the somewhat misleading order, of presentation pursued in DSB, which seems to give pride of place to singular terms rather than sentences, is that the road from the Grundlagen account of numbers to that of the Grundgesetze needed to pass through a more thorough understanding of identity claims. Sluga is quite clear that for Frege, beginning with the Grundlagen, the only concept we have of an object is as that which determines the semantic significance of a singular term. For an expression to play the semantic role of a singular term is for it to make a certain contribution to the inferential potential of sentences containing it, a contribution which is constituted by the appropriate (truth preserving) substitutions which can be made for that expression. The substitution inference potential of a singular term is in tum codified in the endorsed identity claims involving that term. That what we mean by 'object' is according to Frege exhausted by our conception of that the recognition of which is expressed in identity claims in virtue of their licensing of intersubstitu-tion is one genuinely transcendental element in his thought about which Dummett, Sluga, and Bell agree.


In the Grundlagen Frege argued that according to this criterion, number-words are singular terms, so that if statements about them are ever objectively true or false they must be so in virtue of properties of the objects which are identified and individuated in assertions of numerical identities. The logicist thesis that the truths of mathematics are derivable from the truths of logic by logical means alone accordingly entails that numbers are purely logical objects, in the sense that the identities which express the recognition and individuation of these objects are themselves logical truths. Sluga's ingenious suggestion is that Frege's concern in DSB with the nature of synthetic or potentially knowledge-extending identities specifying ordinary objects should be understood as a stage in the working out of his mature account of analytic (logically true) identities required for the adequate specification of the logical objects treated in the Grundgesetze. The specific interpre-tive use to which Sluga puts this general insight is hard to warrant, however.

For he claims that the difference between these two sorts of identities resides in the fact that the identities by which logical objects are identified and individuated express coincidence not just of reference, but also of sense.48 It is not clear what reasons there are to accept this reading, nor what interpretive advantages would accrue from doing so. For Frege explicitly affirms on a number of occasions that the two expressions '22' and '2 + 2' express different senses. And he seems committed to this view by structural principles of his approach, in particular by the compositionality principle as it applies to senses. Different function-expressions appear in these two complex designa-tions, and the senses of components are parts of the senses of com-plexes containing them. Nor does the fact that such identities are to be logically true entail that they express identities of sense rather than merely of reference. Identity of sense would of course be sufficient for identity of reference. But we are often told that logic need be concerned only with truth and reference, and Frege's view seems to be that it can be logically true that two different senses determine the same reference.

This mistake aside, Sluga's tracing of the development of Frege's attempts to define abstract objects of the sort instantiated by logical objects is a valuable contribution, and raises issues of the first impor-tance for our understanding of the constraints on interpretations of Frege's technical concepts. The story begins with the second definition of number which Frege tries out in the Grundlagen. It states that two concepts have the same number associated with them iff the objects


those concepts are true of can be correlated one-to-one.49 He rejects such a definition as inadequate to specify numbers as objects, on the grounds that it will not determine whether, for example, Julius Caesar or England are identical to any number. Such a definition settles the truth-values of identities (and hence the appropriateness of substitutions) only for terms which are the values for some argument expression of the function-expression "the number of the concept ... ". This procedure would be legitimate only if we had independently defined the concept (function from terms to truth values) number signified by this function-expression. But it is not possible simultaneously to specify that function and the objects for which it yields the value True. If objects had been specified by this definition, then there would be a fact of the matter as to whether Julius Caesar was one of them. But the definition does not settle this issue either way. On the basis of this objection, Frege moti-vates his third and final definition of numbers, considered below.

Sluga traces through the later works Frege's efforts to clarify the specification of numbers in such a way that it will not be subject to this objection, culminating in the Grundgesetze account of courses of values. Given the centrality to Frege's project of producing an adequate defini-tion of number this progress is of interest for its own sake. But the task of responding to the objection to the second GL definition of number is made especially urgent for interpreters of Frege by a consideration which Sluga does not mention. For the specifications of the abstract objects in terms of which Frege's semantic analysis proceeds (e.g. sense, reference, thought, truth-value) are of the same objectionable form as the second GL definition of number. Nothing we are ever told about the senses of singular terms or sentences, for instance, settles the question of whether Julius Caesar can be such a sense. Though this may seem like a question of no interest, some interesting questions do take this form. For in interpreting the notion of sense one is concerned both With subdividing the explanatory functional role played by the concept (as exhibited in the discussion of Bell) and with the possibility of identifying senses with things otherwise described - for example, the uses of expressions, sets of possible worlds, mental representations. Frege him�self addresses such issues when he denies that the senses of sentences are to be identified with ideas in people's minds. How is the identity he wishes to deny given a sense?

All that is given is a criterion determining when the senses associated with two expressions are the same (namely if they are intersubstitutable without change of cognitive value - Erkenntniswerte). If something is


not specified as the sense associated with an expression (compare: number associated with a concept) its identity or nonidentity with anything which has been so given is entirely undetermined. Frege's procedure for introducing his technical concepts such as sense is invariably to attempt to specify simultaneously a realm of abstract semantic interpretants and a function which assigns a member of this realm to each expression.

We are, for instance, to associate truth-values with sentences. But we are told only that the truth-value associated with p is the same as the truth-value associated with q just in case for no occurrence of p (either as a free-standing sentence or as a component in a more complex sentence) can a good inference be turned into a bad one by substituting q for that occurrence of p (reading the principle that good inferences never take true premises into conclusions that are not true as defining truth-values in terms of the goodness of inferences). Even conjoining such a specification with the stipulation that the truth-value associated with the sentence '2 + 2 = 4' is to be called 'the True' does not settle the question of whether Julius Caesar is that truth-value. He had better not be, for if the logicist program of GG is to be successful, truth-values must be definable as purely logical objects. The current question is how the identity which is denied here is given a sense so that something could count as justifying that denial. The functions which associate the various kinds of semantic significances with expressions are always of the form: f(x) = fey) iff R(x, y), where x and y range over expressions, and R is some relation defined in terms of the inferential potentials of those expressions. These are exactly the kind of definition Frege found wanting in GL.

Seeing how Frege believes he can overcome the objectionable in-determinateness of concepts such as that determined by the second GL

definition of number is thus a matter of considerable importance for the appraisal of his success in specifying his own technical concepts, as well as for the narrower project of introducing numbers as logical objects. The third and final definition of number which Frege offers in GL is: "the Number which belongs to the concept F is the extension of the concept 'equal (Gleichzahlig) to the concept F' "50 The number three is thus identified with the extension of the concept, for example, "can be correlated one-to-one with the dimensions of Newtonian space". This definition does not have the form Frege had objected to. However, it essentially involves a new concept, 'extension', which has not previously


appeared in GL, nor indeed anywhere else in Frege's writings. In a footnote to the definition, Frege says simply "I assume that it is known what the extension of a concept is." Sluga points out that this defini-tionally unsatisfactory situation is not remedied in the remainder of the book. The result is scarcely up to the standards of definition to which Frege held others and himself. The project of GL could not be counted a success until and unless it could be supplemented with an account of the extensions of concepts.

Six years later, in 'Funktion und Begriff', Frege offers such an account. The general notion of a function is explicated, and concepts are defined as functions from objects to truth-values. The extension of a concept is defined as the 'course of values' (Wertheverlauf) of that function. This is the first appearance of the concept of a course of values. Since extensions are reduced to them, the residual definitional burden bequeathed by GL is put off onto this new concept. What Frege tells us here is just that the course of values associated with function F is the same as the course of values associated with function G just in case for every argument the value assigned to that argument by F is the same as the value assigned to it by G. The trouble with such a stipulation, as Sluga says, is that it has exactly the objectionably indeterminate form of the second GL definition of number which it is invoked to correct. Frege wants to associate with each function a new kind of object, a course of values. This domain of objects and the function which assigns one to each function are introduced simultaneously. The result is that it has not been determined whether Julius Caesar is the course of values of any function. A given course of values has only been individuated with respect to other objects specified as the courses of values asso-ciated with various functions. In sum, the courses of values in terms of which the extensions of concepts are defined suffer from exactly the defect of definition which extensions of concepts were introduced to rectify or avoid.

In the Grundgesetze when courses of values are introduced this difficulty is explicitly acknowledged and described in the same terms used to raise the original objection in GL (though without reference to the earlier work). Frege introduces the same principle for determining when the courses of values of two functions are identical, and then points out that such a principle cannot be taken to determine any objects until criteria of identity and individuation have been supplied with respect to objects which are not given as courses of values. He


proposes to supplement his definition so as to satisfy this demand. His proposal is that for each object not given as a course of values it be stipulated to be identical to an arbitrary course of values, subject only to the condition that distinct objects be identified with distinct courses of values.

Frege expresses the function which assigns to each function an object which is its course of values by means of an abstraction operator binding a Greek variable. The course of value of a function F is written as ' 0 (F 0). Axiom V of the Grundgesetze tells us that:

(a) 'O(FO)='a(Ga)iff('Vx) [Fx(=) Gxl.

Frege recognizes that this principle alone does not suffice to determine the identity of objects which are courses of values. To show this he points out that if X is a function which yields distinct values if and only if it is applied to distinct arguments (what we may call an "individuation preserving" function), then:

(a') X('O(FO»=X('a(Ga» iff ('Vx) [Fx(=) Gxl

without it's having been settled for instance whether

(a") X('O(FO» = 'a(Ga)

for any F and G (including the case in which F = G). The by now familiar point is that (a) only determines the truth-values of homo�geneous identities, those both terms of which are of the form 'O(FO). And (a') only determines the truth-values of identities which are homogeneous in that both terms have the form X('O(Fo». But (a") asks about heterogeneous identities, whose terms are of different forms. Another identity which is heterogenous and whose truth-value is ac-cordingly not settled by principle (a) is Julius Caesar = 'O(Fo).

To fix up this indeterminateness, which would result from taking Axiom Valone as the definition of courses of values, Frege proposes to supplement it by stipulating the truth-values of the heterogenous iden-tities. Actually, he is required to specify the inferential behavior of course of value expressions in all contexts in which they can appear. In Frege's terminology such contexts are functions, so this requirement is equivalent to the demand that it be determined for every single-argu-ment function-expression what value is designated by the substitution of any course of values expression in its argument place. Doing so will then determine all of the properties of the objects designated by expression


of the form for those properties just are concepts, that is, functions whose values are truth-values. Among those properties are individuative properties, the facts corresponding to identity contexts involving course of values expressions. Thus the Grundlagen require-ment that to introduce a new set of objects one must settle all identities involving them is in the Grundgesetze motivated by the omnicontextual condition. (It is worth noticing, as Sluga points out, that there is an endorsement of a strong context principle in Frege's claim that what it is to have introduced expressions of the form as the names of definite objects is for the truth-values of all sentential contexts in which those expressions can be substituted to have been settled.) In fact, in the spare environment of GG it turns out that it is not only necessary to settle the truth-values of all identities involving course of value expres-sions in order to satisfy the omnicontextual requirement, but sufficient as well.

Indeed, in the system of the Grundgesetze at the time courses of values are introduced the only objects already defined are the two truth-values, and so the only heterogenous identities Frege explicitly addresses are those involving a course of values and a truth-value. But he must justify the general procedure of stipulating truth-values for heterogenous identities, and not just his application of it. For if he does not, then the GG definition of number will still be open to the objection to the second GL account of number (that it has not been settled whether Julius Casesar is one) which drove him to define the extensions of concepts and hence courses of values to begin with. Indeed, the concept "logical object" will not have been defined if it has not been settled whether Julius Caesar is one. Further, as we have seen, Frege's own definitions of his technical terms in general suffice only to deter-mine the truth-values of homogeneous identities, for example, identities of two truth-values, or two senses, or two references, but not the heterogeneous identities which would be required to make the claim that Julius Caesar = the Bedeutung of the expression 'Julius Caesar', or that a certain linguistic role is the sense of some expression.

In particular, Frege's substitutional-inferential methodology deter-mines only the nonrelational sense of 'Bedeutung', according to which expressions are sorted into substitutional equivalence classes as having the same Bedeutung. For Frege to add to this determination of homo-geneous identities (both of whose terms are of the form "the Bedeutung of the expression t") the relational sense of reference in which these

286 FREGE'S TECHNICAL CONCEPTS

Bedeutungen are identified with objects suitably related to all and only the members of the nonrelational substitutional equivalence class of expressions is precisely to stipulate the truth-values of the heterogene�

ous identities. The question of whether such a procedure can be justified on Frege's own terms is thus exactly the question of whether the two notions of Bedeutung can be made into "two aspects of one notion" as Dummett claims and Frege is committed to, or whether they are simply conflated without warrant, as Bell claims. Following Sluga's develop-ment of Frege's attempted definition of terms which refer to logical objects thus leads to the argument which must justify the identification of the things playing the two explanatory roles which Bell has shown must be distinguished under the heading "Bedeutung".

In Section 10 of GG Frege offers his justification of the procedure of stipulating the heterogeneous identities, in an argument which Currie has called ''brilliantly imaginative".SJ The argument is a difficult one, and we shall have to examine it with some care. What is to be shown is that it is legitimate to stipulate (a) above, determining the homogeneous identities involving courses of values, together with the following stipu-lation for heterogenous identities:

(b) 'T(LT) = tl and 'a(Ma) = 0.,

where t} 0. and (3x) (Lx Mx). L and M are to be arbitrary functions, and t} and 0. are terms which are not of the form 'a(Pa). For the purposes of the GG argument, the terms in question are "the True" and "the False". In the context of Sluga's point that Frege's defense of his own view against his objection to the second attempted definition of number in GL must be traced through the account of extension to the account of courses of values, it will be worth keeping in mind that for this purpose the argument must apply equally to the case in which t} is "Julius Caesar" and 0. is "England". To emphasize this requirement, the exposition of Frege's argument which follows will use those values for t} and 0. rather than the truth-values which Frege employed. In any case the poinI is that distinct objects which are not given as courses of values are stipulated to be identical to the courses of values a like number of arbitrary distinct functions. The task is to show that such a stipulation is legitimate.

The strategy of the argument is to construct a domain of objects of which (a) and (b) can be proven to hold. To start, suppose it has been stipulated that:


(c) -lJ(FlJ)= -y(Gy)iff('1x) [Fx(=) Gxl,

that is, we stipulate the homogeneous identities for terms of the form - lJ(FlJ), where the function which associates objects so denominated with functions F is unknown except that principle (c) holds. As was pointed out above by means of (a') and (a"), the fact that both (a) and (c) hold does not in any way settle the heterogenous identities one of whose terms is a course of values and the other of which is of the form -lJ(FlJ). The next step is to use the arbitrary distinct functions Land M

of (b) to construct an individuation preserving function X as above. The function Xis defined by five clauses:

(1) X(Julius Caesar) = -lJ(LlJ)

(2) X( - lJ(LlJ» = Julius Caesar (3) X(England) = - y(My) (4) X( - y(My) = England (5) For all other y, X(y) = y.

The function X is constant except when it is applied to either the two objects which are not specified as the result of applying - -abstraction to some function (Julius Caesar and England, or the True and the False) or to the result of applying - -abstraction to the arbitrarily chosen functions L and M. In these special cases, the function X simply permutes the distinguished values.

X is constructed to be individuation preserving, so that a correlation is preserved between distinctness of its arguments and distinctness of its values. It follows then that:

(d) X( - lJ(FlJ» = X( - y( Gy» iff (V x) [Fx (=) Gxl.

In these terms we could now define the course of values notation (which has not previously appeared in this argument) by agreeing to let:

(e) 'a(Fa) =df X( -lJ(FlJ» for all functions F.

Given the definition (e) and the truth of (d), principle (a) for courses of values follows immediately. The truth of (d), as we have seen, follows from (c), together with Clauses (1)-(5) defining the function X But Clauses (2) and (4) of that definition, together with (e), entail principle (b) concerning courses of values (with the substitution of Julius Caesar for t1 and England for Thus given only the homogeneous identities in (c) we have constructed courses of values in such a way that their homogeneous identities in (a) can be shown to hold and in such a way


that heterogeneous identities can be proven for two of them, since 'a(La) = Julius Caesar (= X( - 1J(L1J») and 'DeMo) = England (= X( - y(My»). The legitimacy of stipulating heterogeneous identities in the context of a principle determining homogeneous ones has been shown by reducing the questionable stipulation to the composition of two obviously acceptable forms of stipulation: the specification of the values which the function X is to take for various arguments (in particular in Clauses (2) and (4», and the introduction of the expression "'a(Fa)" (previously without a use) as a notational abbreviation of "X( -1J(F1J»".

This imaginative argument is Frege's ultimate response defending his account of number and of logical objects generally against the objec-tions he had raised but not answered in the Grundlagen. Seen in that context, the argument is fallacious. The problem concerns the extremal Clause (5) of the definition of the individuation preserving function X If that clause is expanded to make explicit what is contained in the condi-tion ''for all other y" it becomes:

(5') (lty) [(y Julius Caesar & y - 'fj(L'fj) & Y England & y - y (My» = > X(y) = yJ.

It may then be asked whether it is appropriate at this point in the argument to make use of a condition such as y - y(My). If the term substituted for 'y' is also represented as the product of applying - -abstraction to some function, then clause (c) will settle the truth-value of the resulting identity. For it settles just such homogeneous identities. But what of the case in which the identity is heterogeneous?

All that has been fixed concerning - -abstraction is principle (c), which says nothing about such identities. Indeed, the whole strategy of the argument depends upon starting from a specification of purely homo-geneous identities with one sorf of abstractor (-) and using the function X to construct an abstractor (') for which the heterogenous identities are specified. Nothing which has been said, or, given the strategy just indicated, could be said, settles a truth-value for hetero-geneous identities such as

(t) Julius Caesar = - y(My)

and

(g) England = - 'fj(L'fj).


For all that principle (c) concerning - -abstraction and the distinctness of the functions L and M settle, (f) could be true and (g) false. Given the truth of (f), substituting in Clause (4) would yield that x(Julius Caesar) = England, and so by clause (1) that England = - rJ(L-fJ), that is, that (g) is true. So the definition of X presupposes valuations for hetero-geneous identities which it is in no way entitled to.

Matters are just as bad if we consider some other object, say the direction of the Earth's axis (also discussed in GL). It has nowhere been determined whether it is identical to - rJ(LrJ) and so falls under Clause (2), or identical to - y(My) and so falls under Clause (4), or to neither and hence falls under Clause (5). The definition of X, in terms of which it is to be shown acceptable to stipUlate heterogeneous identities for - -abstraction, is well-formed only if the heterogeneous identities involving - -abstraction have already been settled. They have not been settled. Further, to add to the argument the assumption that truth-values for such heterogeneous identities involving expressions of the form - o(Fo) have been settled is to assume exactly what the argument as a whole is supposed to show, namely that such matters are open for stipulation in the first place (so long as suitable care is taken to match distinct objects with the result of abstracting distinct functions). If more is supposed about - -abstraction than principle (c) fixing homogeneous identities, the question will be begged. And without some supposition about heterogeneous identities the argument does not go through.

The intent of the offending extremal clause is to deal with all objects which can be represented by expressions of the form - o(Fo), where F

L and F M. Distinct objects not so representable are each to be dealt with by a pair of clauses, letting the function X permute them with the result of abstracting from corresponding arbitrarily chosen distinct functions. There is nothing in general wrong with such a definitional strategy. It may not be used in the context of this argument, however. The distinction between the cases which are to be dealt with by paired specific stipulations and those which remain to be dealt with by the extremal stipulation cannot be made precise without begging the ques-tion. For that distinction corresponds to the distinction between hetero-geneous identities and homogeneous ones, in the sense of stipulations for objects not representable by expressions of the form - o(Fo) and those which are so representable. This distinction is not one which a principle like (c) specifying the purely homogeneous identities permits us to make, and we are entitled to presuppose no more than such a principle.


Put otherwise, the form of definition essentially requires that there be a pair of specific clauses dealing with every object whose individuation with respect to the results of applying - -abstraction to functions has not been settled by principle (c). But this class of objects cannot be described or specified in the terms permitted for the definition if it is to play its appointed role in the larger argument.

The only way in which this situation might be remedied would be if there were some property available which could be independently appealed to in order to distinguish the two kinds of cases. Thus if to (C) were added:

(c') (\fy)[P(y)(=) (3F)(y= -o(Fo»]

then the extremal clause in the definition of X could be amended to

(5") -y(My»(=)X(y)=y]

In the context of (c'), (5") will have the desired effect of applying only to objects which can be designated by expressions of the form - o(Fo),

where F L and F M. More important, (c') would ensure that the identities in (5") are homogeneous with respect to - -abstraction, and hence have had their truth-values settled by (c). It was the failure to ensure the homogeneity that was responsible for the inadequacy of the original definition of X

The trouble with this way out is that no such independently specifi-able property is available. Already in the Grundlagen Frege had pointed out that the account of when the numbers associated with two concepts were identical (settling identities homogeneous with respect to the form: the number of the concept F) could be defended against his objection if the concept " •.. is a number" were available. For then the truth-values of the heterogeneous identities (such as those involving Julius Caesar) could be settled by specifying that for any t, if t is not a number, then it is not identical to the number of any concept. But the problem the desired definition was to solve was precisely that of specifying the concept " ... is a number", as the current task is to specify the concept " ... is a course of values". It would be circular to use for the property P

" ... is an x such that there is an F such that x = - o(Fo)" For that would precisely presuppose that the heterogeneous identities have some-how already been settled, rather than independently settling them. Nor could some property such as " ... is not in the causal order" be used, for there are other logical objects (such as the True and the False) whose


individuation with respect to objects specified by - -abstraction has not been determined. Nor in any case would such a property be available to a logicist.

Frege's argument does not work, then, and it cannot be made to work. If the Grundgesetze is meant to offer an account of number which will meet the demands set by the Grundlagen, then it is a failure by Frege's own standards. Further this failure is not a matter of the incon-sistency of the later system. Although Axiom V is the CUlprit in both cases, it is different features of that principle which are found objection-able in the two cases. The current complaint is that settling the truth-values of the homogeneous identities alone, as that principle does, is definitionally too weak to meet the requirements imposed by the discussion of GL. Those demand the justification of the stipulative extension of the definition to heterogeneous identities. That it leads to inconsistency, on the other hand, shows that that Axiom is inferentially too strong. Putting aside the question of inconsistency which makes the claim counterfactual, even if the account of courses of values in GG were technically adequate, it would not be philosophically adequate as a specification of its objects and concepts. For it has not settled whether Julius Caesar is the number three, nor shown that stipulating an answer in the case of logical objects such as the truth-values is a legitimate procedure. Nor can this be shown with the materials at hand.

I take it that this definitional inadequacy has not been remarked upon for two connected reasons. In the purely technical context of the Grundgesetze the stipulation of the two heterogeneous identities con-cerning the truth-values and arbitrary distinct courses of values is in fact perfectly acceptable. Further, provided that it is stipulated that neither of the truth-values is identical to the result of applying - -abstraction to any function, Frege's argument shows that his procedure is in order. It is only in the larger philosophical context provided by Sluga's historical tracing of the stages in Frege's development of an answer to his own objections to the second attempted definition of number in GL, from the invocation of the extension of a concept in the third and final GL definition, via the reduction of concepts to a special kind of function and of extensions to courses of values in 'Funktion und Begriff, to the {inal attempt to define courses of values adequately in the early sections of the Grundgesetze that it can be seen that satisfying the purely techni-cal constraints will not suffice to render the definition of courses of values (and hence of logical objects generally) adequate by the philoso-phical standards Frege has insisted upon.


But the result is significant not only for the appraisal of the success in its own terms of Frege's account of the logical objects which were his explicit subject matter in GG. For as we have seen, the technical philosophical concepts Frege developed to use in that discussion, such as reference, and sense, and truth-value, are all given the same form of definition as courses of values are, which individuates them only homo-geneously. Thus " ... we cannot say what the sense of an expression is. The closest we may approach to this is to say that the sense of a given expression E1 is the same as the sense of another expression, E

2

."52 It follows that so far as interpretation (rather than further development) of Frege's concept of sense is concerned, one can only subdivide the explanatory roles played by his concept, but cannot identify anything as playing those roles. Thus it is legitimate and valuable to distinguish the cognitive role from the semantic role played by senses, or sense as content from sense as character, or input and output senses as Bell does. But to entertain hypotheses about whether thoughts are mental pictures (as Frege did by denying this) or sets of possible worlds, or denizens of some extra-causal realm is to consider claims which have been given no sense by Frege's purely homogeneous specification of the entities in question. Truth values are similarly immune from heterogeneous iden-tification, from identification in any other form than as the truth-value associated with some expression.

Probably most important is the case of singular term reference. Here Frege tried explicitly to supplement the purely homogeneous sorting into semantic equivalence classes of the reference associated with various expressions (the nonrelational sense of 'Bedeutung') with the stipulation of heterogeneous identities involving the references of expressions and ordinary objects. In accord with his inferential/substitutional method-ology, these stipulations are grounded in the intersubstitutability for all terms t of the term itself and the expression 'the Bedeutung of t'. Bell has shown how much of Frege's conceptual scheme depends upon the assumption that such heterogeneous identities are determined (and hence a relational sense of reference applies) for other parts of speech, given only the determination of the homogeneous identities (settling a non-relational sense of reference) which is all that is available for expres-sions of these other categories. Pursuing further a line of thought Sluga initiated has shown that this assumption is indeed unwarranted, and that even Frege's attempted stipulation of coincidence of relational and nonrelational senses of 'reference' in the case of singular terms has not


been justified by Frege's own standards. Thus extending Sluga's argu-ment permits better understanding of the philosophical status of Frege's technical concepts in general, and in particular of the two sides of the concept of reference which Bell, following Dummett, has so usefully distinguished.

University of Pittsburgh

I See [19].

2 [8]. 3 Dummetl's [10]. 4 [10], pp. xii-xvi.

NOTES

5 The most influential proponents of the view were Grossmann in [16), and Marshall, in in [24] and [25). 6 p. 65 in [14]. 7 [3). Bell is Lecturer in Philosophy at the University of Sheffield. 8 As in the title of Dummett's book, [8). 9 [10), pp. 476-95. 10 In Section 3 of BGS, reprinted in [14], Frege says that the begriffliche Inhalt of two judgements is the same just in case "all inferences which can be drawn from the first judgement when combined with certain other ones can always also be drawn from the second when combined with the same other judgements." 11 [19], pp. 9-46. 12 For instance by Resnik in [28] and [29], and Angelelli in [1]. 13 [3], pp. 139-40. 14 I have my say in [4]. IS [3], p. 42. 16 [10], pp. 47S-9. 17 in [32]. 18 [10], p. 479. 19 I have argued that a purely intralinguistic anaphoric account of such facts can be offered by construing 'refers' and its cognates as complex pronoun-forming operators, in [5]. 20 Kripke in [23], and Putnam in [27]. 21 Most prominently, Perry in [26], and Kaplan in [20] and [21]. 22 [3], p. 112. 23 [3], p. 115. 24 [3], p. 5l. 25 [3], p. 64. 26 [3], p. 65. 27 First in [S], pp. 293-4, then at greater length as chapter 13 of [10]. Citations here from p. 251 of the latter.


28 in [15]. 29 [10],pp.251-2. 30 [19], p. 119. Bell's other citations are in [3], p. 72. 31 [3], pp. 74-8. 32 See [31]. Sluga is Professor of Philosophy at the University of California at Berkeley. 33 Dummett in [7], p. 225, quoted by Sluga in [31], p. 8. 34 In Dummett's defense it should be said that in the final chapter of [8] the main historical significance of Frege's work is taken to be precisely his anti-empiricist and anti-psychologist shifting of concern from the acquisition of concepts to what such mastery consists in - from how the cognitive trick is performed [e.g., by material beings of our sort] to what counts are performing it. The injudicious invocation of a dominant Hegelianism as the psychologistic culprit is explicitly made subsidiary to this central point. 35 [31], p. 59. 36. Dummett, in [9]. 37 Bierich in [2]. 38 [31], pp. 53, 192. 39 [31], pp. 59-60. 40 [31], pp. 44-5, and 106. 41 [31], p. 60. 42 [31], p. 60. pp. 43)[31], p. 91. 44 cf. [31], p. 181. 45 [31], p. 95. 46 [19], p. 253. 47 [31], p. 134 and Note 21 to chapter 4. 48 e.g. in [31],p. 156. 49 Sections 62,63 of [12]. 50 Section 68 of [12]. Equality of concepts in the sense invoked here has been defined as obtaining iff the objects of which the concepts are true can be put into one-to-one correspondance. 51 in [6], p. 69. 52 Bell in [3], p. 55 [emphasis in original]. He-goes on to point out an analogy with the Fregean concept of concept reference: "concept words refer, but we cannot stipulate what it is they refer to". But in this case the reasons are purely substitutional, since expressions like "the concept horse" will never be intersubstitutable with predicative function-expressions. This shows that all heterogeneous identities involving function-expressions on one side and singular terms on the other must be false.

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Phenomenological Research 27,356-365. [29] Resnik, M.: 1967, 'Frege's Context Principle Revisited', included in [30] pp.

35-49. [30] Schirn, M. (ed.): 1967, Studien zu Frege, Vol. III, Frommann-Holzboog, Stuttgart

and Bad Cannstatt. (31) Sluga, H.: 1980, Gottlob Frege, Routledge and Kegan Paul, London. [32] Tugendhat, E.: 1970, 'The Meaning of "Bedeutung" in Frege', Analysis 30,

177-189. [33) Uehling, T., Wettstein, H., and French, P. (eds.): 1978, Contemporary Perspectives

on Philosophy of Language, Midwest Studies in Philosophy, University of Minnesota Press, Minneapolis, Minnesota.

PARTlY

PHILOSOPHY OF MATHEMATICS

PHILIP KITCHER

FREGE, DEDEKIND, AND THE PHILOSOPHY OF

MATHEMA TICS*

1. INTRODUCTION

In the 1880s, two men who had both been trained as rpathematicians wrote short books defending the idea that arithmetic has an intimate relation with logic. Neither work was exactly a commercial or intellectual success. Frege's Grundlagen (Frege 1884) went virtually unnoticed, and Frege recorded his disappointment and frustration in the introduction to his Grundgesetze (Frege, 1893, p. xi; Furth, 1967, p. 8). Dedekind's monograph, Was Sind und was Sollen die Zahlen? (Dedekind, 1888), fared only a little better. Before Dedekind published it, he had been encouraged by the interest of other mathematicians in his project. In 1878, for example, Heinrich Weber urged him not to postpone his planned study on the number concept (Dugac, 1976, p. 273). However, when the book appeared, it made comparatively little stir: certainly, Dedekind's discussion of the natural numbers aroused nothing like the interest excited by his study of the real numbers (Dedekind, 1872). Although many of Dedekind's contemporaries viewed him as an impor�tant mathematician, they did not rank Was Sind und was Sollen die Zahlen ? among his major achievements.1

Today, Frege's Grundlagen is widely appreciated as a philosophical masterpiece. In retrospect, the mathematicians who ignored it appear as men who failed to recognize a pioneering work. Yet, in the intervening century, Frege's study of the foundations of arithmetic has been applied in a number of different philosophical enterprises. In the early decades of the century, for example, Frege's defense of logicism was used to support a program with quite different motivations from his own? Faced with the problem of accounting for the status of arithmetic, a traditional bastion of rationalist epistemology, the logical positivists and their empiricist successors, sought comfort in Frege's claim that arith�metic is disguised logic. More recently, philosophers of mathematics have approached Frege's work not as an arsenal that can be raided in the defense of empiricism but as a treasure trove of philosophical insights about mathematics. Despite the fact that Frege's own solutions

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to the problems he posed are not widely accepted, his conception of the central questions of the philosophy of mathematics is frequently viewed as definitive. If the history of philosophy in general is a series of footnotes to Plato, then the history of the philosophy of mathematics in the last thirty years appears as a series of footnotes to Frege.

Because our current understanding of what the philosophy of mathe-matics ought to be is so dominated by Frege's view of the field, it is important to discuss the way in which Frege's enterprise was born. My aim will be to show how Frege's own philosophical preconceptions led him to identify, and to bequeath to his successors, a misguided picture of the central problems of the philosophy of mathematics. To achieve this, I shall use Dedekind as a foil. Nearly a century after it was published, Dedekind's monograph on the concept of natural number persists in its state of benign neglect. Dedekind appears to us as a lesser Frege, a man who groped toward some Fregean insights but who only saw dimly what Frege saw clearly. I shall attempt to show that Dedekind's work does not deserve to be ignored, that it responds to a different set of philosophical problems than those identified by Frege, and that a bedekindian view of mathematics might provoke new and profitable philosophical inquiries.

2. THE ROUTE TO LOGICISM: FREGE

Frege's logicism was the product of his philosophical reflections on the state of late nineteenth century mathematics, reflections that were by no means the innocent philosophical thoughts of a philosophically untu-tored mathematician. In his earliest writings we can already discern a philosophical view, inherited from Kant. Frege's inaugural dissertation, presented in 1873, is devoted to the problem of giving a geometrical representation of imaginary points and figures. The young mathemati-cian who regaros this problem as an important question for his research is already concerned with the general project of showing that new developments within mathematics are compatible with the epistemo-logical ideals of the discipline. The first sentence provides a clear view of Frege's intentions:

When one considers that the whole of geometry ultimately rests upon axioms, which receive their validity from the nature of our faculty of intuition, then the question of the sense of imaginary figures appears to be well-justified, since we often attribute to such figures properties which tontradict our intuition. (Frege/ Angelelli, 1967, p. 1)3

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These are not the words of a philosophical naif. The linking of geometrical axioms to our faculty of intuition reveals the influence of Kant, and it is the background of Kantian epistemology that gives significance to Frege's mathematical enterprise.

Yet, as every reader of Frege knows, he was to break explicitly with Kant over the status of arithmetic. Grundlagen is devoted to defending a non-Kantian account of arithmetic, and it contains a respectful critique of Kant's own sketchy ideas about the subject. The divergence from Kant's specific proposal for arithmetic was already clear by 1874. In the opening section of the dissertation he presented on his arrival at Jena, Frege writes.

It is thus clear that for so comprehensive and abstract a concept as that of the concept of quantity there can be no intuition. Because of this there is a significant difference between geometry and arithmetic in the way in which their fundamental propositions are grounded. Because the object of arithmetic is not intuitable (keine Anschaulichkeit

hat), the fundamental propositions of the subject cannot stem from intuition. (Fregel

Angelelli, 1967, p. 50)4

Significantly, while rejecting one part of Kantian epistemology, Frege makes explicit his commitment to other parts. Within Kant's epistemo-logy for mathematics we can distinguish two main doctrines. Many prominent sections of the Critique reflect a particular type of apriorism about mathematics, the idea that mathematics is a corpus of a priori knowledge. Kant is committed to what I shall call an apriorist program.

He believes that it is possible to give an epistemological reconstruction of mathematics which will make it clear that the truths asserted by mathematicians can be (and perhaps are) known a priori. This general idea is articulated in the second main doctrine, a specific proposal about the way in which the apriorist program will go. On Kant's view, all mathematical truths can be known by means of inferences from funda-mental propositions which are themselves knowable a priori through pure intuition. Frege rejects this specific proposal, while honoring the more general commitment. Moreover, his dissent about the details is oply partial: throughout his career, Frege continues to espouse Kant's explanation of the a priori knowability of geometry.

Let me make this interpretative claim more precise by defining the notion of an apriorist program. I shall assume that a necessary condition for a person to know that p is for that person's belief that p to have been produced in an appropriate way. This very general assumption about

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what knowledge is has not only been ably defended in the recent epistemological literature; it is also an assumption that both Kant and Frege would have accepted.s With respect to any item of knowledge, there must be a tree-like structure which represents the causal process through which the belief in question was produced. The ultimate causal nodes in this structure are basic beliefs, beliefs which can be arrived at without inference from other beliefs (Kornblith, 1980). To reconstruct a body of knowledge is to reveal the causal networks underlying the statements that are known, showing what items of knowledge are basic, and what chains of inference lead from these pieces of basic knowledge to the rest of the corpus. To reconstruct a body of a priori knowledge one must demonstrate how that knowledge results from statements that are knowable a priori without inference from other statements by means of chains of inference along which apriority is transmitted. Someone who believes that mathematics is a body of a priori knowledge is committed to the possibility of a reconstruction of this kind. Such a person must hold that there is a feasible apriorist program for mathe-matics, a program which is to be carried out by completing the following tasks: (a) identifying a set of basic a priori statements (statements that can be known a priori without inference from other statements); (b) explaining the nature of the processes through which a priori knowledge of the basic a priori statements can be gained; (c) identifying a set of apriority-preserving rules of inference, rules which license inferential transitions that always yield a priori knowledge of the conclusion given a priori knowledge of the premises; (d) showing that all accepted mathe-matical statements can be obtained from basic a priori statements through a sequence of inferences each of which is licensed by an apriority-preserving rule of inference.

Kant's influential contributions to the philosophy of mathematics consist in his general account of what a priori knowledge is and his suggestions about how such knowledge can be obtained. As I have indicated elsewhere, (Kitcher, 1980a), the Kantian idea of a priori knowledge as knowledge which is independent of experience can be developed in a coherent way. Given this development, we are able to define the notions of a basic a priori statement and of an apriority-preserving rule of inference, thus making clear the concept of an apriorist program and identifying the claims to which someone wllO believes in the apriority of mathematical knowledge is committed. But Kant not only offered this very general characterization of a priori

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knowledge. He also proposed that there are two ultimate sources of a priori knowledge; conceptual analysis and pure intuition. Kant's own philosophy of mathematics proceeds from the identification of what an apriorist program is to the task of explaining the nature of the processes - processes of pure intuition - through which he takes basic a priori mathematical knowledge to be obtained.(Thus Kant focusses on task (b) above.) Kant does not try to specify the statements which are basic a priori statements or the rules of inference that preserve apriority. His attitude seems to be that that is the business of the mathematician, perhaps that mathematical proofs are in order as they are.

The similarities and differences between Kant and Frege can now be described more precisely. Like Kant, Frege is committed to apriorism and his work reflects acceptance of the Kantian idea of an apriorist program. Moreover, Frege endorses Kant's conception of the possible sources of a priori knowledge. The most obvious disagreement concerns the way in which areas of mathematics are taken to be related to basic sources of a priori knowledge. Frege contends, of course, that Kant was wrong to hold that arithmetical knowledge is based' ultimately on pure intuition. Instead, Frege believed that our knowledge of arithmetic is to be traced to the source of our knowledge of logical truths. This shift in the specifics of the apriorist program causes Frege to emphasize differ-ent parts of the enterprise. Thinking that our source of a priori knowl-edge of logic can do more than Kant had given it credit for, Frege sees the importance of identifying the basic a priori statements - basic laws of logic - on which arithmetical knowledge depends, of specifying the apriority-preserving rules of inference, and of showing that the state-ments and rules so identified suffice for the reconstruction of arithmetic. Thus tasks (a), (c), and (d) receive a great deal of attention from Frege, while task (b) is ignored. From Frege's perspective, the basic epistemo-logical issue, the question of explaining the ultimate sources of a priori knowledge, had already been settled. But for Frege (unlike Kant) mathematics itself needs reform. The subject will not stand forth as an a priori science until the ordinary proofs of mathematicians are improved.

Frege's inaugural dissertation addresses one of the obvious problems that arise for Kant's favored apriorist program. Late nineteenth century geometry had advanced beyond the discipline whose epistemological foundations Kant had considered. Frege's project is to show how geo-metrical ideas that initially appear beyond the scope of pure geometrical intuition can be accommodated within the Kantian program. But

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Frege's early reflections on arithmetic must have raised for him a more serious difficulty. Having arrived at the judgment that it is impossible to trace arithmetical knowledge to propositions known through pure intuition, Frege set for himself a new apriorist program, one that would require a revision of the usual ways of presenting proofs within arith-metic. Arithmetic must be constructed from the fundamental laws of logic. So Begriffsschrift was born. Having undertaken to provide arith-metic ''with the most secure foundation," Frege formulated the problem as follows:

.. , I first had to ascertain how far one could proceed in arithmetic by means of inferences alone, with the sole support of those laws of thOUght that transcend all particulars. My initial step was to attempt to reduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from there to the concept of number. To prevent anything intuitive (Anschauliches) from penetrating here unnoticed, I had to bend every effort to keep the chain of inferences free of gaps. In attempting to comply with this requirement in the strictest possible way I found the inadequacy of language to be an obstacle; no matter how unwieldy the expressions I was ready to accept, I was less and less able, as the relations became more and more complex, to attain the precision that my purpose required. This deficiency led me to the idea of the present ideography. (Frege/van Heijenoort, 1967, pp. 5-6)

Formalization entered the philosophy of mathematics in the service of Frege's epistemological ideals.

When we reconstruct Frege's route to logicism, one question stands out. Why did he believe that the logical principles he identified were a priori and the rules of inference he set forth apriority-preserving? I shall have more to say about this in what follows, but, for the moment, let me use my comparison of Frege with Kant to offer a quick explanation of why Frege is so silent about fundamental epistemological issues. At the one place in his published writings where Frege confronts the question of ''why and with what right" we acknowledge a law of logic to be true, Frege announces that, as a logician, the issue is none of his business (Frege, 1893, p. xvii; Furth, 1967, p. 15). I suggest that Frege saw his own work as the counterpoise to Kant's. Kant had characterized the notion of a priori knowledge, thereby making clear the idea of an apriorist program. Kant had also identified the sources of a priori knowledge, one that would generate knowledge of analytic truths and one that would yield knowledge of synthetic truths. Kant's philosophy of mathematics was incomplete in its failure to show in detail how the whole of mathematics could be traced to these sources. Where it was in

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error was in its assignment of arithmetic to the wrong source, to pure intuition instead of to the source of a priori knowledge of analytic truths. Frege's enterprise is directed at remedying the incompleteness and correcting the error, but, in this enterprise, certain basic features of the Kantian framework are taken for granted.6

Frege's work is intermediate between the projects of general epis-temologists (like Kant) and those of practicing mathematicians (like Frege's contemporaries who saw no point in what he was doing). There is a pervasive philosophical myth, to which Frege himself contributed, to the effect that his probing of the foundations of arithmetic is an out-growth of the nineteenth century tradition of making analysis rigorous.1 A careful look at the history of research in the foundations of nineteenth century analysis explodes this myth. The work of the great mathemati-cians - men like Cauchy, Weierstrass, and Dedekind - who struggled with the foundations of analysis was motivated by a quite different set of interests from those which provoked Frege's metamorphosis from mathematician to philosopher.

Consider the case of Cauchy. Cauchy was not moved to offer new definitions of 'limit', 'continuity', 'convergence', and 'derivative' because he yearned to show that analysis was genuinely a body of a priori knowledge. His aims were far more pragmatic. Certain important research problems of nineteenth century analysis could not be resolved using the techniques that Cauchy and his contemporaries had inherited. The mix of algebra and geometry constructed by Euler, d'Alembert and Lagrange does not make it possible to determine whether the sum of an infinite series of continuous functions is always continuous. A decision on this issue was needed if the merits of various methods for solving partial differential equations were to be clearly understood. Cauchy's new analysis was born from a desire to tackle the urgent problems of mathematical research, and Cauchy was quite happy to rely on the older ideas of the eighteenth centJry when he thought he could get away with it. When we read Cauchy (or Abel, or Dirichlet) through Fregean spectacles, their texts are extremely puzzling. There are constant lapses from the ideal of giving full algebraic derivations, periodic invocations of geometrical evidence, and references to eighteenth century notions (like that of infinitesimal) which we might have thought Cauchy and his contemporaries were in the business of eliminating. Removing the Fregean spectacles improves our vision. Cauchy et al. developed mathe-matical concepts and methods for the purpose of resolving technical

306 PHILIP KITCHER

questions, and they used those concepts and methods where they found them useful (or even indispensable). They had no general interest in showing the apriority of analysis, or in eschewing old-fashioned tech-niques that might more readily be applied in some contexts.

Ironically, Cauchy'S own reform of analysis showed for the first time how certain old-fashioned techniques, specifically the method of infini�tesimals, might lead to false conclusions. Of course, from the seven-teenth century on, mathematicians had been aware that they did not fully understand why infinitesimalist reasoning works. But no critic -not even Berkeley - suggested that some conclusions generated by such reasoning were false. Although Cauchy's work was rightly admired for its ability to yield recognizably correct answers to certain traditional questions, his employment of some traditionally sanctioned shortcuts generated anomalies. The late nineteenth century rigorization of analysis - whose full flowering is the work of Weierstrass and Dedekind -was a response to mathematical difficulties which Cauchy'S work had unearthed. Because some of Cauchy'S techniques gave demonstrably false results and because Weierstrass needed reliable methods to tackle recherche questions in elliptic function theory , Weierstrass was forced to develop the austere style of analysis which has dominated the subject ever since. Similarly, Dedekind's work on continuity was motivated by the desire to establish the existence of limits in cases where Cauchy had had to appeal to geometric analogies, analogies which were, by 1850, recognizably faulty. The program of rigorization of nineteenth century analysis was kept in motion by the mathematical difficulties left over by the latest achievement.8

Frege arrived on the scene at a time when the search for rigor had temporarily come to a halt and when mathematicians were eager to put to work the tools that they had acquired from Weierstrass. (Frege was to appreciate that this was the attitude of his mathematical contemporaries - see [Frege, 1893, p. xii; Furth, 1967, pp. 9-10; also Frege/Long et

al., 1979, pp. 165-66]- although he continued to attack it with wither-ing scorn.) His campaign for attention to the foundations of arithmetic fell on deaf ears, precisely because it was felt that the available concepts and methods of mathematics were adequate to the important research problems and that little would come of foundational ventures in an area which had yielded no anomalies. If Frege's contemporaries did indeed say of his work "Metaphysica sunt, non leguntur." then they were only remaining true to the motivations of Cauchy and Weierstrass.9


Frege's enterprise thus falls between the very general project of the epistemologist, whose interest in mathematics is one of integrating the subject within a global account of human knowledge, and the activity of mathematicians, whose concern with foundational issues is fueled by their recognition of current inability to solve interesting technical ques�tions. If successful, an enterprise like Frege's could point in either (or both) of two directions: it could show the mathematicians that certain types of inquiry need to be reformulated as well as reveal to philoso�phers that certain epistemological goals mayor may not be attainable. But an enterprise that borrows so heavily from general epistemology is vulnerable to the possibility that, if some of the epistemology that is taken over is misguided then the project will lack the epistemological significance that is attributed to it. As I shall suggest below, despite the rich heritage of Frege's work in logic, his philosophy of mathematics is flawed in precisely this way.

3. THE ROUTE TO LOGICISM: DEDEKIND

Like Frege, Dedekind was led to investigate the foundations of arith�metic for philosophical reasons; however, his motivation was interest�ingly different from Frege's. The reconstruction of arithmetic offered in Was Sind und was Sollen die Zahlen? grew out of Dedekind's earlier work on continuity and irrational numbers. As we shall see, all of Dedekind's foundational work is pervaded by a particular conception of mathematics, which differs from Frege's not only in its characterizations but in the questions that it seeks to resolve.

Yet it is easy to assimilate Dedekind to the Fregean program, to see him as a pygmy working beside a giant. When we read Dedekind after reading Frege, we find cryptic phrases that remind us of Frege's goals. Consider Dedekind's account of how he was led to his investigation of continuity. He reports his early experience of trying to teach differential calculus:

In discussing the notion of the approach of a variable magnitUde to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such recourse to geometric intuition in a first presentation of the calculus, I regard as extremely useful from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. (Dedekind/Beman, 1901, p. 1)

308 PHILIP KITCHER

Here we might discern the Fregean project of carrying out an apriorist program that would correct the Kantian mistake of tracing our knowl-edge of arithmetic and analysis to pure intuition. However, I think that Dedekind has something different in mind. Although geometrical dia-grams may sometimes help us to see ·that results about the continuity of functions or the convergence of sequences are true (indeed they may be the best means of convincing the beginner that these results are true) they have two deficiencies. First, the transition between the algebraic notion of continuity and the geometrical notion of continuity is not as simple as analysts like Cauchy had believed; the uncritical use of that transition (and others like it) can generate mistakes. Second, even if the transition were to produce correct results, it would be a detour. We venture into geometry because we lack any algebraic way of formulating that continuity of the real numbers which is so easily represented geometrically, and because, as a result of this lack, we are unable to establish on a purely algebraic basis theorems about the existence of limits. 10

Why should we adopt this construal of Dedekind's intent rather than that which assimilates him to the Fregean enterprise? My answer is drawn from the structure of the argument advanced in Dedekind's memoir on continuity. Dedekind sets himself the task of providing an algebraic characterization of the continuity of the real numbers, and showing that this characterization suffices for the derivation of standard theorems about the existence of limits. Far from proposing that the principle of continuity which he introduces is evident a priori, Dedekind recommends it by explaining how it can be used to demonstrate results about limits which had baffled his predecessors. As I interpret him, there is a body of prior mathematical work which stands in need of algebraic systematization - systematization which is intended not to provide us with a priori knowledge but to improve our mathematical understanding and our ability to solve mathematical problems - and his proposal to define continuity should earn our acceptance because of its ability to answer to that need. (For reasons to be given below, I suggest that this style of argumentation is an important pattern through which mathematical knowledge develops, and that our understanding of mathe-matical knowledge will be advanced by taking such patterns seriously.)

Nevertheless, even though Dedekind's work on continuity is free from commitment to the kind of apriorist views which motivate Frege, it does contain philosophical views about mathematics, ideas which, as we

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shall see, motivate his subsequent studies of arithematic. Quite early in his memoir, Dedekind adopts what appears to be a primitive construc-tivist approach to what mathematics is about: the human mind "creates" the numbers, and thereby fashions "an exceedingly useful instrument" for itself. This constructivist attitude erupts ih the substantive mathe-matics. As Russell pointed out, Dedekind does not identify the real numbers with sets of rationals. Rather, he "creates" real numbers to correspond to certain partitions of the rationals (Dedekind cuts), thereby provoking Russell's gibe that he has obtained "all the advan-tages of theft over honest toil." However, this taunt is unfair to Dedekind's intentions. Dedekind does not interpret mathematics as describing some pre-existent realm of abstract objects (sets) among which the real numbers are to be found. Rather, he suggests that the numbers are products of acts of construction. His project is to charac-terize the way in which the real numbers are constructed, by giving a new description of a process which mathematicians have carried out since ancient times, but which has hitherto been specified only geometri-cally, in a fashion that does not allow for algebraic derivation of results in analysis. The real worries that we should have about Dedekind's enterprise concern the coherence of his constructivist picture and the limitations that may need to be placed on the mathematician's power to create. Can Dedekind allow for the objective content of mathematical statements? Can he meet the complaint that arithmetical statements were true before humans engaged in acts of mathematical "creation"? Are our creative powers sufficient to generate the mathematics that was already beginning to emerge in the late nineteenth century and that has flowered in our own times? These are serious and important questions, and I shall indicate below how I think they can be answered. My present purpose is simply to note that it was a precise version of the problem about the potential limits on human mathematical "creation" that led Dedekind from the study of real numbers to the foundations of arith-metic.

Was Sind und was Sollen die Zahlen? contains a picture of the construction of the natural numbers which is at odds with the account Dedekind had previously offered. Earlier, he had assumed that there is a primitive iterative act through which the mind generates numbers (Dedekind/Beman, 1901, p. 4). In his later work, however, arithmetical construction proceeds via the characterization of the notion of an w-sequence, and arithmetic is regarded as unfolding the properties of

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w-sequences. Dedekind's new approach enables him to solve two problems, one of which is philosophical, the other mathematical. The philosophical problem is to explain the centrality of mathematics to our thinking. Arithmetic is to stand revealed as a discipline which studies fundamental operations of the mind, operations without which "no thinking is possible" (Dedekind/Beman, 1901, p. 32; also Dedekind/van Heijenoort, 1967, p. 100). The mathematical problem is to turn back the challenge that the alleged constructions of the early memoir on continuity are beyond the constructive power of finite minds.

Let us consider the mathematical problem first. Given Dedekind's constructivist picture of mathematics, the characterization of the real numbers offered in the memoir on continuity presupposes the possi-bility of constructing arbitrary collections of rational numbers. One influential mathematician, who shared the view of the construction of the natural numbers indicated in Dedekind's early memoir, denied the legitimacy of mathematical references to arbitrary infinite totalities, on the gounds that such totalities are beyond the constructive powers of finite beings. Kronecher's critique cut equally at the accounts of real number suggested by Weierstrass, Cantor, and nedekind, but it had special force against Dedekind because of his predilection for a con-structivist view. Two significant footnotes to Was Sind und was Sollen

die Zahlen? show that Dedekind was aware of Kronecker's arguments, and concerned to resist their conclusion. In effect, the new construction of the natural numbers carried out in that essay provides a reply to Kronecker. If the construction of infinite sets is itself presupposed in the development of the natural numbers, then there can be no basis for allowing the construction of the natural numbers and balking at the construction of infinite sets in the development of the reals. Dedekind's response imitates a Kantian strategy for dealing with certain forms of skepticism: a challenge to the legitimacy of a notion is turned back by showing that conceptions used in formulating the challenge presuppose the very notion whose legitimacy is in question.

Dedekind's response rests on his ability to cope with a deeper and more general issue, how we determine the limits of our mathematical constructive powers, Kronecker, like the intuitionists after him, believes that he can prescribe limits to the ability of the mind to engage in mathematical construction. There is a strong temptation to believe that because our constructions are ours we must be able to discover a priori how they proceed and where they are limited. But Dedekind does not

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THE PHILOSOPHY OF MA THEMA TICS 311

subscribe to this apriorist epistemology. He divorces the constructivist ontological thesis, the thesis that mathematics is concerned with human constructions, from its typical accompaniment, the constructivist epis-temological thesis that declares that the scope and limits of mathemati-cal construction can be settled a priori. Along with the epistemological view that I discerned in the memoir on continuity, Dedekind believes that we discover our capacities for mental construction by reflecting on and systematizing the mathematics which has already been developed. In a letter in which he attempts to correct a misunderstanding of his views, Dedekind remarks that his essay on natural numbers is "a syn-thesis, constructed after protracted labor, based upon a prior analysis of the sequence of natural numbers just as it presents itself, in experience, so to speak, for our consideration" (Dedekind/van Heijenoort, 1967, p. 99). The Preface to Was Sind und was Sollen die Zahlen? also reveals Dedekind's rejection of the idea that our constructions are transparent to us.

But I feel conscious that many a reader will scarcely recognise in the shadowy forms which I bring before him his numbers which all his life long have accompanied him as faithful and familiar friends; he will be frightened by the long series of simple inferences corresponding to our step-by-step understanding, by the matter-of-fact dissection of the chains of reasoning on which the laws of numbers depend, and will become impatient at being compelled to follow out proofs for truths which to his supposed inner conscious-ness seem at once evident and certain. (DedekindlBeman, 1901, p. 33)

The message is clear. The deliverances of supposed inner consciousness are illusory. We must learn the nature and limits of our constructions by analysis of the mathematics which we have already come to know.

Thus I regard Dedekind as having developed, at least in outline, a strategy for responding to the charge that his constructivism cannot accommodate the whole of mathematics, and I see him as implementing this strategy in response to Kronecker's attack on the Dedekindian construction of the reals. However, Dedekind's probing of the founda-tions of arithmetic was not simply a response to mathematical problems. He is also concerned with the philosophical problem of understanding the centrality of mathematics to our thinking. In his early essay on continuity, Dedekind remarks that the construction of numbers forms "an extremely useful instrument for the human mind" (Dedekind/ Beman, 1901, p. 4), and I suggest that his later study is designed to explain why this is so. As we shall see, this explanatory project, like Frege's enterprise, is related to Kantian themes. Although the most

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obvious doctrine about mathematics in the Critique is Kant's claim that mathematics is synthetic a priori, Kant also endeavors to explain (in the Aesthetic) why mathematical truths are necessary and (in the Axioms of Intuition) why all experiences must be describable in quantitative terms,u Dedekind's investigations into the foundations of arithmetic pursue these philosophical projects. While he rejects Kant's claim that the truths of arithmetic are necessary in virtue of properties of the forms of intuition, Dedekind seeks to explain why any experience of which we can conceive must be one to w:hich the arithmetical concepts are applicable and one of which the principles of arithmetic must be true. Dedekind's version of the logicist thesis - expressed in his claim that the number concept is "an immediate result from the laws of thought," is his attempt to provide the explanation.

Thus far, four possible motivations for studying the foundations of mathematics have been developed. When these motivations are ex-amined, the contrasting ways in which Frege and Dedekind were led to their investigations becomes clear.

(1) to refine mathematical methods and concepts, so as to meet the research needs of mathematics;

(2) to show how we obtain a priori knowledge (or how we could

obtain a priori knowledge) of the parts of mathematics under study;

(3) to show why the mathematical statements whose credentials are investigated must be true in any world of which we can conceive, and why the mathematical concepts figuring in those statements must be applicable to any such world;

(4) to defeat skeptical challenges which charge that part of "mathe-matics" transgresses the boundaries of proper mathematics.

The program of making analysis rigorous, a program exemplified in the work of Cauchy, Weierstrass, and the Dedekind of the memoir of continuity, is driven by (1). Frege's investigations are prompted by the Kantian project (2), although, as we saw, Frege emphasizes different aspects of this project from those which had occupied Kant. Finally, Dedekind's later work in the foundations of mathematics is motivated by (3) and (4).


4. VARIETIES OF APRIORISM

Frege and Dedekind thus came to investigate the foundations of arith�metic and to advocate logicism by different routes. In itself, that fact would be of little philosophical significance. Yet, understanding the different aetiologies will help us achieve a clearer conception of their philosophical work, and even to arrive at a new perspective on the philosophy of mathematics. A necessary step in making use of the history will be to disentangle various conceptions of the a priori, con�ceptions that descend from Kant and which live on in the ideas of Frege, Dedekind, and their successors.

The basic Kantian concept of the a priori, the concept which Kant introduces at the beginning of the Critique, construes 'a priori' as a predicate of items of knowledge. To declare that someone knows a pri�ori that p is to say that that person knows that p in a way that is independent of experience. I explicate this idea as follows. Items of a priori knowledge are items of knowledge produced by special types of processes (a priori warrants). An a priori warrant for a proposition p is a type of process such that, given any experience which would be sufficient to enable a person to entertain the proposition that p, some process of that type would be available to the person, would warrant belief that p, and would produce true belief that p. (These conditions are discussed in my [1980a]. For a defense ofthe claim that they capture the traditional idea that a priori knowledge is knowledge that is independent of experience, see Chapter 5 of my [1983a].) In what follows, I shall suppose that a priori knowledge is knowledge that meets the conditions just given, and I shall identify one version of apriorism about mathe�matics as maintaining the following thesis.

(5) Mathematical knowledge is a priori knowledge. For each mathe�matical proposition, there is a type of process which would be available given any sufficiently rich experience, and which, given any sufficiently rich experience, would produce warranted true belief in the proposition.

(5) is a thesis to which both Frege and Kant subscribe, a thesis which is to be articulated by carrying out an apriorist program such as that undertaken by Frege.

But there are other apriorist traditions. When Kant, and his succes�sors, talk of a priori propositions or a priori concepts, it is not always

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appropriate to understand them as claiming that the truth of those propositions or the applicability of those concepts can be known a priori in the sense already given. Sometimes, what seems to be intended is that the concepts or propositions in question are prerequisites for

thought (or experience) in the sense that, without use of those concepts or acceptance of those propositions, thought (or experience) would not be possible. The following analyses capture intentions of this kind:

(6) (a) C is a conceptual precondition for thought if and only if necessarily, for any rational being S, if S thinks (forms beliefs) then Shas some belief, part of whose content is the concept C. (b) P is a propositional precondition for thought if and only if necessarily, for any rational being S, if S thinks (forms beliefs) then S must believe that p.

The point of explicitly recognizing these notions is twofold: it makes it possible for us to express clearly what has often been presented under the rubric of an undifferentiated concept of apriority, and it enables us to show that there is no obvious connection between the propositions that meet (6b) and those which are a priori knowable.

Consider, for example, the traditional claim that the law of non-contradiction is an a priori proposition. Often this doctrine is defended by pointing out that, if the principle is abandoned, the possibility of rational thought and discourse vanishes. That defense has no bearing on the issue of whether the principle is a priori knowable, for it does nothing to establish the existence of any process through which belief in the proposition could be warranted independently of experience. My suggestion is that the traditional claim is indeed one to which the traditional defense is directly relevant: its champions lavish their atten-tion on showU'g that, were we to abandon the principle of non-contra-diction we would not be able to have any beliefs, precisely because they take the principle to be a priori in the sense of being a propositional precondition of thought.12

The example just noted brings into the open some tricky issues which emerge when the analyses given in (6) are put to work. How exactly are we to understand the notion of a person's having certain beliefs? Quite evidently, the mere disposition to assent to a sentence expressing the content whose ascription is in question will be neither necessary nor sufficient. We can easily conceive of situations in which we would maintain that someone who claimed to have abandoned the principle of


non-contradiction continued to subscribe to it. In such cases we deny that the person believes the negation of the principle, despite the presence of a disposition to assent to sentences expressing that negation. Other cases may pose problems because the person in whose beliefs we are interested does not have the conceptual repertoire to express those beliefs explicitly. Nonetheless, we may legitimately claim that certain "tacit beliefs" guide our subject's reasoning, decision and action.

Because of difficulties of these kinds, further analysis is needed before the explications begun in (6) can stand forth as characterizations of a notion of apriority. However, without pursuing the analysis further, I think that we can see that the notion of a priori truth generated from (6) is not likely to be equivalent to that of a proposition which is a priori knowable. For consider the following points. First, there is no guarantee that we have any way of knowing propositions which are propositional preconditions for thought. Certainly, there is no obvious reason why these propositions should be knowable in a way which would survive any experience. If there are propositional preconditions of thought, then it is eminently possible that these should be identified by reflection on and analysis of our thought and discourse (as, for example, Dedekind suggests that he was led to identify the "laws of thought" on which arith-metic rests), and an unkind experience should override our right to suppose that our analysis had led us to correct conclusions. Second, unless the notion of belief is construed in such a way that any person automatically counts as believing the logical consequences of her beliefs (in which case, of course, the doctrine that logical truths are propositional preconditions of thought approaches the status of a truism), it seems unlikely that the set of propositional preconditions of thought will be closed under logical consequence. For, intuitively, if p is a propositional precondition of thought, there will be recherche logical consequence of p which are not propositions that a person needs to believe if she is to think at all. By contrast, 9n most traditional concep-tions of a priori knowledge, the class of propositions which are a priori knowable will be taken to be closed under logical consequence. If the propositions in a set r can be known a priori, and if p is a logical consequence of r then there will be a formal derivation of p from r, and, it will be claimed, someone should come to know a priori that p by following whatever processes lead to a priori knowledge of the members of r and then rehearsing the formal derivation of p. (This argument makes some assumptions about logical consequence, derivability, and

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the epistemological status of long proofs that might be called into ques�tion. Nevertheless, I do not think that refinement of these assumptions would help someone who hoped to show that the set of propositional preconditions of thought is identical to the set of a priori knowable propositions.) So the set of propositional preconditions of thought cannot be the same as the set of a priori knowable propositions, because the latter is closed under logical consequence and the former is notY

Given that the embryonic analyses offered in (6) indicate a concep�tion of apriority which is different from that provided in my reconstruc�tion of Kant's idea of a priori knowledge, we can consider the possibility of a different apriorist thesis about mathematics.

(7) Some mathematical concepts are conceptual preconditions for thought and some mathematical truths are propositional precon�ditions for thought. Among these propositional preconditions for thought are the truths of mathematics that genuinely deserve to be called fundamental.

A doctrine very close to that expressed in (7) can be discerned in Kant's discussion of the Axioms of Intuition, and this theme is elaborated in Dedekind's work. Dedekind believes that in some sense all subjects of belief must employ certain mathematical concepts and must believe some mathematical truths. He would deny that subjects must be able to give an explicit formulation of the concepts or propositions in question, so that (7) is compatible with a certain type of explicit mathematical ignorance. Emphasizing that the basic truths of mathematics may not be immediately accessible to consciousness, Dedekind would contend that belief in these propositions nevertheless guides any subject in his representation of the world.

To articulate fully the type of apriorism about mathematics that Dedekind wishes to defend, we must identify a third strand in Kant's thinking about the a priori. Kant introduces a special notion of neces�sity, by focusing on propositions which are true in any world of which we can have experience. Believing that all and only necessary proposi�tions are a priori knowable, he frequently casts doctrines about the I'\ecessity of a proposition as claims that that proposition is a priori.14

The assimilation is made easier by Kant's distinctive view about the way in which the necessity of synthetic propositions is to be explained. Kant believes that there are synthetic necessary truths because, in represent�ing the world, the mind imposes certain of its features upon the data, so

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that any world of experience must bear these features. Thus there arises the idea of the a priori as mind-imposed. To say that a proposition is a priori may be to claim that it is true in any world of which we can have experience, and that its being true in any such world is the result of the structure which the mind imposes on experience.

I suggest that this third conception of apriority also lives on in Dedekind's philosophy of mathematics. Dedekind believes that all thinkers (or, at least, all human thinkers - I shall consider the possible difference below) must have certain properties, and that, in virtue of these properties, arithmetical concepts are inevitably applicable to our experience and arithmetical propositions are true in any world of which we can have experience (or, indeed, any world about which we can think). In other words, in virtue of our status as thinkers our experience is inevitably stamped with arithmetical structure. If the book of nature is written in the language of mathematics it is because we have framed the terms in which that book is to be written.

Integrating this third apriorist theme with the previous idea of a priori propositions as preconditions of thought, we can finally arrive at the apriorist position that Dedekind wishes to defend, and thereby recognize the differences between his views and those of Frege. Because of the nature of thought (or our thought) human experience is neces-sarily experience to which mathematical concepts are applicable and of which mathematical propositions are true. Even though we may not have explicitly identified these concepts and explicitly given our assent to these propositions, we nevertheless tacitly believe and use them, in that they are always employed in our representation of the world. The explicit formulation of the concepts and the explicit knowledge of the propositions is not gained through some a priori procedure. If we come to explicit mathematical knowledge - and it is not necessary that we should do so - then our route is an empirical one.

There is a partial analogy with a position that someone might take concerning our grammatical beliefs. Let us suppose that, following Chomsky and others, language learning is a process in which we use innate knowledge of principles of universal grammar and sensory stimulations to arrive at tacit knowledge of the grammar of a natural language, knowledge which is then put to work in communicating with others. The structure of universal grammar is inevitably to be found in any language that we are capable of learning. We have tacit knowledge of the principles describing that structure. But if we become lingUIsts

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and eventually obtain explicit knowledge of the principles of universal grammar, then our explicit knowledge is thoroughly empirical.

Armed with some understanding of the varieties of apriorism, we can now begin to see why, although Frege and Dedekind both advance an apriorist thesis in claiming that there is an important relationship between arithmetic and logic, their positions are quite different. Frege wants to show that arithmetical knowledge is a priori by uncovering a route from basic principles of logic, conceived as basic a priori state-ments, through apriority-preserving inferences, to the theorems of arithmetic. Dedekind's apriorism is focused on the possibility of show-ing why our representations are inevitably arithmetically structured by exposing arithmetical structure as a consequence of the applicability of the fundamental operations of thinking. As far as explicit mathematical knowledge is concerned, Dedekind is prepared to allow that that knowl-edge is thoroughly empirical, gained by reflection on our experience and on the body of mathematical knowledge that we have inherited.

I can now state my main thesis more precisely. It is important to distinguish between the foundational studies of Frege and Dedekind because, while it has dominated the philosophy of mathematics for a century, the Fregean apriorist enterprise is hopeless. On the other hand, the approach to mathematical knowledge which is encapsulated in Dedekind's brief comments on the subject and, even more, in his prac-tice of mathematics, seems to me to be correct. Moreover, Dedekind's apriorism raises new questions for the philosophy of mathematics, questions which I believe to be well worth investigating. The remainder of this essay will be devoted to developing and defending this thesis. Let us begin by returning to Frege.

5. WHAT'S WRONG WITH FREGE'S PHILOSOPHY OF MATHEMATICS?

There is a very obvious difficulty with Frege's philosophy of mathe-matics. Frege's system of logic turned out to be inconsistent. Hence, quite trivially, Frege's Grundgesetze did not reveal a route to a priori knowledge of the truths of arithmetic. But this is usually not thought to be important. After all, we now have systems (such as ZF) that avoid the paradoxes on which Frege's own theory foundered and that enable us to preserve many features of the Fregean derivation of arithmetic. So it seems that all is well. In the standard set-theoretic reduction of arith-

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metic (analysis and so forth) we have a vindication of what might be called neo-Fregeanism.

Yet, if the neo-Fregean answer to the question of "the foundations of mathematics" is to inherit the significance that Frege attributed to resolving that question, then it must be true that the basic principles from which the derivation of arithmetic begins are basic a priori statements and the rules of inference employed in the derivations are apriority-preserving. If these conditions are not satisfied then the neo-Fregean approach does not do what Frege thought that a foundation for arithmetic should do, and we ought to ask about the significance of the derivations with which neo-Fregeans provide us. What is the point of deriving arithmetic within set theory, if not to provide us with a priori knowledge of arithmetic?

Neo-Fregeanism is born by reflecting on certain aspects of Frege's program. It takes seriously the Fregean theme that numbers are objects, continues by suggesting that the objects in question cannot be spatio-temporal objects, interprets Frege's Grundgesetze as containing, in part, a faulty theory about the properties of these objects, and concludes that some set theory (such as ZF) provides a correct theory. I have brought to the fore a different Fregean idea, one which, as I have argued above, was crucial to his transition from mathematics to philosophy. That idea provokes the question of whether the axioms of the set theory within which the neo-Fregean derives arithmetic (analysis, and so forth) are basic a priori statements. I think that the answer to this question is "No." To recapitulate briefly an argument I have given in more detail else-where (1983a, Chapter 3), there is no reason to believe that the processes through which we know the axioms of set theory (say, for the sake of definiteness, the axioms of ZF) provide us with a priori knowl-edge of those axioms. One might think that the processes which produce our belief that the axioms of ZF are true are somewhat elusive, that, at least in the case of great mathematicians, they might be amenable to study, and that, by studying them, we might discover that they meet the conditions that are demanded of a priori warrants. But such easy optimism is shortsighted. One of the conditions that must be met by an a priori warrant is that processes of that type must have the power to warrant belief in the proposition in question against the background of any sufficiently rich experience, no matter how recalcitrant that experi-ence may be. Nebulous processes of set-theoretic intuition, processes which we think might go on in us at those moments when we feel that

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''the axioms of set theory force themselves upon us", are incapable of meeting this condition, precisely because of their elusive character. We know, from Frege's unfortunate experience, that what seems for all the world like a self-evident truth may be shown to be false, and this knowledge is the entering wedge from which our belief in the axioms of ZF can be undermined. Given an experience in which some axiom of ZF is called into question - perhaps in an indirect way through the testimony and sophistry of someone whom we have every reason to think of as vastly more knowledgeable than ourselves, perhaps by means of more direct chicanery - we would be unwarranted in proceeding as we normally do to form the belief that the axiom is true. For we know that we are fallible, we know that we (and great figures of the past, like Frege, Dedekind, and Cantor) have been inclined to attribute self-evidence to contradictory propositions, we know that we know so little about the process through which our belief in the axiom is formed that we would be unable to distinguish it from the kind of process which we conjecture to have gone on in the minds of our great, but misguided, predecessors. So, just as Frege would have been dogmatic in using some supposed deliverance of inner intuition to override Russell's famous letter, we too would be unwarranted in turning back a sophisticated (but, I assume, sophistical) challenge to axioms of current set theory. The inaccessibility of the processes through which, on the neo-Fregean account, our knowledge of the axioms of set theory is gained is itself responsible for the inability of those alleged processes to warrant belief given suitably recalcitrant experiences.

Thus neo-Fregeanism will not solve what Frege took to be the most fundamental problem for the philosophy of mathematics, the problem of showing that mathematical knowledge is a priori. Nor will Frege's tum be served by adopting a different approach which might also claim credit from his writings. Inspired by the Fregean claim that arithmetic is analytic (a claim which, for Frege, amounts to the derivability of arithmetic from logic), one might try to assert the apriority of arithmetic by maintaining that arithmetic is analytic and that analytic truths are knowable a priori. Quine's well-known critique of analyticity has some-what dampened the ardor of proponents of this version of apriorism, but, since it continues to linger on, I think it is worth explaining why I think that the appeal to analyticity also fails to live up to Frege's demands. Suppose that we take seriously the idea of an individual mathematician coming to know the truths of arithmetic through some


process of grasping the meanings of arithmetical terms.15 Once again, we must ask whether the process in question could warrant belief against the background of any experience, however recalcitrant. A fundamental Quinean insight indicates, I think, that the answer is "No." For us to use our understanding of our language (grasp of concepts, or whatever) to yield warranted belief, we must have no reason for thinking that our languages (concepts) are ill-suited to the description and explanation of the world and its workings. Reliance on items of language which have been shown to be poor vehicles for the investigation of nature is as unreasonable as the refusal to countenance violations of one's favored generalizations. Hence, given experiences which suggested to us that certain of our mathematical concepts were misguided (either because of inconsistency or because of fruitlessness), we could not warrant belief in mathematical propositions by appealing to our grasp of those concepts. Even if the notion of analyticity can be rescued from Quine's reiterated challenges to give a clear account of the conditions under which a statement counts as analytic, one should deny that analytic truths are a priori knowable. (For a more detailed elaboration of this argument, see my 1981a and Chapter 4 of my 1983a.)

Frege was motivated to defend logicism because of his conviction that the Kantian apriorist program breaks down in the case of arith-metic. If I am right, then neither of the two programs which can trace their pedigree to Frege's original enterprise will satisfy his apriorist intentions. This means that the type of mathematical apriorism common to Kant and Frege is bankrupt. There is no reason to think that we can give a coherent account which will show mathematical knowledge to be priori. I draw the conclusion that mathematical knowledge is not a priori.

This leaves us with a question that I posed above. What is the value of the usual set theoretic reductions of arithmetic and analysis if they do not give us a priori knowledge of the theorems that are derived? One suggestion, involving a relatively modest departure from the Fregean program, would be to claim that the derivation exposes the route through which we have knowledge of the truths of arithmetic, analysis, and so forth. The axioms of ZF are knowable (not a priori knowable but knowable nonetheless) without inference from other statements, and we gain knowledge of arithmetic by following a chain of inferences from these items of basic knowledge. But, once we have abandoned the Frege-Kant apriorism, this modest departure is unmotivated. Our reason

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for thinking that certain principles of set theory might be knowable without inference from other statements lay in the idea that arithmetic was a priori knowable and that an account of its a priori knowability would have to derive from something epistemologically more funda-mental. Given Frege's critique of Kant's apriorist program and the vicissitudes of his own approach, the axioms of ZF emerged from the historical process as the most likely candidate for the more fundamental something. There is no independent source of the view that we have some direct knowledge of the ZFaxioms. Rather, the facts of history point the other way. Our knowledge of arithmetic flourished long before our knowledge of set theory. Moreover, the principles of set theory introduced by Zermelo, were explicitly formulated to systematize a prior body of mathematical knowledge while avoiding the difficulties on which the systems of Frege, Dedekind and Cantor had foundered. By what token do we believe that this historical introduction does not provide the source of our knowledge of set theory? It is worth listening here to the voice of Dedekind in his memoir on continuity, the authentic voice of the mathematician, concerned with a mathematical problem of foundations: the new account of continuity of the real numbers is defended on the basis of its ability to systematize the previous results of real analysis.

Hence the collapse of Frege's apriorism ought to prompt us to a more radical departure from Frege's epistemology. For Frege, Kant, and a host of other philosophers of mathematics, past and present, a reconstruction of mathematical knowledge is to treat the knowledge of an individual independently of her contemporaries and the history of mathematics. One is to look for the basic items of knowledge which the mathematician has and the chains of inference through which the rest of her knowledge is generated. If it is pointed out that, for most of us, mathematical knowledge begins with the testimony of teachers and textbooks and is extended by applying rules which have been explicitly taught, then that observation is not usually taken as epistemologically relevant. For, it is claimed, individuals have ways of knowing mathe-matical truths independently of their interactions with authorities. I believe that this claim offers a misguided view of mathematical knowl-edge. We should not only give up the idea that there is a possible reconstruction of our mathematical knowledge as a priori knowledge, but also the claim that the mathematical knowledge of the individual can be explained in ways that are independent of the community of knowers, past and present. Each of us comes to mathematical knowl-


edge by relying on our predecessors and contemporaries. A few of us extend that knowledge by advancing truths on the basis of processes which accord with the principles of the methodology of mathematics. The major problems of mathematical knowledge are to explain how the chain of mathematical knowers got started and to understand the ways in which mathematical knowledge is extended and transformed. There is no Archimedean - Fregean - point from which we can derive our mathematical knowledge, without reliance on the past and present practices of mathematicians.16

The difficulty of finding a via media between the apriorism of Frege and Kant and the epistemological position that I have just recommended is revealed in Frege's own rejection of the idea that mathematical knowledge is empirical. It is clear from Frege's critique of Mill that he does not regard himself as showing how we can obtain a priori knowl-edge of a discipline (arithmetic) which we can also know empirically. He denies that our arithmetical knowledge could be based on experience. The denial has a number of sources. First, Frege's conception of the methods of .empirical science is a very narrow one. His rejection of empiricism about arithmetic is cast under the rubric of opposing the idea that arithmetical laws are "inductive truths." But there is a second, and I think more significant, presupposition of his critique. The empiri-cist is envisaged as claiming that the mathematical knowledge of each individual is built up by the individual on the basis of his childhood perceptual experiences. Any philosopher who is sensitive to the com-plexity of nineteenth or twentieth century mathematics ought to find that empiricist picture unpersuasive. If empiricism about mathematical knowledge is to be made plausible, it must attribute to perceptual experience only an initiating role, one of generating very elementary mathematical knowledge, which is extended and transformed through the history of mathematics. Because Frege underrates the possibility of a significant methodology of mathematics, a set of methods for extending mathematical knowledge which have been put to work in the historical development of the subject, he overrates the importance of the role which perception will play in an empiricist account of mathematical knowledge. Emphasizing that perceptual experience cannot deliver what he takes to be demanded from it, Frege regards empiricism about mathematics as a non-starter.

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of the subject. If we make this radical break, we shall not be able to defend the view that the usual set-theoretic reductions of arithmetic and analysis are significant because they expose the sources of our knowl-edge (not a priori knowledge, but knowledge nonetheless) of arithmetic. Instead, these reductions will be important for the same reasons that reductions of scientific theories are sometimes important. The incor-poration of arithmetic and analysis within set theory is explanatory: it increases our understanding of results that we previously knew. We accept the axioms of set theory precisely because they enable us to systematize a prior corpus of knowledge. The point of exhibiting the derivations is not to provide a warrant for the conclusions, but to justify us in adopting the premises from which those derivations begin. One of the central problems of the epistemology of mathematics is to under-stand the conditions under which this type of adoption is warranted.17

To summarize: neo-Fregeanism is deeply flawed, because it cannot satisfy the Fregean demand to exhibit mathematics as a body of a priori knowledge; moreover, when this shortcoming is revealed we should change our views about the character of the epistemology of mathe-matics, modifying our ideas about the central problems of the field, and we should re-evaluate the significance of the "set-theoretic foundations" of mathematics. I want to conclude this section by making a historical point. Frege himself was no neo-Fregean. Although he had the oppor-tunity to endorse the idea of a set-theoretic foundation for arithmetic, he did not do so. Why not?

One clue to Frege's attitude towards the idea of giving a set-theoretic foundation for arithmetic can be discovered in his explicit comparison of his own work with that of Dedekind:

Herr Dedekind, like myself, is of the opinion that the theory of numbers is a part of logic; but his work hardly contributes to its confirmation, because the expressions "system" and "a thing belongs to a thing" are not usual in logic and are not reduced to acknowledged logical notions. (Frege, 1893, p. viii; Furth, 1967, p. 4)

This passage is puzzling because of the fact that a reformer in logic, like Frege, hardly seems entitled to criticize another on the grounds that the bounds of traditional logic have been overstepped. Quite evidently, Frege believes that there is a collection of tasks which the discipline of logic ought to undertake, that these tasks are not carried out adequately by classical logic, that his own logical system provides a more adequate approach to them, and that Dedekind's work does not really address


them. We can come to understand Frege's refusal to adopt a set-theoretic foundation for arithmetic by exposing the conception of the goal of logic that underlies his dismissal of Dedekind's work.

'System' is Dedekind's term for sets, so that, in Dedekind's essay, Frege had the opportunity to inspect an outline of the mathematical work of a neo-Fregean program. Frege supports his negative assessment of Dedekind's study by making two different points. The more straight-forward of these, which is presented in a passage following that which I have quoted, is that Dedekind's proofs do not meet Frege's criterion of gaplessness. The other, which underlies the complaint that Dedekind has not used acknowledged logical notions, is that Dedekind's premises are not candidates for basic a priori statements. Frege believes that Dedekind has not contributed to showing the apriority of mathematics because his proofs do not begin with basic a priori statements and do not proceed by apriority-preserving inferences.

One of these complaints could easily be met. If a neo-Fregean were to produce a derivation of arithmetic within formal ZF, there would be no Fregean worry that the chains of inference did not satisfy the criterion of gaplessness on which Frege insists. However, the problem that the axioms of ZF would not meet Frege's standards for basic a priori statements would remain.

Although he does not have a detailed epistemology which would explain how some principles can be known directly a priori, Frege seems confident that laws of logic are a priori, and that he can identify some such laws which are knowable without inference from other statements. The source of one of his complaints about Dedekind is the idea that, if our first principles draw only on "acknowledged logical notions" - that is, if they are principles which, given Frege's conception of the province of logic, belong to a proper formulation of the discipline - then we can be confident of their apriority. If they do not do so, then their a priori status is controversial, because it is unclear that they can be traced to either of the sources of a priori knowledge, viz. the source of logical knowledge and pure intuition. The general judgment that logic is a priori knowable stems, as I have indicated above, from Frege's endorsement of the general features of Kant's epistemology. Yet Frege's critique of Dedekind requires a much more specific claim. Frege believes that he has himself correctly identified the basic principles of logic, and that Dedekind has not done so. By what right does he believe this?

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As I suggested above, Frege's confidence rests ultimately on his conception of the goal of logic. Again and again, in his published and unpublished writings, logic is characterized as a normative discipline, which prescribes to us the way in which we are to think if we are to attain the truth or even if we are to think at all. For example, in contrasting the laws of logic with laws of psychology, Frege writes as follows:

The (laws of. logic) have a special title to the name "laws of thought" only if we mean to assert that they are the most general laws, which prescibe universally the way in which one ought to think if one is to think at all. (Frege, 1893, p. xv; Furth, 1967, p. 12)

In an unpublished essay, (written in 1897), Frege is even more explicit:

Like ethics, logic can also be called a normative science. How must I think in order to reach the goal, truth? We expect logic to give us the answer to this question, but we do not demand of it that it should go into what is peculiar to each branch of knowledge and its subject-matter. We must assume that the rules for our thinking and for our holding something to be true are prescribed by the laws of truth. Consequently we can also say: logic is the science of the most general laws of truth. (Frege/Long et al., 1979, p. 128)

These passages, and many others, indicate the criterion which I con-jecture that Frege used in satisfying himself that he had identified the basic laws of logic, laws which are genuine basic a priori statements. Like Kant, Frege used the notion of apriority without disentangling the quite separate ideas that I have distinguished in Section 4. In the quest for principles which would be a priori knowable, he was led to search for the basic laws of logic. Conceiving of these basic laws of logic as laws that prescribe to all thinking, he asked himself whether the prin-ciples that he had chosen could be denied without our losing the capacity for coherent thought, or whether, "they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace" (Frege, 1893, p. xvi; Furth, 1967, p. 13). I suggest that, putting his logical laws to this test, Frege became convinced that they had the status of basic prescriptions to all thought. From this he concluded that they were properly to be counted as laws of logic, and holding that the basic laws of logic are directly knowable a priori, he therefore regarded himself as having found an appropriate basis for the reconstruction of arithmetic.

This justification for the status of his principles is faulty, not only because it led Frege to endorse his unfortunate Basic Law V,18 but


because it conflates the notion of an a priori knowable proposition with that of a propositional precondition of thought. However, if this flawed justification is correctly attributed to Frege then we can begin to understand his response to Dedekind, his rejection of set-theoretic foundations for arithmetic, and his despair at the collapse of Grundge�setze. Dedekind's foundation for arithmetic, and the neo-Fregean approach of basing arithmetic on set theory, construe our arithmetical knowledge as being knowledge of the properties of particular abstract objects, sets. Frege has no reason to think that we can have any a priori knowledge of the properties of such objects. Given his rejection of the possibility of tracing our arithmetical knowledge to pure intuition, he supposes that a priori arithmetical knowledge must be based on knowl-edge of basic logical laws. But explicit axioms of set theory will not meet his criterion for basic logical laws. Why should there be a prescription to our thought that we countenance the existence of a universe of sets? By contrast, in Frege's own formulation, the commitment to the objects that mathematics demands is accomplished by means of what, in his inno-cent youth, Frege saw as a basic principle of correct thought: we are always able to express the equivalence of functions as an identity, an identity between courses of values, and this license is encapsulated in the unfortunate Basic Law V. (See [Sluga, 1980jpp. 108-111; [Resnik 1980] pp. 206-208 for useful discussion.) With the collapse of this law, Frege saw no way to introduce objects for mathematics and simultane-ously to satisfy his criterion for identifying basic a priori statements, and this, I believe, led him, in the years after 1903, to reject the available neo-Fregean ways of patching up his program. Thus, Frege disdained neo-Fregeanismnot because he saw that the axioms of set theory did not count as basic a priori statements, but because he accepted a conflation of two notions of apriority, due to Kant, and, on this basis, denied the a priori knowability of set-theoretic axioms.

6. DEDEKIND REVISITED

Frege dismissed Dedekind's work because the Dedekindian study of natural numbers did not advance his own apriorist program. But, as I admitted in Section 3, there are genuine worries about the coherence of Dedekind's constructivist picture. I now want to argue that those worries can be allayed, and that we can find in Dedekind's monograph important insights and suggestions for the philosophy of mathematics. I

328 PHILIP KITCHER

shall begin by offering an interpretation of the opening pages of Was

Sind und was Sollen die Zahlen?, in which Dedekind sets out his philosophical position.

The monograph starts with a discussion of the foundations of mathe-matics, in which, from a Fregean perspective, a number of quite distinct themes are muddled together.

In speaking of arithmetic [algebra, analysis] as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought. My answer to the problems propounded in the title of this paper is, then, briefly this: numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. It is only through the purely logical process of building up the science of numbers and by thus acquiring the continuous number-domain that we are prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind. If we scru-tinize closely what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. Upon this unique and therefore absolutely indispensable foundation, ... , must, in my judgment the whole science of numbers be established. (DedekindlBeman, 1901, pp. 31-32)

The apparent confusions of this passage result from Dedekind's linking of three ideas: the idea that logic studies fundamental operations of the mind, the idea that we create the numbers, and the idea that our creation of the numbers serves as a useful instrument for the investiga-tion of nature.

Let us begin where Dedekind does, in opposition to Kant's claim that arithmetic is linked to the forms of intuition. I interpret Dedekind as rejecting this specific Kantian claim while accepting the more general idea that arithmetic describes a formal structure that we impose upon experience. On Kant's view, the truths of arithmetic and geometry would have to be true of any world of which we had sensory experience. Dedekind's conception is more abstract. The truths of arithmetic have to be true of any world of which we can think This is because, in thinking, we inevitably perform certain mental operations, and the truths of arith-metic owe their truth to the performance of these operations. Thus the link between constructivism and the utility of arithmetic is forged by contending that the operations whose performance makes the truths of arithmetic true are so fundamental that they must be performed when-ever we think at all about any aspect of any experience. Dedekind's


logicism stems from his conception of logic as the science that studies those operations of our thought without which no thinking is possible. At first glance, this conception of logic not only appears remote from that which we derive from Frege but also seems to invoke the kind of psychologism against which Frege campaigned. But I think that Dede-kind is innocent of the charge that he has wrongfully conflated laws of logic with laws of psychology. It is quite possible to read him as maintaining that there are laws which prescribe to us the way in which we have to think if we are to think at all, and that these laws are included among the basic laws oflogic. Construed in this way, Dedekind is a proponent of that conception of logic which appears in Frege's writings, and, even though his account of logic may differ from our own, Dedekind's logicism should not be dismissed as the product of muddled psychologism.

Nevertheless there are still grounds for skepticism about the con-structivist picture on which Dedekind's account of arithmetic depends. The most naive reading takes him as supposing that, in performing the mental operations he identifies as fundamental, each of us brings into being mathematical objects - sets, functions, and numbers. Mathemati-cal entities are the products of mental operations. But, given this interpretation, there are straightforward reasons to doubt the objectivity of mathematics and the truth of mathematical statements that appear to be true. Newton and Leibniz, and you and I, perform numerically distinct operations, so that we all produce different numbers. Should we then say that the meaning of a simple arithmetical identity varies according to the subject whose mental operations we consider? Should we allow that such identities are true prior to the time at which humans first came to engage in the relevant mental operations? Are there true arithmetical statements which assert the existence of very large num-bers, larger than those which any human has ever "created," larger than those which, given our finite cognitive abilities, any human is ever able to "create?" These are serious questions, (questions that Frege himself raised about constructivist philosophies of mathematics), and Dedekind's constructivism is doomed unless we can find adequate answers to them.

My first suggestion is that we give up the misleading and problematic idea that our mental operations bring into being certain objects as their products. Instead I propose that mathematical statements be construed as describing the properties of the operations themselves. So, for example, instead of saying that there is a set (or system) whose members

330 PHILIP KITCHER

have a particular common property, a set which is regarded as brought into being through our performance of a mental operation of collecting the things with the property in question, I suggest that we simply claim that there is an operation of collecting those things. As I have shown in some detail elsewhere (1983a, Chapter 6), it is possible to make sense of the idea of iterated collective operations and thus to give an account of the ontology of set theory and of arithmetic which does not presup-pose the existence of products of our constructive opefations. Adopting this approach solves one of the problems raised in the last paragraph: we are no longer faced with the threat of a variable content of mathe-matical statements, because we no longer construe those statements as true in virtue of the properties of the products of constructive activity, products which are different for different individuals. Instead, mathe-matics is taken to describe the general properties of constructive operations, operations which may be performed by any subject. The assertion that 2 is prime does not commit us to the claim that some definite product of somebody's construction (Newton's? Leibniz'? Yours? Mine?) has a particular property. Rather, it is implicitly general, claiming that all operations of a particular kind (two-operations) have a particular property.

So far, nothing has been done to alleviate the difficulties posed by the facts that arithmetic was true prior to the time at which humans first engaged in constructive activity (and would be true in a world like ours in which humans never evolved) and that there are truths of mathe-matics (perhaps including truths of arithmetic) which, on the construc-tivist picture, seem to call for operations that lie beyond the limits of our capacities. I propose to address these problems by thinking of arithmetic (and mathematics generally) as an idealizing theory. It is quite correct to note that there are some conceivable constructive operations which no human ever carries out. The claim that such operations exist is literally false. But we should not think of arithmetic (mathematics) as describing the actual operations that humans actually perform. Rather, arithmetic is the description of the constructive operations of a nonexistent being, an ideal subject, who is recognizably an idealization Of ourselves. We understand our own finite performances by representing this being and telling the story of its performances in just the same way that we understand the behavior of actual gases by introducing the notion of a perfect gas and devising the theory of perfect gases. The existence claims that mathematics requires are to be construed not as


asserting the existence of an actual human constructive operation but as maintaining that an appropriate operation would belong to the per�formance of an ideal being who is recognizably an idealization of ourselves.

Consider now the problem of worlds in which no intelligent life evolves or times in our own world prior to the evolution of intelligent life. In the spirit of the response to the difficulty of superhuman construction, we should not suppose that the truth of arithmetical statements in such situations is to depend on the actual performance of constructive operations. Rather the constructivist claim should be that any description of or theorizing about such situations must represent them through the performance of constructive operations which are properly idealized in the same way that arithmetic idealizes our actual operations. Suppose that we envisage a world in which no intelligent beings evolve and, in consequence, no constructive operations are ever performed. The constructivist claims that the truths of arithmetic are nonetheless true at this world, and articulates the claim by contending that, were we to envisage any adequate description of or theorizing about the world in question, that representation would inevitably require the performance of mental operations whose idealized descrip�tion would be given by arithmetic. From the constructivist perspective, arithmetic is true of the envisaged world because arithmetic is part of the best theory about that world. At this point, we can begin to see how to develop Dedekind's theme that arithmetic is central to our thinking.

So far, I have been attempting to show how the constructivist picture introduced at the beginning of Dedekind's monograph can be rescued from some obvious threats. I now want to use my interpretation to understand the project that Dedekind undertakes and show how the constructive operations of which arithmetic provides an idealized description are related to the fundamental operations of thinking. Those fundamental operations are the operation of collecting ("relating things to things," or forming objects into "systems") and the operation of pairwise ordering ("letting a thing correspond to a thing"). The former operation is introduced at the beginning of the monograph:

It very frequently happens that different things, a, b, c, ... for some reason can be considered from a common point of view, can be associated in the mind, and we say that they form a system 5.19 (Dedekind/Beman, 1901, p. 45)

Thus the first fundamental operation is that of associating objects of

332 PHILIP KITCHER

thought, considering them from a common point of view, or, more simply, that of collecting them. Dedekind explores what he takes to be the nature of our collective powers for several pages before introducing the second fundamental operation:

By a transformation (Abbildung) ¢ of a system S we understand a law according to which to every determinate element s of S there belongs a determinate thing which is called the transform of s . ... (Dedekind/Beman, 1901, p. 50)

The operation which corresponds to this second crucial notion is that of correlating or pairwise ordering. From the interplay between the notions of collecting and correlating Dedekind proposes to characterize the constructions which underlie arithmetic.

In terms of standard set theory, the derivation of arithmetic runs as follows. Given the notions of set and function, we can develop the notion of a one-one correspondence, and we can define infinite sets as those sets which can be set in one-one correspondence with one of their proper subsets. Dedekind then shows how to characterize the notion of anw-sequence (in his terms, the notion of a "simply infinite system"). Having argued that there are infinite sets and that every infinite set contains a subset which is an w-sequence (with respect to a suitably chosen relation), Dedekind proceeds to the "creation" of the natural numbers:

If in the consideration of a simply infinite system N set in order by a transformation ¢ we entirely neglect the special character of the elements; simply retaining their distin-guishability and taking into account only the relations to one another in which they are placed by the order-setting transformation ¢, then are these elements called natural

numbers or ordinal numbers or simply numbers, and the base-element 1 is called the base-number of the number-series N. With reference to this freeing the elements from every other content (abstraction) we are justified in calling numbers a free creation of the human mind. The relation or laws which are derived entirely from the conditions a, p, y, 0 in (71) and therefore are always the same in all ordered simply infinite systems whatever names may happen to be given to the individual elements ... form the first object ofthe science of numbers or arithmetic. (DedekindlBeman, 1901, p. 68)

To make sense of this passage, in which Dedekind does not define the numerals but claims to describe the creation of the numbers, we need to reinterpret his entire derivation according to the constructivist picture developed earlier. Think of the entire monograph as a description of the performance of an ideal subject. The subject has two fundamental powers, that of collecting and that of correlating. Dedekind shows how to define the notion of a correlating operation as being one of fashioning


a one-one correspondence, and how we can thus distinguish between finite and infinite operations of collecting. The next step is to charac-terize a particular type of constructive operation, which we would identify as that of setting things in an w-sequence. The theorem that every infinite set contains a simply infinite subset, will now be reinter-preted as the result that, with respect to any operation of infinite collect-ing, we identify an operation of setting things in an w-sequence. Similarly, Dedekind's theorem that there are infinite sets translates into the theorem that the ideal subject performs infinite collectings. On this basis, we identify the subject as performing operations of setting things in w-sequences, and, abstracting from the objects upon which such operations are performed, we suppose that arithmetic is the science which sets forth the common properties of these operations.

What exactly does this show? Dedekind's emphasis that he intends to trace arithmetic to fundamental operations of thought, operations with-out which no thinking is possible, can be interpreted as trying to show that, given any world about which we can think, arithmetical concepts must apply to that world and arithmetical theorems must be true of it. The derivation of arithmetic just outlined is intended as the explanation. How does it work?

Consider first Dedekind's acceptance of the general Kantian idea that worlds of which we can form a conception must ipso facto exemplify that structure which the mind imposes. That does not mean, of course, that any being experiencing such a world will explicitly formulate to itself true principles describing the structure that its mind imposes. However, imagine a dedicated theorist, a being whose investigations of her world were sufficiently advanced to expose to herself this mentally imposed structure. Then the claim of the inevitable applicability of arithmetical concepts and the inevitable truth of arithmetical theorems can be formulated as the thesis that arithmetic would necessarily belong to the theory of any such theorist no matter what world she was attempting to fathom. The thesis is justified by claiming first that any representation of the world must involve the performance of collective and correlative operations. Without grouping and pairing, no thought about the world will be possible. Hence the theorist must perform at least some operations of collecting and correlating, and an adequate theory of her world must set forth the principles governing the per-formance of these operations. As a result, any such theory must contain the constructivist version of minimal set theory (some of which is developed at the beginning of Dedekind's monograph).

334 PHILIP KITCHER

So far we have an elaboration of Dedekind's basic idea that the operations of collecting and correlating are fundamental to all thought. But now we must ask if any world which we can think about is a world to which the full resources of arithmetic must be applicable. When we reflect on our own experience, we recognize ourselves as performing a vast array of collective operations, and, because there seems to be no determinate bound on the number of objects that we can collect, we describe and explain our own collecting by hypothesizing an ideal extension of ourselves who engages in infinite collecting. The question that Dedekind must face is whether this idealization is always necessary. Might there be a world so impoverished that an adequate theory about it called for no such idealization, a world that did not require the full resources of arithmetic for its description?

As I interpret him, Dedekind thinks that he can eliminate this apparent possibility. Whatever the contents of the world external to the theorizing subject, that subject must find in herself a domain of objects to which arithmetic is inevitably applicable. To put the point concisely, if there are to be objects of thought then it must be appropriate to describe them using a theory which hypothesizes infinite collecting.2o

With respect to any such infinite collecting there will be operations of setting objects in w-sequences, so that, in any world which serves as an object of our thought (or experience) the laws of arithmetic must apply.

Like most efforts at transcendental argument, this line of reasoning is slippery. My interest here is not to assess the argument, but merely to suggest that it is the reasoning that underlies Dedekind's treatment of the natural numbers. That interpretative claim rests on its ability to make sense of a number of puzzling features of Dedekind's monograph. First, it shows how the substantive mathematics elaborates the project set out in the opening sentences of the Preface, quoted at the beginning of this section. Second, it allows for an appreciation of the continued emphasis on the creation of the numbers, a creation which is to serve important practical purposes. Finally, it enables us to see why, in his efforts to show that there are infinite systems, Dedekind adduces the totality of thoughts.

Dedekind tries to carry out an apriorist enterprise which is different from that pursued by Frege. Was Sind und was Sol/en die Zahlen? is supposed to describe the sequence of operations performed by a theorist, reflecting on a world which is susceptible to his reflections, irrespective of the specific features that are found in that world. The


significance of this project is to lie in its explanation of the impossibility of a world that we can represent to ourselves, an adequate theory of which does not include arithmetic.

My interpretation of Dedekind is ambiguous between two readings. For we can either conceive of the explanandum as claiming that any world of which humans can conceive is an arithmetical world, or as the stronger claim that this impossibilty of conceiving of an unarithmetical world is something that holds for all rational beings. Construed in the former way, Dedekind would only be committed to the idea that any being with the kinds of cognitive capacities distinctive of Homo sapiens must perform the operations he regards as fundamental, if that being is to think at all. The stronger version of his view is that thought itself, the thought of any rational being whatsoever, requires the performance of these operations in such a way that any representation of a world will represent a world an adequate theory of which must contain arithmetic. Quite evidently, these latter claims have considerable affinity with the grand theses of Kant's Analytic, although Dedekind approaches them from a very different point of view.

Dedekind's monograph, as I have construed it, is a response to a commonplace phenomenon. When we try to conceive of an unarith-metical world, a world in which the principles of arithmetic are violated, or a world to which the concepts of arithmetic do not apply, we inevitably fail. If there are objects of thought - even if they are only space-time points - then they can be collected and counted, and the operations of collecting and counting will, it seems, always accord with the familiar laws of arithmetic.21 One response to this might be to dismiss it as a failure of our imagination, something from which we (or our descendants) may some day be liberated when the course o(science shows us how to expand our conceptual horizons. The alternative is to view the impossibility as genuine, and to attempt to trace it to its source. That alternative is pursued by Dedekind, and from it develops his distinctive form of apriorism.

Frege was also aware of the phenomenon, and there is a remarkable expression of it in the Grundlagen. After noting that we are able to conceive of (but not render intuitable) worlds in which the laws of Euclidean geometry do not hold, Frege continues as follows:

The fact that this is possible shows that the axioms of geometry are independent of one another and of the primitive laws of logic, and consequently are synthetic. Can the same be said of the fundamental propositions of the science of number? Here, we have only

336 PHILIP KITCHER

to try denying anyone of them, and complete confusion ensues. Even to think at all seems no longer possible. The basis of arithmetic lies deeper, it seems, than that of any of the empirical sciences, and even that of geometry. The truths of arithmetic govern all that is numerable. This is the widest domain of all; for to it belongs not only the actual, not only the intuitable, but everything thinkable. Should not the laws of number, then, be connected very intimately with the laws of thought? (Frege, 1884, p. 21)

Here, Frege gives a lucid exposition of the thesis that Dedekind labored to explain. Yet, because Frege assimilated that thesis to the different issue of the a priori knowability of arithmetic, he never elaborated the explanation that is indicated in the passage just quoted. For him, the unimaginability of an unarithmetical world signalled the unthinkability of such a world, and that unthinkability was, in its turn, an indicator of the a priori knowability of arithmetic (a status grounded in the a priori knowability of logic). In Dedekind, by contrast, we find another version of apriorism. Holding no brief for the idea that arithmetic is a priori knowable, Dedekind suggests that we do need to understand the impos-sibility of thinking of an unarithmetical world. Was Sind und Was Sollen

die Zahlen? is his attempt to explain the phenomenon. I suggest that it is not a poor man's version of the Grundlagen but an interesting attack on a problem to which Frege gives only passing attention.

7. AFTER FREGE

Many philosophers have drawn from Frege's work the ideal of provid-ing mathematics with a foundation, and they have, tacitly or explicitly, linked this to some apriorist doctrine. But the notions traditionally run together under the heading of the a priori have received little attention, and, insofar as there has been any divergence from the conflation of apriorist claims traced in Frege, it has been connected with the rise of an epistemological picture that has made the notion of apriority even more obscure. As I remarked in Section 2, Frege and Kant share a basic view about human knowledge: for a state of belief to count as an item of knowledge, that state must be produced in the right way. Ironically, Frege's own work, specifically his attack on a bundle of psychologistic doctrines, inspired early twentieth century epistemology to abandon the Frege-Kant insight. Epistemology became apsychologistic, and it was assumed that one could assess the episternic credentials of a true belief by identifying the logical relations between the content of that belief and other propositions. This was especially unfortunate for treatment of the


epistemological issues in mathematics. All too frequently in works in the philosophy of mathematics it was taken for granted that various propo-sitions are a priori, and that, no matter how a person may come to believe one of these propositions, that person knows the proposition in question a priori. Not only is this epistemological view mistaken,22 it has also prevented many philosophers from identifying clearly what is inadequate about the claim that mathematics is a priori knowable. There are no processes that will meet the requirements of a priori warrants for the statements hailed as the starting points for mathe-matical knowledge, so that the type of program to which the apriorist is committed cannot be successfully completed.

Thus I think that much of the work that has been done in "laying the foundations" of mathematics thrives on the idea that the enterprise is significant because something is being shown about the a priori status of mathematics. But, when we adopt an adequate epistemological per-spective, it becomes clear that the simplest version of mathematical apriorism - the doctrine that mathematical knowledge is a priori -cannot be sustained, and the character of mathematical knowledge is importantly different from what it has traditionally been taken to be. If my diagonsis in Section 5 is correct, ithen do minor tinkhing will mend the problems with Fregean ideas about the epistemological structure of mathematics. Even if we abandon the claim that the derivation of mathematics within ZF exposes a route to a priori knowledge, there is little plausibility to the idea that it reveals the way in which we do in fact know the truths of arithmetic, analysis, and other parts of classical mathematics. Instead, I suggest that we take seriously the idea that mathematical knowledge has grown through the history of the subject: that it began with very simple items of elementary practical knowledge, acquired by our remote ancestors, and it has been extended and trans-formed according to principles which it is the task of the philosopher to identify. So conceived, the epistemology of mathematics will be treated as part of the epistemology of science. (For a first attempt to study the epistemology of mathematics in this way, see Chapters 7-10 of my 1983a.)

One model for this view of mathematical knowledge is Dedekind. Although his remarks about the nature of mathematical knowledge are scanty, Dedekind's own mathematical works provide a clear example of the ways in which mathematical knowledge is extended and trans-formed. Both in the memoir on continuity and in his study of the natural

338 PHILIP KITCHER

numbers, Dedekind introduces principles (the principle which charac�terizes the continuity of the reals, the "Peano" postulates) which are legitimately accepted because they enable us to systematize items of prior mathematical knowledge. Instead of thinking (with Kant and Frege) that mathematical axioms are known directly and then wonder�ing how we know them directly, we should regard our knowledge of the axioms as based upon our knowledge of some of the theorems that we shall subsequently derive from them. Our epistemological question should be "Under what conditions is the introduction of axioms justi�fied?" Of course, this is only one of the questions that arises within the epistemology of mathematics once we have abandoned the epistemo�logical framework bequeathed to us by Kant and Frege. The general project is to understand the ways in which elementary mathematical knowledge begins, and how, from that elementary knowledge, sophisti�cated mathematics grows. The latter part of that project will focus on a multitude of ways through which mathematics can justifiably be extended, only one of which is to be found in the examples I have taken from Dedekind's mathematical work.

Yet there may seem to be something important that is missed in this refocusing of the epistemology of mathematics. If we had a complete account of the origins and growth of mathematical knowledge which showed us how very simple mathematical truths can be known on the basis of perception, and how, from these items of primitive mathe�matical knowledge, it is possible to develop knowledge of sophisticated mathematics in a way that corresponds (roughly) to the historical development of the discipline, one may still feel that something very important about mathematics has not been explained. For nothing in my envisaged story of the development of mathematical knowledge would offer us insight into the centrality of mathematics in our thinking. The phenomenon which Frege mentions in passing in the Grundlagen,

namely the apparent inescapability of mathematics in our representation of experience, deserves an explanation. (The simplest, and crudest, empiricist explanation is, of course, to explain it away as mere appear�ance, a failure of imagination from which we may someday be liberated.) My comparison of Frege and Dedekind is meant to show that this is a different apriorist theme from that of the a priori knowability of mathematics, and it can be treated without commitment to the kind of epistemology which underlies Frege's work.


Irrespective of the merits of the transcendental argument which I have tried to identify in Was Sind und was Sollen die Zahlen?, it is worth pursuing questions like those which I interpret Dedekind to have been addressing. Are there mathematical beliefs which are tacitly used in all our representations? Are there mathematical concepts which are conceptual preconditions of human thought? Or conceptual precondi-tions of all thought? These are significant questions for the philosophy of mathematics, questions which should be separated from the question of explaining our mathematical knowledge. When we give up our view of mathematical knowledge as a priori, we may begin to speculate that adequate answers to them may depend on general results gathered by workers in psychology and artifical intelligence. If the legacy of the epistemological investigations of Frege and Kant belongs to the phi-losopher of science, then Dedekind's problem belongs perhaps to the philosophy of psychology.

So much for a speculative overview of the questions that a post-Fregean philosophy of mathematics might address. In conclusion, I want to place my critique of Frege in context. I have been arguing that Frege's enterprise is misguided, and I have suggested that, when we recognize its shortcomings we might modify our conception of the philosophy of mathematics and its problems. Nobody who makes such claims should wish to deny the magnitude of Frege's logical accomplishments or the utility of the concepts he introduced for posing philosophical questions (including questions within the philosophy of mathematics). Frege's contributions to logic were designed to help him in completing a project that failed, and which he recognized as having failed. But they are enormously valuable for other enterprises, both philosophical and mathematical.

Hempel concludes his critique of Carnap's philosophy of science (Hempel, 1963) by re-telling a well-known parable. On his death bed, an old wine-grower instructs his sons to "dig for a treasure hidden in the family vineyard. In untiring search, his sons turn over the soil and thus stimulate the growth of the vines: the rich harvest they reap proves to be the true and only treasure in the vineyard" (Hempel, 1963, p. 707). If the parable applies to Carnap, then, I claim, it applies even more dramatically to Frege.

University of Minnesota

340 PHILIP KITCHER

NOTES

* I am grateful to Michael Resnik for helpful comments. 1 Of course, there were exceptions. Both Hilbert and Peano had praise for Dedekind's monograph. Nevertheless, for most of his contemporaries, Dedekind's primary achieve-ments lay in his contributions to algebra and his treatment of continuity.

This is perhaps the place to correct a historical inaccuracy that occurs in Hans Sluga's interesting study of Frege (Sluga, 1980). Sluga suggests that Frege's "fate" was like that of "two other philosopher mathematicians, Cantor and Dedekind," and he characterizes Dedekind's ''fate'' as that of being "buried alive in an insignificant post at the Technical College at Braunschweig" (ibid. 8). It is quite wrong to suggest that Dedekind's work was ignored by his fellow mathematicians, and that he had no opportunity to leave the Brunswick Polytechnic. Unlike Frege, Dedekind was often warmly praised by other mathematicians, and he received several offers to teach at German universities. It seems that he refused these offers because he enjoyed excellent facilities for research at Brunswick and because he did not wish to leave his family home (see Dugac, 1976). Thus, while it is true that Dedekind's study of the natural numbers was not influential, other parts of his work won the applause of his peers. 2 A number of writers have argued that Frege's aims and interests are antithetical to logical positivism. For a comparison of Frege's view about meaning and those of the positivists, see (Dummett, 1973). The antiempiricist character of Frege's ideas about mathematics is described in (Sluga, 1980), in (Benacerraf, 1981) and in my own (1979). 3 This translation, like others from Frege's Kleine Schriften, is my own. 4 It is noteworthy that, in this passage and throughout the dissertation, Frege adopts the view of his contemporaries that arithmetic is the science of quantity. Thus, at this stage, his use of 'arithmetic' covers the arithmetic of negative, real, and complex numbers, as well as the arithmetic of natural numbers. 5 For defenses of the assumption, see (Harman, 1973), (Goldman, 1979) and (Korn-blith, 1980). I have argued that the assumption applies equally to cases of a priori knowledge (1980a), and I have tried to show that both Frege and Kant accepted it (1979, 1975a). 6 This interpretation of Frege's intentions is defended in my (1979). For further discus-sion of Frege's opposition to empiricist accounts of mathematical knowledge, see below pp. 323-325. For his reasons for adopting the principles he hails as basic laws, see pp. 325-327. 7 For Frege's version see his (1884) p. 1. Frege's account of the history is endorsed by several recent commentators: see, for example, (Dummett, 1973, p. xvi) and (Coffa, 1982). One writer who recognizes the difference between Frege's aims and those of the nineteenth century analysts is Gregory Currie (1982, pp. 10-11). 8 These two parargraphs summarize an account of the history of nineteenth century analysis that is defended in some detail in Chapter 10 of my (1983a). 9 I should note that Frege's concerns with rigor in analysis do not seem to be the hilttorical starting point of his investigations. His early research appears occupied with different developments in nineteenth century mathematics than those of the Cauchy-Weierstrass tradition. As a mathematician, Frege's interests seem to lie with geometry and algebra, rather than with analysis. 10 This second source of dissatisfaction recapitulates Bolzano's complaint, made explicitly in his (1817), to the effect that appeals to geometry in analysis run counter to


the correct order of dependency among the sciences. Bolzano freely admits (as Dedekind hints) that the geometrical argument may be the best way of convincing somebody that the theorem is true. He insists, however, that it doesn't show us why the theorem is true. So long as we have only the geometrical argument, our understanding of the theorem is deficient. Interestingly, until it became clear that geometrical analogies are untrustworthy - i.e. until the first source of dissatisfaction was added to the second - mathematicians did not take Bolzano's complaints seriously. For a reconstruction of Bolzano's philosophical position and an account of its influence on his mathematics, see my (1975b). II As I have tried to show in my (1981a), the latter enterprise forms part of Kant's project of showing that mathematical knowledge is a priori. Unless Kant could argue for the inevitability of the use of quantitative concepts in describing experience, he would be vulnerable to the charge that experiences not describable in quantitative terms would undermine the ability of processes of pure intuition to generate mathematical knowl-edge. (I have attempted to reconstruct Kant's account of pure intuition in my [1975aJ.) In discussing Dedekind's enterprise, I shall be concerned only with the claim that quantitative description is inevitable, a claim which (while it may be necessary for the thesis that mathematical knowledge is a priori) can be detached from commitment to the a priori knowability of mathematics. 12 The idea of a priori propositions as propositional preconditions of thought has been articulated by Hilary Putnam (1979) and Manley Thompson (1981). I think that there is no doubt that this idea, like that of a priori knowledge, has a Kantian pedigree, and that some of Kant's remarks in the Critique require interpretation according to one of these ideas, others according to the other. For slightly different considerations which suggest that the two characterizations of apriority are not equivalent, see my (1983b). 13 The most controversial assumption involved in this argument seems to me to be the view that apriority is preserved along very long chains of logical inference. The signifi-cance of recent computer-generated proofs (discussed in Tymoczko, 1979, and Teller, 1981) seems to me to be that very long sequences of individually compelling inferences do not suffice to transmit apriority. I have tried to explain why this is so in Chapter 2 of my (1983a). Given either of the accounts outlined there, there is little chance of showing that the boundary of the set of a priori knowable propositions coincides exactly with that of the set of propositional preconditions of thought. Hence I do not believe that my use of a controversial assumption about long proofs can be turned to advantage by someone who hopes to defend the equivalence that I am criticizing. 14 I have defended this interpretation in my (1975a). As a number of people -originally and most notably Saul Kripke - have shown, the notions of necessity and apriority are not equivalent. I have tried to show that there is little hope for the traditional (Kantian) attempts to connect these notions in my (1980b). 15 A famous argument of Quine's casts doubt on the coherence of this idea. (See the end of his 1936.) The treatment of analyticity and its relevance to a priorist views about mathematics which is presented in the text ignores certain well-known Quinean criticisms in the interests of showing that, even given a charitable understanding of those views, they are still deficient. 16 This is a major theme of my (1983a). However, I think it only right to acknowledge that some of the epistemological ideas sketched here and worked out in more detail in that work were first formulated in various papers by Quine and in Hilary Putnam's (1975). In particular, Putnam argues forcefully that our knowledge of the axioms of ZF

342 PHILIP KITCHER

is gained by "quasi-empirical inference" from prior results. I have discussed a different example of the same phenomenon at greater length in my (1981b). 17 Putnam (1975) also sees this as a crucial epistemological problem, as does Imre Lakatos (1976). 18 Of course, Frege was more hesitant about his Basic Law V than about his other basic laws. He admits (Frege, 1893, p. vii; Furth, 1967, p. 4) that controversy may arise with respect to this law. However, his considered view is that it is a genuine basic law of logic, and that distrust of it will be based on the fact that, while it is something logicians have had in mind, it is likely to strike them as unfamiliar. 19 In glossing this passage as I do, I am of course ignoring Dedekind's suggestion that the acts of collecting have a product - a system. This is in accordance with the interpretative strategy adopted on p. 329 above. 20 This is my construal of the argument for the existence of infinite systems that Dedekind draws from Bolzano and presents in Paragraph 66 of his monograph (Dedekind/Beman, 1901, p. 64). 21 In a sense, Dedekind's reconstruction of arithmetic as the general theory of w-sequences makes life more difficult than it needs to be. For the applicability of any number concept is thus made to depend on the applicability of an arithmetical system committed to infinitely many natural numbers. If number-operations were characterized "from below" for example, in the iterative manner of Dedekind's early memoir or in a constructivist analogue of some of the standard set-theoretic proposals, this apparent difficulty would be avoided, and Dedekind could attempt to lIJgue for the weaker thesis that, in any world about which we can think, some number concepts (possibly only a very few) must be applicable. 22 See the references given in Note 5.

REFERENCES

Benacerraf, P.: 1981, 'Frege: The Last Logicist.' In Midwest Studies in Philosophy, VI, eds. P. French, T. Uehling, H. Wettstein. Minneapolis: University of Minnesota Press.

Coffa, A.: 1982, 'Kant, Bolzano, and the Emergence of Logicism.' Journal of Philosophy 79:679-89.

Currie, G.: 1982, Frege: An Introduction to His Philosophy. Sussex: Harvester Press. Dedekind, J. W. R: 1872, Stetigkeit und Irrationale Zahlen. Braunschweig: Vieweg. Dedekind, J. W. R: 1888, Was Sind und was Sollen die Zahlen? Braunschweig: Vieweg. Dedekind, J. W. R/Beman, W. W.: 1901, Dedekind's Essays on the Theory of Numbers

(translation of Dedekind's 1872 and 1888). LaSalle: Open Court. Reprinted by Dover Books, 1963.

Dedekind, J. W. R/van Heijenoort, J.: 1967, Letter to Keferstein in From Frege to Godel: A Sourcebook in Mathematical Logic. Cambridge, MA: Harvard University Press.

Dugac, P.: 1976, Richard Dedekind et les Fondements des Mathematiques. Paris: Vrin. Dummett, M. A. E.: 1973, Frege: Philosophy of Language. New York: Harper and Row. Frege, G.: 1884, Die Grundlagen der Arithmetik. Breslau: Koebner. Translated by J. L.

Austin. Blackwell, 1950.


Frege, G.: 1893, Die Grundgesetze der Arithmetik Vol. I. Jena: Verlag Hermann Pohle. Reprinted by Olms, 1962.

Frege, G./ Angelelli, I.: 1967, Kleine Schriften. Darmstadt: Wissenschaftlicher Buch-gesellschaft.

Frege, G./van Heijenoort, J.: 1967, Begriffsschrift. In From Frege to Godel: A Source Book in Mathematical Logic. Cambridge, MA: Harvard University Press.

Frege, G.lLong et al.: 1979, Posthumous Writings. Chicago: University of Chicago Press.

Furth, M.: 1976, The Basic Laws of Arithmetic. Translation of parts of Frege 1893. Berkeley: University of California Press.

Harman, G.: 1973, Thought. Princeton: Princetion University Press. Hempel, C. G.: 1963, 'Carnap and Philosophy of Science.' In The Philosophy of Rudolf

Carnap, ed. P. Schilpp, LaSalle: Open Court. Goldman, A. I.: 1979, 'What is Justified Belief?' In Justification and Knowledge, ed. G.

Pappas, Dordrecht: Reidel, pp. 1-23. Kitcher, P. S.: 1975a, 'Kant and the Foundations of Mathematics.' Philosophical Review

84: pp. 23-50. Kitcher, P. S.: 1975b, 'Bolzano's Ideal of Algebraic Analysis.' Studies in the History and

Philosophy of Science 6: 229-71. Kitcher, P. S.: 1979, 'Frege's Epistemology.' Philosophical Review 88: 235-62. Kitcher, P. S.: 1980a, 'A Priori Knowledge.' Philosophical Review 89: 3-23. Kitcher, P. S.: 1980b, 'Apriority and Necessity.' Australasian Journal of Philosophy 58:

89-101. Kitcher, P. S.: 1981a, 'How Kant Almost Wrote "Two Dogmas of Empiricism" (And

Why He Didn't)'. Philosophical Topics 12:217-49. Kitcher, P. S.: 1981b, 'Mathematical Rigor - Who Needs It?' Nous 15: 469-93. Kitcher, P. S.: 1983a, The Nature of Mathematical Knowledge. New York: Oxford

University Press. Kitcher, P. S.: 1983b, 'Kant's Philosophy of Science.' In Midwest Studies in Philosophy

VIII, eds. P. French, T. Uehling, H. Wettstein. Minneapolis: University of Minnesota Press.

Kornblith, H.: 1980, 'Beyond Foundationalism and the Coherence Theory.' Journal of Philosophy 77: 597-612.

Lakatos, I.: 1976, Proofs and Refutations. Cambridge: Cambridge University Press. Putnam, H.: 1975, 'What is Mathematical Truth?, In Philosophical Papers Volume I.

Cambridge: Cambridge University Press. Putnam, H.: 1979, 'Analyticity and Apriority: Beyond Wittgenstein and Quine.' In

Midwest Studies in Philosophy IV, eds. P. French, T. Uehling, H. Wettstein. Minnea-polis: University of Minnesota Press, pp. 423-441.

Quine, W. V.: 1936, 'Truth by Convention.' Reprinted in The Ways of Paradox, W. V. Quine. Cambridge, MA: Harvard University Press, 1979.

Resnik, M. D.: 1980, Frege and the Philosophy of Mathematics. Ithaca: Cornell Univer-sity Press.

Sluga, H.: 1980, Gottlob Frege. London: Routledge. Teller, P.: 1980, 'Computer Proof.' Journal of Philosophy 77: 797-803. Thompson, M.: 1981, 'On A Priori Truth.' Journal of Philosophy 78: 458-82. Tymoczko, A. T.: 1979, 'The Four-Color Problem and Its Philosophical Significance.'

Journal of Philosophy 76: 57-83.

GREGORY CURRIE

CONTINUITY AND CHANGE IN FREGE'S

PHILOSOPHY OF MATHEMATICS*

INTRODUCTION

We think of Frege's philosophy of arithmetic as constituted by a number of illuminating (and contentious) assertions about the natural numbers; that a statement of number is a statement about a concept, that numbers are logical objects, that truths about the arithmetical properties of numbers are logical truths. These are the central theses of his most accessible work, Die Grundlagen der Arithmetik But Frege's writings on arithmetic, both before and after the Grundlagen, include discussion of the real and, on occasions, the complex numbers as well. My primary aim here is to provide an analysis, largely historical, of Frege's theory of real numbers. In order to see how the theory developed and to under-stand its ramifications in other areas of Frege's philosophy I shall embed the discussion of real numbers in a more general framework which will include Frege's concept of a course of values - for this is central to his logical construction of number theory and analysis - and his views on the relation between arithmetic and geometry. This should lead to a better appreciation of some of the changes which his views underwent du}'ing fifty years of intense occupation with the nature of mathematics.

My strategy will be to look for connections and contrasts between the works of four different periods in Frege's career; his early work on magnitudes and geometry, the period of the Grundlagen, the period which includes the writing of Die Grundgesetze der Arithmetik and ends with Frege's initial response to Russell's paradox, and the period during which Frege abandoned the logical foundation of arithmetic in favour of a geometrical approach. But a strictly chronological development will not serve to make the points I intend. Instead I shall begin by outlining three principles which the discovery of Russell's paradox eventually caused him to abandon.! I shall then say something about the role that each of them played in the development of his thought during these periods.

The three principles are:

(1) To every concept there corresponds an object, the extension of that concept.2

345


346

(2)

(3)

GREGORY CURRIE

The applicability of the real numbers to measurement marks a theoretically important distinction between them and the natural numbers.3

There is a sharp distinction between the sources of arith�metical and of geometrical knowledge.4

1. COURSES OF VALUES

It can be argued that Frege's rejection of the first assumption was not a dramatic step for him. His commitment to extensions had never been strong. In the Grundlagen he had proposed that the natural numbers be identified with the extensions of concepts, but remarked on two occa�sions in that work that he did not set much store by the definition, reference to extensions being in principle eliminable in favour of reference to concepts alone.s This was written before he introduced (in 1891) the notion of the course of values of a function, assimilated concepts to functions and extensions to courses of values.6 Courses of values are introduced via a specification of their identity conditions:

If(E) = ag(a) == Vx(jx= gxf.

This became Axiom V of the logical system of the Grundgestze. It figures essentially in the proof of (1), and hence in the derivation of the paradox.s

Although this principle is logically independent of the other two it has a kind of methodological primacy, for it is almost certain that Frege would never have come to question (2) and (3) on independent grounds. He rejects them because the refutation of (1) called into question his whole philosophical programme. Why was the refutation of Axiom V such a blow to Frege's system? Hans Sluga has argued that it jeopardized not merely the logical foundation of arithmetic, but the very semantical theory in the light of which the system is interpreted. The semantical theory says that reference is secured for a singular term when we provide a truth value for any identity statement involving it.9 The function of Axiom V is exactly to lay 'down conditions under which any identity statement involving an expression supposed to denote a value�range would be true or false'. Russell's discovery showed thClt this could not be done, and so the antinomy was 'a threat to the whole semantic theory, and not merely to one of the axioms of the formal system'.

FREGE'S PHILOSOPHY OF MATHEMATICS 347

But this explanation is objectionable on two independent grounds. 1O

First, the axiom does not provide truth conditions for all identity state-ments involving a course of values name; it requires supplementation by a number of stipulations about the references of terms.!! Secondly it cannot be that Russell's paradox refuted both Axiom V and the seman-tic theory of the Grundgesetze. If the discovery shows that Axiom V is false and that not all course of values terms have reference, it cannot also refute the criteria which we use to decide whether a term has reference. That would be like saying that the discovery of Mercury's anomalous perihelion refuted not only Newton's theory but also the methodological requirement that acceptable theories survive empirical tests. What that discovery showed is that Newton's theory does not meet this requirement, and Russell's paradox shows, similarly, that course of values terms do not meet the requirements laid down in Grundgesetze

for a term to refer, as a close analysis of the proof in Section 31 will show.12

The reason why the refutation of Axiom V was such a great blow to Frege - one which he described in the most dramatic terms!3 - was that it revealed a fundamental epistemological weakness in the pro-gramme; a failure to show that we have an adequate grasp of the concept of a logical object. Frege's aim had always been to give a firm foundation to arithmetic; to show that the ultimate justification for believing in arithmetic is that it is derivable from evident logical truths by methods of inference which are purely logical in character. But in what sense is Axiom V, upon which the proof of the existence of courses of values (and therefore numbers) depends, an evident logical truth? Frege was never able to persuade himself that it was absolutely self-evident, and his worry was based, I suspect, upon the idea that we do not have any 'fix' on the concept of a course of values independent of the axiom; the only thing we know about courses of values is that they obey Axiom V,t4 The stipulations concerning the referents of names of truth values which Frege makes in order to determine the truth value of any identity sentence involving a course of values expression do not improve this situation, for arbitrary stipulations about the uses of expressions cannot tell us anything substantive about a concept. Modem logicians who think that the intuitive notion of set is embodied in the idea of the cumulative hierarchy can claim to be setting forth axioms which more or less closely capture this independent intuition. Frege, after Russell's discovery, was in the doubly serious position, not only of

348 GREGORY CURRIE

having taken a logical falsehood for a logical truth, but also having, as a consequence of this discovery, no justification left for claiming that we are epistemically related to logical objects.15

It is unclear exactly when Frege abandoned the identification of numbers with conceptual extensions. For some time at least he seems to have been prone to underestimate the impact of the paradox. In an essay of 1906 he refers to Axiom V as the means by which we infer the existence of an extension for each concept, without acknowledging that the axiom in its original form is definitely shown to be false by Russell's discovery. He says merely:

Of course it is not as self-evident as one would wish for a law of logic. And if it were possible for there to be doubts previously, those doubts have been reinforced by the shock the law has sustained from Russell's paradox. 16

But probably the principle had ceased to play a constructive role in Frege's thinking after 1903; he did not try to reconstruct his theory of numbers as logical objects, preserving as much as possible of the old principle, nor did he pursue further his investigations into the logical foundations of the real numbers (about which I shall say more in the next section)P

I shall not discuss Frege's abandonment of extensions further here.18

In terms of the concerns of this essay, the rejection of (1) is of interest primarily because it provides the necessary background for a discussion of (2) and (3). For we must understand the importance in Frege's think-ing of the doctrine that numbers are logical objects before we can discuss his theory of real numbers and assess the significance of his eventual decision to base arithmetic on geometry.

2. REAL NUMBERS AND MAGNITUDES

(a) From the Grundlagen to the Grundgesetze

While Frege in the Grundlagen is not wholly committed to (1), he does tacitly presuppose it in his identification of the cardinal numbers with extensions. What, at that time, was his attitude towards (2)? (I have stated this principle somewhat vaguely. Its content will become apparent from the discussion below.) At the end of the book he turns his atten-tion briefly to other kinds of numbers 'to make some use in this wider


field of what we have learnt in the narrower' (Section 92). He raises the same question here that he asked earlier about the natural numbers.

How are complex numbers to be given to us then, and fractions and irrational numbers? ... everything in the end will come down to the search for a judgement-content which can be translated into an identity whose sides precisely are the new numbers. In other words, what we must do is fix the sense of a recognition-judgement for the case of these numbers. In doing so, we must not forget the doubts raised by such transformations, which we discussed in §§63-68. If we follow the same procedure as we did there, then the new numbers are given to us as extensions of concepts (CI., Section 104).

These remarks should give us certain expections about Frege's inten-tions; in particular that he planned to treat the rational, real and complex numbers as successive extensions of the domain of the natural numbers. There does not seem to be any evidence that these numbers are to be regarded either as belonging to an importantly distinct kind from the natural numbers, or as being constructed by a different meth-od. Yet the standpoint of the later Grundgesetze presupposes a sharp distinction - both methodological and ontological - between the natural and real numbers. (2) plays an important role in that work. Slightly more than one volume is devoted to a detailed construction of the natu-ral numbers, after which Frege begins a similar treatment of the real numbers, including, after the fashion of the Grundlagen, an extended analysis (though not a very sympathetic or instructive one) of the defects in other theories of real number.19 He then makes it clear that he wants to treat the natural and real numbers separately:

... it is not possible to extend the domain of the natural numbers to the real numbers; it is a quite separate domain. The natural numbers tell us how many objects of a certain kind there are, while the real numbers can be looked on as measurement numbers, and tell us how large a magnitude is compared to a unit of magnitude (Cz., vol.2, Section 157).20

The question I want to ask is this: to what extent did Frege's views on the status of the real numbers undergo modification between his writing of the Grundlagen and of the Grundgesetze? I shall develop an answer in three stages. First I shall give a brief description of the theory of real numbers at which Frege seems to be aiming in the Grundgesetze, and of its connection with the concept of magnitude. Then I shall look back to the period before the Grundlagen and examine Frege's early views on the nature of magnitudes. Finally I shall try to place the Grundlagen in

350 GREGORY CURRIE

relation to Frege's early and late views on the correct form for a theory of arithmetic.

( b) The Grundgesetze Theory of Real Numbers

Given Frege's tireless insistence on the distinction between pure and applied mathematics and his polemical use of it against his opponents, the remarks I have quoted directly above may seem surprising. Frege draws a conclusion about the nature of pure mathematics - the real numbers are of a quite different kind from the natural numbers - from a premise about the applications of these different kinds of numbers. Although he argued that we ought not to conflate numbers with the entities to which they are applicable, it seems that he did think a philosophically adequate theory of pure mathematics should explain the conditions for their application. In volume 2 of the Grundgesetze he says:

We avoid the deficiency of [formalism]: that measurement either makes no appearance at all, or is externally patched on without any internal connection being established with the essence of number itself. In that case how each magnitude kind is to be measured and how a number would be obtained by this means must be given an individual explanation. In this way we lack a general indication of when a number can be used as a measurement number and what determines its applicability.

We may expect, therefore, that, on the one hand, the means by which numbers are applied in particular fields of knowledge will not escape our grasp, without, on the other hand, contaminating arithmetic with the objects, concepts and relations of these sciences, endangering its characteristic essence and independence. We can expect that arithmetic will provide us with such means even though it is not itself concerned with application (Section 159).

Having decided that natural numbers are properly applicable to con-cepts, Frege construes these numbers as extensions of concepts of the form equinumerate with the concept F The number two, then, is the extension of the concept under which fall all concepts having exactly two instances. Frege can then give precise conditions for the applicabil-ity of a number to a concept: n is applicable to F if and only if F falls under the concept of which n is the extension?l

The real numbers are treated in a similar way, though their applica-tion is to measurement. When we measure a quantity we assign it a magnitude relative to a unit of magnitude. Real numbers thus apply to magnitude pairs, and the same real number will apply to pairs (s, t), (u, v)


if and only if s:t = u:v. Real numbers are taken to be the extensions of relations between magnitudes, and we can again give abstract condi-tions for their applicability: a real number r is applicable to a pair of magnitudes (s, t) if and only if (s, t) falls under the relation of which r

is the extension.22 Seen in this light, the Grundgesetze theory of real numbers shares many features of the Grundlagen theory of natural numbers.

The insistence on building an account of the applicability of the var-ious numbers into the mathematical theory itself led Frege to reject the prevailing foundational programme: the Cantor-Dedekind-Weierstrass programme for the arithmetization of analysis. Its guiding heuristic principle was to take the natural numbers as given and to extend the domain successively to include the rational, real and complex numbers. Magnitudes, considered as entities distinct from the numbers them-selves, have no place in the construction (and must, Frege says, be 'externally patched on'). Also, the real numbers are defined in terms of the rationals; we end up with a class of rational numbers and a distinct

class of real numbers, containing 'copies' of all the rationals. But on Frege's account, since the application of rational and irrational numbers is to measurement, they ought to receive a single, unified treatmentP

How does Frege construct a theory of magnitude? He begins by adopting a suggestion of Gauss, that 'positive and negative numbers can find an application only where the numbered thing has an opposite that annihilates it when the two are thought together. Detailed examination shows that this assumption applies not to substances (objects thinkable in themselves) but to relations between any two objects that are numbered'.24 (For further remarks on Frege's method see below, section 2 (e).)

Frege accordingly treats magnitudes as (extensions of) relations, relations which belong to a certain structure called a 'magnitude field'; the closure under inversion of a class of relations, together with the null relation. A structure called a positive class is introduced, and the real numbers are to be defined as relations between the members of the magnitUde field of a positive class. (Such a magnitude field is isomorphic to the real line.) Associativity of the composition (addition) of relations is proved. Frege then proves the Archimedean Axiom and shows that composition within the magnitude field of a positive class is commuta-tive. The task which remains at the end of volume two is to show that there exists a positive class and to show, consequently, that the real

352 GREGORY CURRIE

numbers, defined as ratios of magnitudes, exist. (For further details of Frege's programme see the Appendix.) Precisely because of the dis-covery of Russell's paradox, this task was not carried out.

Frege concluded the second volume of the Grundgesetze with these remarks:

The next task will be to show, as indicated in Section 164, that there is a positive class. This will make it possible to define the real numbers as ratios between the magnitudes in the field of a positive class. We will then be able to prove that the real numbers are themselves magnitudes in the field of a positive class (ibid., Section 245).

In Section 164 of the same work Frege had outlined his plan for carrying out this task. The crucial importance of the passage requires an extensive quotation .

. . . from where do we get the magnitudes whose ratios are irrational numbers [?] ... We ... need a class of objects which stand to one another in the relations of our magnitude field, and this class must encompass an infinite number of objects. This brings us to the concept finite natural number, which has an infinite number - what we have called Endless [Endlos). But this infinity will not suffice. We will call the extension of a concept which is subordinate to the concept finite natural number, a class of finite

natural numbers. To the concept class of finite natural numbers there belongs an infinite number greater than Endless ...

We would have to show the existence of relations between classes of finite numbers which could then be understood as belonging to a magnitude field. Actually things will turn out somewhat differently, as we shall see.

Let us assume for the moment an understanding of the irrational numbers. Every positive number a can be represented in the form

k-ro { 1 } r+ I -;;;-

k-i 2

where 'r' is understood to be a positive whole number or 0, and 'n i ', 'n/ etc. are understood to be positive whole numbers, the number of which we assume to be infinite. In this way there belongs to every positive rational or irrational number a a pair, the first member of which (r) is a positive whole number or 0, and the second member a class of positive whole numbers (the class of nk's). For whole numbers we can substitute natural numbers, so that to every positive real number there belongs a pair, the first member of which is a natural number, and the second member of which is a class of natural numbers to which 0 does not belong. If a, b and c are positive numbers and a + b = c,

then for every b, a relation obtains between the pairs which belong to a and to c. And this relation can be defined without reference to the real numbers a, b, c, hence without presupposing any knowledge of the real numbers. So we have relations each of which is characterised by a pair (that which belongs to b). To these we add the inverses. The extensions of these relations (relation classes) correspond uniquely to the positive and negative real numbers. The addition of numbers b and b' corresponds to the composi-


tion of the associated relation classes. The class of these relation classes is now a field with which we can execute our plan. These indications may be sufficient for the moment to dispel doubts about the feasibility of the plan. This is not to say that we are going to stick strictly to this path .... In the execution of this programme there must first of all be no mention of irrational numbers. Instead we shall begin with classes of natural numbers, between which we will define certain relationships, without mentioning the connection with the addition of real numbers. Thus we will succeed in defining the real numbers purely arithmetically or purely logically as ratios of demonstrably available magnitudes, so that no uncertainty can remain concerning the existence of irrational numbers (ibid., Section 164).

Frege's plan seems to be this. He will introduce pairs (r, Nk ) where ris a natural number and Nk an infinite set of positive natural numbers. If a, b and c are such pairs, 6 is the extension of the relation which holds between a and c if and only if c - a = b. The class of 6's will be shown to be a positive class. The members of the magnitude field of this positive class will be the magnitudes out of which real numbers are constructed. Geometrically, the magnitude 6 can be represented as the class of intervals of length b on a line, where those intervals are taken as positive or negative, depending on their position with respect to the origin. Real numbers proper will then be defined as the extensions of relations (i.e. as ratios) between the relations in the magnitude field of this positive class.

(c) Earlier Views

Let us return to the question of the continuity between the Grundlagen and the Grundgesetze. While describing Frege's Grundgesetze theory of real numbers, I noted a way in which it bears a similarity to the theory of natural numbers presented in the Grundlagen. Further evidence of continuity is to be found in Frege's earliest work on the philosophy of arithmetic, his Habilitationsschrift of 1874; 'Methods of Calculation based on an Extension of the Concept of Magnitude'.

In this very early work Frege does not suggest any separate treatment of the real and natural numbers, but he does suggest that the founda-tions of arithmetic are to be connected with the concept of magnitude. The following is an important passage from the introduction.

To what do these principles refer, from which arithmetic germinates and grows as if from a seed? To addition: from this come all other kinds of calculations. So intimate is the connection between the concepts of addition and of magnitude that the one cannot be grasped without the other. Quite generally speaking the process of addition is the following: we replace a group of things by one single thing of the same type. Here is

354 GREGORY CURRIE

given a detennination of the concept of magnitude equality. If we can judge in every case when objects agree in a property, then we clearly have the correct concept of that property. Thus by giving the conditions under which equality of magnitude holds, we determine thereby the concept of magnitude. A magnitude of a certain kind - for instance a length - would then be a property in which a group of things, independently of their internal structure, can agree with a single thing of the same kind (K. S., p. 51 ).25

A number of ideas are crowded together here in a somewhat obscure way. Amongst them are these: the truths of arithmetic are (in some sense) derivable from the laws governing additions; a definition of addition gives us necessary and sufficient conditions for equality between magnitudes, and a grasp of these conditions constitutes a grasp of the concept of magnitude.

In the later Grundgesetze the first of these ideas recurs. (Real) numbers are to be defined in terms of magnitudes, and we characterise a magnitude field in terms of the laws of addition:

The demarcation of a magnitude field results from the demand that the essential1aws of addition, known as the commutative and associative principles, hold. The question can now be asked: what properties must a class of relation classes have in order for the commutative and associative laws to hold for the composition of relation classes? (Gz., Section 165).26

The second is implicity abandoned, since magnitude equality is there defined in terms of ratios.

In the third we have the probable origin of Frege's views on the connection between identity conditions and concepts; views which were important to him both in the Grundlagen and the Grundgesetze. Frege's claim here is a little unclear, partly because it is unclear what meaning 'agreement in a property' has for him. 'a and b agree in the property F'

may mean 'a and b both have F', 'a and b have Fto the same degree' or 'a and b are the same F'. The second seems to me the most likely, because from 'a and b have Fto the same degree' we can infer (A): 'the magnitude of a's Fness = the magnitude of b's fuess', and it is probably a judgment of this latter form which Frege has in mind when he says 'by giving the conditions under which equality of magnitude holds, we determine thereby the concept of magnitude'. At the very least then, Frege seems to be committed to the view that to know the condition under which (A) is true is to grasp the concept of magnitude. In the Grundlagen he gave considerable attention to showing that analogous claims for statements of numerical identity and identity of direction are


false. And the same argument clearly applies to magnitudes. To know the condition for the truth of (A) does not tell us whether Julius Caesar is a magnitude. In the elaborately dialectical development of Grundlagen Section 62-68 - beginning a process of definition which is then declared inadequate - Frege is teaching his readers a lesson about the relation between concepts and identity conditions which he had to learn at the expense of his own earlier views.

If the passage just cited is also taken as providing a definition, or even an informal explanation, of the concept magnitude, it falls very short of the standard of clarity we are familiar with in Frege's later work. Whatever other changes there were in his views between here and the Grundgesetze, Frege certainly realized by the time he was planning the latter work that a more precise account of that notion was required. In the Grundgesetze he prefaced his analysis witli a complaint which has obvious applications to his own earlier exposition:

What a magnitude is has never indeed been satisfactorily explained. When we examine the attempts at explanation we often come up against the phrase 'same kind' or some-thing similar .... The phrase 'same kind' is obviously empty; for things can be of the same kind in one respect but different in others. Then the question whether an object is of the same kind as another cannot be answered 'yes' or 'no' (Gz., vol. 2, Section 160).

I have said something already about Frege's later attempt to give precision to the concept of magnitude. Further information is contained in the Appendix.

There seems, then, to be at least some continuity between the early remarks on magnitudes and arithmetic in the Habilitationsschrift and the theory of real numbers in the Grundgesetze. While Frege has, in the later work, sharpened his explanation of what it takes for something to be a magnitude and has introduced the crucial idea of a positive class, he continues to insist that an analysis of the laws which magnitudes obey must proceed by way of an analysis of the laws of addition, and that arithmetic, at least the arithmetic of real numbers, is connected with the comparison of magnitudes.

The only significant discussion of magnitudes in the Grundlagen occurs in Section 19 where Frege refers briefly to Newton's view that number is 'not so much a multitude of unities, as the abstracted ratio of any quantity to another quantity of the same kind, which we take for

356 GREGORY CURRIE

unity'.27 He concedes that this may be an appropriate view of number 'in the wider sense in which it includes also fractions and irrationals'. But it presupposes the concept of magnitude ratio, and this in tum presup�poses natural number. So far this is close to the construction given in the Grundgesetze. But Frege goes on to reject the view of Newton, on the grounds that it involves a geometrical definition and is therefore unable to explain the varied application of number, though Newton's words do not suggest that he understands magnitudes to be geometric objects. Frege notes, finally, that Newton may take magnitudes, more generally, to be sets. But 'relation between a set and the unit of set' tells us no more than 'number by which a set is determined.'

What is causing Frege to distance himself from Newton's view is not, I think, any hostility to the connection of numbers and magnitudes, but an inability to see how the connection can be made without favouring one particular kind of magnitude (say, length) over all other magnitude kinds with which measurement numbers are implicated. It is this prob�lem which Frege finally conceived himself as having solved by the introduction, in the Grundgesetze, of abstract magnitude fields.

A remark positively favouring the connection of real numbers and magnitudes occurs the next year. In 1885 Frege published a review of Hermann Cohen's book The Prfnciple of the Infinitesimal Method and its History28, and took Cohen to task for, amongst other things, wanting to draw a distinction between intensive and extensive magnitudes. Frege s'ays:

Now the distinction between intensive and extensive magnitudes has no sense in pure arithmetic, and, come to that, in mathematics at all. The number 3, for example, can serve as a measurement number for a distance with respect to a unit of length. It can serve also as the measurement number for an intensive magnitude, for example a light strength measured against a unit of luminosity. The calculation proceeds in both cases according to the same laws. Thus the number 3 is neither an intensive nor an extensive magnitude but rather stands above this contrast. (K.s., p, 101)

This might seem to be a repudiation of the connection between real numbers and magnitudes. But I think it is best understood as insisting that real numbers cannot be straightforwardly identified with partiCUlar kinds of magnitudes, because in that case we would not be able to explain the connection between real numbers and magnitudes of every kind. The passage is, in fact, strikingly similar to one from the Grundge�setze which the detailed working out of the exact nature of the relationship between real numbers and magnitudes:


Exactly the same magnitude ratio which is found in the case of distance, occurs also in the case of temporal intervals, mass, luminosity, etc. In this way the real numbers are freed from these particular kinds of magnitudes and, as it were, suspended above them.29

We might summarize the development, up to about 1902, of Frege's views about the relations between the natural numbers, the real num�bers and the concept of magnitude in the following way. Around 1874 he was thinking of arithmetical concepts as in every case inseparable from the concept of magnitude. Later, some time before 1884,30 he devised a way of providing a purely logical foundation for the arithmetic of natural numbers, without appeal to the concept of magnitude.

At this point, and as a result of undertaking a radically new and powerful foundational programme, the notion of magnitude probably faded from Frege's thoughts while he gave exclusive attention to the theory of the natural numbers. He may even have assumed, while writing the Grundlagen, that the natural numbers could be construed as a subclass of the real numbers. This would explain why, at the end of the Grundlagen, Frege uses the term 'positive whole numbers' to refer to the natural numbers, whereas in the later Grundgesetze he reserves this expression for a subclass of the real numbers, given in his notation as 1, 2, 3, ... etc., which is disjoint from the natural numbers, denoted by terms of the form 1, 1,3, ... etc.3 !

But soon afterwards, perhaps asa result of confronting Cohen's views on magnitudes in his review of 1885, the need for a connection between real numbers and measurement reasserted itself in Frege's mind. Rethinking the problem, Frege probably went back to his early work on magnitudes and realized that his programmatic remarks on the dependence of arithmetic on concepts and operations associated with magnitudes, though inappropriate for the natural numbers, provided the basic for a construction of the real numbers. At this point Frege has adopted (2). But by this time he has added a further constraint which vastly complicates his task. He wants to show that the arithmetic of real numbers ultimately derives, like that of the natural numbers, from purely logical principles. For anyone working within the Dedekind�Weierstrass programme for the arithmetization of analysis this would seem to be a natural conclusion: the real numbers simply inherit the logical status of the natural numbers. But from Frege's point of view the conclusion that the theory of real numbers is part of pure logic is not so immediate. What was the reasoning which led Frege to it?

358 GREGORY CURRIE

Certainly, not all of Frege's arguments for the logical status of the natural numbers carry over to the real numbers. In Grundlagen, Section 14 he argues that while Euclidean Geometry is applicable to the restricted domain of the intuitable, the arithmetic of natural numbers applies to 'the widest domain of all'; that which is numerable. While the status of the claim that everything is countable is unclear on reflection,32 it has greater intuitive appeal than the corresponding claim that every-thing is measurable.33 But in the same passage Frege gives a slightly different (but not distinguished) argument for the logical status of the laws of the natural numbers: that 'we have only to try to deny anyone of them, and complete confusion ensues. Even to think at all seems no longer possible'. This seems to be a more hopeful candidate for transfer to the real numbers. The denial of, say, the associative law for the addition of magnitudes seems as likely to induce confusion of thought as the denial that every natural number has a successor.

Having decided, on whatever grounds, that the real numbers must be treated purely logically, Frege's problem was then to establish the connection between real numbers and magnitudes without recourse to extra-logical assumptions. Frege's solution was the complex construc-tion I have outlined above. In order to establish the existence, on purely logical assumptions, of objects which constitute a magnitude field, Frege appeals to the existence of the only objects he has yet been able to introduce purely logically, the natural numbers. These will be the raw material from which magnitudes and ultimately real numbers are con-structed. Just as the lengthy and complex working out of this idea was begun, Russell's paradox was discovered.

(d) Reduction and Explanation

The constraints which Frege sets himself in the construction of a theory of real numbers tell us something about the way in which he would have approached the problem of multiple reductions. The problem, presented in a classic paper by Benacerraf, arises because there are a number of different ways, all satisfactory from the mathematical point of view, of defining arithmetical entities. But .since there is no mathematical reason to prefer one of these reductions to any of the others, no one of them can be correct; mathematical adequacy being our only legitimate stand-ard of correctness. Benacerraf concludes that our concept of number is, properly understood, the concept of a certain structure rather than the


concept of a particular sequence of objects.34 But we can see that Frege would have wanted to insist that there are conditions of adequacy on a reductive theory of arithmetical objects (natural and real numbers) which go beyond the demand that the reduction preserve their arith�metical properties. Indeed, it is part of Frege's plan in the second volume of the Grundgesetze to show that there is a class of objects satisfying all the mathematical properties of the real line but which he does not identify with the real numbers; the magnitude field of a positive class. This structure is then used as the basis of the construction of the real numbers proper, which appear as the extensions of relations between objects in this magnitude field. For Frege, the difference between the real numbers proper and the objects in the magnitude field of a positive class is not mathematical but explanatory. Using the definition of the real numbers proper we get, so he believed, natural looking, necessary and sufficient conditions for the applicability of a real number to a magnitude, relative to a unit of magnitude (see above, p. 351). And we do not get, so he believed, the same thing from the definition of the magnitude field of a positive class.

(e) The Concept of Magnitude

In view of this it comes as something of a surprise that Frege takes the concept of magnitude itself to be a purely structural one:

Instead of asking, what properties must an object have in order to be a magnitude? we must ask, how must a concept be constituted so as to have as its extension a magnitude field? ... What properties must a class have in order to be a magnitude field? Some�thing is not a magnitude in itself, but only in so far as it belongs, with other objects, to a class which is a magnitude field.35

Now this is clearly a useful assumption for Frege to make. By construing the conditions on a thing's being a magnitude as purely structural he can build a magnitude field out of purely abstract objects, with no appeal to contingent entities. In this way Frege can base his construction of a magnitude field on material already given to him by his earlier work; the natural numbers. As we have seen, the members of the magnitude field are relations between pair (r, N

k

), with r a natural number and Nk a class of natural numbers.

But the question is whether the concept of a magnitude is perhaps essentially connected with the concept of some qualitatively given

360 GREGORY CURRIE

characteristic, the intensity of which it is the purpose of a given kind of magnitude to measure. As we have seen, Frege's own examples of magnitude kinds are things like length, time, mass, luminosity, etc. Any adequate theory of the real numbers must, he tells us, show how real numbers are equally applicable to all these kinds. But these sorts of magnitudes play no role in Frege's construction. Their relation to the abstract 'magnitude field' which Frege planned to use as the basis for the real numbers is left obscure, and so, consequently, is the relation of Frege's real numbers to these intuitive kinds of magnitudes.

The natural solution to this problem would be to show that the magnitude field which Frege constructs - let us call it MF* - can go proxy for all magnitude kinds in the sense that all magnitude kinds can be treated as structures isomorphic to MF* when we neglect those properties and relations of and between members of the kind which are to do with the way in which the kind is given to us empirically, and concentrate on the order relations between them. The explanation for the applicability of real numbers to empirically given magnitudes would then depend upon the existence of this isomorphism.

But this idea suggests a quite different and decidedly simpler route to Frege's goal, once we realise that the field of Fregean real numbers is itself isomorphic to MF*. For if MF* is adequate in the sense that every magnitude kind is a structure isomorphic to it, then every such kind is isomorphic to the field of real numbers. Frege could then proceed as follows. First, characterise a magnitude field structurally and provide an argument to the effect that it is adequate in the sense explained above. Then construct the real numbers arithmetically in the style of Dedekind and Weierstrass and show that they form a magnitude field. Then we can \ say that the real numbers so construed are applicable to any system of quantities having the structure of a magnitude field.36 Thus the real numbers receive a direct construction from the natural and rational numbers without interposing MF*, and we have at the same time an explanation of their applicability.

Of course Frege did not have the clear model theoretic perspective which enables us to talk about structure preserving maps, but he was aware that the field of real numbers itself constitutes a magnitude field, and must have thought that any magnitude kind can also be treated as a magnitude field. There ought to have been no conceptual difficulty for him in formulating the point as follows. The real numbers apply to all magnitude kinds because the real numbers and any such kind fall under


the concept being a magnitude field. And the real number r applies to the particular magnitude Mk of kind K when compared to the unit of magnitude i

k

in K if and only if r occupies the same role in the structure of real numbers as the pair (Mk i

k

) occupies in K. Looked at from this point of view we see an essential difference

between the account of the applicability of the natural numbers and that of the real numbers. The explanation of the applicability of a natural number n is local; it depends only on what concepts fall under the concept of which n is the extension. But the explanation of the applica�bility of a real number r is global; it depends upon the existence of a mapping from the real numbers into the structure which is the particular magnitude kind we are considering.

3. ARITHMETIC AND GEOMETRY

In Frege's last papers the distinguished status of the natural numbers with respect to their application is taken to be a philosophically unim�portant fact about the way in which we first learn arithmetical concepts through counting37 and the natural numbers are consequently to be identified with the positive whole numbers. Arithmetic is now to be based on a geometric construction of the complex numbers. I shall briefly examine this later theory in order to see what relation it bears to Frege's earlier work, but before I do we should ask: what reason did Frege 'have for abandoning (2)? The most probable answer is, unfor�tunately, not that Frege has discovered some principled reason for giving up the idea that the difference between the application of the natural and real numbers is conceptually important, but that the result�ing distinction is not capable of being made in a natural way in the geometrical setting which he has now adopted. As we shall see, numbers are to be identified with ratios between intervals in the complex plane in such a way that each point corresponds to a complex number. There is no way to distinguish, given these resources, the natural number N from the real number n. (2) is thus a casualty in the retreat from (3); Frege had no independent reason for thinking that it was false, though he was ready enough to describe it as a psychologistic mistake once it had become inconvenient.38 The question then is, why did Frege feel obliged to adopt a geometrical construction?

Frege's claim that arithmetic is deducible from logic was intended to solve an epistemological problem, which arises for him in the following

362 GREGORY CURRIE

way. Arithmetical knowledge essentially involves knowledge of the existence of a completed infinity, and this is not something we are given through experience.39 What, then, is its source? Frege's answer is that the objects of arithmetic are really 'logical objects', or conceptual extensions, and our grasp of logical truth tells us that to every concept there corresponds an extension. The possibility of endlessly forming distinct concepts then enables us to grasp the existence of an infinite series of such objects.4o

But the supposition that to every concept there corresponds an extension is exactly what is refuted by Russell's paradox. If we are to have arithmetical knowledge at all its source must be non-logical, and it must be such as to provide us with knowledge of the infinite. Kant had thought that this source was the pure intuition of time, but Frege does not seem to have considered adopting this view. Instead he decided that its source was our pure intuition of spaceY 'Arithmetic and geometry have developed on the same basis - a geometrical one in fact - so that mathematics in its entirety is really geometry'.42 It is at this point that Frege has abandoned (3).43

In contrast with (1) and (2), Frege's commitment to (3) in the Grundlagen was unambiguous. There he wrote:

The basis of arithmetic lies deeper, it seems, than that of any of the empirical sciences, and even than that of geometry.'The truths of arithmetic govern all numerable .... Should not the laws of number, then, be connected very intimately with the laws of thought? (GI., Section 14)

Frege had already formulated this view by the time of his Habilitations�schrifr.

There is ... a notable difference between the ways in which geometry and arithmetic ground their principles. The elements of all geometrical constructions are intuitions, and geometry points to intuition as the source of its axioms. Just as the objects of arithmetic do not have any intuitive qualities, sO'its principles cannot proceed from intuition. How indeed should it be able to guarantee propositions which hold true for magnitudes of quite different kinds, of which perhaps some categories are still unknown to us (K.S., p.50).

It looks, then, as if Frege at the age of 76 was embarking on a programme of arithmetical construction for which he could not possibly have been prepared by the work of his creative years. I shall try to show that in fact there are elements of continuity with his earlier thought even here.44


In the Grundgesetze Frege had defined the real numbers as ratios between magnitudes, but as we have seen, the magnitudes which are used are not constructed out of geometric objects. Having defined a positive class Frege intended to prove that there exists a class of relations which belong to the magnitude field of a positive class, using the natural numbers as the basis for his construction (see the remarks from Section 164 of the Grundgesetze in Section 2 (b) above). This was natural enough. Having shown (so he thought) that the existence of the natural numbers could be proved from pure logic, Frege hoped to use them as the basis for the completion of this programme. When he later turned to geometry as the foundation of arithmetic, he was free to define numbers as ratios between spatial distances. If he had intended to construct only the real numbers this is no doubt the method which would have appealed to him. But instead he wanted to 'go straight to the final goal, the general complex numbers'.45 Ratios between Euclidean intervals will not take us out of the class of real numbers, if the ratios between the intervals are taken to be independent of how the intervals are oriented with respect to one another. Frege had, therefore, to find magnitudes in a related but richer structure.

At this point Frege may have found help in some of his own earlier work. In a brief note of 1878 he had suggested that the shape of a triangle, relative to a unit of shape, could be represented as a complex number.46 Now there is an obvious connection between the shape of a triangle and a ratio between intervals. For if triangles ABC and DEF are similar, then AB :AC = DE :DF. Using this idea, strengthened in a natural way, Frege shows in his last paper how ratios between intervals in the plane can be defined in terms of similarity between triangles, and how complex numbers can be identified with ratios between (oriented) intervals in the plane.

Taking line and point as undefined concepts, and the symmetry of

one point with another with respect to a line as an undefined relation, Frege defines similarity between triangles in the plane.47 Further

(1) AB:AC=DE:DF

is taken to mean the same as

(2) Triangle ABC is similar to triangle DEF.

Here we have, in the style of the Grundlagen, the transformation of an equality between ratios into an equiValence relation which displays the

364 GREGORY CURRIE

truth conditions of the equality.48 It will now tum out that every ratio between intervals in the plane is represented by a unique point in the plane. Let OA be a unit interval in the plane. Then for each triangle PQR there is a unique point C such that OCA is similar to PQR or, what is now the saUle thing, PQ:PR = OC:OA. So complex numbers correspond to relations between intervals. (u, v) and (s, t) belong to the same relation when u: v = s: t, clear cut identity conditions having now been given for ratios.

Note that the stipulated equivalence of (1) and (2) does not corre-spond to our intuitive notion of the condition for equality of ratios between intervals. (2) would normally be taken to entail (1) but not conversely. And this just reflects the fact that the ratio between inter-vals, as Frege now understands that concept, is not invariant under changes in the orientation of the lines with respect to one another. It is by using this extended notion of ratio that Frege is able to set up a correspondence between ratios and points in the complex plane. And his method draws upon three aspects his earlier work; his use of the transformation of a relational statement into an identity statement, his representation of triangle shapes as complex numbers, and his use of ratios between magnitudes in the construction of the real numbers.

The last of these is important, because it is unclear whether Frege is entitled to claim that complex number appear as ratios between geo-metric magnitudes in any intuitive sense, given that the representation of these ratios is sensitive to changes in the orientation of the geometric objects (intervals) involved. What Frege was seeking to show in this last project is thus better expressed in terms of his note of 1878: that complex numbers correspond to the shapes of triangles. Given only the demand for a geometric foundation for the complex number system this seems to be a satisfactory result. Why was Frege not content with it? Because of the continuing influence on his thought of the idea that measurement numbers must be intrinsically connected with magnitudes. But for the full complex domain this can be implemented only in an arbitrary and unconvincing way.

A second difficulty is this. Frege had stressed earlier the requirement that real numbers be not closely tied to any particular magnitudes, because we should respect the circumstance that real numbers are equally applicable to all magnitudes, geometric and non-geometric.49 The method we are now considering violates this, since it takes geometric magnitudes as the basis of the construction.


4. CONCLUSIONS

I have tried to display the problems which Frege faced in his attempt to found analysis on logical principles, and how he went about solving them. I have argued that the principles at work in that construction may be traced back to his early work on magnitudes, and that they were helpful to him in his very late project for a geometric foundation. I tried also to show why Frege rejected the programme for the arithmetization of analysis; because of its failure to connect with the concept of magnitude and because. of its employment of a two step generalization of the natural to the real numbers via the rationals. I argued that at least the first of these objections was misconceived, once we realize that the applicability of real numbers to magnitudes like length and temperature must appeal to the idea of a common structure underlying the real numbers and these kinds of magnitudes. I suggested that Frege aban-doned (2) - that there is a theoretically important distinction between the natural and real numbers - because, in opting for a geometrical foundation (thus abandoning (3» he was no longer able to make the distinction. And (3) was rejected because, after Russell's paradox, geometry seemed to be the only alternative to a logical foundation for arithmetic which could accommodate its essentially infinitistic character. So if 'x -> y' represents the relation the rejection of x brought about the rejection ofy, we have (1) -> (3) -> (2).

Finally, I shall say something about the relative importance of the three principles in Frege's thinking. (1), concerning the existence of conceptual extensions, he had tentatively adopted while writing the Grundlagen, and finally abandoned sometime after 1906.50 It was thus the most short-lived of his principles. (2), according to which the natural numbers are to be distinguished from the real numbers, may have been adopted around the time of the Grundlagen. There is no sign of it in any earlier work, including the Habilitationsschrift. Unlike the case of (1) we have no reason to think that Frege abandoned it until very late, perhaps not until 1924. The abandonment of (3) marks the most profound change in Frege's thinking. It appears in the Habilitationsschrift and, like (2), there is no firm evidence of its rejection until 1924. If we were to put the principles in an increasing order acccording to the tenacity with which they were held by Frege, it would correspond to the numerical ordering I have given them here.

We would have reached different conclusions about the ordering if

366 GREGORY CURRIE

we had adopted a different perspective from which to judge the im�portance of the propositions concerned. If we had analyzed them, not in terms of the duration and intensity of Frege's commitment to them, but in terms of their centrality within his research programme, we would have to conclude, I think, that (1) was considerably more important than (2), and that (3) stands in a relation of incommensurability to the others.

For all Frege's unwillingness to commit himself to it, (1) was crucial to his programme. He was unable, evidently, to dispense with extensions and treat numbers as concepts, as he had suggested might be done in the Grundlagen. Frege's technique, tacitly in the Grundlagen and explicitly in the Grundgesetze, was to exploit (1) in order to prove the existence of infinitely many natural numbers. If we cannot pass from a concept to its extension, almost everything in these works collapses.

By comparison, (2) is of peripheral importance for the programme. Frege's attempt to establish logicism would not itself have failed if he had been unable to base the real numbers on the concept of magnitude. It would have been possible for him to take the course he had pre�viously rejected; that of constructing the real numbers directly from the natural numbers, without interposing magnitudes between them.51

It is impossible, I believe, to place (3) in relation to (1) and (2) from this standpoint. The reason is that (3) plays no direct role within Frege's logicist programme. Its importance lies in its providing the original motivation for that programme. Frege thought that Kant had established the foundation of geometry in synthetic a priori intuition. The idea that mathematical knowledge has such an intuition as its source had no intrinsic repugnance for him. He simply felt driven to the conclusion that there happens to be no such intuition corresponding to arithmetic. Accordingly he had to find the epistemic basis for arithmetic elsewhere. His first idea was that the basis was to be found in the theory of magnitudes,52 and . later that it was pure logic, the theory of magnitudes now being seen as providing a bridge between logic and the theory of real numbers.

But one need not adopt (3) in order to undertake a logicist pro�gramme. In Russell's version of the programme geometric truths appear as logically true conditionals of the form G::J T, where G is the conjunction of the geometric axioms and T a theorem. Conversely, (3) does not entail any version of logicism; as we have seen, Frege held (3) at a time (1874) when he wished to base arithmetic on the concept of magnitude rather than on logical principles.53

Thus (3) bears no deductive or heuristic relation to logicism. It is not a presupposition of that programme, nor does it suggest methods which would enable Frege to implement the programme. The sort of consi-derations with which I have been concerned in this paper, considera-tions of continuity, change and commitment may not be the best guides to the objective significance of principles in a research programme. But logicism as an autonomous research programme has not been my concern here.

University of Otago

APPENDIX:

Magnitude Fields in the Grundgesetze, Volume II.

We shall treat relations as classes of ordered pairs. If ris a relation, then the inverse of r, written r-1, is {(y, x) : (x, y) E r}. The composition of relations p and q, written p + q, is {(x, z) : 3y«x, y) E P and (y, z) E q)}. We write p + q-l as p - q. D(r) = {x: (x, y) E r} and R(r) = {y: (x, y) E r}. The null relation 0 = {(x, y) : x = y}. We consider only one-one relations.

THEOREM 1: Composition is associative (Section 166).

DEFINITION 1: If S is a class of relations then the magnitude field associated with S, written as, = {x: XES or X-I E S or x = O} (Section 173).

DEFINITION 2: S is a positival class, written ,9r(S), if and only if for any p, qE S

(1) D(p) = R(q); (2) p+qE S; (3) p-pIJ S; (4) P - q E oS (Section 175).

THEOREM 2: If,9r( S) and pES then p - p = 0 (Section 178).

DEFINITION 3: P < s q if and only if q - pES (Section 187).

368 GREGORY CURRIE

'THEOREM 3: Ifp <s qandp+ t E Sthen q+ t E S(Section 188).

THEOREM 4: If p, q E S then either p < s q, p = q or q < s P (Section 190).

In Section 175 Frege writes:

If the magnitude field [associated with a positive class] is to be continuous, the following must be the case. If a magnitude in this field has a property cI> which is possessed by all smaller magnitudes, and there is a magnitude in the field which does not have cI>, then there must be an upper bound to all the magnitudes in the field which do have cI>. That is, there must be a magnitude such that all smaller magnitudes have cI>, and all larger ones are larger than at least one magnitude without cI>.

To that end we have:

DEFINITION 4: d is an S-limit on cp if and only if (1) 9>(S); (2) dE S; (3) IfpE Sandp <sdthenpE CP; (4) If pES and d < s P then 3q( q E S 1\ q < s P 1\ q if. cp)

(Section 193).

THEOREM 5: If p and q are S-limits on cp then p = q (Section 196).

DEFINITION 5: Sis a positive class, written peS), if and only if (1) 9>(S); (2) Shas no least element; (3) If (p E S 1\ '1r(r E S 1\ r < s p::J r E cp) 1\ 3q(q E S 1\

q if. $» then there is an S-limit on cp (Section 197).

THEOREM 6: (Archimedean Axiom) If two relations belong to a positive class then there is a multiple of the one not less than the other (Section 199-214).

THEOREM 7: Composition is commutative for members of the magni-tude field of a positive class (Sections 215-244).54


NOTES

.. 'I shall use the following abbreviations when referring to Frege's work, 'GL' is Die

Grundlagen der Arithmetik, first published in 1884, English translation by J. L. Austin with German text on facing pages, Oxford: Blackwell, 1953. 'Gz.' is Die Grundgesetze

der Arithmetik, two volumes, 1893 and 1903, reprinted as one volume, Hildesheim: Georg Olms, 1962. 'K. S.' is Kleine Schriften, edited by I. Angelelli, Hildesheim: Georg Olms, 1967. 'N. S.' is Nachgelassene Schriften, edited by H. Hermes et al., Hamburg: Felix Meiner, 1969. 'w. B: is Wissenschaftliche Briefwechsel, edited by G. Gabriel et al.,

Hamburg: Felix Meiner, 1976. 'G. B: is Translations from the Philosophical Writings

of Gottlob Frege, edited by P. T. Geach and M. Black, Oxford: Blackwell, third edition, 1980. 'Po w.' is Posthumous Writings, translated by P. Long and R. White, Oxford: Blackwell, 1979. 'Po M. c: is Philosophical and Mathematica1 Correspondence, edited by B. McGuinness and translated by H. Kaal, Oxford: Blackwell, 1980. Occasionally I have modified the existing translations. Quotations from works of which there is no published English version are my own translations. I am grateful to Loraine Hawkins, Graham Oddie and Pavel Tichy for their comments on an earlier version of this paper, and to August Obermeyer for checking my translations of Frege's German.

Doug Jesseph, of Princeton University, is preparing a translation of some material from the Grundgesetze. 1 have sometimes employed the phrasing that he gives. 1 See the discussion in N.S., pp. 286-302, P. w., pp. 267-281; also letter to Honigswald, 1925, W.B., pp. 85-87, P.M.c., pp. 54-56 and letter to Zsigismondy, W.B., pp. 271-273, P.M. c., pp. 176-8. 2 See Theorem 1 to Gz., vol. 1, Section 55. The result is stated generally for courses of values, of which extensions are special cases. 3 See Gz., vol. 2, Section 157. 4 See K.S., p. 50, GI., Section 14. 5 See' GL, Section 68, Note and Section 107. 6 A preliminary draft of 'On Concept and Object' contains a footnote according to which 'the question whether one should simply put "the concept" for "the extension of the concept" is in my view one of expediency' (N.S., p. 116, P. w., p. 106). The printed version omits this. The note, together with several lines of text also missing in the printed version gives the impression that Frege still holds the position of agnosticism with respect to extensions which he had expressed in the Grundlagen. This impression is not conveyed by the wording as it appedfed in print. 7 See K.S., p. 130, G.B., p. 26 and Gz., vol. 1, Section 3. 8 See Gz., vol. 1, Section 54-55. In an Appendix to Gz. Frege proposed a modification of the axiom as a way of avoiding the paradox. The modification is distinctly ad hoc and cannot take the place of Axiom V in the proof that there is an infinity of natural numbers. Frege would quickly have realized this. (See M. Dummett, Frege. The Philo�

sophy 0/ Language (London: Duckworth, 1973), p. xxiii.) It was later shown to lead to a new contradiction when combined with the other axioms of the system. 9 See Sluga, Review of N. S., Journalo/Philosophy, 1971, pp. 269 and 272. 10 In his book Gottlob Frege (London: Routledge and Kegan Paul, 1980) Sluga offers a different explanation for the effect of Russell's paradox on Frege's work: Frege, follow-ing Kant, insisted that logical principles may not be existential. Russell's paradox

370 GREGORY CURRIE

showed that no non-existential logical basis for arithmetic is possible, so the whole programme of logicism must be at fault (p. 110). Sluga is probably right in thinking that Frege would not have allowed existential principles to count as logical, but his commit-ment to Kantian ideas does not, I believe, go to the heart of the matter. Rather, Frege would have objected to putatively logical principles which have the form of assertions of existence because such assertions seem particularly susceptible to sceptical doubt. In other words, it was Frege's mathematical epistemology which blocked all the alternative paths, and this can be understood independently of a Kantian theory of the form of logical truths. In addition it is difficult to know how much Kantianism we are entitled to attribute to Frege (see my review of Sluga's book, British Journal for the Philosophy of

Science 32 (1981), pp. 200-206). 11 See Gz., Section 10. This section is confusing because Frege raises an objection to the claim that Axiom V fixes the references of course of values terms based on the possibility of there being a mapping of the universe into itself with some course of values not a fixed point under the mapping. Stipulations about the references of truth value names are then introduced, but it is clear that the same objection can still be applied (see Michael Dummett, The Interpretation of Frege's Philosophy, pp. 421-4 for a discussion of this point). Frege seems to be confused between two different concepts of reference; absolute and relative. The argument from the mapping of the universe into itself can be taken as showing that reference cannot be established extra-systematically. No matter how rich a theory is it can never identify its models more closely than up to isomorphism. So if the only constraints on reference are given by the axioms and definitions within the theory itself, distinct objects within distinct models have equal right to be the referents of a given term. Absolute reference cannot be secured. But within a theory we can secure reference for a term by making sure that every sentence of the theory in which the term occurs has a determinate truth value .. It is reference in this relative sense which Frege tries to secure for course of values terms (see Gz., Section 31), but he was mistaken if he thought that absolute reference could be secured in this way. This problem has some affinities with Putnam's recent 'model theoretic' attack on realism. The possibility of securing reference in this relative sense corresponds closely to the position which Putnam endorses and calls 'internal realism'. (For an attempt to modify Putnam's argument see my 'A Note on Realism', Philosophy of Science 49 (1982), pp. 263-268.) 12 See Charles Parsons, 'Frege's Theory of Number' in M. Black (ed.) Philosophy in

America, (London: George Allen and Unwin, 1965), pp. 189-190. 13 'After the completion of the Basic Laws of Arithmetic the whole edifice collapsed around me' (letter to Honigswald, May 4th 1925, W.B., p. 87, P.M.C, p. 55). 14 See my 'The Origin of Frege's Realism', Inquiry 24, pp. 448-454, 1981. 15 I have argued elsewhere that Frege's epistemological position precluded him from supposing that we have direct epistemic access to 'logical objects'. Our grasp of them is indirect, mediated by the recognition of the truth of certain Thoughts (Gedanken). See 'The Origin of Frege's Realism'. 16 N.S. p. 198, P. w., p. 182. In a note written in 1910 to Jourdain's account of his work, Frege says 'We can, perhaps, regard arithmetic as a further developed logic' (P.M.C,p.191). 17 Though Jourdain, in a letter to Frege of 29 March 1913, says 'In your last letter to me you spoke about working at the theory of irrational numbers' (W.B., p. 124, P.M. C, p. 76). This letter of Frege's has been lost. The rest of Jourdain's letter indicates that he


takes Frege to be offering a new theory of the logical foundation of the real numbers. But he may simply be mistaken about the character of Frege's investigations. In lectures delivered at Jena around this time Frege showed how definitions of continuity and other arithmetic notions could be expressed in his Begriffsschrift, without, however, offering any theory of the nature of the real numbers. It may be that Frege was referring to this work in his letter to Jourdain. (I am grateful to Richard Creath for allowing me to read Carnap's notes of Frege's lectures in the Carnap Archive at the University of Pittsburgh in 1978.) 18 See Charles Parsons, 'Some Remarks on Frege's Conception of Extension' in M. Schirn (ed.) Studies on Frege, vol. 1 (Stuttgart-Bad Cannstatt: Friedrich Frommann, 1976), pp. 265-277. 19 See Gz., vol. 2, III, 1: 'Kritik der Lehren von den Irrationalzahlen'. 20 See also letter to Russell, 21 May 1903, (W.B., p. 239, P.M C, p. 156). 21 See GI., Section 19. 22 See Gz., vol. 2, Section 164. (See also the Appendix to this paper.) 23 See letter to Russell, (W.B., p. 239, P.M.C, p. 156). 24 Gauss, 'Theoria residuorum biquadraticorum, commentatio secunda', Werke, vol. 2, pp. 169-179. Quoted in Gz., vol. 2, Section 162. See also letter to E. V. Huntington, WB., p. 89, P.M. C, p. 58. 25 Frege's method in his Habilitationsschrift is to associate magnitudes with systems of functions in such a way that the magnitude of the function f( x) = x is the null magnitude, and if g(f(x» = x then g has a magnitude equal and opposite to that of f(K.S. pp. 52-53). 26 Cf. 'Only general properties of addition remain, and they emerge as the essential characteristics of magnitude' (K.S., p. 50). 27 'Arithmetica Universalis', in D. T. Whiteside (ed.): The Mathematical Works of Isaac Newton, vol. 2. New York: Johnson Reprint Corporation, 1967, p.7. 28 Das Prinzip der Infinitesimal Methode und seine Geschichte, Berlin: F. Dummler, 1883. 29 Gz., vol. 2 Section 158. GI., Section 12 also points to the variety of magnitudes as a reason for not identifying numbers with magnitudes, but suggests nothing positive about their relation. 30 I have not attempted here to answer the question of how much of Frege's logicist programme was implieit in his Begriffsschrift of 1879. Since the central mathematical concept explored there is the concept of induction, it seems likely that he was thinking of the application of his new logic to the natural numbers only. But there is no hint in the central theorems of that work of a purely logical proof that the natural numbers exist. 31 See GI., Section 109, also Section 4, Note 1, and Gz., vol. 1, Section 42, and vol. 2, Section 157.lt would also explain why Frege wrote in Section 19 of GI., that 'operations with negative, fractional and irrational numbers can be reduced to operations with the natural numbers'. 32 Partly because, on Frege's view, a statement of number is a statement about a concept; numbers are not directly applicable to objects at all. And they are applicable only to those concepts for which the answer to the question 'how many ... l' is some finite number. 33 We can count thoughts and theorems, but can we measure them? 34 'What Numbers could not be', Philosophical Review 74 (1965), pp. 47-73.

372 GREGORY CURRIE

35 GZ., vol. 2, Section 161. Frege's approach to magnitudes is similar to Hilbert's way of introducing the real numbers via a set of postulates. (See Hilbert's Uber den Zablbe-griff', lahresberichte der Deutschen Mathematiker- Vereinigung, 8 (1899), pp. 180-184). The difference is, of course, that while Hilbert seeks to supplement the axioms with a consistency proof, Frege hopes to display a model of the axioms. (See Frege's letter to Hilbert of January 6th, 1900 (WB., p. 71, P.M. c., p. 44)). 36 Cf. H. H. Field, Science without Numbers (Oxford: Blackwell, 1980). 37 'Za.llien und Arithmetik', N.S., pp. 295-297, P. W., pp. 275-277. 38 'Counting, which arose psychologically out of the demands of active life, has led the learned astray' (N.S., p. 297, P. W, p. 277). 39 See K.S., p. 166. 40 See N.S., p. 277, P. W, p. 257. 41 In a very late paper Frege gives a brief account of his epistemological position. There are three sources of knowledge: the empirical, the logical and the geometric-temporal. Frege says only this concerning the temporal source: 'Besides the spatial, the temporal must also be recognised. A source of knowledge corresponds to this too, and from this also we derive the infinite. Time stretching to infinity in both directions is like a line stretching to infinity in both directions.' (N.S., p. 294, P. W, p. 274.) Having rejected both the empirical and now the logical sources as foundations for arithmetic, and bearing in mind his earlier rejection of the association between arithmetic and geometry, we might expect Frege to follow Kant in associating arithmetic with a pure intuition of time. Yet he does not consider this possibility, and instead, abandoning his earlier conviction, opts for geometry as the foundation for arithmetic. The claim that arith-metical knowledge is based upon a pure intuition of time never had the plausibility of the corresponding claim for geometry and the pure intuition of space. 42 N.S., p. 296, P. W, p. 277. 43 In a draft letter to Zsigismondy, probably written shortly after 1918, Frege says that the discovery of Russell's paradox 'has acted as a constant stimulus which would not let the question [what is a number?] rest inside me. It continued to operate in me even though I had officially given up my efforts in the matter. And to my surprise, this work, which went on· in me independently of my will, suddenly cast a full light over the question' (WB., p. 272, P.M.c., p. 176). Is Frege here referring to his idea offounding arithmetic on geometry? If so, we could date its origin rather earlier than the notes of 1924. 44 Here I speak of mathematical rather than philosophical continuity. For the latter see Philip Kitcher, 'Frege's Epistemology', Philosophical Review 88 (1979), pp. 235-262. 45 N.S., p. 299, P. W, p. 279. 46 'Uber eine Weise, die Gestalt eines Dreieck;s als Komplexe Grosse aufzufassen', K.S., pp. 90-1. Triangles are given by triples of vectors (a, b, c), with a + b + c = O. If a =

b/cthen

i (2 + 3a - 3a

2 - 2a

3 )

n= 3,/3 a

2 + a

defines the magnitude -of the shape of the triangle. The unit of shape is the equilateral triangle obtained by rotation from 1 to i. For the degenerate case, where the vertices lie


on one line, n is purely imaginary. n = 0 for a degenerate triangle with two sides equal to each other and half the length of the other side. For an isosceles triangle with a = eiO,

nis real. 47 See N.S., p. 307, P. W., p. 281. 48 See Ct., Section 64. 49 ' ... it does not seem appropriate to let our way of thinking [about real numbers] be too firmly fixed to geometric images' (Cz., vol. 2, Section 158). See also the remarks on Frege's geometrical programme in Sluga's review of N.s. 50 'Uber Schoenflies: Die togischen Paradoxien der Mengenlehre', 1906, is the last work in which Frege wrote as if he assumed the existence of conceptual extensions. See above, text to Note 16 (He does not here or in any later work refer to courses of values. His late, sceptical remarks refer to extensions, but of course he meant them to apply to courses to values generally.) 51 Frege described Russell's method of constructing real numbers as classes of rationals as 'logically unassailable', even though he rejected it on other grounds. (See letter to Russell, 1903, W.B., p. 239, P.M. c., p. 156.) 52 Frege denied that the concept of magnitude is ultimately an intuitive one. See K.S., p. 51. 53 See ibid. 54 For further details of the mathematical aspects of Frege's theory see Franz Kut-schera, 'Freges Begriindung der Analysis', Archiv fUr mathematische Logik und Crundlagenforschung 9, 1966, pp. 102-11: Reprinted in M. Schirn (ed.), ibid., pp. 301-12.

A. W. MOORE AND ANDREW REIN

GRUNDGESETZE, SECTION 10*

1

Section 10 of Frege's Grundgesetze exhibits a number of puzzling fea-tures. I Our goal here is to resolve one central puzzle, which surrounds the appearance in the Section of what we will call, following Dummett, Frege's permutation argument.2

Before stating the puzzle, we need to outline the context in which the permutation argument appears. In Section 3 of Grundgesetze, Frege made the following stipulation:

'I use the words "the function (s) has the same value-range as the function '¥(s)" generally to denote the same as the words "the functions (s) and '¥ (s) have always the same value for the same argument".'3

(This is the stipulation which is later to be incorporated into Frege's formal system by means of his notorious Axiom V.)4 At the beginning of Section 10, Frege writes of this stipulation that it 'by no means fixes completely the denotation of a name like "t<D(e)". We have only a means of always recognizing a value-range if it is designated by a name like "t<D(e)", by which it is already recoginzable as a value-range. But we can neither decide, so far, whether an object is a value-range that is not given us as such, and to what function it may correspond, nor decide in general whether a given value-range has a given property unless we know that this property is connected with a property of the correspond-ing function.'

It is at this point that Frege invokes his permutation argument, which can be presented as follows. Suppose is the intended assignment of objects to value-range terms. And suppose there is a permutation h of . all objects within the domain of Frege's system, which is such that for some value-range, say t<P(e), h(t<P(e» of t<P(e). Now consider the assignment of objects to value-range terms which is related to as follows: if x is assigned by to a given value-range term and y = h(x),

then y is assigned by to that value-range term. It follows that S is an assignment of objects to value-range terms which differs from but

375

376 A. W. MOORE AND ANDREW REIN

which satisfies Axiom V if !1 does. There must then be at least one value-range term such that Axiom V is powerless to decide which of two objects its referent is. This shows, Frege concludes, that if such a permutation as h exists, then 'by identifying the denotation of "t(e) = a'lf ( a)" with that of ( 0) = 'If ( Q)", we have by no means fully deter-mined the denotation of a name like" t ( e )".'

Immediately after giving this argument, Frege asks, 'How may this indefiniteness (Unbestimmtheit) be overcome?' And he replies, 'By its being determined for every function when it is introduced, what values it takes on for value-ranges as arguments, just as for all other arguments.' Now the puzzle is twofold.s First, Frege's permutation argument appears to introduce a deeper problem of indeterminacy than that involved in the initial statement of his objeections to the stipulation introduced in Section 3. Second, Frege's prescription for eliminating the kind of indeterminacy of reference established by the permutation argument appears wholly inadequate. What the permutation argument seems to show is that if there is one assignment of objects to value-range terms which satisfies Axiom V then there are others. In order to eradicate this indeterminacy, then, it would seem that we would have to find a means of ensuring that a unique assignment of objects to value-range terms satisfies Axiom V. But by rendering it determinate what values the primitive functions of Frege's system are to have for value-ranges and other objects (if such there be) as arguments, we do nothing towards ensuring that a unique assignment of objects to value-range terms satisfies Axiom V. Not only is a permutation of the kind in question which respects the truth of Axiom V still possible after Frege's prescription has been filled, but such a permutation will respect the truth-value of every formula of Frege's formal system. This is true because the value of any given function of Frege's system for a value-range as argument will depend solely on this function and the function whose value-range is the argument. For example, the referent of'F( t( e»'

will depend solely on the concept and the function A realignment of referents to value-range terms determined by an appro-priate permutation cannot thus affect the truth-value of this expression.

The nature of the indeterminacy underlying the permutation argu-ment therefore seems to be this. Beginning with a conception of a domain of objects, we suppose that each value-range term is assigned some object from this domain as its referent. The permutation argument (generalized somewhat) then shows that different such assignments will

GRUNDGESETZE, SECTION 10 377

respect the truth-values of all sentences of the formal system and thus that there is no telling, for a given value-range term, which object it refers to. Now suppose we conceive of Frege's formal theory, in accord-ance with a more modem conception, as having a model with a specialized domain. (For the purpose of this discussion, we are pretend-ing that the theory is consistent.) We can then, by strengthening the permutation argument, establish a more radical kind of indeterminacy. For the purpose of this strengthened argument, we require a correlation which maps the objects of this domain onto the objects of a quite dis-joint domain. Such a correlation is what Quine has called a proxy func-tion.6 The language of the theory is then reinterpreted, in the familiar way, grounded in the new domain, in such a way that the new interpre-tation provides an alternative model of the theory. The more radical form of indeterminacy established by this strengthened argument is just this: there is no distinguishing, from within a given theory, II between isomorphic models of that theory. Thus a very natural progression from Frege's permutation argument leads directly to this radical indetermin-acy thesis.

Evidently, something has gone wrong. Frege appears to be advancing an argument which invokes a radical kind of indeterminacy to which contemporary logicians have only recently drawn attention. This kind of indeterminacy appears to bear no significant relation either to Frege's prescription for eliminating it or to his initial complaint that the denota-tions of value-range terms have not been fixed by the stipulation of Section 3. Is there a way of viewing the permutation argument which reconciles it both with Frege's prescription for eliminating the indeter-minacy established by means of it and with the reservations he expresses, prior to stating the permutation argument, concerning his crucial stipulation?

II

These reservations are most naturally seen in the following terms. The fact that 't<l>(e) = aW(a)' is generally to have the same truth-value as '.s. ( Q) = 'II ( Q )', which is so far all that has been claimed on behalf of the abstraction function,? leaves unanswered a whole range of questions which can be couched with the formalism of Frege's system. Our view is that the permuation argument is correctly construed, not as introducing a different (let alone deeper) problem, but simply as a semi-formal


demonstration that at least some of these questions really do remain unanswered. The puzzle which we are attempting to solve stems from the temptation, entirely natural to anyone versed in contemporary model theory, to view the permutation argument as relating to questions which arise in some metalanguage in which the referents of the names in Frege's system are capable of being discussed - questions which have no bearing on the truth-value of any sentence belonging to the system. It then presupposes, in Dummett's phrase, 'a connection between proper names and their referents that transcends what is required for a deter-mination of the truth-values of sentences containing those proper names.'8 But there is no need to construe it in this somewhat anachron-istic way. It can quite satisfactorily be construed as anticipating what Frege will eventually be able to highlight as the sole problem, namely that there is, as yet, no way of ruling out the possibility that the True and the False are themselves value-ranges, and, if they are, no way of telling to which particular functions they correspond.9

How, then, according to this view, does the permutation argument work? (In the ensuing discussion, we shall again pretend that Frege's system is consistent.) Having stated that the stipulation made in Section 3 fails to settle a range of questions, Frege attempts to establish this rigorously for at least some of these questions, by showing that, for all we know so far, quite different fucntions may count as the abstraction function. But in our view, 'different' here should be taken to mean 'recognizably different in terms of what can be proved within the formalism itself;lO otherwise, there really is a danger that the permuta-tion argument will be taken to involve the kind of transcendent connec-tion between proper names and their referents mentioned by Dummett. (Furthermore - though this is something which Frege will only sub-sequently show - to say that such functions are recognizably different in terms of what can be proved within the formalism is just to say that they differ with respect to which arguments, if any, take the True and the False as values.) Consequently, the permutation h which is involved in the argument has to be one which is expressible in the formal language and which bears on the truth-values of sentences which are so far neither provable nor refutable within the formalism, for example the sentence, 'tee = e) = = a)'. This perhaps explains the otherwise remarkable fact that Frege adds as a rider to the conclusion of the permutation argument 'at least if there does exist such a function .. .' -for the existence of a bare permutation could surely not be in doubt.


Now Dummett objects that Frege cannot be concerned with permuta-tions definable within the formalism on the grounds that he must be interested only in permutations which respect the truth of everything provable within the formalism. I I This objection holds only if every identity-statement within the formalism is either provable or refutable. For if there were any identity-statement neither provable nor refutable within the formalism, say 'a = b', then the permutation which inter-changes the referent of 'a' with the referent of 'b' would respect the truth of everything provable within the formalism and yet be definable within the formalism. But Frege has made his stipulation con-cerning the truth-values, there are identity-statements neither provable nor refutable within the formalism, for example the sentence, 't(E = E) = (..?rQ = a),. Certainly, after these stipulations have been made, the identity-relation is fully determined; and it must be from that vantage-point that Dummett is viewing things. The example he cites to illustrate his objection is a permutation that interchanges the referents of what he writes, in modern notation, as '{a, I}' and '{O}', and indeed we do have the resources to show, within the formalism, that these referents are not identical. But we have the resources to show this prior to making any special stipulations. A more pertinent example, on our account, would have been the one alluded to above. For, before Frege has laid down his stipUlations concerning the True and the False, his formalism is power-less to decide the truth or falsity of the given identity-statement.

Dummett, of course, recognizes that his objection may be countered by an appeal of permutations which alter the truth-values of sentences neither provable nor refutable with the formalism. He insists, however, that such an appeal is of no use, on the grounds that 'Frege would have claimed his axioms, taken together with the additional informal stipula-tions not embodied in them, as yielding a complete theory: to impute to him an awareness of the incompieteness of higher-order theories would be an anachronism.'12 But to insist thus is again to bypass the crucial dis-tinction between what can be proved or refuted within the formalism before any supplementary stipUlations have been laid down and what can be proved or refuted with the formalism afterwards - the very distinction which gives those stipulations their raison d'etre. Before the supplementary stipulations have been made, that is, while the per-mutation argument still has force, Frege's concern is precisely with permutations which alter the truth-values of sentences neither provable nor refutable within the formalism. Thus, according to this view, Frege


did indeed recognize that, prior to his making the supplementary stipulations, his formalism contained undecidable sentences. But we need have no qualms about crediting Frege with awareness of that. As long as we keep in mind when these sentences are undecidable, it is obvious that the undecidability in question has nothing whatsoever to do with Godelian incompleteness.

Because he views the permutation argument as essentially involving a permutation which does not affect the truth-value of any sentence of the formal language, Dummett is forced to admit that it is an anomaly. On his view, 'if the permutation argument proved that the references of value-range terms were not fixed by Axiom V, it also proves that they are not fixed by the addition of the new stipulation, or by any that could be expressed in the formal language.'13 On our construal of the per-mutation argument, on the other hand, the argument gets a grip prior to Frege's stipulations but not after.

On our view, then, the permutation argument, or more strictly its subsequent elaboration, which we shall consider in due course, is thought by Frege to show the following: that it is compatible with every-thing which has so far been laid down that the abstraction function should have the True as value for any given argument and the False as value for any other argument. By stipulating for which arguments it does take the True and the False as values, Frege meets the objection. But it is interesting to note parenthetically that the permutation argument, in itself, may not be able to show as much as Frege presumed. It is certainly able to show that, on the strength of what we know prior to his stipulations, we are powerless to prove that any given object is not a value-range; for it shows us how to effect a reductio from the as-sumption that we have a counterexample, namely by considering a permutation h which interchanges the given object with some particular value-range v and observing that nothing we know so far rules out the possibility that the function whose value-range is v should have h(v) as its value-range instead. In addition, the permutation argUment is able to show that, prior to the stipulations, we are powerless to prove that any object given to us independently of value-ranges is identical to some particular value-range; once again, in order to rule out any putative counter-example, it suffices to consider a suitable permutation which interchanges the given object with some other value-range. But does the permutation argument show that we are powerless to prove, prior to the stipulations, that such an object quite simply is a value-range (that is,


some value-range or other)? No. It is quite compatible with the permuta-tion argument that we should in fact be able to prove this, as it is quite compatible with the permutation argument that, even on the strength of what we know prior to the stipulations, we should have the resources to prove that everything is a value-range. No permutation can be used to refute this conclusion, save on the strength of the question-begging assumption that there are some objects which are not value-ranges (with which an appropriate permutation can be effected). Consequently, Frege's claim that we cannot decide so far whether an object is a value-range that is not given us as such is stronger than anything that can be shown by means of the permutation argument. But this does not really matter. All the permutation argument is supposed to establish is that certain questions of identity concerning value-ranges remain to be answered once they have been introduced by means of the stipUlation in Section 3. And this follows at once from the conclusion of the permutation argument: that there is no telling, for any given function, whether or not an object not given as a value-range is in fact the value-range of that function.

Let us review the position of the permutation argument in Section 10. Frege opens the discussion by objecting that the way in which value-ranges have been introduced into the system 'by no means fixes com-pletely' the denotations of value-range terms; that is, it leaves us incapable of answering a whole range of questions involving value-ranges. The permutqtion argument is then introduced as a semi-formal demonstration that, in particular, we are incapable of answering certain questions of identity involving value-ranges. Frege rightly takes this argument as sufficient on its own to establish that the denotations of value-range terms have not been fully determined. He then asks how the indeterminacy caused by the introduction of value-ranges into the system is to be eliminated; that is, how the denotations of value-range terms are to be completely fixed. The obvious answer is, 'By its being deteimined for every function when it is introduced, what values it takes on for value-ranges as arguments.' This procedure will ensure that the entire range of questions alluded to by Frege in his opening remarks in Section 10 and, in particular, the questions of identity with which the permutation argument is concerned, will be answerable. As it turns out, this entire range of questions turns on just the questions of identity which are the sole concern of the permutation argument. This becomes evident when Frege carries out the procedure for eliminating the indeterminacy. Wnat is required is that all primitive first-level functions


introduced so far into the system be defined for value-ranges as argu-ments. (If new functions are subsequently introduced, not reducible to these, then further determination will be required.) These three func-tions are:

But Frege shows that if the first of these is determined, the remaining two will ipso facto be determined. Since the only objects which have proper names in the formalism are the two truth-values and the value ranges, it is in turn simply a matter of determining whether or not each of the truth-values is the value-range corresponding to any given function. 14

It is at this point that Frege elaborates the permutation argument so as to emphasize what can now be recognized as its real import, namely that there is no telling, so far, whether or not either truth-value is the value-range of any given function. This elaboration also serves to show how the difficulty may be resolved; for it shows that we are free to stipulate that each truth-value is the value-range of an arbitrary function, subject only to the obvious constraint that we do not identify both with the value-range of extensionally equivalent functions. The argument proceeds thus. Let fbe any function which satisfies all that we know so far about the abstraction function, and let a and b be any two of its values. Now consider any permutation h, such that h(a) = the True, h(b) = the False, h(the True) = a, h(the False) = b, and, for any other object x, h( x) = x. Then the composition of h and f also satisfies all that we know so far about the abstraction function. In view of this, Frege stipulates that each truth-value is to be identified with its own unit class. And this simple stipulation ensures that the problem underlying Section 10, illustrated most graphically by the permutation argument, has now been solved. In Frege's own words, 'with this we have determined me value-ranges so far as is here possible.' It is thus possible, after all, to view the permutation argument as establishing an indeterminacy which is in line with the reservations expressed by Frege at the beginning of section 10 and which the filling of his subsequent prescription serves to e'liminate.

III

This resolves one central difficulty of Section 10. However, the means


Frege adopts for eliminating the indeterminacy introduced into his system by the stipulation in Section 3 gives rise to another puzzle, which we shall simply record. Frege stipulates that each truth-value is to be identified with its unit class. But with what right does he do this? For Frege, the truth-values and value-ranges inhabit a Platonic realm, whose existence and structure are quite independent of our capacity to appre-hend them. Surely, then, it is a matter of objective fact whether or not each of the truth-values is a value-range and thus not something which can be decided by stipulation. In short, Frege's stipulation identifying the truth-values with their unit classes appears to be in direct conflict with his avowed Platonism.

This conflict, interestingly, is conspicuous in a footnote within Sec-tion 10.15 There Frege considers the possibility of generalizing his stipulation so that all and only those objects which are not given as value-ranges are identified with their unit classes. He rejects this pro-posal on the grounds that 'the way in which an object is given must not be regarded as an immutable property of it, since the same object can be given in a different way.' The objection seems to be that an object not given as a value-range may yet be a value-range and, in particular, a value-range other than that with which our stipulation would identify it. This objection is quite in keeping with Frege's Platonic viewpoint. But if it is valid then it undermines not only the programme for generalizing the stipulation concerning the truth-values but also that very stipulation itself. For, according to the objection, the truth-values themselves may yet be value-ranges (since they are not given us as such) and, indeed, value-ranges other than those with which we have identified them. Perhaps, at this point, the demands of Frege's formalism outweigh the demands of his philosophy of mathematics. However that may be, Frege does not appear to have a consistent position here.

University of Oxford

NOTES

* Our thanks are due to Gordon Baker, John Campbell, Pamela Tate and especially Michael Resnik for their valuable comments. 1 Frege: 1962, Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, Georg Olms, Hildesheim. 2 See Dummett: 1981, The Interpretation of Frege's Philosophy, Duckworth, London, p. 408.


3 Quotations from Grundgesetze are all from Furth's translation: 1967, The Basic Laws of Arithmetic, Exposition of the System, University Press, California, except that his expression 'course-of-values' has been replaced throughout by 'value-range'. 4 A note on Frege's notation: Frege employs the expressions 'ecfl(e)' and 'e1Jl(e)' to indicate the value-ranges of the functions and respectively - 'cfl' and '1JI' here being purely schematic. In Section 8, he defines '-!lrcfl(a)' in the following way: '" a cfl(Q)" is to denote the True if for every argument the value of the function is the True, and otherwise is to denote the False.' Axiom V in Frege's notation is hence '(ef(e) = ag(a» = (Sf(Q) = g(a»', where '/' and 'g' are assumed to be bound by initial universal quantifiers. (We omit the assertion sign which Frege prefixed to the Axiom.) 5 Cf. Dummett, op. cit., pp. 422-424, from which our account of the puzzle derives. 6 Quine: 1976, Ontological Reduction and the World of Numbers in The Ways of Paradox and Other Essays, Harvard University Press, Cambridge, pp. 199-208. 7 The abstraction function, is just the second-level function which maps first-level functions onto their value-ranges. 8 Dummett, op .. cit., p. 423. 9 That Frege's eventual concern in this regard extends only to the two truth-values and not to other objects has puzzled many commentators. Was Frege in fact operating with a conception of a domain of objects restricted to just the value-ranges and the two truth-values, contrary to the impression he gives, when introducing his primitive first-level functions, that the domain encompasses all objects? We leave this question unanswered here and simply assume that Frege restricted his attention to the True and the False because he was concerned only with questions which could be stated within his formalism. Since his formalism contained proper names only for value-ranges and the two truth-values, questions involving other objects (if there are such objects) cannot be formulated within the system. 10 This notion of what is provable with the formalism should not be confused with the notion of what is provable within the formal system, the latter comprising what can be demonstrated by means of Frege's axioms and rules of inference, the former comprising anything that can be stated in the formal language and proved, semi-formally, by appeal to those axioms together with relevant stipulations and background knowledge con-cerning the semantics of the system. There can be no doubt that Frege would have acknowledged such a distinction, since he wanted to embody in his axioms only what would be necessary to prove the various mathematical theorems that were, after all, his chief concern; and, relative to those theorems, the question of whether or not the True or. the False is some particular value-range (which, as we have remarked, will emerge as the sole question yet to be answered) is an irrelevance. Yet he felt the need to provide stipulations, not incorporated into the axioms of his system, which would decide such questions. Of course, unbeknown to Frege, what can be proved in the formalism is in fact precisely what can be proved in the formal system, namely everything, because of the inconsistency inherent in Axiom V. 11 Dummett, op. cit., p. 423. 12 ibid. 13 ibid. 14 Cf. Note 9. 15 Furth, op. cit., p. 48, footnote 17.

INDEX OF NAMES

Abel, N. H. 305 Abelard, Peter 238,239 Angelelli, Ignacio 155, 170 Aristotle 67,156,157,158,162,171 Austin,J.L. 93-95,153

Bachmann, Friedrich 42 Baker, Gordon 383 Bartlett,J. 187,194 Bell, David 7, 164, 255-261, 263-

267, 269, 276, 277, 279, 281, 286, 292-294

Benacerraf, Paul 340,358 Berkeley, George 306 Black, Max 64, 93 Bolzano, B. 341 Boole, G. 67,257 Brandom, Robert 7 Buchner, L. F. 270 Burge, Tyler 6,63, 133, 142, 145, 149,

153

Campbell, John 383 Cantor, G. 310,320,322,340 Carnap, Rudolf 339,198 Carroll, Lewis 83 Cauchy,A.L. 305,306,308,312 Chomsky, Noam 317 Church, Alonzo 97, 104, 108, 111,

151,221-223 Cocchiarella, Nino B. 7 Coffa, t\. 340 Cohen, Hermann 356-357 Currie, Gregory 8, 286, 340 Czolbe, H. C. 270

d'Alembert, J. Le R. 305 Dant, Mary 153 Davidson, Donald 150

385

De Morgan, Augustus 4 Dedekind, Richard 7-8, 299, 300,

305-308, 310-313, 316-318, 320, 322, 324-325, 327-329, 331-335,338-342·

Dirichlet, L. 305 Dreben, Burton 26, 93, 95 Dudman, V. H. 95 Dugac, P. 299,340 Dummett, Michael 7, 10, 54, 61-64,

94, 97, 115-116, 123-124, 126-128, 144, 146, 147, 148, 150, 152-153,155, 253-257,259,260,267, 268, 276, 279, 286, 293, 294, 340, 369-370,375,378-380,383

Elgin, Catherine 195 Euler, L. 305

Field, H. H. 372 F0llesdal, Dagfinn 63 Friedman, Michael 93 Furth, Montgomery 64, 93, 146, 153,

342,383

Gabriel, G. 94 Gauss, K. F. 351,371 Geach, Peter 64, 93, 268 Gettier, Edmund 249 Godel, Kurt 97, 104, 108, 121, 122,

151 Goldfarb, WarrenD. 26,93-95; 170 Goldman, A. L. 340 Grossmann, R. 293 Gruppe, O. F. 270

Haaparanta, L. 6, 171 Haeckel, E. 270 Harman G. 340

386 INDEX OF NAMES

Hawkins, Loraine 369 Heijenoort, Jean van 3, 5, 64, 94, 159,

171 Hempel, C. G. 339 Hermann, Gottfried 156 Hermes, Hans 27,93 Hilbert, David 17-18, 27, 94, 134,

138,153,193-194,340,372 Hintikka, Jaakko 3-4, 6, 155-159,

170-171 Hume,D. 55 Husserl, E. 54,153 Hylton, Peter 93

Jensen, R. 228, 229, 230, 233, 242 Jourdain, Phillip E. B. 140,170,370

Kahn, Charles 155, 158, 170 Kant, I. 16, 27, 55, 74, 81, 149, 256,

270,274,301-304,316,341 Kaplan, David 262 Kaplan, Mark 26, 262 Kerry, Benno 85,208 Kitcher, Philip 7,10,26,195,372 Kluge, W. 27 Knuuttila, Simo 170 Kornblith, H. 302, 340 Kotarbinski, T. 275 Kreisel, Georg 43 Kripke, Saul 64,262,341 Kronecker, L. 310 Kutschera, Franz 373

Lakatos, Imre 342 Lange, L. 276 Leibniz, G. W. 167,270,271 Lesniewski, S. 191,194,207 Long, Peter 27 Lotze, Hermann 55-58,64,270-275

Marshall, W. 293 Martin,E. 187,188,189,190,194 Mates, Benson 155, 156 McGuinness, Brian 94 Mill, John Stuart 4,16,233 Moleschott, J. 270

Montague, Richard 250 Moore, A. W. 8

Newton, I. 347,355-356

Oddie, Graham 369 Owen, G. E. L. 156

Parmenides 122 Parsons, Charles 187,194,370-371 Peano, G. 33, 35, 38-39, 95, 166,

170,179,194,340 Plato 156 Poincare, H. 36 Punjer, Bernard 161 Putnam, Hilary 26, 150, .262, 341-

342,370

Quine, W. V. 41-42,76,94-95,150, 229,231,242,320-321,341,377

Rein, Andrew 8, 195 Resnik, Michael D. 6,327,383 Ricketts, Thomas G. 5, 26 Russell, Bertrand 6-7, 35-36, 41-

42, 47, 66, 76, 101, 105-107, 111, 151, 155: 191, 194, 197-198, 200-203, 206-207, 212-215, 218-223,233-236,238,244,250, 320

Schirn, Matthias 155, 170-171 Schroeder, E. 163,193,221,222 Searle, John 54,63 Sluga, Hans 3,5,7,10,26,64,93,190,

194,256-257,261,269,270-281, 283, 285-286, 291, 293-294, 340, 346,369-370

Specker, Ernst 232,233 Stuhlmann-Laeisz, R. 171

Tarski, Alfred 67, 76, 145, 153, 180, 193

Tate, Pamela 383 Teller, P. 341 Thompson, Manley 341

Tichy, Pavel 369 Trendelenburg, A. 270,271 Tugendhat, E. 260,261 Tymoczko, A. T. 341

Vogt, Karl 270

Wallace, John 150 Wang, Hao 242 Weber, Heinrich 299

INDEX OF NAMES 387

Weierstrass, K. T. W. 305, 306, 310, 312

Weiner, Joan 4,27,93 Whitehead, A. N. 42 Williams, C. J. F. 171 Wittgenstein, L. 10, 67, 155, 164, 165,

168,182,256

Zermelo, E. 322

INDEX OF SUBJECTS

a priori 313-320,322,324-327,337, 341

knowability of geometry 301 knowlege 302,303,305,312 warrant 313 vs. a posteriori 12, 13 apriorist program 301-304, 307,

317 apriorism 313,316,318,321-323,

335-336 Abelardian thesis 238,239,246 abstraction 194 abstraction operator 181, 183, 186,

187,189,192 analysis (higher analysis) 305-306,

312 analytic, analyticity (see also synthetic)

205,320-321,341 truths 305 vs. synthetic 12-14

Aristotelianlogic 4,11,13,14 arithmetic, arithmetical 12, 21, 301,

310,360-361 concepts 158 construction 309 equations 56 knowledge 304,318,362 truth 15-17,55-56 arithmetization of analysis 365 Dedekind's derivation of arithmetic

332 foundations of arithmetic 307 unarithmetical world 335-336

assertion 71-72,92,98,125 assertive force 120,148,258

axiom Archimedean 368 set for standard second order predi-

cate logic with identity 205

388

of set theory 319-320,327 of choice 233 of complete typ"ical ambiguity 232 of infinity 197,207,233,248 ofreducibility 197,248-249 of geometry 24, 335 V (Frege) 59,60,61,136,137,138,

141, 178, 190, 191-192, 194, 208, 210-211, 228, 284, 326-327, 341, 346, 347, 369, 370, 375,376,380,384

Basic Laws of Arithmetic, see Die

Grundgesetze der Arithmetik

Bedeutung (see also reference) 4, 19, 100,277

Begriffsschrift 14, 15, 17, 57, 58, 63, 66,97,128-131,157-159,165, 255-256,272,275,304

Begriffsschrift, see concept notation 'Uber Begriffund Gegenstand' 25,208 berng 156,159,161-163 belief 315

basic 302 bivalence 36

universal bivalence 5 Boole's logical calculus 257

categories ontological 5,66, 84, 89 Aristotelian 157 cross-categorical predication 269

causal, causation 271 conditions 272 explanation of person's beliefs 74 theories of reference 66

character/content distinction 262,266 Church-Godel argument 104, 108-

109,110,111,151


class 59,212,231-232,239 as many or as one 212 offinite cardinal numbers 352

cognitive capacities 245-247,249 distinctions 53 notion of sense 262, 266

collecting, operation of 334 completeness-incompleteness distinction

117 completude 35

principle of 5,32-34,161,186 composition principle 99, 100-102,

104-105,108-110,115,151 comprehension principle 205, 218-

219,224,227,240 concept, conceptual 87, 89, 118, 163,

164,236,246,283 analysis 303 class 193 correlate 200, 202, 208-212, 223,

229, 231-232, 234, 236, 238, 240,244

notation 22-23, 86-87, 90, 177, 257,271

Platonism 245,248-249 precondition 314 script, see concept notation words 155 of existence 6 of identity 165 of magnitude 353-354, 355, 357,

359,365-366,373 of number 15,16,21,358

concepts 114,127,149,155, 166, 167, 200, 202, '204, 208, 215, 223, 229, 234-235, 244-245, 247, 249,256-257,269

asfunctions 272,278 of existence 170

conceptualism 246-247 consistency 137,189

proof 190 ofAHST* 227

construction, construct\ve 309-311, 330-332

constructivism 311,328-329

content stroke 128 contents

of judgment 73 of terms 128

context, contextual principle 103, 124, 126, 199,210,

257,266,275-279,285 definitions 191,279 stipulation 192 theory of abstract entities 178 theoryofreference 184-185,192 contextualist principles 126 of utterances 264

continuity 307-308,372 correlating operation 332,334 correspondence

theory oftruth 77-78 with reality 77

course-of-values (see also value range) 7, 134-136, 140-141, 143, 148-149, 281, 283-287, 291, 345-347

criteria of eliminability and noncreativity 191

criteria of referentiality 181, 183-185, 187,189,192,194

Critique of Pure Reason 13, 65

definition, definitions 17-22,78-79 conditional 33-34,191 contextual 191,279 creative 191,194 implicit 191 may contradict a mathematical theory

138 of number 280-282,285-286 Peanoon 33 piecemeal 32-33

denotation 99-101, 108, 148, 152, 183

-less nominalized predicates 239 -less singular terms 237 of a compound name 180 of a part of a sentence 103 of a sentence 103-104, 109-110,

115,117,147 of complete sentences 116

390 INDEX OF SUBJECTS

of predicates 116 of terms 109,125,147,151 of words 104

'Dialog mit Punjer iiber Existenz' 158, 160, 164

distinction between identity and predication

166-169 between objects and concepts 166 between predicates and singular

terms 199 between sense and reference 254 between subjective and objective 72 between the act and content of judg-

ment 72-73 between the natural and real num-

bers 349,365 double correlation thesis 210-211,

219, 223-225, 230-231, 236, 240

empirical objects 274 realism 273

enthymatic reasoning 74 essential properties 166-167 existence 158, 160, 161 existence postulate 194 existential posits 205 experience 12 extension

of concepts 118-119, 132, 134-136, 139, 142, 208, 211-212, 229,232,283,346,349

extensionality principle of 228-229

fact 106 false, falsity

The False 91,182,378 formalism 191 frege

style theory of meaning 47,52-54 his permutation argument 8 his theory of judgment 7,255

function 90, 114, 117-118, 134, 155, 259,282

names 155 functionality 204 functional abstraction 183 first level 149,187

'Funktion und Begriff 59-61,90,100

'Der Gedanke' 272 geometry, geometrical 300-301, 308,

361-362 axioms of 24,335 Euclidean 358 foundations of 366 foundation for the complex number

364-365 intuition 307 knowledge of 11 objects 356,363-364 truths 274,366 construction 361 non-Euclidean 274

gleichbedeutend 61 Die Grundgesetze der Arithmetik 6-8,

59, 61-62, 68, 97-98, 106, 112, 124, 126-129, 131-132, 139, 145, 148, 177-179, 181, 183,255,318-319

Die Grundlagen der Arithmetik 8, 12, 15-21, 23, 125, 137, 139, 145, 158,160, 166, 255, 257, 299, 345,348

'Uber die Grundlagen der Geometrie' 17

hierarchy of individuals 221-222 of unsaturated concepts 225

higher-order entities 171 horizontal function 143-146,182

ideal language of thought 142 idealism

Hegelian 270 transcendental 273-274 idealization 330, 334 ideas 245 identity 165,169,182,211,219

conditions 355


involving value ranges 381 of reference 280 of sense 280 principle of 55 symbol 168 substitutivity of identicals 166

identity statements 56, 58-60, 165, 167-169,237,272,379

heterogeneous 284-292 homogeneous 284-289,292 indiscernibility of identicals 219 informative 48, 50-52, 57 incompleteness 115 numerical 280 trivial 51-52 of higher order theories 379

impredicative comprehension principle 247

impredicative concept-formation 247 incomplete symbols 235 indeterminacy, indeterminateness 32,

37,40,282,377,382-383 indexical (reflexive) expression 262 indirect discourse 92 individuals 201-202,212,234

abstract 222 concrete 231 individuals, hierarchy of 221-222 individuation, failure of 35

induction, mathematical 31,371 inductive over 31 inference

patterns 85-87 polyadic quantificational 86 rules of 83

infinity 27 infinitesimals, method of 306 input sense 263-266 intension, intensionality 229

principle of 248 of a proposition 234

intersubjectivity 275 intuition 27,74,274,301,303-304

pure 7,12,24,327 set-theoretic 319 of space 307 oftime 362

is ambiguity of 4,158-160,168 of class-inclusion 157 of existence 157, 168 of identity 156,157,166-168 of predication 156-157, 167 -168

judgment 66, 70-72, 76-78, 91-92, 125, 129, 256-258, 275, 277-278,293

judgmental force 130 justification 17, 18, 271

Kantian epistemology 4, 10, 11, 12, 15

knowledge 302 mathematical 313,317, 320-323,

337-339 sources of 372 tacit 317 of arithmetic 307 of geometry 307 of logical truth 303 of the infinite 362

laws of logic 80-81, 177, 198, 202,

204-205, 221-223, 228-229, 304,326

of number 362 of psychology 326 of thought 271,315,326,362 of truth 80, 106, 125, 139, 144,

147,150, Leibniz's law 85,92,217 levels of generality 89 lingua characterica 159-160 logic 75,271

as a universal language 159 as language 3 logica magna 43 logica utens 43

logical aliens 68-69 categories 5,66,86,88-89 concepts 159 consequence 109-110


laws 76, 89, 327 notions 324-325 objects 98, 111, 122-123, 132,

135, 140, 144, 148-149, 152, 208, 280, 282, 285, 291-292, 347-348,362

principles 79 logical positivists 299, 340 logicism 7, 122, 135, 197-202, 204,

212, 215, 219, 235-236, 239, 242, 244, 246, 249, 299-300, 313,321,329,366-367,369

logicist program 134, 159

magnitude 351-352, 356, 358, 363, 371

fields 359-361,367 non-geometric 364

mathematics, mathematical classical 236 empiricism 323 entities 329 foundations of 325 functionality 203 knowledge 313, 317, 320-323,

337-339 truths 312

meaning 229 ofa word 102

mental images 245 modus ponens 82 mutual subordination 211

names complex proper 124 proper 87,155,167,199 forming 179 simple 184,186

natural kinds 262 naturalism 271-273 necessity 341 neo-Fregeanism 320,324-325,327 no class theory 191,193,212 noema 54 nondenotingterms 101,113,190 number 100,139

complex 363-364

conceptof 15-16,21,358 concepts 312 creation of 332 definition of 233, 280-282, 285-

286 lawsof 362 natural 346, 350-351, 353, 358,

361,365-366,371 real 309, 345, 350-354, 356-360,

363,365-366,373

object, objects 87, 115-117, 119-120,131,134,156,166,168

abstract 280, 309 improper 221,223 logical (see logical objects) as metaphysical units 170 as we know them 170 of arithmetic 362

objective, objectivity 22, 23, 37-38, 92,275

difference 25 and subjective 65,67,69

objectivist epistemology 273 'On Concept and Object' see 'Begriff

und Gegenstand' 'On Sense and Reference' see 'Sinn und

Bedeutung' 'On the Foundations of Geometry' see

'Grundlagen der Geometrie' output sense 263-266

permutation argument 375-382 philosophy of mathematics 300, 313,

339 Platonism 273-275 positive class 351-353, 359, 363,

367-368 predicates 87,114,214

complex 207 first level 87 nominalized 198-202, 211-212,

214, 216-219, 227, 237-239, 244,246

predicative nature of 200 sense of 269 vagueness of 31


primitive, primitiveness 23 names 177 property 79 terms 17,20,22-23 truth 17

proposition 106 unity of 204

propositional calculus 141-143 functions 201-204, 212, 214,

234-235,249 preconditions of thought 314-315,

327 psychologism 67-70, 81, 270-272,

274,329

realism 132,134,136 internal 370 Platonic 274-275

reference 4,19,31,259-261 non-relational notion of 285,292

l:eferential, referentiality 178, 189-191

concepts 246 function name 186-188 objectname 188 proper names 186 proof of 189,192 of names 177 of truth functors 186 of the universal quantifiers 188 criteria of 188

regimentation 42 reism 275 relations 155 rigid, rigidity

concepts 235-236 principle of 233, 236, 243-244,

248 relation 235

Russell-style theories of meaning 47, 52-54

Russell's paradox 119, 122, 184, 190, 193, 197, 202, 212, 215, 217-218, 225-226, 237, 239, 345, 347-348, 352, 362, 365, 369, 372

second order logic of nominalized predi-cates 226

second-level variables 89 semantic

content 49-52,54,57,60 distinctions 53 notion of sense 262 rules 180 semantically informative and trivial

sentences 53 semantics

ineffability of 159 sensation 27 sense 38-40, 61, 92, 99, 169, 229,

262,277-278,281-282 cognitive notion of 262, 266 determines reference 265 and reference 3, 18, 20, 62, 92, 99,

101,167 of a function 267-269 ofaname 92,166-167,266 of a predicate 268-269 of a sentence 267 ofa term 267

sentences 124,126-127,259 denotations of 151

settheory 134-135,232,239,322 NF 229,231

set-theoretic foundation for arithmetic 324

sets as composed of their members 240 sharpness (see also vagueness) 35

of concepts 5 singular term 201,214,239,259

abstract 198-199 complex 215 substitution inference potential of

279 Sinn (see sense) 'tIber Sinn und Bedeutung' 9, 48, 58-

59,62,90,100-101, 104-105, 119,147,254

skepticism 149 solipsism 70 stratification

cumulative 224-225 heterogeneous 224-226


homogeneous 241 of the unsaturated concepts 224

subject-predicate distinction 130 subjective and objective 68 subjective difference 25 substitutional-referential approach to

quantification and abstraction 185

syllogism, Aristotelian 40 synthetic, syntheticity, see also analytic

a priori 12, 56 necessary truths 316

Tarski truth schema 145 Tarski approach to semantics 180 term 22,126-127

denotation of 109, 125, 127, 147, 150-151

non-denoting 101,113,190 prinritive 17,20,22-23 sense of 268

'The Thought', see 'Der Gedanke' thought 71-72,79,90-92,129,278

compound 75 formal rules of 4,11,13 laws of 271,315,326,362

transformitional grammar 217 truth, true

conditions and meaning 182 laws of 80,106,125,139,144,147,

150 maximally general 79-80 normative notion of 6, 107, 110,

150 predicate 145-146, 153 prinritive 17 redundancy theory of 119-120,

130,142,144-146 The True 91,140-141,143,146-

147, 149, 182, 186, 190, 254, 378

truth-values 103, 105, 111-112, 115-117, 119-120, 131-132, 135-136, 138-139, 142-144, 150,152,229,292,383

in arithmetic 318,328,331,336 in geometry 274 in logic 56 in mathematics 316 of identity statements 347

understanding 71-72 universal

characteristic 271 class 222 domain 36 grammar 317-318 instantiation 237 language 162 quantifier function 182

universals 238 universe

of discourse 37 of sets 36

unsaturatedness 115, 179, 199-201, 204,212,223,269

ure1ements 231-232

vagueness 35,37,39,43,132-133 value-range (see also course of values) 7,

8, 59, 190, 192-193,208-211, 236,378,380-384

notation 214 term 8,376-377

Werthverlauf (see value-range)

Zerme10 set theory 228

Symbols and Abbreviations

A-operator 207,215 A-abstracts 240 A-conversion 227,237

principle of 207

AHST* + (Ext*) 229-230,232,233 lliST* 226-228,241,243 w-sequence 332-334,342 DlliST* + (DExt*) + (PR) 236

DAHST* + (DExt*) 235 (CP) 207 (CP*) 220,236 (HSCP*) 236 (LL*) 219 A1* 226 A2* 226 A3* 227 Axiom (A3) 217,237 HSCP* 227

INDEX OF SUBJECTS

HST* 241-244 HST*!p 248 HST* A 245 Jd* A 227 LL* 227 NF 231,233 NFU 233,236,250 NFU-Sets 229,232,236 NFU 230

395

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